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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..d7b82bc --- /dev/null +++ b/.gitattributes @@ -0,0 +1,4 @@ +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/13309-pdf.pdf b/13309-pdf.pdf Binary files differnew file mode 100644 index 0000000..c3841e1 --- /dev/null +++ b/13309-pdf.pdf diff --git a/13309-t/13309-t.tex b/13309-t/13309-t.tex new file mode 100644 index 0000000..633fbbc --- /dev/null +++ b/13309-t/13309-t.tex @@ -0,0 +1,9919 @@ +\documentclass{book}
+\usepackage{amsmath, amssymb, amsthm, makeidx, hhline}
+\usepackage{graphicx}
+\usepackage{soulutf8} %May fix some issues
+\pagestyle{plain}
+\begin{document}
+\thispagestyle{empty}
+\small
+\begin{verbatim}
+The Project Gutenberg EBook of A First Book in Algebra, by Wallace C. Boyden
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever. You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: A First Book in Algebra
+
+Author: Wallace C. Boyden
+
+Release Date: August 27, 2004 [EBook #13309]
+
+Language: English
+
+Character set encoding: TeX
+
+*** START OF THIS PROJECT GUTENBERG EBOOK A FIRST BOOK IN ALGEBRA ***
+
+
+
+
+Produced by Dave Maddock, Susan Skinner
+and the PG Distributed Proofreading Team.
+
+
+\end{verbatim}
+\normalsize
+\newpage
+
+\begin{titlepage}
+ \begin{center}
+ {\LARGE\bfseries A FIRST BOOK IN ALGEBRA}\\[2.5cm]
+ BY\\[2.5cm]
+ {\Large WALLACE C. BOYDEN, A.M.}\\[3cm]
+ {\large SUB-MASTER OF THE BOSTON NORMAL SCHOOL}\\[1cm]
+ \textsc{1895}
+ \end{center}
+\end{titlepage}
+
+\section*{PREFACE}
+\addcontentsline{toc}{section}{\numberline{}Preface}
+
+In preparing this book, the author had especially in mind classes
+in the upper grades of grammar schools, though the work will be
+found equally well adapted to the needs of any classes of
+beginners.
+
+The ideas which have guided in the treatment of the subject are
+the following: The study of algebra is a continuation of what the
+pupil has been doing for years, but it is expected that this new
+work will result in a knowledge of \textit{ general truths} about
+numbers, and an increased power of clear thinking. All the
+differences between this work and that pursued in arithmetic may
+be traced to the introduction of two new elements, namely,
+negative numbers and the representation of numbers by letters. The
+solution of problems is one of the most valuable portions of the
+work, in that it serves to develop the thought-power of the pupil
+at the same time that it broadens his knowledge of numbers and
+their relations. Powers are developed and habits formed only by
+persistent, long-continued practice.
+
+Accordingly, in this book, it is taken for granted that
+the pupil knows what he may be reasonably expected to
+have learned from his study of arithmetic; abundant
+practice is given in the representation of numbers by
+letters, and great care is taken to make clear the meaning
+of the minus sign as applied to a single number,
+together with the modes of operating upon negative numbers;
+problems are given in every exercise in the book;
+and, instead of making a statement of what the child is
+to see in the illustrative example, questions are asked
+which shall lead him to find for himself that which he
+is to learn from the example.
+
+BOSTON, MASS., December, 1893.
+
+\cleardoublepage \tableofcontents
+
+\cleardoublepage
+
+
+\part*{A FIRST BOOK IN ALGEBRA.}
+
+\cleardoublepage \chapter*{ALGEBRAIC NOTATION.}
+\addcontentsline{toc}{chapter}{\numberline{}ALGEBRAIC NOTATION.}
+\addcontentsline{toc}{section}{\numberline{}PROBLEMS}
+
+
+\textbf{ 1.} Algebra is so much like arithmetic that all that you
+know about addition, subtraction, multiplication, and division,
+the signs that you have been using and the ways of working out
+problems, will be very useful to you in this study. There are two
+things the introduction of which really makes all the difference
+between arithmetic and algebra. One of these is the use of
+\textit{ letters to represent numbers}, and you will see in the
+following exercises that this change makes the solution of
+problems much easier.
+
+\subsubsection*{Exercise I.}
+
+\textit{ Illustrative Example}. The sum of two numbers is 60, and
+the
+greater is four times the less. What are the numbers?\\
+
+\begin{center}
+\textbf{ Solution}.\\
+\end{center}
+\begin{center}
+\begin{tabular}{lr@{=}ll}
+\text{Let } &$x$ & \text{ the less number};\\
+\text{then } &$4x$ & \text{ the greater number,}\\
+\text{and } &$4x + x$ & 60,\\
+\text{or } &$5x$ & 60; \\
+\text{therefore } &$x$ & 12,\\
+\text{and } &$4x$ & 48. \text{ The numbers are 12 and 48.}
+\end{tabular}
+\end{center}
+
+\begin{enumerate}
+
+\item The greater of two numbers is twice the less, and the sum of
+the numbers is 129. What are the numbers?
+
+\item A man bought a horse and carriage for \$500, paying three
+times as much for the carriage as for the horse. How much did each
+cost?
+
+\item Two brothers, counting their money, found that together they
+had \$186, and that John had five times as much as Charles. How
+much had each?
+
+\item Divide the number 64 into two parts so that one part shall
+be seven times the other.
+
+\item A man walked 24 miles in a day. If he walked twice as far
+in the forenoon as in the afternoon, how far did he walk in the
+afternoon?
+
+\item For 72 cents Martha bought some needles and thread, paying
+eight times as much for the thread as for the needles. How much
+did she pay for each?
+
+\item In a school there are 672 pupils. If there are twice as
+many boys as girls, how many boys are there?
+
+\textit{ Illustrative Example}. If the difference between two
+numbers is 48, and one number is five times the other, what are
+the numbers?
+
+\begin{center}
+\textbf{ Solution}.
+\end{center}
+\begin{center}
+\begin{tabular}{lr@{=}ll}
+\text{Let } &$x$ & \text{ the less number;} \\
+\text{then } &$5x$ & \text{ the greater number,}\\
+\text{and } &$5x - x$ & 48, \\
+\text{or } &$4x$ & 48;\\
+\text{therefore } &$x$ & 12, \\
+\text{and } &$5x$ & 60.\\
+\end{tabular}
+\end{center}
+
+The numbers are 12 and 60.
+
+\item Find two numbers such that their difference is 250 and one
+is eleven times the other.
+
+\item James gathered 12 quarts of nuts more than Henry gathered.
+How many did each gather if James gathered three times as many as
+Henry?
+
+\item A house cost \$2880 more than a lot of land, and five times
+the cost of the lot equals the cost of the house. What was the
+cost of each?
+
+\item Mr. A. is 48 years older than his son, but he is only three
+times as old. How old is each?
+
+\item Two farms differ by 250 acres, and one is six times as large
+as the other. How many acres in each?
+
+\item William paid eight times as much for a dictionary as for a
+rhetoric. If the difference in price was \$6.30, how much did he
+pay for each?
+
+\item The sum of two numbers is 4256, and one is 37 times as
+great as the other. What are the numbers?
+
+\item Aleck has 48 cents more than Arthur, and seven times
+Arthur's money equals Aleck's. How much has each?
+
+\item The sum of the ages of a mother and daughter is 32 years,
+and the age of the mother is seven times that of the daughter.
+What is the age of each?
+
+\item John's age is three times that of Mary, and he is 10 years
+older. What is the age of each?
+\end{enumerate}
+
+\subsubsection*{Exercise 2.}
+\textit{ Illustrative Example.} There are three numbers whose sum
+is 96; the second is three times the first, and the third is four
+times the first. What are the numbers?
+
+\begin{center}
+\textbf{ Solution}.
+\end{center}
+
+\begin{center}
+\begin{tabular}{lr@{=}ll}
+\text{Let } &$x$ & first number,\\
+ &$3x$ & second number,\\
+ &$4x$ & third number.\\
+&$x + 3x + 4x$& 96\\
+ &$8x$ & 90\\
+ &$x$ & 12\\
+ &$3x$ & 36\\
+ &$4x$& 48\\
+\end{tabular}
+\end{center}
+The numbers are 12, 36, and 48.
+
+\begin{enumerate}
+\item A man bought a hat, a pair of boots, and a necktie
+for \$7.50; the hat cost four times as much as the necktie,
+and the boots cost five times as much as the necktie. What
+was the cost of each?
+
+\item A man traveled 90 miles in three days. If he traveled
+twice as far the first day as he did the third, and
+three times as far the second day as the third, how far
+did he go each day?
+
+\item James had 30 marbles. He gave a certain number
+to his sister, twice as many to his brother, and had three
+times as many left as he gave his sister. How many did
+each then have?
+
+\item A farmer bought a horse, cow, and pig for \$90. If
+he paid three times as much for the cow as for the pig, and
+five times as much for the horse as for the pig, what was
+the price of each?
+
+\item A had seven times as many apples, and B three times
+as many as C had. If they all together had 55 apples, how
+many had each?
+
+\item The difference between two numbers is 36, and one
+is four times the other. What are the numbers?
+
+\item In a company of 48 people there is one man to each
+five women. How many are there of each?
+
+\item A man left \$1400 to be distributed among three
+sons in such a way that James was to receive double
+what John received, and John double what Henry received.
+How much did each receive?
+
+\item A field containing 45,000 feet was divided into three
+lots so that the second lot was three times the first, and
+the third twice the second. How large was each lot?
+
+\item There are 120 pigeons in three flocks. In the second
+there are three times as many as in the first, and in the
+third as many as in the first and second combined. How
+many pigeons in each flock?
+
+\item Divide 209 into three parts so that the first part
+shall be five times the second, and the second three times
+the third.
+
+\item Three men, A, B, and C, earned \$110; A earned
+four times as much as B, and C as much as both A and B.
+How much did each earn?
+
+\item A farmer bought a horse, a cow, and a calf for \$72; the cow
+cost twice as much as the calf, and the horse three times as much
+as the cow. What was the cost of each?
+
+\item A cistern, containing 1200 gallons of water, is emptied by
+two pipes in two hours. One pipe discharges three times as many
+gallons per hour as the other. How many gallons does each pipe
+discharge in an hour?
+
+\item A butcher bought a cow and a lamb, paying six times as much
+for the cow as for the lamb, and the difference of the prices was
+\$25. How much did he pay for each?
+
+\item A grocer sold one pound of tea and two pounds of coffee for
+\$1.50, and the price of the tea per pound was three times that of
+the coffee. What was the price of each?
+
+\item By will Mrs. Cabot was to receive five times as much as her
+son Henry. If Henry received \$20,000 less than his mother, how
+much did each receive?
+\end{enumerate}
+
+\subsubsection*{Exercise 3.}
+
+\textit{ Illustrative Example}. Divide the number 126 into two
+parts such that one part is 8 more than the other.
+
+\begin{center}
+\textbf{ Solution}\\
+\end{center}
+\begin{center}
+\begin{tabular}{lr@{=}ll}
+\text{Let }&$x$& \text{less part,}\\
+&$x+8$& \text{greater part.}\\
+&$x+x+8$ & 126\\
+&$2x + 8$& 126 \\
+&$2x$& 118\footnotemark\\
+&$x$&59 \\
+&$x + 8$ & 67
+\end{tabular}
+\end{center}
+
+\footnotetext{Where in arithmetic did you learn the principle
+applied in transposing the 8?} The parts are 59 and 67.
+
+\begin{enumerate}
+
+\item In a class of 35 pupils there are 7 more girls than
+boys. How many are there of each?
+
+\item The sum of the ages of two brothers is 43 years, and
+one of them is 15 years older than the other. Find their
+ages.
+
+\item At an election in which 1079 votes were cast the successful
+candidate had a majority of 95. How many votes
+did each of the two candidates receive?
+
+\item Divide the number 70 into two parts, such that one
+part shall be 26 less than the other part.
+
+\item John and Henry together have 143 marbles. If I
+should give Henry 15 more, he would have just as many
+as John. How many has each?
+
+\item In a storehouse containing 57 barrels there are 3
+less barrels of flour than of meal. How many of each?
+
+\item A man whose herd of cows numbered 63 had
+17 more Jerseys than Holsteins. How many had he of
+each?
+
+\item Two men whose wages differ by 8 dollars receive
+both together \$44 per month. How much does each
+receive?
+
+\item Find two numbers whose sum is 99 and whose difference
+is 19.
+
+\item The sum of three numbers is 56; the second is 3
+more than the first, and the third 5 more than the first.
+What are the numbers?
+
+\item Divide 62 into three parts such that the first part
+is 4 more than the second, and the third 7 more than the
+second.
+
+\item Three men together received \$34,200; if the second
+received \$1500 more than the first, and the third \$1200
+more than the second, how much did each receive?
+
+\item Divide 65 into three parts such that the second part
+is 17 more than the first part, and the third 15 less than the
+first.
+
+\item A man had 95 sheep in three flocks. In the first
+flock there were 23 more than in the second, and in the
+third flock 12 less than in the second. How many sheep
+in each flock?
+
+\item In an election, in which 1073 ballots were cast, Mr.
+A receives 97 votes less than Mr. B, and Mr. C 120 votes
+more than Mr. B. How many votes did each receive?
+
+\item A man owns three farms. In the first there are
+5 acres more than in the second and 7 acres less than in
+the third. If there are 53 acres in all the farms together,
+how many acres are there in each farm?
+
+\item Divide 111 into three parts so that the first part
+shall be 16 more than the second and 19 less than the
+third.
+
+\item Three firms lost \$118,000 by fire. The second firm
+lost \$6000 less than the first and \$20,000 more than the
+third. What was each firm's loss?
+\end{enumerate}
+
+\subsubsection*{Exercise 4.}
+
+\textit{ Illustrative Example.} The sum of two numbers is 25, and
+the larger is 3 less than three times the smaller. What are the
+numbers?
+
+\begin{center}
+\textbf{ Solution.} \\
+\end{center}
+\begin{center}
+\begin{tabular}{lr@{=}l}
+Let & $x$ & smaller number,\\
+ & $3x - 3$ & larger number. \\
+ & $x+3x-3$ &$25$ \\
+ & $4x-3$ &$25$ \\
+ & $4x$ &$28$\footnotemark \\
+ & $x$ &$7$ \\
+ & $3x-3$ &$18$ \\
+\end{tabular}
+\end{center}
+
+The numbers are 7 and 18.
+
+\begin{enumerate}
+\item Charles and Henry together have 49 marbles, and
+Charles has twice as many as Henry and 4 more. How
+many marbles has each?
+
+\item In an orchard containing 33 trees the number of
+pear trees is 5 more than three times the number of apple
+trees. How many are there of each kind?
+
+\item John and Mary gathered 23 quarts of nuts. John
+gathered 2 quarts more than twice as many as Mary. How
+many quarts did each gather?
+
+\item To the double of a number I add 17 and obtain as
+a result 147. What is the number?
+
+\item To four times a number I add 23 and obtain 95.
+What is the number?
+
+\item From three times a number I take 25 and obtain 47.
+What is the number?
+
+\footnotetext{Is the same principle applied here that is applied on page 12?}
+
+\item Find a number which being multiplied by 5 and having 14
+added to the product will equal 69.
+
+\item I bought some tea and coffee for \$10.39. If I paid for the
+tea 61 cents more than five times as much as for the coffee, how
+much did I pay for each?
+
+\item Two houses together contain 48 rooms. If the second house
+has 3 more than twice as many rooms as the first, how many rooms
+has each house?
+
+\textit{ Illustrative Example}. Mr. Y gave \$6 to his three boys.
+To the second he gave 25 cents more than to the third, and to the
+first three times as much as to the second. How much did each
+receive?
+
+\begin{center}
+\textbf{ Solution.}\\
+\begin{tabular}{lr@{=}l}
+Let &$x$& number of cents third boy received,\\
+&$x + 25$& number of cents second boy received,\\
+&$3x + 75$& number of cents first boy received.\\
+&$x + x + 25 + 3x + 75$& 600\\
+&$5x + 100$& 600\\
+&$5x$& 500\\
+&$x$& 100\\
+&$x+ 25$& 125\\
+&$3x + 75$& $375$\\
+\end{tabular}
+\begin{tabular}{c}
+1st boy received \$3.75,\\
+2d boy received \$1.25,\\
+3d boy received \$1.00.\\
+\end{tabular}
+\end{center}
+
+\item Divide the number 23 into three parts, such that the second
+is 1 more than the first, and the third is twice the second.
+
+\item Divide the number 137 into three parts, such that
+the second shall be 3 more than the first, and the third
+five times the second.
+
+\item Mr. Ames builds three houses. The first cost \$2000
+more than the second, and the third twice as much as the
+first. If they all together cost \$18,000, what was the cost
+of each house?
+
+\item An artist, who had painted three pictures, charged
+\$18 more for the second than the first, and three times as
+much for the third as the second. If he received \$322 for
+the three, what was the price of each picture?
+
+\item Three men, A, B, and C, invest \$47,000 in business.
+B puts in \$500 more than twice as much as A, and C puts
+in three times as much as B. How many dollars does each
+put into the business?
+
+\item In three lots of land there are 80,750 feet. The
+second lot contains 250 feet more than three times as
+much as the first lot, and the third lot contains twice as
+much as the second. What is the size of each lot?
+
+\item A man leaves by his will \$225,000 to be divided
+as follows: his son to receive \$10,000 less than twice as
+much as the daughter, and the widow four times as much
+as the son. What was the share of each?
+
+\item A man and his two sons picked 25 quarts of berries.
+The older son picked 5 quarts less than three times as
+many as the younger son, and the father picked twice as
+many as the older son. How many quarts did each pick?
+
+\item Three brothers have 574 stamps. John has 15 less than Henry,
+and Thomas has 4 more than John. How many has each?
+\end{enumerate}
+
+\subsubsection*{Exercise 5}.
+
+\textit{ Illustrative Example}. Arthur bought some apples and
+twice as many oranges for 78 cents. The apples cost 3 cents
+apiece, and the oranges 5 cents apiece. How many of each did he
+buy?
+
+\begin{center}
+\textbf{ Solution}.
+\end{center}
+\begin{center}
+\begin{tabular}{l r c l}
+Let & $x$ &=& \mbox{number of apples}, \\
+& $2x$ &=& \mbox{number of oranges},\\
+& $3x$ &=& \mbox{cost of apples},\\
+& $10x$ &=& \mbox{cost of oranges}.\\
+& $3x + 10x $&=& 78\\
+& $13x$ &=& 78\\
+& $x$ &=& 6\\
+& $2x$ &=& 12\\
+\end{tabular}
+\end{center}
+
+Arthur bought 6 apples and 12 oranges.
+
+\begin{enumerate}
+\item Mary bought some blue ribbon at 7 cents a yard, and three
+times as much white ribbon at 5 cents a yard, paying \$1.10 for
+the whole. How many yards of each kind did she buy?
+
+\item Twice a certain number added to five times the double of
+that number gives for the sum 36. What is the number?
+
+\item Mr. James Cobb walked a certain length of time at the rate
+of 4 miles an hour, and then rode four times as long at the rate
+of 10 miles an hour, to finish a journey of 88 miles. How long did
+he walk and how long did he ride?
+
+\item A man bought 3 books and 2 lamps for \$14. The price of a
+lamp was twice that of a book. What was the cost of each?
+
+\item George bought an equal number of apples, oranges, and
+bananas for \$1.08; each apple cost 2 cents, each orange 4 cents,
+and each banana 3 cents. How many of each did he buy?
+
+\item I bought some 2-cent stamps and twice as many 5-cent stamps,
+paying for the whole \$1.44. How many stamps of each kind did I
+buy?
+
+\item I bought 2 pounds of coffee and 1 pound of tea for \$1.31;
+the price of a pound of tea was equal to that of 2 pounds of
+coffee and 3 cents more. What was the cost of each per pound?
+
+\item A lady bought 2 pounds of crackers and 3 pounds of
+gingersnaps for \$1.11. If a pound of gingersnaps cost 7 cents
+more than a pound of crackers, what was the price of each?
+
+\item A man bought 3 lamps and 2 vases for \$6. If a vase cost 50
+cents less than 2 lamps, what was the price of each?
+
+\item I sold three houses, of equal value, and a barn for
+\$16,800. If the barn brought \$1200 less than a house, what was
+the price of each?
+
+\item Five lots, two of one size and three of another, aggregate
+63,000 feet. Each of the two is 1500 feet larger than each of the
+three. What is the size of the lots?
+
+\item Four pumps, two of one size and two of another, can pump 106
+gallons per minute. If the smaller pumps 5 gallons less per minute
+than the larger, how much does each pump per minute?
+
+\item Johnson and May enter into a partnership in which Johnson's
+interest is four times as great as May's. Johnson's profit was
+\$4500 more than May's profit. What was the profit of each?
+
+\item Three electric cars are carrying 79 persons. In the first
+car there are 17 more people than in the second and 15 less than
+in the third. How many persons in each car?
+
+\item Divide 71 into three parts so that the second part shall be
+5 more than four times the first part, and the third part three
+times the second.
+
+\item I bought a certain number of barrels of apples and three
+times as many boxes of oranges for \$33. I paid \$2 a barrel for
+the apples, and \$3 a box for the oranges. How many of each did I
+buy?
+
+\item Divide the number 288 into three parts, so that the third
+part shall be twice the second, and the second five times the
+first.
+
+\item Find two numbers whose sum is 216 and whose difference is
+48.
+\end{enumerate}
+
+\subsubsection*{Exercise 6}.
+
+\textit{ Illustrative Example}. What number added to twice itself
+and 40 more will make a sum equal to eight times the number?
+
+\begin{center}
+\textbf{ Solution}.
+\end{center}
+
+\begin{center}
+\begin{tabular}{l r c l}
+Let & $x$ &=& the number. \\
+& $x + 2x + 40$ &=& $8x$\\
+& $3x + 40$ &=& $8x$\\
+& 40 &=& $5x$\\
+& $8 $&=& $x$\\
+\end{tabular}
+\end{center}
+
+The number is 8.
+
+\begin{enumerate}
+
+\item What number, being increased by 36, will be equal to ten
+times itself?
+
+\item Find the number whose double increased by 28 will equal six
+times the number itself.
+
+\item If John's age be multiplied by 5, and if 24 be added to the
+product, the sum will be seven times his age. What is his age?
+
+\item A father gave his son four times as many dollars as he then
+had, and his mother gave him \$25, when he found that he had nine
+times as many dollars as at first. How many dollars had he at
+first?
+
+\item A man had a certain amount of money; he earned three times
+as much the next week and found \$32. If he then had eight times
+as much as at first, how much had he at first?
+
+\item A man, being asked how many sheep he had, said, "If you will
+give me 24 more than six times what I have now, I shall have ten
+times my present number." How many had he?
+
+\item Divide the number 726 into two parts such that one shall be
+five times the other.
+
+\item Find two numbers differing by 852, one of which is seven
+times the other.
+
+\item A storekeeper received a certain amount the first month; the
+second month he received \$50 less than three times as much, and
+the third month twice as much as the second month. In the three
+months he received \$4850. What did he receive each month?
+
+\item James is 3 years older than William, and twice James's age
+is equal to three times William's age. What is the age of each?
+
+\item One boy has 10 more marbles than another boy. Three times
+the first boy's marbles equals five times the second boy's
+marbles. How many has each?
+
+\item If I add 12 to a certain number, four times this second
+number will equal seven times the original number. What is the
+original number?
+
+\item Four dozen oranges cost as much as 7 dozen apples, and a
+dozen oranges cost 15 cents more than a dozen apples. What is the
+price of each?
+
+\item Two numbers differ by 6, and three times one number equals
+five times the other number. What are the numbers?
+
+\item A man is 2 years older than his wife, and 15 times his age
+equals 16 times her age. What is the age of each?
+
+\item A farmer pays just as much for 4 horses as he does for 6
+cows. If a cow costs 15 dollars less than a horse, what is the
+cost of each?
+
+\item What number is that which is 15 less than four times the
+number itself?
+
+\item A man bought 12 pairs of boots and 6 suits of clothes for
+\$168. If a suit of clothes cost \$2 less than four times as much
+as a pair of boots, what was the price of each?
+\end{enumerate}
+
+\subsubsection*{Exercise 7}.
+
+\textit{ Illustrative Example}. Divide the number 72 into two
+parts such that one part shall be one-eighth of the other.
+\begin{center}
+\textbf{ Solution}.
+\end{center}
+\begin{center}
+\begin{tabular}{l r c l}
+Let & $x$ &=& greater part,\\
+& $\frac{1}{8}x$ &=& lesser part.\\
+& $x + \frac{1}{8}x$ &=& $72$\\
+& $\frac{9}{8}x $&=& $72$\\
+& $\frac{1}{8}x $&=& $8$\\
+& $x$&=& $64$\\
+\end{tabular}
+\end{center}
+
+The parts are 64 and 8.
+
+\begin{enumerate}
+\item Roger is one-fourth as old as his father, and the sum of
+their ages is 70 years. How old is each?
+
+\item In a mixture of 360 bushels of grain, there is one-fifth as
+much corn as wheat. How many bushels of each?
+
+\item A man bought a farm and buildings for \$12,000. The
+buildings were valued at one-third as much as the farm. What was
+the value of each?
+
+\item A bicyclist rode 105 miles in a day. If he rode one-half as
+far in the afternoon as in the forenoon, how far did he ride in
+each part of the day?
+
+\item Two numbers differ by 675, and one is one-sixteenth of the
+other. What are the numbers?
+
+\item What number is that which being diminished by one-seventh of
+itself will equal 162?
+
+\item Jane is one-fifth as old as Mary, and the difference of
+their ages is 12 years. How old is each?
+
+\textit{ Illustrative Example}. The half and fourth of a certain
+number are together equal to 75. What is the number?
+
+\begin{center}
+\textbf{ Solution}.
+\end{center}
+\begin{center}
+\begin{tabular}{l r c l}
+Let & $x$ &=& the number.\\
+& $\frac{1}{2}x + \frac{1}{4}x $ &=& 75.\\
+& $\frac{3}{4}x$ &=& $75$\\
+& $\frac{1}{4}x $&=& $25$\\
+& $x$&=& $100$\\
+\end{tabular}
+\end{center}
+
+The number is 100.
+
+\item The fourth and eighth of a number are together equal to 36.
+What is the number?
+
+\item A man left half his estate to his widow, and a fifth to his
+daughter. If they both together received \$28,000, what was the
+value of his estate?
+
+\item Henry gave a third of his marbles to one boy, and a fourth
+to another boy. He finds that he gave to the boys in all 14
+marbles. How many had he at first?
+
+\item Two men own a third and two-fifths of a mill respectively.
+If their part of the property is worth \$22,000, what is the value
+of the mill?
+
+\item A fruit-seller sold one-fourth of his oranges in the
+forenoon, and three-fifths of them in the afternoon. If he sold in
+all 255 oranges, how many had he at the start?
+
+\item The half, third, and fifth of a number are together equal to
+93. Find the number.
+
+\item Mr. A bought one-fourth of an estate, Mr. B one-half, and
+Mr. C one-sixth. If they together bought 55,000 feet, how large
+was the estate?
+
+\item The wind broke off two-sevenths of a pine tree, and
+afterwards two-fifths more. If the parts broken off measured 48
+feet, how high was the tree at first?
+
+\item A man spaded up three-eighths of his garden, and his son
+spaded two-ninths of it. In all they spaded 43 square rods. How
+large was the garden?
+
+\item Mr. A's investment in business is \$15,000 more than Mr.
+B's. If Mr. A invests three times as much as Mr. B, how much is
+each man's investment?
+
+\item A man drew out of the bank \$27, in half-dollars, quarters,
+dimes, and nickels, of each the same number. What was the number?
+\end{enumerate}
+
+\subsubsection*{Exercise 8}.
+
+\textit{ Illustrative Example}. What number is that which being
+increased by one-third and one-half of itself equals 22?
+
+\begin{center}
+\textbf{ Solution}.
+\end{center}
+\begin{center}
+\begin{tabular}{l r c l}
+Let & $x$ &=& the number.\\
+& $x + \frac{1}{3}x + \frac{1}{2}x $ &=& 22.\\
+& $1\frac{5}{6}x$ &=& $22$\\
+& $\frac{11}{6}x $&=& $22$\\
+& $\frac{1}{6}x $&=& $2$\\
+& $x$&=& $12$\\
+\end{tabular}
+\end{center}
+
+The number is 12.
+
+\begin{enumerate}
+
+\item Three times a certain number increased by one-half of the
+number is equal to 14. What is the number?
+
+\item Three boys have an equal number of marbles. John buys
+two-thirds of Henry's and two-fifths of Robert's marbles, and
+finds that he then has 93 marbles. How many had he at first?
+
+\item In three pastures there are 42 cows. In the second there are
+twice as many as in the first, and in the third there are one-half
+as many as in the first. How many cows are there in each pasture?
+
+\item What number is that which being increased by one-half and
+one-fourth of itself, and 5 more, equals 33?
+
+\item One-third and two-fifths of a number, and 11, make 44. What
+is the number?
+
+\item What number increased by three-sevenths of itself will
+amount to 8640?
+
+\item A man invested a certain amount in business. His gain the
+first year was three-tenths of his capital, the second year
+five-sixths of his original capital, and the third year \$3600. At
+the end of the third year he was worth \$10,000. What was his
+original investment?
+
+\item Find the number which, being increased by its third, its
+fourth, and 34, will equal three times the number itself.
+
+\item One-half of a number, two-sevenths of the number, and 31,
+added to the number itself, will equal four times the number. What
+is the number?
+
+\item A man, owning a lot of land, bought 3 other lots adjoining,
+-- one three-eighths, another one-third as large as his lot, and
+the third containing 14,000 feet, -- when he found that he had
+just twice as much land as at first. How large was his original
+lot?
+
+\item What number is doubled by adding to it two-fifths of itself,
+one-third of itself, and 8?
+
+\item There are three numbers whose sum is 90; the second is equal
+to one-half of the first, and the third is equal to the second
+plus three times the first. What are the numbers?
+
+\item Divide 84 into three parts, so that the third part shall be
+one-third of the second, and the first part equal to twice the
+third plus twice the second part.
+
+\item Divide 112 into four parts, so that the second part shall be
+one-fourth of the first, the third part equal to twice the second
+plus three times the first, and the fourth part equal to the
+second plus twice the first part.
+
+\item A grocer sold 62 pounds of tea, coffee, and cocoa. Of tea he
+sold 2 pounds more than of coffee, and of cocoa 4 pounds more than
+of tea. How many pounds of each did he sell?
+
+\item Three houses are together worth six times as much as the
+first house, the second is worth twice as much as the first, and
+the third is worth \$7500. How much is each worth?
+
+\item John has one-ninth as much money as Peter, but if his father
+should give him 72 cents, he would have just the same as Peter.
+How much money has each boy?
+
+\item Mr. James lost two-fifteenths of his property in
+speculation, and three-eighths by fire. If his loss was \$6100,
+what was his property worth?
+\end{enumerate}
+
+\subsubsection*{Exercise 9}.
+
+\begin{enumerate}
+\item Divide the number 56 into two parts, such that one part is
+three-fifths of the other.
+
+\item If the sum of two numbers is 42, and one is three-fourths of
+the other, what are the numbers?
+
+\item The village of C---- is situated directly between two cities
+72 miles apart, in such a way that it is five-sevenths as far from
+one city as from the other. How far is it from each city?
+
+\item A son is five-ninths as old as his father. If the sum of
+their ages is 84 years, how old is each?
+
+\item Two boys picked 26 boxes of strawberries. If John picked
+five-eighths as many as Henry, how many boxes did each pick?
+
+\item A man received 60-1/2 tons of coal in two carloads, one load
+being five-sixths as large as the other. How many tons in each
+carload?
+
+\item John is seven-eighths as old as James, and the sum of their
+ages is 60 years. How old is each?
+
+\item Two men invest \$1625 in business, one putting in
+five-eighths as much as the other. How much did each invest?
+
+\item In a school containing 420 pupils, there are three-fourths
+as many boys as girls. How many are there of each?
+
+\item A man bought a lot of lemons for \$5; for one-third he paid
+4 cents apiece, and for the rest 3 cents apiece. How many lemons
+did he buy?
+
+\item A lot of land contains 15,000 feet more than the adjacent
+lot, and twice the first lot is equal to seven times the second.
+How large is each lot?
+
+\item A bicyclist, in going a journey of 52 miles, goes a certain
+distance the first hour, three-fifths as far the second hour,
+one-half as far the third hour, and 10 miles the fourth hour, thus
+finishing the journey. How far did he travel each hour?
+
+\item One man carried off three-sevenths of a pile of loam,
+another man four-ninths of the pile. In all they took 110 cubic
+yards of earth. How large was the pile at first?
+
+\item Matthew had three times as many stamps as Herman, but after
+he had lost 70, and Herman had bought 90, they put what they had
+together, and found that they had 540. How many had each at first?
+
+\item It is required to divide the number 139 into four parts,
+such that the first may be 2 less than the second, 7 more than the
+third, and 12 greater than the fourth.
+
+\item In an election 7105 votes were cast for three candidates.
+One candidate received 614 votes less, and the other 1896 votes
+less, than the winning candidate. How many votes did each receive?
+
+\item There are four towns, A, B, C, and D, in a straight line.
+The distance from B to C is one-fifth of the distance from A to B,
+and the distance from C to D is equal to twice the distance from A
+to C. The whole distance from A to D is 72 miles. Required the
+distance from A to B, B to C, and C to D.
+\end{enumerate}
+
+\section*{MODES OF REPRESENTING THE OPERATIONS.}
+\addcontentsline{toc}{section}{\numberline{}MODES OF REPRESENTING
+THE OPERATIONS.}
+\subsection*{ADDITION.}\addcontentsline{toc}{subsection}{\numberline{}Addition.}
+
+\begin{tabular}{cll}
+\textbf{ 2.} &ILLUS. 1. & The sum of $y$ + $y$ + $y$ + etc.
+written
+seven times is 7$y$. \\
+\\
+&ILLUS. 2. & The sum of $m$ + $m$ + $m$ + etc. written $x$ times is $xm$.\\
+\\
+\end{tabular}
+
+The 7 and $x$ are called the coefficients of the number
+following.
+
+The \textbf{ coefficient} is the number which shows how many times
+the number following is taken additively. If no coefficient is
+expressed, \textit{ one} is understood.
+
+Read each of the following numbers, name the coefficient,
+and state what it shows:
+\[
+6a,\; 2y,\; 3x,\; ax,\; 5m,\; 9c,\; xy,\; mn,\; 10z,\; a,\; 25n,\;
+x,\; 11xy.
+\]
+
+\begin{tabular}{cll}
+&ILLUS. 3. & If John has $x$ marbles, and his brother gives him
+5\\
+& & marbles, how many has he?\\
+\\
+&ILLUS. 4. &If Mary has $x$ dolls, and her mother gives her $y$\\
+& &dolls, how many has she?\\
+\end{tabular}
+
+\textit{ \textbf{Addition is expressed by coefficient and by sign
+plus(+).}}
+
+When use the coefficient? When the sign?
+
+\subsubsection*{Exercise 10.}
+
+\begin{enumerate}
+\item Charles walked $x$ miles and rode 9 miles. How far did he
+go?
+
+\item A merchant bought $a$ barrels of sugar and $p$ barrels of
+molasses. How many barrels in all did he buy?
+
+\item What is the sum of $b$ + $b$ + $b$ + etc. written eight
+times?
+
+\item Express the sum of $x$ and $y$.
+
+\item There are $c$ boys at play, and 5 others join them. How many
+boys are there in all?
+
+\item What is the sum of $x$ + $x$ + $x$ + etc. written $d$ times?
+
+\item A lady bought a silk dress for $m$ dollars, a muff for $l$
+dollars, a shawl for $v$ dollars, and a pair of gloves for $c$
+dollars. What was the entire cost?
+
+\item George is $x$ years old, Martin is $y$, and Morgan is $z$
+years. What is the sum of their ages?
+
+\item What is the sum of $m$ taken $b$ times?
+
+\item If $d$ is a whole number, what is the next larger number?
+
+\item A boy bought a pound of butter for $y$ cents, a pound of
+meat for $z$ cents, and a bunch of lettuce for $s$ cents. How much
+did they all cost?
+
+\item What is the next whole number larger than $m$?
+
+\item What is the sum of $x$ taken $y$ times?
+
+\item A merchant sold $x$ barrels of flour one week, 40 the next
+week, and $a$ barrels the following week. How many barrels did he
+sell?
+
+\item Find two numbers whose sum is 74 and whose difference is 18.
+\end{enumerate}
+
+\subsection*{SUBTRACTION.}
+\addcontentsline{toc}{subsection}{\numberline{}Subtraction.}
+
+\begin{tabular}{cll}
+\textbf{ 3}. &ILLUS. 1. & A man sold a horse for \$225 and gained
+\$75.\\
+& & What did the horse cost?\\
+\\
+&ILLUS. 2. & A farmer sold a sheep for $m$ dollars and gained\\
+& & $y$ dollars. What did the sheep cost? \textit{Ans.} $m - y$
+dollars.\\
+\end{tabular}
+
+
+\textit{\textbf{Subtraction is expressed by the sign minus}}
+($-$).
+
+
+\begin{tabular}{cll}
+&ILLUS. 3. & A man started at a certain point and traveled\\
+& & north 15 miles, then south 30 miles, then north 20 miles,\\
+& &then north 5 miles, then south 6 miles. How far is he\\
+& &from where he started and in which direction?\\
+\\
+&ILLUS. 4. & A man started at a certain point and traveled east $x$\\
+& &miles, then west $b$ miles, then east $m$ miles, then east $y$\\
+& &miles, then west $z$ miles. How far is he from where he started?\\
+\end{tabular}
+
+
+We find a difficulty in solving this last example, because we do
+not know just how large $x, b, m, y$, and $z$ are with reference
+to each other. This is only one example of a large class of
+problems which may arise, in which we find direction east and
+west, north and south; space before and behind, to the right and
+to the left, above and below; time past and future; money gained
+and lost; everywhere these opposite relations. This relation of
+oppositeness must be expressed in some way in our representation
+of numbers.
+
+In algebra, therefore, numbers are considered as increasing from
+zero in opposite directions. Those in one direction are called
+Positive Numbers (or + numbers); those in the other direction
+Negative Numbers (or - numbers).
+
+In Illus. 4, if we call direction east positive, then direction
+west will be negative, and the respective distances that the man
+traveled will be $+x, - b, + m, + y$, and $-z$. Combining these,
+the answer to the problem becomes $x - b + m + y - z$. If the same
+analysis be applied to Illus. 3, we get 15 - 30 + 20 + 5 - 6 = +4,
+or 4 miles north of starting-point.
+
+\textbf{\textit{ The minus sign before a single number makes the
+number negative, and shows that the number has a subtractive
+relation to any other to which it may be united, and that it will
+diminish that number by its value. It shows a relation rather than
+an operation.}}
+
+\textit{Negative numbers} are the second of the two things
+referred to on page 7, the introduction of which makes all the
+difference between arithmetic and algebra.
+
+NOTE.---Negative numbers are usually spoken of as less than zero,
+because they are used to represent losses. To illustrate: suppose
+a man's money affairs be such that his debts just equal his
+assets, we say that he is worth nothing. Suppose now that the sum
+of his debts is \$1000 greater than his total assets. He is worse
+off than by the first supposition, and we say that he is worth
+less than nothing. We should represent his property by $-1000$
+(dollars).
+
+\subsubsection{Exercise 11.}
+
+\begin{enumerate}
+\item Express the difference between $a$ and $b$.
+
+\item By how much is $b$ greater than 10?
+
+\item Express the sum of $a$ and $b$ diminished by $c$.
+
+\item Write five numbers in order of magnitude so that $a$ shall
+be the middle number.
+
+\item A man has an income of $a$ dollars. His expenses are $b$
+dollars. How much has he left?
+
+\item How much less than $c$ is 8?
+
+\item A man has four daughters each of whom is 3 years older than
+the next younger. If $x$ represent the age of the oldest, what
+will represent the age of the others?
+
+\item A farmer bought a cow for $b$ dollars and sold it for $c$
+dollars. How much did he gain?
+
+\item How much greater than 5 is $x$?
+
+\item If the difference between two numbers is 9, how may you
+represent the numbers?
+
+\item A man sold a house for $x$ dollars and gained \$75. What did
+the house cost?
+
+\item A man sells a carriage for $m$ dollars and loses $x$
+dollars. What was the cost of the carriage?
+
+\item I paid $c$ cents for a pound of butter, and $f$ cents for a
+lemon. How much more did the butter cost than the lemon?
+
+\item Sold a lot of wood for $b$ dollars, and received in payment
+a barrel of flour worth $e$ dollars. How many dollars remain due?
+
+\item A man sold a cow for $l$ dollars, a calf for 4 dollars, and
+a sheep for $m$ dollars, and in payment received a wagon worth $x$
+dollars. How much remains due?
+
+\item A box of raisins was bought for $a$ dollars, and a firkin of
+butter for $b$ dollars. If both were sold for $c$ dollars, how
+much was gained?
+
+\item At a certain election 1065 ballots were cast for two
+candidates, and the winning candidate had a majority of 207. How
+many votes did each receive?
+
+\item A merchant started the year with $m$ dollars; the first
+month he gained $x$ dollars, the next month he lost $y$ dollars,
+the third month he gained $b$ dollars, and the fourth month lost
+$z$ dollars. How much had he at the end of that month?
+
+\item A man sold a cow for \$80, and gained $c$ dollars. What did
+the cow cost?
+
+\item If the sum of two numbers is 60, how may the numbers be
+represented?
+\end{enumerate}
+
+\subsection*{MULTIPLICATION}.
+\addcontentsline{toc}{subsection}{\numberline{}Multiplication.}
+
+\begin{tabular}{cll}
+\textbf{ 4.} &ILLUS. 1. &$4 \cdot 5 \cdot a \cdot b \cdot c$, $7
+\times 6$,
+$x \times y$.\\
+\\
+&ILLUS. 2. &$abc$, $xy$, $amx$.\\
+\\
+&ILLUS. 3. &$x \cdot x = xx = x^2$.\\
+&&$x \cdot x \cdot x = xxx = x^3$.
+\end{tabular}
+
+These two are read ``$x$ second power,'' or ``$x$ square,'' and
+``$x$ third power,'' or ``$x$ cube,'' and are called powers of
+$x$.
+
+A \textbf{ power} is a product of like factors.
+
+The 2 and the 3 are called the exponents of the power.
+
+An \textbf{ exponent} is a number expressed at the right and a
+little above another number to show how many times it is taken as
+a factor.
+
+\textbf{ \textit{Multiplication is expressed (1) by signs, i.e.
+the dot and the cross; (2) by writing the factors successively;
+(3) by exponent}.}
+
+The last two are the more common methods.
+
+When use the exponent? When write the factors successively?
+
+\subsubsection*{Exercise 12}.
+
+\begin{enumerate}
+\item Express the double of $x$.
+
+\item Express the product of $x, y$, and $z$.
+
+\item How many cents in $x$ dollars?
+
+\item Write $a$ times $b$ times $c$.
+
+\item What will $a$ quarts of cherries cost at $d$ cents a quart?
+
+\item If a stage coach, goes $b$ miles an hour, how far will it go
+in $m$ hours?
+
+\item In a cornfield there are $x$ rows, and $a$ hills in a row.
+How many hills in the field?
+
+\item Write the cube of $x$.
+
+\item Express in a different way $a \times a \times a \times a
+\times a \times a \times a \times a \times a$.
+
+\item Express the product of $a$ factors each equal to $d$.
+
+\item Write the second power of $a$ added to three times the cube
+of $m$.
+
+\item Express $x$ to the power $2m$, plus $x$ to the power $m$.
+
+\item What is the interest on $x$ dollars for $m$ years at 6 \%?
+
+\item In a certain school there are $c$ girls, and three times as
+many boys less 8. How many boys, and how many boys and girls
+together?
+
+\item If $x$ men can do a piece of work in 9 days, how many days
+would it take 1 man to perform the same work?
+
+\item How many thirds are there in $x$?
+
+\item How many fifths are there in $b$?
+
+\item A man bought a horse for $x$ dollars, paid 2 dollars a week
+for his keeping, and received 4 dollars a week for his work. At
+the expiration of $a$ weeks he sold him for $m$ dollars. How much
+did he gain?
+
+\item James has $a$ walnuts, John twice as many less 8, and Joseph
+three times as many as James and John less 7. How many have all
+together?
+\end{enumerate}
+
+
+\subsection*{DIVISION}.
+\addcontentsline{toc}{subsection}{\numberline{}Division.}
+
+\begin{tabular}{cll}
+\textbf{5.} & ILLUS. & $a \div b,\; \frac{\displaystyle x}{\displaystyle y}$\\
+\end{tabular}
+
+\textbf{ \textit{Division is expressed by the division sign, and
+by writing the numbers in the fractional form}.}
+
+\subsubsection*{Exercise 13}.
+
+\begin{enumerate}
+\item Express five times $a$ divided by three times $c$.
+
+\item How many dollars in $y$ cents?
+
+\item How many books at $a$ dimes each can be bought for $x$
+dimes?
+
+\item How many days will a man be required to work for $m$ dollars
+if he receive $y$ dollars a day?
+
+\item $x$ dollars were given for $b$ barrels of flour. What was
+the cost per barrel?
+
+\item Express $a$ plus $b$, divided by $c$.
+
+\item Express $a$, plus $b$ divided by $c$.
+
+\item A man had $a$ sons and half as many daughters. How many
+children had he?
+
+\item If the number of minutes in an hour be represented by $x$,
+what will express the number of seconds in 5 hours?
+
+\item A boy who earns $b$ dollars a day spends $x$ dollars a week.
+How much has he at the end of 3 weeks?
+
+\item A can perform a piece of work in $x$ days, B in $y$ days,
+and C in $z$ days. Express the part of the work that each can do
+in one day. Express what part they can all do in one day.
+
+\item How many square feet in a garden $a$ feet on each side?
+
+\item A money drawer contains $a$ dollars, $b$ dimes, and $c$
+quarters. Express the whole amount in cents.
+
+\item $x$ is how many times $y$?
+
+\item If $m$ apples are worth $n$ chestnuts, how many chestnuts is
+one apple worth?
+
+\item Divide 30 apples between two boys so that the younger may
+have two-thirds as many as the elder.
+\end{enumerate}
+
+
+\section*{ALGEBRAIC EXPRESSIONS.}
+\addcontentsline{toc}{section}{\numberline{}ALGEBRAIC
+EXPRESSIONS.}
+
+\begin{tabular}{cll}
+\textbf{ 6.} & ILLUS. & $ a,\; -c,\; b + 8,\; m-x+2c^2.$
+\end{tabular}
+
+An \textbf{ algebraic expression} is any representation of a
+number by algebraic notation.
+
+\begin{tabular}{lll}
+\textbf{ 7.} & ILLUS. 1.& $-3a^2b,\; 2x+a^2z^3-5d^4.$
+\end{tabular}
+
+$-3a^2b$ is called a term, $2x$ is a term, $+a^2z^3$ is a term,
+$-5d^4$ is a term.
+
+A \textbf{ term} is an algebraic expression not connected with any
+other by the sign plus or minus, or one of the parts of an
+algebraic expression with its own sign plus or minus. If no sign
+is written, the plus sign is understood. By what signs are terms
+separated?
+
+\begin{tabular}{l r r}
+ILLUS. 2.&$a^2bc$&$3x^2y^3$\\
+&$-7a^2bc$&$-x^2y^3$\\
+&$5a^2bc$&$\frac{1}{2}x^2y^3$
+\end{tabular}
+
+The terms in these groups are said to be similar.
+
+\begin{tabular}{l r r r}
+ILLUS. 3.& $x^2y$&$xy$&$x^2y$\\
+&$3a^2b$&$3x^2y$&$3ab$\\
+\end{tabular}
+
+The terms of these groups are said to be dissimilar.
+
+\textbf{ Similar terms} are terms having the same letters affected
+by the same exponents.
+
+\textbf{ Dissimilar terms} are terms which differ in letters or
+exponents, or both.
+
+How may similar terms differ?
+
+\begin{tabular}{ccr@{....}c@{....}l}
+ILLUS. & 4.& $abxy$ & fourth degree & $7x^2y^2$\\
+ && $x^3$ & third degree & $abc$\\
+ &&$3xy$ & second degree & $a^2$\\
+ && $2a^2bx^3$ & sixth degree & $4a^5b$\\
+\end{tabular}
+
+The \textbf{ degree of a term} is the number of its literal
+factors. It can be found by taking the sum of its exponents.
+
+\begin{tabular}{lr}
+ILLUS. 5. & $2x^4$ \\
+& $-a^3y$\\
+& $5x^2y^2$\\
+\end{tabular}
+
+How do these terms compare with reference to degree? They are
+called \textit{homogeneous} terms.
+
+What are homogeneous terms?
+
+\[
+\begin{array}{llll}
+\textbf{8.}& \text{ILLUS.}& 3x^2y& \text{called a monomial.}\\
+&& \left. \begin{array}{ll}
+7x^3 - 2xy \\
+3y^4 - z^2 & +3yz^2
+\end{array} \right\}
+& \text{ called polynomials.}
+\end{array}
+\]
+
+A \textbf{monomial} is an algebraic expression of one term.
+
+A \textbf{polynomial} is an algebraic expression of more
+than one term.
+
+A polynomial of two terms is called a \textbf{binomial}, and
+one of three terms is called a \textbf{trinomial}.
+
+The \textbf{degree of an algebraic expression} is the same
+as the degree of its highest term. What is the degree
+of each of the polynomials above? What is a homogeneous
+polynomial?
+
+\subsubsection*{Exercise 14.}
+\begin{enumerate}
+\item Write a polynomial of five terms. Of what degree is it?
+
+\item Write a binomial of the fourth degree.
+
+\item Write a polynomial with the terms of different degrees.
+
+\item Write a homogeneous trinomial of the third degree.
+
+\item Write two similar monomials of the fifth degree which shall
+differ as much as possible.
+
+\item Write a homogeneous trinomial with one of its terms of the
+second degree.
+
+\item Arrange according to the descending powers of $a$:
+$-80a^3b^3 + 60a^4b^2 + 108ab^5 + 48a^5b + 3a^6 - 27b^6 -
+90a^2b^4$.
+
+What name? What degree?
+
+\item Write a polynomial of the fifth degree containing six terms.
+
+\item Arrange according to the ascending powers of $x$:
+
+\[ 15x^2y^3 + 7x^5 - 3xy^5 - 60x^4y + y^7 + 21x^3y^2 \]
+
+What name? What degree? What is the degree of
+each term?
+
+When $a = 1$, $b = 2$, $c = 3$, $d = 4$, $x = O$, $y = 8$, find
+the value of the following:
+
+\item $2a + 3b + c$.
+
+\item $5b + 3a - 2c + 6x$.
+
+\item $6bc - 3ax + 2xb - 5ac + 2cx$.
+
+\item $3bcd + 5cxa - 7xab + abc$.
+
+\item $2c^2 + 3b^3 + 4a^4$.
+
+\item $\frac{1}{2}a^3c - b^3 - c^3 - \frac{3}{4}abc^3$.
+
+\item $2a - b - \frac{2ab}{a + b}$.
+
+\item $2bc - \frac{3}{4}c^3 + 3ab - 2a - x + \frac{4}{15}bx$.
+
+\item $\frac{a^2bx + ab^2c + abc^2 + xac^2}{abc}$.
+
+\item Henry bought some apples at 3 cents apiece, and twice as
+many pears at 4 cents apiece, paying for the whole 66 cents. How
+many of each did he buy?
+
+\item Sarah's father told her that the difference between
+two-thirds and five-sixths of his age was 6 years. How old was he?
+\end{enumerate}
+
+\chapter*{OPERATIONS.}
+\addcontentsline{toc}{chapter}{\numberline{}OPERATIONS.}
+
+\section*{ADDITION.}
+\addcontentsline{toc}{section}{\numberline{}ADDITION.}
+
+\textbf{ 9.} In combining numbers in algebra it must always be
+borne in mind that negative numbers are the opposite of positive
+numbers in their tendency.
+
+\begin{tabular}{l r r}
+
+ILLUS. 1.&$3ax$&$-7b^2y$\\
+&$5ax$&$-3b^2y$\\
+&$2ax$&$-4b^2y$\\
+&$\overline{10ax}$&$\overline{-14b^2y}$
+\end{tabular}
+
+\textit{ To add similar terms with like signs, add the
+coefficients, annex the common letters, and prefix the common
+sign.}
+
+\begin{tabular}{l r r}
+
+ILLUS. 2.&$5a^2b$&$3x^2y^2$\\
+&$-3a^2b$&$8x^2y^2$\\
+&$-4a^2b$&$-5x^2y^2$\\
+&$6a^2b$&$-7x^2y^2$\\
+&$\overline{4a^2b}$&$\overline{-x^2y^2}$
+
+\end{tabular}
+
+\textit{ To add similar terms with unlike signs, add the
+coefficients of the plus terms, add the coefficients of the minus
+terms, to the difference of these sums annex the common letters,
+and prefix the sign of the greater sum.}
+
+\begin{tabular}{l c c}
+ILLUS. 3.&$a$&$2x$\\
+&$b$&$-5y$\\
+&$c$&$-3a$\\
+&$\overline{a + b + c}$&$\overline{2x - 5 - 3a}$\\
+\end{tabular}
+
+\textit{ To add dissimilar terms, write the terms successively,
+each with its own sign}.
+
+\begin{tabular}{lrrrr}
+ILLUS. 4. &$2ab$ & $-3ax^2 $ & $+ 2a^2x$\\
+&$-8ab $ & $-ax^2 $ & $- 5a^2x $&$+ ax^3$\\
+&$12ab $&$+10ax^2 $ & $- 6a^2x$\\
+\cline{2-5}\\
+&$6ab $&$+6ax^2$ & $ - 9a^2x$ & $ + ax^3$\\
+\end{tabular}
+
+\textit{ To add polynomials, add the terms of which the
+polynomials consist, and unite the results}.
+
+\subsubsection*{Exercise 15.}
+
+Find the sum of:
+\begin{enumerate}
+
+\item $3x$, $5x$, $x$, $4x$, $11x$.
+
+\item $5ab$, $6ab$, $ab$, $13ab$.
+
+\item $-3ax^3$, $-5ax^3$, $-9ax^3$, $-ax^3$.
+
+\item $-x$, $-5x$, $-11x$, $-25x$.
+
+\item $-2a^2$, $5a^2$, $3a^2$, $-7a^2$, $11a^2$.
+
+\item $2abc^2$, $-5abc^2$, $abc^2$, $-8abc^2$.
+
+\item $5x^2$, $3ab$, $-2ab$, $-4a^2$, $5ab$, $-2a^2$.
+
+\item $5ax$, $-3bc$, $-2ax$, $7ax$, $bc$, $-2bc$.
+
+Simplify:
+
+\item $4a^2 - 5a^2 - 8a^2 - 7a^2$.
+
+\item $x^5 + 5a^4b - 7ab - 2x^5 + 10ab + 3a^4b$.
+
+\item $\frac{1}{3}a - \frac{1}{2}a + \frac{2}{3}a + a$.
+
+\item $\frac{2}{3}b - \frac{3}{4}b - 2b - \frac{1}{3}b +
+\frac{5}{6}b + b$.
+
+\item A lady bought a ribbon for $m$ cents, some tape for $d$
+cents, and some thread for $c$ cents. She paid $x$ cents on the
+bill. How much remains due?
+
+\item A man travels $a$ miles north, then $x$ miles south, then 5
+miles further south, and then $y$ miles north. How far is he from
+his starting point?
+
+Add:
+
+\item $a + 2b + 3c$, $5a + 3b + c$, $c - a - b$.
+
+\item $x + y - z$, $x - y - z$, $y - x + z$.
+
+\item $x + 2y - 3z + a$, $2x - 3y + z - 4a$, $2a - 3x + y - z$.
+
+\item $x^3 + 3x^2 - x + 5$, $4x^2 - 5x^3 + 3 - 4x$, $3x + 6x^3 -
+3x^2 + 9$.
+
+\item $ca - bc + c^3$, $ab + b^3 - ca$, $a^3 - ab + bc$.
+
+\item $3a^m - a^{m-1} - 1$, $3a^{m-1} + 1 - 2a^m$, $a^{m-1} + 1$.
+
+\item $5a^5 - 16a^4b - 11a^2b^2c + 13ab$, $-2a^5 + 4a^4b +
+12a^2b^2c - 10ab$, $6a^5 - a^4b - 6a^2b^2c + 10ab$, $-10a^5 +
+8a^4b + a^2b^2c - 6ab$, $a^5 + 5a^4b + 6a^2b^2c - 7ab$.
+
+\item $15x^3 + 35x^2 + 3x +7$, $7x^3 + 15x - 11x^2 + 9$, $9x - 10
++ x^3 - 4x^2$.
+
+\item $9x^5y - 6x^4y^2 + x^3y^3 - 25xy^5$, $-22x^3y^3 - 3xy^5 -
+9x^5y - 3x^4y^2$, $5x^3y^3 + x^5y + 21x^4y^2 + 20xy^5$.
+
+\item $x - y - z - a - b$, $x + y + z + a + b$, $x + y + z + a -
+b$, $x + y - z - a - b$, $x + y + z - a - b$.
+
+\item $a^2c + b^2c + c^3 - abc - bc^2 - ac^2$, $a^2b + b^3 - bc^2
+- ab^2 - b^2c - abc$, $a^3 + ab^2 + ac^2 - a^2b - abc - a^2c$.
+
+\item A regiment is drawn up in $m$ ranks of $b$ men each, and
+there are $c$ men over. How many men in the regiment?
+
+\item A man had $x$ cows and $z$ horses. After exchanging 10 cows
+with another man for 19 horses, what will represent the number
+that he has of each?
+
+\item In a class of 52 pupils there are 8 more boys than girls.
+How many are there of each?
+\end{enumerate}
+
+What is the sum of two numbers equal numerically
+but of opposite sign? How does the sum of a positive
+and negative number compare in value with the positive
+number? with the negative number? How does the
+sum of two negative numbers compare with the numbers?
+Illustrate the above questions by a man traveling
+north and south.
+
+
+
+
+\section*{SUBTRACTION.}
+\addcontentsline{toc}{section}{\numberline{}SUBTRACTION.}
+
+\textbf{ 10.} How is subtraction related to addition? How are
+opposite relations expressed?
+
+Given the typical series of numbers
+
+$\mathbf{- 4a,\; - 3a,\; - 2a,\; - a,\; - 0,\; a,\; 2a,\; 3a,\;
+4a,\; 5a.}$
+
+What must be added to $2a$ to obtain $5a$? What then must be
+subtracted from $5a$ to obtain $2a?$ $5a - 3a = ?$
+
+What must be added to $-3a$ to obtain $4a$? What then must be
+subtracted from $4a$ to obtain $-3a$? $4a - 7a = ?$
+
+What must be added to $3a$ to obtain $-2a$? What
+then must be subtracted from $-2a$ to obtain $3a$?
+$(-2a)-(-5a)=?$
+
+What must be added to $-a$ to obtain $-4a$? What
+then must be subtracted from $-4a$ to obtain $-a$?
+$(-4a)-(-3a)=?$
+
+Examine now these results expressed in another form.
+
+\begin{tabular}{llrlr}
+1.&From&$5a$&To&$5a$\\
+ &take&$3a$&add&$-3a$\\
+ & & $\overline{\;2a}$& &$\overline{\;2a}$\\
+ \\
+2.&From&4a&To&4a\\
+ &take&$7a$&add&$-7a$\\
+ & &$\overline{-3a}$& &$\overline{-3a}$\\
+ \\
+3.&From&$2a$&To&$2a$\\
+ &take&$5a$&add&$-5a$\\
+ & &$\overline{-3a}$& &$\overline{-3a}$\\
+ \\
+4.&From&$-4a$&To&$-4a$\\
+ &take&$-3a$&add&$3a$\\
+ & &$\overline{- a}$& &$\overline{- a}$
+\end{tabular}
+
+The principle is clear; namely,
+
+\textbf{ \textit{The subtraction of any number gives the same
+result as the addition of that number with the opposite sign.}}
+
+\begin{tabular}{cc}
+ILLUS.& \\
+&$ 6a+3b- c $ \\
+&$-4a+ b-5c$\\
+&$\overline{10a+2b+4c}$
+\end{tabular}
+
+\textit{To subtract one number from another, consider the sign of
+the subtrahend changed and add.}
+
+What is the relation of the minuend to the subtrahend
+and remainder? What is the relation of the
+subtrahend to the minuend and remainder?
+
+\subsubsection*{Exercise 16.}
+\begin{enumerate}
+\item From $5a^{3}$ take $3a^{3}$.
+
+\item From $7a^{2}b$ take $-5a^{2}b$.
+
+\item Subtract $7xy^{3}$ from $-2xy^{3}$.
+
+\item From $-3x^{m}y$ take $-7x^{m}y$.
+
+\item Subtract $3ax$ from $8x^{2}$.
+
+\item From $5xy$ take $-7by$.
+
+\item What is the difference between $4a^{m}$ and $2a^{m}$?
+
+\item From the difference between $5a^{2}x$ and $-3a^{2}x$ take
+the sum of $2a^{2}x$ and $-3a^{2}x$.
+
+\item From $2a + b + 7c$ take $5a + 2b - 7c$.
+
+\item From $9x - 4y + 3z$ take $5x - 3y + z$.
+
+\item Subtract $3x^{4} - x^{2} + 7x - 14$ from $11x^{4} - 2x^{3} -
+8x$.
+
+\item From $10a^{2}b^{2} + 15ab^{2} + 8a^{2}b$ take $-10a^{2}b^{2}
++ 15ab^{2} - 8a^{2}b$.
+
+\item Subtract $1- x + x^{2} - 3x^{3}$ from $x^{3} - 1 + x^{2} -
+x$.
+
+\item From $x^{m} - 2x^{2m} + x^{3m}$ take $2x^{3m} - x^{2m} -
+x^{m}$.
+
+\item Subtract $a^{2n} + a^{n}x^{n} + x^{2n}$ from $3a^{2n} -
+17a^{n}x^{n} - 8x^{2n}$.
+
+\item From $\frac{2}{3}a^{2} - \frac{5}{2}a - 1$ take
+$-\frac{2}{3}a^{2} + a -\frac{1}{2}$.
+
+\item From $x^{5} + 3xy^{4}$ take $x^{5} + 2x^{4}y + 3x^{3}y^{2} -
+2xy^{4} + y^{5}$.
+
+\item From $x$ take $y - a$.
+
+\item From $6a^3 + 4a + 7$ take the sum of $2a^3 + 4a^2 + 9$ and
+$4a^3 - a^2 + 4a - 2$.
+
+\item Subtract $3x - 7x^3 + 5x^2$ from the sum of $2 + 8x^2 - x^3$
+and $2x^3 - 3x^2 + x - 2$.
+
+\item What must be subtracted from $15y^3 + z^3 + 4yz^2 - 5z^2x -
+2xy^2$ to leave a remainder of $6x^3 - 12y^3 + 4z^3 - 2xy^2 +
+6z^2x$?
+
+\item How much must be added to $x^3 - 4x^2 + 16x$ to produce $x^3
++ 64$?
+
+\item To what must $4a^2 - 6b^2 + 8bc - 6ab$ be added to produce
+zero?
+
+\item From what must $2x^4 - 3x^2 + 2x - 5$ be subtracted to
+produce unity?
+
+\item What must be subtracted from the sum of $3a^3 + 7a + 1$ and
+$2a^2 - 5a - 3$ to leave a remainder of $2a^2 - 2a^3 - 4$?
+
+\item From the difference between $10a^2b + 8ab^2 - 8a^2b^2 - b^3$
+and $5a^2b - 6ab^3 - 7a^2b^2$ take the sum of $10a^2b^2 + 15ab^2 +
+8a^2b$ and $8a^2b - 5ab^2 + a^2b^2$.
+
+\item What must be added to $a$ to make $b$?
+
+\item By how much does $3x - 2$ exceed $2x + 1$?
+
+\item In $y$ years a man will be 40 years old. What is his present
+age?
+
+\item How many hours will it take to go 23 miles at $a$ miles an
+hour?
+\end{enumerate}
+
+\subsection*{PARENTHESES}.
+\addcontentsline{toc}{subsection}{\numberline{}PARENTHESES.}
+
+\begin{tabular}{lcl}
+\textbf{11.}& ILLUS. 1.& $5 (a + b)$.\\
+&ILLUS. 2.& $(m + n) (x + y)$.\\
+&ILLUS. 3.& $x - (a + y - c)$.\\
+\end{tabular}
+
+The parenthesis indicates that the numbers enclosed are considered
+as one number.
+
+Read each of the above illustrations, state the operations
+expressed, and show what the parenthesis indicates.
+
+Write the expressions for the following:
+
+\begin{enumerate}
+\item The sum of $a$ and $b$, multiplied by $a$ minus $b$.
+
+\item $c$ plus $d$, times the sum of $a$ and $b$,---the whole
+multiplied by $x$ minus $y$.
+
+\item The sum of $a$ and $b$, minus the difference between two $a$
+and three $b$.
+
+\item $(x - y) + (x - y) + (x - y) +$ etc., written $a$ times.
+
+\item The sum of $a + b$ taken seven times.
+
+\item There are in a library $m + n$ books, each book has $c - d$
+pages, and each page contains $x + y$ words. How many words in all
+the books?
+\end{enumerate}
+
+\begin{tabular}{ll}
+ILLUS. 4.& $a + (b - c - x) = a + b - c - x$.\\
+&(By performing the addition.)\\
+ILLUS. 5.& $a + c - d + e = a + (c - d + e)$.
+\end{tabular}
+
+\textbf{\textit{Any number of terms may be removed from a
+parenthesis preceded by the plus sign without change in the
+terms.}}
+
+And conversely,
+
+\textbf{\textit{Any number of terms may be enclosed in a
+parenthesis preceded by the plus sign without change in the
+terms}.}
+
+\begin{tabular}{lc}
+ILLUS. 6. &$x - (y + z - c) = x - y - z + c$.\\
+&(By performing the subtraction.)\\
+ILLUS. 7. &$a - b - c + d = a - (b + c - d)$.
+\end{tabular}
+
+\textbf{\textit{Any number of terms may be removed from a
+parenthesis preceded by the minus sign by changing the sign of
+each term}.}
+
+And conversely,
+
+\textbf{\textit{Any number of terms may be enclosed in a
+parenthesis preceded by the minus sign by changing the sign of
+each term}.}
+
+\subsubsection*{Exercise 17}.
+
+Remove the parentheses in the following:
+\begin{enumerate}
+\item $x + (a + b) + y + (c - d) + (x - y)$.
+
+\item $a + (b - c) - b + (a + c) + (c - a)$.
+
+\item $a^{2}b - (a^{3} + b^{3}) - a^{3} - (ab^{2} - a^{2}b)
+-(b^{3} - a^{3})$.
+
+\item $xy - (x^{2} + y^{2}) - y^{2} - (x^{2} - 2xy) - (y^{2} -
+x^{2})$.
+
+\item $(a + b - c) - (a - b + c) + (b - a - c) - (c - a - b)$.
+
+\item $(x - y + z) + (x + y + z) - (y + x + z) - (z + x + y)$.
+
+\item $a - (3b - 2c + a) - (2b - a - c) - (6 - c + a)$.
+
+\item $\frac{1}{2}a - \frac{1}{2}c - (\frac{2}{3}b - \frac{1}{2}c)
+- (a + \frac{1}{4}c - \frac{1}{3}b) - (\frac{2}{3}b - \frac{1}{4}c
+- \frac{1}{2}a)$.
+
+In each of the following enclose the last two terms in a
+parenthesis preceded by a plus sign:
+
+\item $x - y + 2c - d$.
+
+\item $2 a^2 + 3a^3x-ab^2 + by^2$.
+
+\item $10 m^3 +31m^2-20m-21$.
+
+\item $ax^4 - x^3 + 2x - 2ax^2$.
+
+In each of the following enclose the last three terms in a
+parenthesis preceded by a minus sign:
+
+\item $a^4 + a^3x + a^2x^2 - ax^3- 4x^4$.
+
+\item $a^4 + a^3-6a^2+a + 3$.
+
+\item $6a^3 - 17 a^2x + 14 ax^2 - 3x^3$.
+
+\item $ax^3 + 2ax^2 + ax + 2a$.
+
+\item A man pumps $x$ gallons of water into a tank each day, and
+draws off $y$ gallons each day. How much water will remain in the
+tank at the end of five days?
+
+\item Two men are 150 miles apart, and approach each other, one at
+the rate of $x$ miles an hour, the other at the rate of $y$ miles
+an hour. How far apart will they be at the end of seven hours?
+
+\item Eight years ago A was $x$ years old. How old is he now?
+
+\item A had $x$ dollars, but after giving \$35 to B he has
+one-third as much as B. How much has B?
+\end{enumerate}
+
+\section*{MULTIPLICATION.}
+\addcontentsline{toc}{section}{\numberline{}MULTIPLICATION.}
+
+\begin{align*}
+\textrm{\textbf{12.} ILLUS. } 1.\quad & 8=2 \cdot 2 \cdot 2 & a^3 = a \cdot a \cdot a\\
+& 6= 2 \cdot 3 & b^2 = b \cdot b \\
+&\overline{48 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} &
+\overline{a^3b^2 = a \cdot a \cdot a \cdot b \cdot b}\\
+\\
+\textrm{ILLUS. } 2.\quad & 2 a^2b^3c\\
+& 3a^4b^2c^3\\
+&\overline{6a^6b^5c^4}
+\end{align*}
+
+In arithmetic you learned that multiplication is the addition of
+equal numbers, that the multiplicand expresses one of those equal
+numbers, and the multiplier the number of them. In algebra we have
+negative as well as positive numbers. Let us see the effect of
+this in multiplication. We have four possible cases.
+
+\begin{enumerate}
+
+\item Multiplication of a plus number by a plus number.
+
+\begin{tabular}{ccc}
+& $+7$&\\
+ILLUS. &$\underline{+ 4}$& This must mean \textit{four sevens}, or
+28.
+\end{tabular}
+
+\item Multiplication of a minus number by a plus number.
+
+\begin{tabular}{ccc}
+&$-7$ &\\
+ILLUS. &$\underline{+ 4}$ &This must mean \textit{four
+minus-sevens}, or $-
+28$.\\
+\end{tabular}
+
+\item Multiplication of a plus number by a minus number.
+
+\begin{tabular}{ccl}
+&$+7$&\\
+ILLUS. &$\underline{- 4}$ &\\
+\end{tabular}
+This must mean the opposite of what $+4$ meant as a multiplier.
+Plus four meant add, minus four must mean subtract.
+\textit{Subtracting four sevens} gives $-28$.
+
+\item Multiplication of a minus number by a minus number.
+
+\begin{tabular}{ccl}
+&$-7$&\\
+
+ILLUS. &$\underline{-4}$& This must mean \textit{subtract four
+minus-sevens}, or 28.
+\end{tabular}
+\end{enumerate}
+
+\begin{tabular}{ccccc}
+ILLUS. 3. &$+b$&$-b$&$+b$&$-b$\\
+& $\underline{+a}$& $\underline{+a}$& $\underline{-a}$& $\underline{-a}$\\
+&$ab$ &$-ab$&$-ab$&$ab$\\
+\end{tabular}
+
+
+\textit{To multiply a monomial by a monomial, multiply the
+coefficients together for the coefficient of the product, add the
+exponents of like letters for the exponent of the same letter in
+the product, and give the product of two numbers having like signs
+the plus sign, having unlike signs the minus sign.}
+
+\subsubsection*{Exercise 18.}
+
+Find the product of:
+\begin{enumerate}
+\item $5x$ and $7c$.
+
+\item $51cy$ and $-xa$.
+
+\item $3x^{3}y$ and $7axy^{2}$.
+
+\item $5a^{2}bc$ and $2ab^{2}c^{3}$.
+
+\item $-3x^{2}y$, $-2ax$, and $3cy^{2}$.
+
+\item $5a^{2}$, $-3bc^{3}$, and $-2abc$.
+
+\item $15x^{2}y^{2}$, $-\frac{2}{3} xz$, and $\frac{1}{10}
+yz^{2}$.
+
+\item $20a^{3}b^{2}$, $\frac{2}{5} ab^{2}$, and $-\frac{1}{8}
+bc^{2}$.
+
+\item $\frac{1}{2} xy$, $-\frac{2}{3} cx^{2}$, $-\frac{2}{5}
+a^{2}y$, and $-\frac{5}{3} x^{2}y^{2}$.
+
+\item $-\frac{2}{3} a^{2}b^{2}$, $\frac{1}{4} c^{2}$,
+$-\frac{6}{5}ac$, and $-\frac{3}{4} b^{3}c$.
+
+\item In how many days will $a$ boys eat 100 apples if each boy
+eats $b$ apples a day?
+
+\item How many units in $x$ hundreds?
+
+\item If there are $a$ hundreds, $b$ tens, and $c$ units in a
+number, what will represent the whole number of units?
+
+\item If the difference between two numbers is 7, and one of the
+numbers is $x$, what is the other number?
+\end{enumerate}
+
+\begin{tabular}{cr}
+\textbf{13.} ILLUS. 1.&\\
+&$a - b + c$\\
+& $x$\\
+&$\overline{ax-bx+cx}$
+\end{tabular}
+
+\textit{To multiply a polynomial by a monomial, multiply
+each term of the polynomial by the monomial,
+and add the results.}
+
+ILLUS. 2. \[\begin{matrix}
+ \;x^3 + 2x^2 + 3x \\
+ \underline{3x^2 - 2x + 1} \\
+ 3x^5 + 6x^4 + 9x^3 \\
+ \qquad\qquad\quad -2x^4 - 4x^3 -6x^2 \\
+ \qquad\qquad \underline{\qquad \qquad\qquad\quad x^3 + 2x^2 +3x} \\
+ \qquad\qquad\quad 3x^5 + 4x^4 +6x^3 - 4x^2 + 3x
+ \end{matrix}\]
+
+\textit{To multiply a polynomial by a polynomial, multiply
+the multiplicand by each term of the multiplier,
+and add the products.}
+
+How is the first term of the product obtained? How is
+the last term obtained? The polynomials being arranged
+similarly with reference to the exponents of some number,
+how is the product arranged?
+
+\subsubsection*{Exercise 19.}
+
+Multiply:
+
+\begin{enumerate}
+\item $ x^2 + xy + y^2 \mbox{by} x^2y^2 $.
+
+\item $ a^2 - ab + b^2 \mbox{by} a^2b $.
+
+\item $ a^3 - 3a^2b + b^3 \mbox{by} -2ab $.
+
+\item $ 8x^3 + 36x^2y + 27y^3 by 3xy^2 $.
+
+\item $ \frac{5}{6}a^4 - \frac{1}{5}a^3b - \frac{1}{3}a^2b^2
+\mbox{by} \frac{6}{5}ab^2 $.
+
+\item $ x^2 - xy + y^2 \mbox{by} x + y $.
+
+\item $x^4 - 3x^3 + 2x^2 - x + 1$ by $x-1$.
+
+\item $x^3 - 2x^2 + x$ by $x^2 + 3x + 1$.
+
+\item $xy + mn - xm - yn$ by $xy - mn + xm - yn$.
+
+\item $x^4 - x^3 + x^2 - x + 1$ by $2 + 3x + 2x^2 + x^3$.
+
+\item $a^5 - a^4b + a^3b^2 - a^2b^3 + ab^4 - b^5$ by $a + b$.
+
+\item $x^2 - xy + y^2 - yz + z^2 - xz$ by $x + y + z$.
+
+\item $x^6 + x^5y + x^4y^2 + x^3y^3 + x^2y^4 + xy^5 + y^6$ by $x -
+y$.
+
+\item $x^4 - 4a^2x^2 + 4a^4$ by $x^4 + 4a^2x^2 + 4a^4$.
+
+\item $a^3 - 3a^2y^2 + y^3$ by $a^3 + 3a^2y^2 +y^3$.
+
+\item $x^4 + 10x + 12 + 9x^2 + 3x^3$ by $-2x + x^2 - 1$.
+
+\item $3x^2 - 2 + x^3 - 3x + 6x^4$ by $-2 + x^2 - 3x$.
+
+\item If $x$ represent the number of miles a man can row in an
+hour in still water, how far can the man row in 5 hours down a
+stream which flows $y$ miles an hour? How far up the same stream
+in 4 hours?
+
+\item A can reap a field in 7 hours, and B can reap the same field
+in 5 hours. How much of the field can they do in one hour, working
+together?
+
+\item A tank can be filled by two pipes in $a$ hours and $b$ hours
+respectively. What part of the tank will be filled by both pipes
+running together for one hour?
+
+What does $x - y$ express? What two operations will
+give that result? What operations will give $4x$ as a
+result?
+\end{enumerate}
+
+\textbf{14.} ILLUS. 1.
+\settowidth{\arraycolsep}{$\mskip0.5\thickmuskip$}
+\[
+\begin{array}{rclll}
+x &+& 5 &&\\
+x &+& 3 &&\\
+\multispan{3}\hrulefill &&\\
+x^2 &+& 5x &&\\
+&& 3x &+& 15 \\
+\multispan{5}\hrulefill \\
+x^2 &+& 8x &+& 15 \\
+\end{array}
+\]
+
+ILLUS. 2.
+\[
+\begin{array}{rclll}
+x &-& 5 &&\\
+x &-& 3 &&\\
+\multispan{3}\hrulefill &&\\
+x^2 &-& 5x &&\\
+&-& 3x &+& 15 \\
+\multispan{5}\hrulefill \\
+x^2 &-& 8x &+& 15 \\
+\end{array}
+\]
+
+ILLUS. 3.
+\[
+\begin{array}{rclll}
+x &+& 5 &&\\
+x &-& 3 &&\\
+\multispan{3}\hrulefill &&\\
+x^2 &+& 5x &&\\
+&-& 3x &-& 15 \\
+\multispan{5}\hrulefill \\
+x^2 &+& 2x &-& 15 \\
+\end{array}
+\]
+
+ILLUS. 4.
+\[
+\begin{array}{rclll}
+x &-& 5 &&\\
+x &+& 3 &&\\
+\multispan{3}\hrulefill &&\\
+x^2 &-& 5x &&\\
+&& 3x &-& 15 \\
+\multispan{5}\hrulefill \\
+x^2 &-& 2x &-& 15 \\
+\end{array}
+\]
+
+How many terms in the product? What is the first
+term? How is the last term formed? How is the
+coefficient of $x$ in the middle term formed?
+
+The answers to the examples in the following exercise
+are to be written directly, and not to be obtained by the
+full form of multiplication:
+
+\subsubsection*{Exercise 20.}
+
+Expand:
+\begin{enumerate}
+\item $(x + 2)(x + 7)$.
+
+\item $(x + 1)(x + 6)$.
+
+\item $(x - 3)(x - 4)$.
+
+\item $(x - 5)(x - 2)$.
+
+\item $(x + 5)(x - 2)$.
+
+\item $(x + 7)(x - 3)$.
+
+\item $(x - 7)(x + 6)$.
+
+\item $(x - 6)(x + 5)$.
+
+\item $(x - 11)(x - 2)$.
+
+\item $(x - 13)(x - 1)$.
+
+\item $(y + 7)(y - 9)$.
+
+\item $(x + 3)(x + 17)$.
+
+\item $(y+2)(y-15)$.
+
+\item $(y+2)(y+16)$.
+
+\item $(a^2 + 7)(a^2 - 5)$.
+
+\item$(a-9)(a+9)$.
+
+\item $(m^2 - 2)(m^2 - 16)$.
+
+\item $(b^3 + 12)(b^3 - 10)$.
+
+\item $(x - \frac{1}{2})(x - \frac{1}{4})$.
+
+\item$(y + \frac{1}{3})(y + \frac{1}{6})$.
+
+\item $(m + \frac{2}{3})(m - \frac{1}{3})$.
+
+\item $(a - \frac{2}{5})(a + \frac{3}{5})$.
+
+\item $(x - \frac{2}{3})(x - \frac{1}{2})$.
+
+\item $(y + \frac{3}{4})(y + \frac{1}{5})$.
+
+\item $(3-x)(7-x)$.
+
+\item $(5-x)(3-x)$.
+
+\item $(6-x)(7+x)$.
+
+\item$(11-x)(3+x)$.
+
+\item$(x-3)(x+3)$.
+
+\item $(y+5)(y-5)$.
+
+\item Find a number which, being multiplied by 6, and having 15
+added to the product, will equal 141.
+
+\item Mr. Allen has 3 more cows than his neighbor. Three times his
+number of cows will equal four times his neighbor's. How many has
+Mr. Allen?
+\end{enumerate}
+
+\subsection*{INVOLUTION.}
+\addcontentsline{toc}{subsection}{\numberline{}INVOLUTION.}
+
+\textbf{ 15}. What is the second power of 5? What is the third
+power of 4?
+
+\textbf{ Involution} is the process of finding a power of a
+number.
+
+\begin{tabular}{ll}
+ILLUS. 1. & $(5 a^2 b^3)^2 = 25 a^4 b^6$.\\
+ILLUS. 2. & $(3 x y^2 z)^3 = 27 x^3 y^6 z^3$.\\
+ILLUS. 3. & Find by multiplication the 2d, 3d, 4th,
+and 5th powers of $+a$ and $-a$.\\
+& Observe the signs of the odd and of the even powers.
+\end{tabular}
+
+\textit{To find any power of a monomial, raise the coefficient to
+the required power, multiply the exponent of each letter by the
+exponent of the power, and give every even power the plus sign,
+every odd power the sign of the original number}.
+
+\subsubsection*{Exercise 21}.
+
+Expand:
+\begin{enumerate}
+\item $(a^2b)^2$.
+
+\item $(xy^2)^3$.
+
+\item $(-a^4b)^2$.
+
+\item $(-x^3y^2)^3$.
+
+\item $(3a^2y)^3$.
+
+\item $(-7ab^2c^3)^2$.
+
+\item $(xyz^2)^5$.
+
+\item $(-m^2nd)^4$.
+
+\item $(-5x^3y^4z)^3$.
+
+\item $(11c^5d^12x^4)^2$.
+
+\item $(\frac{1}{2}x^2am^3)^2$.
+
+\item $(-\frac{1}{3}ab^3c)^2$.
+
+\item $(-15c^6dx^2)^2$.
+
+\item $(-9xy^5z^2)^3$.
+
+\item $(a^9b^2c^4d^2)^4$.
+
+\item $(-x^8yz^3m^2n)^5$.
+
+\item $(-\frac{2}{3}a^2bc^4)^2$.
+
+\item $(\frac{5}{6}mn^2x^3)^2$.
+
+\item In how many days can one man do as much as $b$ men in 8
+days?
+
+\item How many mills in $a$ cents? How many dollars?
+\end{enumerate}
+
+\textbf{16.} Find by multiplication the following:
+\[
+(a+b)^2,\; (a-b)^2,\; (a+b)^3,\; (a-b)^3,\; (a+b)^4,\; (a-b)^4.
+\]
+
+Memorize the results.
+
+It is intended that the answers in the following exercise
+shall be written directly without going through the multiplication.
+
+\begin{tabular}{cl}
+ILLUS. 1. &$(x-y)^4=x^4-4x^3y+6x^2y^2-4xy^3+y^4$.\\
+ILLUS. 2. &$(x-1)^3=x^3-3x^2+3x-1$.
+\end{tabular}
+
+\begin{displaymath}
+\begin{array}{lll}
+ \text{ILLUS. 3.} &\multicolumn{2}{l}{ (2xy+3y^2)^4 } \\
+\qquad & = (2xy)^4+4(2xy)^3(3y^2) & + 6(2xy)^2(3y^2)^2 \\
+ & &{}+ 4(2xy)(3y^2)^3+(3y^2)^4 \\
+\qquad &\multicolumn{2}{l}{
+ = 16x^4y^4+96x^3y^5+216x^2y^6+216xy^7+81y^8.}
+\end{array}
+\end{displaymath}
+
+
+\subsubsection*{Exercise 22.}
+
+Expand:
+\begin{enumerate}
+\item $(z+x)^3$.
+
+\item $(a+y)^4$.
+
+\item $(x-a)^4$.
+
+\item $(a-m)^3$.
+
+\item $(m+a)^2$.
+
+\item $(x-y)^2$.
+
+\item $(x^2+y^2)^3$.
+
+\item $(m^3-y^2)^2$.
+
+\item $(c^2-d^2)^4$.
+
+\item $(y^2+z^4)^3$.
+
+\item $(x^2y+z)^2$.
+
+\item $(a^2b-c)^4$.
+
+\item $(a^2-b^3c)^3$.
+
+\item $(x^2y-mn^3)^2$.
+
+\item $(x+1)^3$.
+
+\item $(m-1)^2$.
+
+\item $(b^2-1)^4$.
+
+\item $(y^3+1)^3$.
+
+\item $(ab-2)^2$.
+
+\item $(x^2y-3)^2$.
+
+\item $(1-x)^4$.
+
+\item $(1-y^2)^3$.
+
+\item $(2x+3y^2)^2$.
+
+\item $(3ab-x^2y)^3$.
+
+\item $(4mn^3-3a^2b)^4$.
+
+\item $(\frac{1}{2}x-y)^2$.
+
+\item $(1-\frac{1}{3}x^2)^3$.
+
+\item $(x^2-3)^4$.
+
+\item John has $4a$ horses, James has $a$ times as many as John,
+and Charles has $d$ less than five times as many as James. How
+many has Charles?
+
+\item A man bought $a$ pounds of meat at $a$ cents a pound, and
+handed the butcher an $x$-dollar bill. How many cents in change
+should he receive?
+
+\item A grocer, having 25 bags of meal worth $a$ cents a bag, sold
+$x$ bags. What is the value of the meal left?
+
+ \item If $a = 5$, $x
+= 4$, $y = 3$, find the numerical value of
+\[
+\frac{7a}{11x-3y} + \frac{11x}{8x-7y} - \frac{10y}{7a-5x}.
+\]
+
+\item Find the value of
+\[
+a^2b - c^2d - (ab+cd)(ac-bd) - bc(a^2c-bd^2)
+\]
+when $a = 2$, $b = 3$, $c = 4$, and $d = 0$.
+\end{enumerate}
+
+\subsubsection*{Exercise 23. (Review.)}
+
+\begin{enumerate}
+
+\item Take the sum of $x^3 + 3x - 2$, $2x^3 + x^2 - x + 5$, and
+$4x^3 + 2x^2 - 7x + 4$ from the sum of $2x^3 + 9x$ and $5x^3 +
+3x^2$.
+
+\item Multiply $b^4 - 2b^2$ by $b^4 + 2b^2 - 1$.
+
+\item Simplify $11x^2 + 4y^2 - (2xy - 3y^2) + (2x^2 - 3xy) - (3x^2
+- 5xy)$.
+
+\item Divide $\$300$ among $A$, $B$, and $C$, so that $A$ shall
+have twice as much as $B$, and $B$ $\$20$ more than $C$.
+
+\item Find two numbers differing by $8$ such that four times the
+less may exceed twice the greater by $10$.
+
+\item What must be added to $3a^3 - 4a^2 - 4$ to produce $5a^3 +
+6$?
+
+\item Add $\frac{2}{3}a^2 - ab - \frac{5}{4}b^2$, $\frac{2}{3}a^2
++ \frac{1}{3}ab - \frac{1}{4}b^2$, and $-a^2 -\frac{2}{3}ab +
+2b^2$.
+
+\item Simplify $8ab^2c^4 \times (-3a^4bc^2) \times (-2a^2b^3c)$.
+
+\item Expand $\left(-\frac{2}{3}xy^2z^3\right)^4$.
+
+\item Simplify $(x-2)(x+7) + (x-8)(x-5)$.
+
+\item Expand $(2a^2b-3xy)^3$.
+
+\item What must be subtracted from $x^3-3x^2+2y-5$ to produce
+unity?
+
+\item Multiply $x^3+ 3x^2y+3xy^2 + y^3$ by $3xy^2 -
+y^3-3x^2y+x^3$.
+
+\item Expand $(x+1)(x-1)(x^2+1)$.
+
+\item Add $4xy^3-4y^4$, $4x^3y-12x^2y^2+12xy^3-4y^4$,
+$6x^2y^2-12xy^3+6y^4$ and $x^4-4x^3y+6x^2y^2-4xy^3+y^4$.
+
+\item A man weighs 36 pounds more than his wife, and the sum of
+their weights is 317 pounds. What is the weight of each?
+
+\item A watch and chain cost \$350. What was the cost of each, if
+the chain cost $\frac{3}{4}$ as much as the watch?
+
+\item Simplify $3x^2-2x+1-(x^2+2x+3)-(2x^2-6x-6)$.
+
+\item Simplify $(a + 2y)^2-(a-2y)^2$
+
+\item What is the value of $1 + \frac{1}{2}a + \frac{1}{3}b$ times
+$1- \frac{1}{2}a + \frac{1}{3}b$ ?
+\end{enumerate}
+
+\section*{DIVISION.}
+\addcontentsline{toc}{section}{\numberline{}DIVISION.}
+
+\textbf{17}. What is the relation of division to multiplication?
+
+\begin{tabular}{ll}
+ILLUS. & $3x^2 \times 2xy = $? then $6x^3y \div {2xy}= $?\\
+\end{tabular}
+
+\textbf{Division} is the process by which, when a product is given
+and one factor known, the other factor is found.
+
+What is the relation of the dividend to the divisor and quotient?
+What factors must the dividend contain? What factors must the
+quotient contain?
+
+\begin{tabular}{c c}
+ILLUS. 1.& $6a^4 b^4 c^6 \div 2a^3 b^2 c^3 = 3a b^2 c^3$.\\
+\end{tabular}
+\begin{center}
+\begin{tabular}{c c|cc|cc|cc|c}
+ILLUS. 2. & $+ab $&$+ab^2$& $-ab $&$+ab^2$& $+ab $&$-ab^2$& $-ab $&$-ab^2$\\
+\cline{3-3}\cline{5-5}\cline{7-7}\cline{9-9}\\
+\end{tabular}
+\end{center}
+
+
+From the relation of the dividend, divisor, and quotient, and the
+law for signs in multiplication, obtain the quotients in Illus. 2.
+
+\textit{To divide a monomial by a monomial, divide
+the coefficient of the dividend by the coefficient
+of the divisor for the coefficient of the quotient,
+subtract the exponent of each letter in the divisor
+from the exponent of the same letter in the dividend
+for the exponent of that letter in the quotient: if
+dividend and divisor have like signs, give the quotient
+the plus sign; if unlike, the minus sign.}
+
+\subsubsection*{Exercise 24.}
+
+Divide:
+\begin{enumerate}
+\item $15x^2 y$ by $3x$.
+
+\item $39a b^2$ by $3b$.
+
+\item $27a^3 b^3 c$ by $9ab^2 c$.
+
+\item $35x^4 y^4 z$ by $7x^2 yz$.
+
+\item $-51cx^3 y$ by $3cyx^2$.
+
+\item $121x^3 y^3 z$ by $-11y^2 z$.
+
+\item $-28x^2 y^2 z^2$ by $-7x y^2$.
+
+\item $-36a^3 b^2 c^4$ by $-4a b^2 c^2$.
+
+\item $\frac{1}{3} a^4 b^5$ by $\frac{1}{6} a^2 b^2$.
+
+\item $\frac{1}{5} x^3 y^4$ by $-\frac{1}{15} xy^3$.
+
+\item $-45x^5 y^7 z$ by $9x^2 y^4 z$.
+
+\item $60a^4 bc^{11}$ by $-4ab^2 c^7$.
+
+\item $-\frac{2}{3} x^7 y^2$ by $-\frac{5}{6} x^4 y$.
+
+\item $\frac{3}{4} a^5 m^4 n^3$ by $-\frac{1}{4} a^2 mn^3$.
+
+\item $5m^4 n^2 x^5$ by $\frac{5}{8} mn^2 x$.
+
+\item $4x^3 y^2 z^8$ by $-\frac{2}{3} xz^5$.
+
+\item $10(x + y)^4 z^3$ by $5(x + y)^2 z$.
+
+\item $15(a - b)^3 x^2$ by $3(a - b)x$.
+
+\item Simplify $(-x^{2}y^{3}z^{2}) \times (-x^{4}y^{5}z^{4}) \div 2x^{3}y^{3}z^{4}$.
+
+\item Simplify $a^{5}b^{2}c \times (-a^{3}b^{3}c^{3}) \div 3(a^{3}bc^{2})^{2}$.
+
+\item Expand $(x^{3}y^{2}-3xy)^{3}$.
+
+\item If a man can ride one mile for $a$ cents, how far can
+two men ride for $b$ cents?
+
+\item In how many days can $x$ men earn as much as 8
+men in $y$ days?
+
+\item $a$ times $b$ is how many times $c$?
+\end{enumerate}
+
+\textbf{18.} ILLUS.
+\begin{tabular}{c|cccccc}
+$-3ab^{3}$ & &$-6a^{3}b^{3}$&$+$&$15a^{2}b^{4}$&$-$&$3ab^{5}$ \\
+\cline{2-7}
+\multicolumn{2}{c}{} &$2a^{2}$&$-$&$5ab$&$+$&$b^{2}$ \\
+\end{tabular}
+
+\textit{To divide a polynomial by a monomial, divide each
+term of the dividend by the divisor, and add the
+quotients.}
+
+\begin{center}
+\subsubsection*{Exercise 25.}
+\end{center}
+
+Divide:
+
+\begin{enumerate}
+\item $18a^{4}b^{3}-42a^{3}b^{2}+90a^{6}bx$ by $6a^{3}b$.
+
+\item $10x^{5}y^{2}+6x^{3}y^{2}-18x^{4}y^{4}$ by $2x^{3}y$.
+
+\item $72x^{5}y^{6}-36x^{4}y^{3}-18x^{2}y^{2}$ by $9x^{2}y$.
+
+\item $169a^{4}b-117a^{3}b^{2}+91a^{2}b$ by $13a^{2}$.
+
+\item $-2a^{5}x^{3}+\frac{7}{2}a^{4}x^{4}$ by $\frac{7}{3}a^{3}x$.
+
+\item $\frac{1}{2}x^{5}y^{2}-3x^{3}y^{4}$ by $-\frac{3}{2}x^3y^{2}$.
+
+\item $32x^{3}y^{4}z^{6}-24x^{5}y^{4}z^{4}+8x^{4}y^{2}$ by $-8x^{3}y$.
+
+\item $120a^{4}b^{3}c^{5}-186a^{5}b^{7}c^{6}$ by $6a^{3}b^{3}c^{4}$.
+
+\item $4x^{8}-10x^{5}+2x^{6}-16x^{3}-6x^{4}$ by $4x^{2}$.
+
+\item $24y^{3}+32y^{7}-48y^{6}$ by $3y^{3}$.
+
+\item $\frac{1}{4}a^2x-\frac{1}{16}abx - \frac{3}{8}acx$ by $\frac{3}{8}ax$.
+
+\item $-\frac{5}{2}x^2+\frac{5}{3}xy+\frac{10}{3}x$ by $-\frac{5}{6}x$.
+
+\item An army was drawn up with $x$ men in front and $y$
+men deep. How many men were there in the army?
+
+\item In how many minutes will a train go $x$ miles at
+the rate of $a$ miles an hour?
+
+\item How many apples at $x$ cents apiece can be bought
+for $b$ dollars?
+\end{enumerate}
+
+\textbf{ 19.} Since division is the reverse of multiplication, let
+us consider an example in the multiplication of polynomials.
+
+\begin{center}
+\begin{math}
+\begin{array}{ccccccccc}
+x^3 & + & 2x^2 & + & 3x & & & & \\
+3x^2 & - & 2x & + & 3 & & & & \\
+\cline{1-5}
+3x^5 & + & 6x^4 & + & 9x^3 & & & & \\
+ & - & 2x^4 & - & 4x^3 & - & 6x^2 & & \\
+ & & & & 3x^3 & + & 6x^2 & + & 9x \\
+\hline
+3x^5 & + & 4x^4 & + & 8x^3 & & & + & 9x \\
+\end{array}
+\end{math}
+\end{center}
+
+Which of these numbers will become the dividend in
+our division? Take $x^3 + 2x^2 + 3x$ for the divisor. What
+will be the quotient? Write these names opposite the
+different numbers. What is the last operation performed
+in obtaining the dividend? What then is the dividend?
+How are these partial products obtained?
+
+Keeping these facts in mind, we will start on the
+work of division, using these same numbers for convenience.
+
+ILLUS. 1.\footnotemark
+\begin{center}
+\begin{math}
+\begin{array}{rcccclccl}
+x^{3} + 2 x^{2} + 3 x ) & 3 x^{5} & + 4 x^{4} & + & 8 x^{3} & + & 9 x&(3 x^{2} & - 2 x + 3 \\
+ & 3 x^{5} & + 6 x^{4} & + & 9 x^{3}\\
+ \cline{2-6}
+ & &- 2 x^{4} & - & x^{3} & + & 9 x\\
+ & & -2 x^{4} & - & 4 x^{3} & - & 6 x \\
+ \cline{3-8}
+ & & & & 3 x^{3} & + & 6 x^{2}& + 9 x\\
+ & & & & 3 x^{3} & + & 6 x^{2}& + 9 x\\
+\end{array}
+\end{math}
+\end{center}
+
+How was the first term of the dividend formed? How
+can the first term of the quotient be found? Knowing the
+divisor and first term of the quotient, what can be formed?
+Subtract this from the dividend. How was the first term of
+the remainder formed? What can now be found? How?
+What can then be formed? etc., etc. How long can this
+process be continued?
+
+ILLUS. 2.
+\begin{center}
+\begin{math}
+\begin{array}{rccrllccl}
+ x^{2} - 3 x y + 2 y^{2} )& x^{4} & - 6 x^{3} y & + 12 x^{2} y^{2} & - 4 y^{4}& (x^{2} & - 3 x y + y^{2} + \frac{9 x y^{3} - 6 y^{4}}{x^{2} - 3 x y + 2 y^{2}} \\
+ & x^{4} & - 3 x^{3} y & + 2 x^{2} y^{2} \\
+ \cline{2-5}
+ & & - 3 x^{3} y & + 10 x^{2} y^{2} & - 4 y^{4}\\
+ & & - 3 x^{3} y & + 9 x^{2} y^{2} & - 6 x y^{3}\\
+ \cline{3-6}
+ & & & x^{2} y^{2} & + 6 x y^{3} & - 4 y^{4}\\
+ & & & x^{2} y^{2} & - 3 x y^{3} & + 2 y^{4}\\
+ \cline{4-6}
+ & & & & 9 x y^{3} & - 6 y^{4}
+\end{array}
+\end{math}
+\end{center}
+
+Why stop the work at the point given? What is the complete
+quotient? Is the dividend exactly divisible by the divisor? When
+is one number exactly divisible by another?
+
+\footnotetext{It would be well for the teacher to work out this
+example on the board with the class along the line of the
+questions which follow the example.}
+
+\textit{ To divide a polynomial by a polynomial, arrange the terms
+of the dividend, and divisor similarly, divide the first term of
+the dividend by the first term of the divisor for the first term
+of the quotient, multiply the divisor by the quotient and subtract
+the product from the dividend; divide the first term of the
+remainder by the first term of the divisor for the next term of
+the quotient, multiply and subtract as before; continue this work
+of dividing, multiplying, and subtracting until there is no
+remainder or until the first term of the remainder is not
+divisible by the first term of the divisor.}
+
+\subsubsection*{Exercise 26.}
+
+Divide:
+\begin{enumerate}
+\item $x^2+8x-105$ by $x+15$.
+
+\item $x^2+8x-33$ by $x+11$.
+
+\item $x^4+x^2-20$ by $x^2-4$.
+
+\item $y^4-y^2-30$ by $y^2+5$.
+
+\item $x^4-31x^2+9$ by $x^2+5x-3$.
+
+\item $a^4-12a^2+16$ by $a^2-2a-4$.
+
+\item $x^3-y^3$ by $x-y$.
+
+\item $a^3+b^3$ by $a+b$.
+
+\item $16a^4-81b^4$ by $2a-3b$.
+
+\item $81x^8-y^4$ by $3x^2-y$.
+
+\item $x^5-x^4y-2x^3y^2-5x^2y^3-17xy^4-12y^5$ by $x^2-2xy-3y^2$.
+
+\item $a^5+a^4b-14a^3b^2+15a^2b^3-4b^3$ by $a^2-3ab+2b^2$.
+
+\item $x^6 - 5x^3 + 3 + 5x^4 - 10x - x^5 + 10x^2$ by $x^2 + 3 -
+x$.
+
+\item $x^6 - 2x^3 - 2 + x - 3x^5 + 2x^4 - 5x^2$ by $x^3 + 2 + x$.
+
+\item $a^5 - a - 2a^2 - a^3$ by $a + a^3 + a^2$.
+
+\item $x^6 - 2x^3 - x^2 - x^4$ by $1 + xx^2 + x$.
+
+\item $a^{11} - a^2$ by $a^3 - 1$.
+
+\item $a^{12} - a^4$ by $a^2 + 1$.
+
+\item $x^4 + 4y^4$ by $x^2 - 2xy + y^2$.
+
+\item $4a^4 + 81b^4$ by $2a^2 + 6ab + 9b^2$.
+
+\item $\frac{1}{2}x^4 + \frac{3}{4}x^3y - \frac{1}{3}x^2y^2
+ + \frac{7}{6}xy^3 - \frac{1}{3}y^4$
+ by $x^2 - \frac{1}{2}xy + y^2$.
+
+\item $\frac{2}{9}y^3 - \frac{5}{36}x^2y + \frac{1}{6}xy^2 +
+\frac{1}{6}x^3$
+ by $\frac{1}{2}x + \frac{1}{3}y$.
+
+\item $\frac{1}{4}(x - y)^5 - (x - y)^3 - \frac{1}{2}(x - y)^2 -
+\frac{1}{16}(x - y)$
+ by $\frac{1}{2}(x - y)^2 + (x - y) +\frac{1}{4}$.
+
+\item $16x^8 - 81y^4$ by $27y^3 + 18x^2y^2 + 8x^6 + 12x^4y$.
+
+\item $4x^4 - 10x^2 + 6$ by $x + 1$.
+
+\item $4a^4 - 5a^2b^2 + b^4$ by $2a - b$.
+
+\item $a^6 - b^6 + a^4 + b^4 + a^2b^2$ by $a^2 - b^2 + 1$.
+
+\item $x^6 - y^6 - x^4 - y^4 - x^2y^2$ by $x^2 - y^2 - 1$.
+
+\item $81a^{12} - 16b^8$ by $12a^3b^4 - 8b^6 - 18a^6b^2 + 27a^9$.
+
+\item $1 + 3x$ by $1 + x$ to four terms of quotient.
+
+\item $1 - 2a$ by $1 - a$ to four terms of quotient.
+
+\item $4 + a$ by $2 - a$ to four terms.
+
+\item $9 - x$ by $3 + x$ to four terms.
+
+\item If a boy can do a piece of work in $x$ minutes, how many
+hours would it take him to perform $12$ times as much work?
+
+\item A man has $x$ dollars, $y$ acres of land worth $m$ dollars
+an acre, and $c$ houses each worth $b$ dollars. What is my share
+if I am one of $n$ heirs?
+
+\item A storekeeper mixed $m$ pounds of coffee worth $a$ cents a
+pound with $p$ pounds worth $b$ cents a pound. How much is the
+mixture worth per pound?
+
+\item If John is $y$ years old, how old was he 11 years ago?
+\end{enumerate}
+
+
+
+\subsection*{EVOLUTION.}
+\addcontentsline{toc}{subsection}{\numberline{}EVOLUTION.}
+
+\textbf{ 20.} ILLUS. $\sqrt{a}$, $\sqrt[3]{x^2 y}$,
+$\sqrt[4]{x-y}$, $\sqrt{16}$, $\sqrt[3]{5}$, $\sqrt[5]{6 a^2 b
+c}$.
+
+\textbf{The root of a number} is indicated by the radical
+sign and index. When no index is expressed, \textit{two} is
+understood.
+
+Express:
+
+1. The square root of $x$, $2ab$, $7x - 3y^2$, $a^2 bc$.
+
+2. The fifth root of $3y$, $2m - n$, $4x^2 yz^3$.
+
+3. The cube root of 2, $x + y$, $17 x^2 y^4$, $m$.
+
+4. The sixth root of $az$, $5m^2 n - 3xy + 14 - 3 ab^3 c$.
+
+What is the square root of a number? the fifth root?
+the fourth root? the cube root? the eleventh root?
+
+\begin{tabular}{cccc}
+ ILLUS. 1. &$(3a^2 bc^3)^3$ = ? & Then& $\sqrt[3]{27a^6 b^3
+c^9}$ = ? \\
+ILLUS. 2. & $\left. \begin{tabular}{c}
+$(+a)^4 = ? $ \\
+$(-a)^4 = ?$
+\end{tabular} \right\} $ & Hence &$\sqrt[4]{+a^4}$ = ?\\
+ & $(+a)^3$ = ? && $\sqrt[3]{+a^3}$ = ?\\
+ &$(-a)^3$ = ? && $\sqrt[3]{-a^3}$ = ?\\
+\end{tabular}
+
+
+
+
+
+
+\textit{To find the root of a monomial, find the required
+root of the coefficient, divide the exponent of each
+letter by the index of the root for the exponent of
+that letter in the root, give to even roots of plus numbers
+the plus-or-minus sign} ($\pm$) \textit{, to odd roots of plus
+numbers the plus sign, and to odd roots of minus
+numbers the minus sign.}
+
+\subsubsection*{Exercise 27.}
+
+Simplify:
+
+\begin{enumerate}
+
+\item $\sqrt{16 a^{2} b^{6}}$.
+
+\item $\sqrt[4]{81 x^{4} y^{8}}$.
+
+\item $\sqrt[3]{-8 x^{6} y^{3}}$.
+
+\item $\sqrt[5]{-32 a^{10} b^{15}}$.
+
+\item $\sqrt[5]{243 a^{5} b^{10}}$.
+
+\item $\sqrt[3]{27 x^{3} y^{9}}$.
+
+\item $\sqrt[4]{16 x^{8} y^{4}}$.
+
+\item $\sqrt{9 x^{8} y^{6}}$.
+
+\item $\sqrt{\frac{4}{9} m^{6} y^{4}}$.
+
+\item $\sqrt[3]{\frac{8}{27} a^{9} b^{6}}$.
+
+\item $\sqrt[3]{-\frac{27}{64} x^{9} y^{12}}$.
+
+\item $\sqrt[5]{-\frac{32}{243} a^{5} b^{15}}$.
+
+\item $\sqrt[6]{x^{12} \left(a - b \right)^{6}}$.
+
+\item $\sqrt[3]{a^9 \left(x^{2} + y^{2} \right)^{3}}$.
+
+\item $\sqrt{4 a^{2} b^{6} \left(x^{2} - y \right)^{4}}$.
+
+\item $\sqrt{16 x^{4} y^{2} \left(m^{3} + y \right)^{6}}$.
+
+\item $\sqrt{\frac{1}{9} a^{4} b^{6}} - \sqrt[3]{\frac{1}{8} a^{6}
+b^{9}} - \sqrt[5]{\frac{32}{243} a^{10} b^{15}} + \sqrt[3]{a^{6}
+b^{9}}$.
+
+\item $ \sqrt[3]{\frac{1}{8} x^{6} y^{3}}
+ + \sqrt[5]{-\frac{1}{32} x^{10} y^{5}}
+ - \sqrt[3]{-x^{6} y^{3}}
+ - \sqrt{\frac{1}{4} x^{4} y^{2}} $.
+
+\item Multiply $\sqrt{25 a^{4} b^{2} c^{2}}$ by $-\sqrt[3]{-8
+a^{3} b^{6} c^{9}}$.
+
+\item Divide $\sqrt{144 x^{4} y^{4}}$ by $\sqrt[5]{243 x^{10}
+y^{5}}$.
+
+\item Multiply $-\sqrt[3]{-125 x^{3} y^{6} z^{6}}$ by $\sqrt{9
+x^{4} y^{2} z^{6}}$.
+
+\item Divide $\sqrt{100 a^{6} b^{12}}$ by $\sqrt[5]{32 a^{15}
+b^{25}}$.
+
+\item From two cities $a$ miles apart two trains start toward each
+other, the one going $x$ miles an hour, and the other $y$ miles an
+hour. How long before they will meet? How far will each train have
+gone?
+
+\item What number is that whose double exceeds its half by 39?
+
+\item Mr. A. is $m$ years old, and 6 years ago he was one-half as
+old as Mr. B. How old was Mr. B. then? How old is Mr. B. now?
+\end{enumerate}
+
+\textbf{ 21.} $(a + b)^{2} = a^{2} + 2ab + b^{2} = a^{2} + (2a +
+b)b$.
+
+\begin{tabular}{lc}
+ILLUS. 1. & Find the square root of $9x^{2} - 12 xy +4y^{2}$.\\
+\end{tabular}
+
+\begin{tabular} {rrccc}
+ & & \multicolumn{2}{l|}{ $9x^{2} -12xy +4y^{2}$} & $3x-2y$ \\
+$a^{2}=$& & \multicolumn{2}{l|}{$9x^{2}$} \\
+\cline{3-4}
+$2a+b=$ & $6x$&\multicolumn{1}{l|}{$ -2y$ }& $-12xy+4y^{2}$\\
+$(2a+b)b=$ & \multicolumn{2}{c|}{}& $-12xy+4y^{2}$\\
+\cline{4-4}
+\end{tabular}
+
+\begin{center}
+\begin{tabular}{lc}
+ILLUS. 2. & Find the square root of
+$x^{6}-6x^{3}-4x+1+6x^{2}-2x^{5}+5x^{4}$.\\
+\end{tabular}
+\end{center}
+\begin{tabular}{ccccccccc}
+&&$x^{6}$&\multicolumn{5}{r|}{ $ -2x^{5} +5x^{4}-6x^{3}+6x^{2}-4x+1 $}&$ x^{3} - x^{2} +2x-1$ \\
+&&\multicolumn{6}{l|}{$x^{6}$}\\
+\cline{3-8}
+
+&$ 2x^{3}$ & $ -x^{2}$&\multicolumn{5}{|l} {$ -2x^{5} + 5x^{4} - 6x^{3} +6x^{2} -4x +1$ }\\
+&&&\multicolumn{6}{|l}{$ -2x^{5} + x^{4}$ }\\
+\cline{4-8}
+
+$ 2(x^{3}-x^{2})+2x=$ &$ 2x^{3}$ &
+$-x^{2}$ & $ +2x$ &\multicolumn{5}{|l}{ $ 4x^{4} -6x^{3}+6x^{2}-4x+1$} \\
+&&&&\multicolumn{5}{|l}{$ 4x^{4} -4x^{3}+4x^{2}$}\\
+\cline{5-8}
+
+$ 2(x^{3} - x^{2} +2x) + (-1) =$& $ 2x^{3}$ & $ -2x^{2}$ &$ +4x$ &
+$-1$ & \multicolumn{4}{|l}{$ -2x^{3}+2x^{2}-4x+1$}\\
+&&&&&\multicolumn{4}{|l}{$ -2x^{3}+2x^{2}-4x+1$}\\
+\cline{6-8}
+\end{tabular}
+
+\textit{To find the square root of a polynomial, arrange the
+terms with reference to the powers of some number;
+take the square root of the first term of the polynomial
+for the first term of the root, and subtract its
+square from the polynomial; divide the first term of
+the remainder by twice the root found for the next
+term of the root, and add the quotient to the trial
+divisor; multiply the complete divisor by the second
+term of the root, and subtract the product from the
+remainder. If there is still a remainder, consider the
+root already found as one term, and proceed as before.}
+
+\subsubsection*{Exercise 28.}
+
+Find the square root of each of the following:
+\begin{enumerate}
+\item $4x^2-12xy+9y^2$.
+
+\item $x^4+10x^3y^3+25x^2y^6$.
+
+\item $16a^2b^2c^4-56abc^2xy^2z+49x^2y^4z^2$.
+
+\item $\frac{1}{4}x^2-xy^2z+y^4z^2$.
+
+\item $a^2b^6+\frac{2}{3}ab^3c^4+\frac{1}{9}c^8$.
+
+\item $x^4-4x^3+2x^2+4x+1$.
+
+\item $x^4+6x^3+17x^2+24x+16$.
+
+\item $4x^4+9x^2+4-4x-4x^3$.
+
+\item $6x^3-5x^2+1-2x+9x^4$.
+
+\item $x^6+2x^5-x^4+3x^2-2x+1$.
+
+\item $6x^4-4x^3+4x^5-15x^2-8x+x^6+16$.
+
+\item $7x^2-6x-11x^4+10x^3-4x^5+1+4x^6$.
+
+\item A is $x$ years old. If his age is as much above 50 as B's
+age is below 40, what is B's age?
+
+\item If $x$ represents the first digit, and $y$ the second digit,
+of a number, what will represent the number?
+\end{enumerate}
+
+\textbf{ 22.} $(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}
+= a^{3} + (3 a^{2} + 3 a b + b^{2}) b$
+
+\begin{center}
+\begin{tabular}{ll}
+ILLUS. 1. &Find the cube root of $8 m^{6} + 12 m^{4} n^{3} + 6
+m^{2} n^{6}$ $+ n^{9}$.\\
+\end{tabular}
+\end{center}
+
+\begin{tabular}{rclcc}
+&& \multicolumn{2}{l|}{$\scriptstyle 8m^6+12m^4 n^3 +6m^2 n^6 + n^9$} &$\scriptstyle 2m^2 + n^3$ \\
+\cline{5-5}
+$\scriptstyle a^3=$ &&\multicolumn{2}{l|}{$\scriptstyle 8m^6$}\\
+\cline{3-4}
+$\scriptstyle 3a^2+3ab+b^2=$&$\scriptstyle 12m^4+6m^2n^3$&$\scriptstyle +n^6 $&\multicolumn{1}{|c}{$\scriptstyle 12m^4 n^3 +6m^2 n^6 +n^9$}\\
+$\scriptstyle
+(3a^2+3ab+b^2)b=$&&&\multicolumn{1}{|c}{$\scriptstyle 12m^4 n^3
++6m^2 n^6
++n^9$}\\
+ \cline{4-4}
+\end{tabular}
+
+\begin{center}
+\begin{tabular}{ll}
+ILLUS. 2. &Find the cube root of $66 x^{2} + 33 x^{4} + 8 - 36 x$
+$+ x^{6} - 63 x^{3} - 9 x^{5}$.\\
+\end{tabular}
+\end{center}
+\begin{tabular}{rccccccccccc}
+&&&\multicolumn{7}{r|}{$\scriptstyle x^6-9x^5 +33x^4 -63x^3 +66x^2
+-36x +8$}
+& $\scriptstyle x^2 -3x+2$\\
+\cline{11-11}
+&\multicolumn{4}{r}{$\scriptstyle x^6$}&\multicolumn{5}{r|}{}\\
+\cline{4-10}
+
+&&\multicolumn{2}{l}{$\scriptstyle 3x^4 - 9x^3+9x^2$} &
+\multicolumn{6}{|r}{$\scriptstyle -9x^5 +33x^4
+-63x^3 +66x^2 -36x +8$}\\
+&&&&\multicolumn{6}{|l}{$\scriptstyle \quad -9x^5 +27x^4 -27x^3 $}\\
+\cline{5-10}
+
+$\scriptstyle 3(x^2 - 3x)^2 = $& &\multicolumn{2}{r}{$\scriptstyle
+3x^4-18x^3 +27x^2$}
+&&&\multicolumn{4}{|r}{$\scriptstyle 6x^4 -36x^3 + 66 x^2 -36x +8$}\\
+$\scriptstyle 3(x^2 - 3x)^2 \times 2 =$&&&\multicolumn{3}{c}{$\scriptstyle \quad 6x^2 -18x$}&\multicolumn{4}{|c}{}\\
+
+$\scriptstyle 2^2=$ &&&&&$\scriptstyle +4$&\multicolumn{4}{|c}{}\\
+\cline{3-6}
+
+& \multicolumn{5}{r}{$\scriptstyle 3x^4-18x^3 +33x^2 -18x +4$}& \multicolumn{5}{|l}{$\scriptstyle 6x^4 -36x^3 + 66 x^2 -36x +8$}\\
+\cline{7-11}
+\end{tabular}
+
+
+\textit{To find the cube root of a polynomial, arrange
+the terms with reference to the powers of some number;
+take the cube root of the first term for the first
+term of the root, and subtract its cube from the
+polynomial; take three times the square of the root
+already found for a trial divisor, divide the first
+term of the remainder by it, and write the quotient
+for the next term of the root; add to the trial
+divisor three times the product of the first term by
+the second and the square of the second; multiply
+the complete divisor by the second term of the root,
+and subtract the product from the remainder. If
+there are other terms remaining, consider the root
+already found as one term, and proceed as before.}
+
+
+
+\subsubsection*{Exercise 29}.
+
+Find the cube root of each of the following:
+\begin{enumerate}
+\item $27 x^{3} - 27 x^{2} y + 9 x y^2 - y^{3}$.
+
+\item $15 x^{2} - 1 - 75 x^{4} + 125 x^{6}$.
+
+\item $144 a^{2} b^{2} + 27 b^{6} + 108 a b^{4} + 64 a^{3}$.
+
+\item $x^{6} - 8 y^{6} + 12 x^{2} y^{4} - 6 x^{4} y^{2}$.
+
+\item $1 + 9 x + 27 x^{2} + 27 x{3}$.
+
+\item $1 - 21 m - 343 m^{3} + 147 m^{2}$.
+
+\item $3 x^{2} + 64 x^{6} - 24 x^{4} - \frac{1}{8}$.
+
+\item $27 x^{9} + \frac{1}{27} + x^{3} + 9 x^{6}$.
+
+\item $8 a^{6} + 18 a^{4} + 9 a^{2} - 12 a^{5} - 13 a^{3} - 3 a +
+1$.
+
+\item $x^{9} - 7 x^{6} + x^{3} - 3 x^{4} - 3 x^{8} + 6 x^{7} + 6
+x^{5}$.
+
+\item $5 x^{3} - 3 x - 3 x^{5} - 1 + x^{6}$.
+
+\item $\frac{1}{8} a^{6} + \frac{3}{2} a^{5} + 5 \frac{1}{4} a^{4}
+ + 2 a^{3} - 10 \frac{1}{2} a^{2} + 6 a - 1$.
+
+\item A board is $8 x$ inches long and $2 x$ inches wide. What is
+the length of a square board having the same area?
+
+\item If $3 x$ represents the edge of a cubical box, what
+represents the cubical contents of the box?
+
+\item If a cubical cistern contains $64y^3$ cubic feet, how long
+is one edge?
+
+\item A man travels north $a$ miles, and then south $b$ miles. How
+far is he from the starting-point? How far has he traveled?
+\end{enumerate}
+
+\subsubsection*{Exercise 30.}
+
+Find the indicated roots:
+\begin{enumerate}
+\item $\sqrt{2025}$.
+
+\item $\sqrt{9409}$.
+
+\item $\sqrt{20449}$.
+
+\item $\sqrt{904401}$.
+
+\item $\sqrt{70.56}$.
+
+\item $\sqrt{.9025}$.
+
+\item $\sqrt{94864}$.
+
+\item $\sqrt{.00000784}$
+
+\item $\sqrt[3]{59.319}$.
+
+\item $\sqrt[3]{389017}$.
+
+\item $\sqrt[3]{241804.367}$.
+
+\item $\sqrt{7039.21}$.
+
+\item $\sqrt[3]{35.287552}$.
+
+\item $\sqrt{2550.25}$.
+
+\item $\sqrt{34.78312}$ to three decimal places.
+
+\item $\sqrt{7}$ to three decimal places.
+
+\item $\sqrt[3]{.1255}$ to three decimal places.
+
+\item A merchant bought a bale of cloth containing just as many
+pieces as there were yards in each piece. The whole number of
+yards was 1089. What was the number of pieces?
+
+\item A regiment, consisting of 5476 men, is to be formed into a
+solid square. How many men must be placed in each rank?
+
+\item What is the depth of a cubical cistern which contains 5000
+gallons of water? (1 gallon = 231 cubic inches.)
+
+\item A farmer plants an orchard containing 8464 trees, and has as
+many rows of trees as there are trees in each row. What is the
+number of trees in each row?
+\end{enumerate}
+
+\chapter*{FACTORS AND MULTIPLES}
+\addcontentsline{toc}{chapter}{\numberline{}FACTORS AND
+MULTIPLES.}
+
+\section*{FACTORING.}
+\addcontentsline{toc}{section}{\numberline{}FACTORING---Six
+Cases.}
+
+\textbf{23}. When we divide a number into two numbers,
+which multiplied together will give a product equal to
+the given number, we have found the factors of that
+number. This process is called \textbf{factoring}.
+
+Name two factors of 48, 27, 18, 35, 49, 72.
+
+Name three factors of 24, 100, 75, 64, 72, 40.
+
+
+\textbf{24. CASE I. To factor a polynomial which has
+a factor common to all its terms}.
+
+\begin{tabular}{ll}
+ILLUS. & Factor $2 a b + 6 a c + 4 a d$.\\
+\end{tabular}
+
+\begin{center}
+\begin{tabular}{rc rclcl}
+ $2a$ & \multicolumn{1}{|l}{$2ab$ + $6ac$ + $4ad$} \\
+\cline{2-2}
+ & $b + 3c + 2d $ \\
+\multicolumn{2}{c}{$ \therefore 2 a b + 6 a c + 4 a d = 2 a
+\left(b + 3 c + 2 d
+\right) $.}\\
+\end{tabular}
+\end{center}
+
+\textit{Divide the polynomial by the largest factor common
+to the terms. The quotient and divisor are the
+factors of the polynomial}.
+
+\subsubsection*{Exercise 31.}
+
+Factor:
+\begin{enumerate}
+\item $5a^2 - 25$.
+
+\item $16 + 64xy$.
+
+\item $2a - 2a^2$.
+
+\item $15a^2 - 225a^4$.
+
+\item $x^3 - x^2$.
+
+\item $3a^2 + a^5$.
+
+\item $a^2 - ab^2$.
+
+\item $a^2 + ab$.
+
+\item $6a^3 + 2a^4 + 4a^5$.
+
+\item $7x - 7x^3 +14x^4$.
+
+\item $3x^3 - x^2 + x$.
+
+\item $a^3 - a^2 y + ay^2$.
+
+\item $3a(x + y) + 5mb(x + y) - 9d^2 x(x + y)$.
+
+\item $4(a - b) - 15 x y (a - b) + (a - b) - 5 a^2 b(a - b)$.
+
+\item $4x^3 y - 12a x^2 - 8 x y^3$.
+
+\item $6 a x^3 y^5 - 4 a x^2 y^6 + 2 a x y^7 - 2 a^2 x y^9$.
+
+\item $51 x^5 y - 34 x^4 y^2 + 17 x^2 y^4$.
+
+\item $6 a^2 b^2 - 3 a^3 b^3 c - 9 a b^3 c + 3 abc^2$.
+
+\item $3 a x^7 - 24 a x + 9 ax^5 - 3 ax^4 - 9 ax^6$.
+
+\item $27 a^8 b^2 c^3 - 81 a^7 b^3 c^3 + 81 a^6 b^4 c^3 - 27 a^5
+b^5 c^3 - 27 a^5 b^2 c^6$.
+
+\item A lady bought a ribbon for $m$ cents, some tape for $d$
+cents, and some thread for $c$ cents, for which she paid $x$ cents
+on account. How much remains due?
+
+\item Julia has the same number of beads in each hand. If she
+should change two from her left hand to her right hand, the right
+hand would contain twice as many as the left. How many beads has
+she?
+
+\end{enumerate}
+
+Sometimes a polynomial in the form given has no
+factor common to all its terms, but has factors common
+to several terms. It may be possible then to
+group the terms in parentheses so that there will be
+a binomial or trinomial factor common to all the terms
+of the polynomial in its new form.
+
+\begin{tabular}{c}
+\begin{tabular}{ll}
+ILLUS. 1. & Factor $2am+2ax+bm+bx$.\\
+\end{tabular}\\
+
+\begin{tabular}{c}
+$2am+2ax+bm+bx=2a(m+x)+b(m+x).$\\
+\end{tabular}\\
+
+\begin{tabular}{lll}
+$ m+x$ &\multicolumn{1}{|l}{$ 2a(m+x)$}&$ + b(m+x) $\\
+\cline{2-3}
+ & $2a $&$+ b$\\
+\end{tabular}\\
+\begin{tabular}{c}
+$\therefore 2am+2ax+bm+bx=(m+x)(2a+b).$
+\end{tabular}\\
+
+\begin{tabular}{ll}
+ILLUS. 2. &Factor $a^8+a^6-a^5-a^3+a^2+1$.\\
+\end{tabular}\\
+
+\begin{tabular}{c}
+$a^8+a^6-a^5-a^3+a^2+1=a^6(a^2+1)-a^3(a^2+1)+(a^2+1).$
+\end{tabular}\\
+
+\begin{tabular}{clll}
+$ a^2+1$ &\multicolumn{1}{|l}{$ a^6(a^2+1)$}&$ - a^3(a^2+1)$&$ + (a^2+1)$ \\
+\cline{2-4} &$ a^6$ &$- a^3 $&$+ 1$\\
+\end{tabular}\\
+\begin{tabular}{c}
+$\therefore a^8+a^6-a^5-a^3+a^2+1=(a^2+1)(a^6-a^3+1).$
+\end{tabular}
+\end{tabular}
+
+\subsubsection*{Exercise 32.}
+
+Factor:
+\begin{enumerate}
+\item $ax+ay+bx+by$.
+
+\item $x^2+ax+bx+ab$.
+
+\item $ax^2+ay^2-bx^2-by^2$.
+
+\item $x^2-ax+5x-5a$.
+
+\item $ax-bx+ab-x^2$.
+
+\item $x^2+mxy-4xy-4my^2$.
+
+\item $2x^4-x^3+4x-2$.
+
+\item $mx-ma-nx+na$.
+
+\item $x^3+x^2+x+1$.
+
+\item $y^3-y^2+y-1$.
+
+\item $x^5+x^4-x^3-x^2+x+1$.
+
+\item $a^2x+abx+ac+aby+b^2y+bc$.
+
+\item $ax-bx+by+cy-cx-ay$.
+
+\item $3ax+3ay-2bx-2by$.
+
+\item $2ax-3by+cy-2ay+3bx-cx$.
+
+\item $6amx+3amy-6anx-3any$.
+
+\item A man had 250 acres of land and 30 houses. After trading $x$
+of his houses for $y$ acres of land, how many has he of each?
+
+\item What is the number that will become four times as large by
+adding 36 to it?
+
+\item The fifth and seventh of a number are together equal to 24.
+What is the number?
+
+\end{enumerate}
+
+\textbf{ 25. CASE II. To factor the difference of the squares of
+two numbers.}
+
+ILLUS. 1. $a^2-b^2=(a+b)(a-b)$.
+
+ILLUS. 2. $9x^2y^4-49a^4b^{12}=(3xy^2+7a^2b^6)(3xy^2-7a^2b^6)$.
+
+ILLUS. 3.
+\begin{eqnarray*}
+x^8-y^8 & = & (x^4+y^4)(x^4-y^4)=(x^4+y^4)(x^2+y^2)(x^2-y^2) \\
+& = & (x^4+y^4)(x^2+y^2)(x+y)(x-y). \\
+\end{eqnarray*}
+
+\textit{ Write two factors, of which one is the sum and the other
+the difference of the square roots of the terms.}
+
+\subsubsection*{Exercise 33.}
+
+Factor:
+\begin{enumerate}
+\item $x^2-y^2$.
+
+\item $m^2-n^2$.
+
+\item $a^2b^4-c^2d^2$.
+
+\item $m^2p^4-x^6y^4$.
+
+\item $a^6b^2x^4-m^4c^8y^{10}$.
+
+\item $x^2y^4z^4-c^2d^2m^4$.
+
+\item $x^4y^2-a^6y^4$.
+
+\item $g^6c^6-x^6z^8$.
+
+\item $4a^2-9x^2$.
+
+\item $16m^2-9n^2$.
+
+\item $81x^2y^4-25b^2d^2$.
+
+\item $729m^4c^2x^{10}-10,000y^4$.
+
+\item $121m^2-64x^2$.
+
+\item $x^4-y^4$.
+
+\item $m^8 - a^8$.
+
+\item $a^4b^8 - l$.
+
+\item $x^16 - b^16$.
+
+\item $16a^4 - 1$.
+
+\item $a^4 - ax^2$.
+
+\item $5b^4 - 5a^2b^2$.
+
+\item $ax^2 - ay^2 - bx^2 + by^2$.
+
+\item $5ax - 5a^3x + 5ay - 5a^3y$.
+
+\item $m^2 - y^2 - am + ay$.
+
+\item $a^2 - x^2 - a - x$.
+
+\item By how much does $x$ exceed $3y$?
+
+\item Eleven years ago C was three times as old as D whose age was
+$x$ years. What is C's present age ?
+\end{enumerate}
+
+\textbf{ 26. CASE III. To factor the sum of the cubes of two
+numbers}.
+
+ILLUS. 1. Divide $a^3 + b^2$ by $a + b$.
+
+ILLUS. 2. Divide $m^3 + n^3y^3$ by $m + ny$.
+
+ILLUS. 3. Divide $8a^6 + b^3c^9$ by $2a^2 + bc^3$.
+
+Notice in each case above:
+\begin{enumerate}
+\item That the divisor is the sum of the cube roots of the terms
+of the dividend.
+
+\item That the quotient is the square of the first term of the
+divisor, minus the product of the first term by the second, plus
+the square of the second.
+\end{enumerate}
+
+\textit{ Write two factors, one of which is the sum of the cube
+roots of the terms, and the other the quotient obtained by
+dividing the original number by the first factor}.
+
+\subsubsection*{Exercise 34}
+
+Factor:
+
+\begin{enumerate}
+\item $x^5 + y^5$
+
+\item $c^3 + d^3$
+
+\item $a^3 + b^3c^3$
+
+\item $a^3x^3 + y^3$
+
+\item $8a^5b^3c^6 + m^6$
+
+\item $x^6y^3 + 216a^3$
+
+\item $a^6 + b^6$
+
+\item $64x^6 + y^6$
+
+\item $x^3 + 8$
+
+\item $27 + a^3b^6$
+
+\item $y^3 + 1$
+
+\item $1 + b^3c^3$
+
+\item $\frac{1}{8}a^6b^3+ c^0$
+
+\item $\frac{1}{27}x^3+ 1$
+
+\item $(m + n)^3+ 8$
+
+\item $1 + (x - y)^3$
+
+\item $2a^2x^3y^6 + 2a^8b^9$
+
+\item $mx + my - nx - ny$
+
+\item $a^3x + b^3x - a^3y - b^3y$
+
+\item $a^6 - b^6$
+
+\item $729x^6 - 64y^6$
+
+\item $1 - x^8$
+
+\item A had $d$ dollars, but, after giving \$26 to B, he had
+one-third as many as B. How many has B? How many had B at first?
+
+\item How many units in $y$ tens?
+
+\item If the sum of two numbers is $x$, and one of the numbers is
+8, what is the other number?
+
+\item What does $a^3$ mean? What does $3a$ stand for? What is the
+product of $x^4$ and $0$?
+\end{enumerate}
+
+\textbf{27. CASE IV. To factor the difference of the
+cubes of two numbers.}
+
+ILLUS. I. Divide $a^3-b^3$ by $a-b$.
+
+ILLUS. 2. Divide $27a^6b^9 - c^3d^12$ by $3a^2b^3 - cd^4$.
+
+Notice in each case above:
+
+\begin{enumerate}
+\item That the divisor is the difference of the cube roots of the
+terms of the dividend.
+
+\item That the quotient is the square of the first term of the
+divisor, plus the product of the first by the second, plus the
+square of the second.
+\end{enumerate}
+
+\textit{Write two factors, one of which is the difference
+of the cube roots of the terms, and the other the
+quotient obtained by dividing the original number
+by the first factor.}
+
+\subsubsection*{Exercise 35.}
+
+Factor:
+
+\begin{enumerate}
+\item $x^3 - a^3$
+
+\item $c^3 - b^3$
+
+\item $a^3 - x^3y^3.$
+
+\item $m^3n^3 - c^3.$
+
+\item $27m^3 - 8x^3.$
+
+\item $8x^3 -64y^3$
+
+\item $64a^3x^6b^3 - 125m^9c^3y^6.$
+
+\item $27b^3c^6y^3 - 216a^9m^6x^6.$
+
+\item $8x^9y^3 - 125.$
+
+\item $27 - 64m^3x^6y^3.$
+
+\item $a^3-b^6.$
+
+\item $m^6 - x^3.$
+
+\item $x^3 - 1.$
+
+\item $1 - y^3.$
+
+\item $\frac{1}{27}x^3y^6 - b^9.$
+
+\item $8 - \frac{1}{64}m^6n^3.$
+
+\item $1 - (a + b)^3.$
+
+\item $x^3y^3 - (x - y^2)^3.$
+
+\item $x^7 - xy^3.$
+
+\item $ab^3 - ac^3 + mb^3 - mc^3.$
+
+\item $x^6 - y^6$ into four factors.
+
+\item $x^6 - x^5 + x^3 - x^2 - x + 1.$
+
+\item $a^2 - b^2 + a^3 -b^3.$
+
+\item $mx^3 + my^3 - x - y.$
+
+\item How many hours will it take $x$ men to dig 75 bushels of
+potatoes if each man digs $y$ bushels an hour?
+
+\item If there are $x$ tens, $y$ units, and $z$ hundreds in a
+number, what will represent the whole number of units?
+\end{enumerate}
+
+\textbf{ 28. CASE V. To factor trinomials which are perfect
+squares.}
+
+Square $c+b, c-b, x^2-y^2, 3mn^3+y^2, 2a^2bc-3x^2yz^3$. How are
+the first and last terms of these trinomial squares formed? How is
+the middle term formed?
+
+When is a trinomial a square?
+
+Name those of the following trinomials which are squares:
+\begin{enumerate}
+
+\item $m^2-2mp+p^2$.
+
+\item $x^2+2xy-y^2$.
+
+\item $4x^2-4xy+y^2$.
+
+\item $x^4+6x^2y+9y^2$.
+
+\item $a^4-18a^2+9$.
+
+\item $9b^4+12d^2+4$.
+
+\item $16y^4-8y^2+1$.
+
+\item $25c^8d^6+20c^4d^3x+4x^2$.
+\end{enumerate}
+
+ILLUS. 1. $a^2+2ab+b^2=(a+b)(a+b)=(a+b)^2$.
+
+ILLUS. 2. $a-2ab+b=(a-b)(a-b)=(a-b)^2$.
+
+ILLUS. 3. $9a^2-12ab+4b^2=(3a-2b)^2$.
+
+\textit{ Write two binomial factors, each of which consists of the
+square roots of the squares connected by the sign of the middle
+term.}
+
+
+\subsubsection*{Exercise 36.}
+
+Factor:
+\begin{enumerate}
+\item $a^2 + 2ax + x^2$.
+
+\item $c^2 - 2cd + d^2$.
+
+\item $4a^2 + 4ay + y^2$.
+
+\item $a^2 + 4ab + 4b^2$.
+
+\item $x^2 - 6cx + 9c^2$.
+
+\item $16x^2 - 8xy + y^2$.
+
+\item $a^2 + 2a + 1$.
+
+\item $9 - 6x + x^2$.
+
+\item $x^2 - 10x + 25$.
+
+\item $4y^2 - 12y + 9$.
+
+\item $9x^2 + 24x + 16$.
+
+\item $81a^4 - 18a^2 + 1$.
+
+\item $9c^2 + 66cd + 121d^2$.
+
+\item $4y^2 - 36y + 81$.
+
+\item $x^4 - 6x^2y + 9y^2$.
+
+\item $9 - 12x^2 + 4x^4$.
+
+\item $16x^2y^4 - 24a^3xy^2 + 9a^6$.
+
+\item $25b^2 + 30bc^2y + 9c^4y^2$.
+
+\item $49a^2 + 28ax^2y + 4x^4y^2$.
+
+\item $\frac{1}{9}x^4 - \frac{2}{3}x^2y^2 + 4y^4$.
+
+\item $\frac{1}{4}a^2 + 2ab + 4b^2$.
+
+\item $\frac{4}{9}x^6y^2 - \frac{4}{3}x^3yz^4 + z^8$.
+
+\item $x^9 - 4x^6y^4 + 4x^3y^8$.
+
+\item $18a^2y + 24ay^3 + 8y^5$.
+
+\item $3a^3x^5 - 30a^2bx^4 + 75ab^2x^3$.
+
+\item $(x+y)^2 - 2a^2(x+y) + a^4$.
+
+\item $(m^2 - n^2)^2 - 2(m^4 - n^4) + (m^2 + n^2)^2$.
+
+\item $a^2 + 2ab + b^2 + 6a + 6b$.
+
+\item $x^2 + y^2 - 3x - 2xy + 3y$.
+
+\item $x^6 + y^6 - 2x^3y^3$.
+
+What must be added to the following to make them
+perfect squares?
+
+\item $a^2 + 2xy$.
+
+\item $m^6 + 6m^3y$.
+
+\item $c^2 + d^2$.
+
+\item $(x - y)^4 + 2(x - y)^2$.
+
+\item $(c - d)^2 - 6(c -d )$.
+
+\item $9x^4y^2 - 12x^2y^3$.
+
+\item $30xy^4 + 9x^2y^6$.
+
+\item $25a^2b^2 + 36b^4c^2$.
+
+\item $49a^2b^2c^2 + 25a^2b^4d^2$.
+
+\item $a^4 - 22a^2 + 9$ (two numbers).
+
+\item $64a^2 - 177ay + 121y^2$.
+
+\item $a^4 - 10a^2b^2 + 9b^4$.
+
+\item $x^4 + x^2 + 1$.
+
+\item $a^2 + a + 1$.
+
+\item A stream flows at the rate of $a$ miles an hour, and a man
+can row in still water $b$ miles an hour. How far can the man row
+up the stream in an hour? In 6 hours? How far down the stream in
+an hour? In 3 hours?
+
+\item A cistern can he filled by two pipes in 3 hours and 5 hours
+respectively. What part of the cistern will be filled by both
+pipes running together for one hour?
+
+\item Nine years ago Henry was three times as old as Julius. If
+Henry is $b$ years old, how old was Julius then? How old is Julius
+now?
+\end{enumerate}
+
+\textbf{29. CASE VI. To factor trinomials in the form}
+$x^2 \pm cx \pm d.$
+
+This is the reverse of the case under Multiplication
+given in Art. 14.
+
+ILLUS. 1. $x^2 + 14x + 45 = (x + 9)(x + 5)$.
+
+ILLUS. 2. $x^2 - 6x + 5 = (x - 5)(x - 1)$.
+
+ILLUS. 3. $x^2 + 2x - 3 = (x + 3)(x - 1)$.
+
+ILLUS. 4. $x^2 - 8x - 20 = (x - 10)(x + 2)$.
+
+\textit{Write two binomial factors, the first term of each
+being the square root of the first term of the given
+trinomial, and for the second terms of the factors
+find two numbers whose algebraic sum is the coefficient
+of the second term and whose product is the
+last term.}
+
+\subsubsection*{Exercise 37.}
+
+Factor:
+\begin{enumerate}
+
+\item $a^2+3a+2$.
+
+\item $x^2+9x+18$.
+
+\item $x^2-5x+6$.
+
+\item $a^2-7a+10$.
+
+\item $y^2-10y+16$.
+
+\item $c^2-c-6$.
+
+\item $x^2+4x-5$.
+
+\item $x^2+5x-6$.
+
+\item $y^2+8y-65$.
+
+\item $a^2-4a-77$.
+
+\item $x^2-2x-63$.
+
+\item $a^2+10a-75$.
+
+\item $a^2-24a+143$.
+
+\item $x^6-4x^3-117$.
+
+\item $x^6+4x^3-77$.
+
+\item $30+11x+x^2$.
+
+\item $21+10a+a^2$.
+
+\item $35-12x+x^2$.
+
+\item $36-13x+x^2$.
+
+\item $c^2+2cd-3d^2$.
+
+\item $a^2+8ax+15x^2$.
+
+\item $x^2-xy-20y^2$.
+
+\item $x^4+x^2y-12y^2$.
+
+\item $x^4-14x^2y^2+45y^4$.
+
+\item $x^5-23x^4+132x^3$.
+
+\item $a^6-12a^5+35a^4$.
+
+\item $3ax^4-39ax^2+108a$.
+
+\item $3a^3-12a^2b+12ab^2$.
+
+\item $c^5d^3-c^2-a^2c^3d^3+a^2$.
+
+\item $ax^2+bx^2-5bx-5ax+6b+6a$.
+
+\item A field can be mowed by two men in $a$ hours and $b$ hours
+respectively. What part of the field can be mowed by both men
+working together for one hour?
+
+\item In how many days can two men do as much as $x$ men in 7
+days?
+\end{enumerate}
+
+\subsubsection*{Exercise 38. (Review.)}
+
+\begin{enumerate}
+\item Divide $50a+9a^4+24-67a^2$ by $a+a^2-6$.
+
+\item Find the square root of
+\[
+12y^5-14y^3+1-8y^4+4y+9y^6.
+\]
+
+\item Expand $(x+y)(x-y)(x^2+y^2)$.
+
+\item What must be subtracted from $a^3-5a^2+27a-1$ to produce
+$a+1$?
+
+\item Expand $\left(x^2y-4z^3a^4\right)^3$.
+
+Factor:
+
+\item $x^2-11x+10$.
+
+\item $\left(a+b\right)^2-\left(c+d\right)^2$.
+
+\item $6x^3+4x^2-9x-6$.
+
+\item $9x^8-66x^6+121x^4$.
+
+\item $8c^6-d^9$.
+
+\item $1+64x^3$.
+
+\item $a^6-1$.
+
+\item $x^3+2x^2-x-2$.
+
+\item $x^5-y^5$.
+
+\item $27+12x+x^2$.
+
+\item Find the cube root of $96m-64-40m^3+m^6+6m^5$.
+
+\item Simplify $\left(m+2n\right)^2-\left(2m-n\right)^2$.
+
+\item Simplify $(5+7x)(5-7x)-\left(3+2x\right)^2+(x-9)(x-4)$.
+
+\item Simplify $(y+x)(y^2-xy+x^2)-(y-x)(y^2+yx+x^2)$.
+
+\item Find three consecutive numbers whose sum is 57.
+
+\item What number, being increased by three-fifths of itself, will
+equal twice itself diminished by 24?
+
+\item Find the value of $(4a + 5b) (3a+4b)-a^2b+b^3c^2-c^3$ when
+$a = 0$, $b = 1$, and $c = 2$.
+\end{enumerate}
+
+\section*{GREATEST COMMON FACTOR.}
+\addcontentsline{toc}{section}{\numberline{}GREATEST COMMON
+FACTOR.}
+
+
+\textbf{30.} What are the factors of $9a^4b^2+9a^3b^3$? Of $3a^4b-3a^2b^3$?
+Which are factors of each of them? What is the largest
+number that is a factor of each of them? This number
+is called their \textit{Greatest Common Factor}. What factors of
+the numbers does it contain?
+
+ILLUS. Find the G.C.F. of
+
+\begin{center}
+
+$3ac+3bc$ and $6a^2x+12abx+6b^2x$.
+
+\[
+\begin{array}{rcll}
+3ac+3bc & = & 3c(a+b) \\
+6a^2x+12abx+6b^2x & = & 6x(a+b)(a+b) \\
+\cline{3-3}
+G.C.F. & = & \multicolumn{2}{l}{$3(a+b)=3a+3b$}
+\end{array}
+\]
+
+\end{center}
+
+
+\textit{To find the G.C.F. of two or more numbers, find the prime
+factors of each of the numbers and take the product of the common
+factors}.
+
+\subsubsection*{Exercise 39.}
+
+Find the G.C.F. of each of the following:
+
+\begin{enumerate}
+\item $3a^4 b^3 + 6a^2 b^5$ and $9a^3 b^2 + 18 a b^3$.
+
+\item $5 x^5 y - 15 x^2 y^2$ and $15x^3 y^3 - 45 x y^5$.
+
+\item $2x^5 y^4 + 2x^2 y^7$ and $6 x^3 y^3 - 6x y^5$.
+
+\item $3a^6 b - 3a^3 b^4$ and $12a^4 b^4 - 12a^2 b^6$.
+
+\item $mx - ma - nx + na$ and $m^3 - n^3$.
+
+\item $81 x^8 - 16$ and $81x^8-72x^4 + 16$.
+
+\item $x^2 - 20 - x$, $3x^2 - 24 x + 45$, and $x^2 - 5x$.
+
+\item $x^2 +x -6$, $x^2-15 - 2x$, and $3x^2-27$.
+
+\item $a^4 - 16$, $a2 -a -6$, and $(a^2 -4)^2$.
+
+\item $y^3 - y$, $y^3 + 9y^2 - 10y$, and $y^4-y$.
+
+\item $x^2 - y^2$, $xy - y^2 + xz - yz$, and $x^3 - x^2 y + xy^2 -
+y^3$.
+
+\item $3a c^5 d^3 - 3a c^2 - 3a^3 c^3 d^3 + 3a3$ and $9a^2 c^4 -
+9a^6$.
+
+\item $64x^{11} - 8x^2 y^6$ and $12x^8 + 3x^2 y^4 - 12x^5 y^2$.
+
+\item A merchant mixes $a$ pounds of tea worth $x$ cents a pound
+with $b$ pounds worth $y$ cents a pound. How much is the mixture
+worth per pound?
+
+\item If a man bought a horse for $x$ dollars and sold him so as
+to gain $5 \%$, what will represent the number of dollars he
+gained?
+
+\item The difference between two numbers is 6, and if 4 be added
+to the greater, the result will be three times the smaller. What
+are the numbers?
+
+Of how many terms does the expression $x^{3} - 4 x^{2} y + y^{3}$
+consist? How many factors has each of the terms?
+What is the value of a number, one of whose factors is
+zero?
+\end{enumerate}
+
+
+
+\section*{LEAST COMMON MULTIPLE.}
+\addcontentsline{toc}{section}{\numberline{}LEAST COMMON
+MULTIPLE.}
+
+
+\textbf{31}. ILLUS. 1. $12 a^{2} b^{3}$ is the least common
+multiple of $3 a^{2} b$ and $4 a b^{3}$. How many of the factors
+of $3 a^{2} b$ are found in the L.C.M.? How many of the factors of
+$4 a b^{3}$?
+
+ILLUS. 2. Find the L.C.M. of $9 a c + 9 b c$, $3 a^{2} x - 3 b^{2}
+x$, and $a x^{2} - b x^{2}$.
+
+\begin{tabular}{rcl}
+$9 a c + 9 b c$ & = & $9 c (a + b)$ \\
+$3 a^{2} x - 3 b^{2} x$ & = & $3 x (a + b) (a - b)$ \\
+$a x^{2} - b x^{2}$ & = & $x^{2} (a - b)$ \\ \hline
+L.C.M. & = & $9 c x^{2} (a + b) (a - b)$
+\end{tabular}
+
+\textit{To find the L.C.M. of two or more numbers, find
+the prime factors of each number, take the product
+of the different factors, using each the greatest number
+of times it is found in any of the numbers}.
+
+
+
+
+
+\subsubsection*{Exercise 40}.
+
+Find the L.C.M. of each of the following:
+
+\begin{enumerate}
+\item $a^{2} - 4$ and $a^{2} + 3 a - 10$.
+
+\item $x^{3} + 1$ and $x^{2} + 2 x + 1$.
+
+\item $x^{2} - 2 x - 15$ and $x^{2} - 9$.
+
+\item $x^{4} - 4 x^{2} + 3$ and $x^{6} - 1$.
+
+\item $x^2 - 1$, $x^2 - 4x + 3$, and $2x^2 - 2x - 12$.
+
+\item $a^3 + 1 + 3a + 3a^2$ and $am - 2 - 2a + m$.
+
+\item $a^2 - 4$, $a^2 + 3a - 10$, $a^2 - 25$, $a^2 - 9$, $a^2 - 8a
++ 15$, and $a^2 + 5a + 6$.
+
+\item $x^3 - 1 - 3x^2 + 3x$ and $xy - 3 - y +3x$.
+
+\item $x^2 - 1$, $x^2 - 9$, $x^2 - 2x - 3$, $x^2 - 16$, $x^2 - x -
+12$, $x^2 + 5x + 4$.
+
+\item $x^6 + x^3$, $2x^4 - 2x^2$, and $x^2 + x$.
+
+\item $a^4 + 2a^2 + 1$, $1 - 2a^2 + a^4$, and $1 - a^4$.
+
+\item $a^2z - x^2z$, $3ax - 3x^2$, and $2ax + 2a^2$.
+
+\item $am + an - 3m - 3n$, $m^2 - n^2$, and $a^2 - 10a + 21$.
+
+\item $3b - 3b^2$, $1 - b^3$, and $2x - 2b^2x$.
+
+\item $a^2 - x^2 - a - x$, $5a^2b - 10abx + 5bx^2$, and $a^2b -
+abx - ab$.
+
+\item The thermometer now indicates $c$ degrees above zero, but
+yesterday it indicated $y$ degrees below zero. How much warmer is
+it to-day than yesterday?
+
+\item A certain county extends from $b$ degrees north latitude to
+$y$ degrees north latitude. What is the extent of the county from
+north to south?
+
+\item If A can build a wall in $x$ days, what part of the wall
+will he build in two days?
+
+\item How many times can $x^3 + 12$ be subtracted from $x^6 + 24
+x^3 + 144$?
+
+\item Divide \$2142 between two men so that one shall receive six
+times as much as the other.
+\end{enumerate}
+
+\subsubsection*{Exercise 41.\footnotemark}
+\footnotetext{This exercise should be conducted orally, and if, at
+any point, the students do not readily recall the method, the
+examples of that class should be duplicated till the principle is
+clear.}
+
+\begin{enumerate}
+
+\item Reduce to lowest terms:
+
+$\frac{3}{6}$, $\frac{6}{9}$, $\frac{4}{10}$, $\frac{12}{14}$, $\frac{2}{16}$, $\frac{9}{36}$, $\frac{8}{22}$, $\frac{8}{20}$, $\frac{7}{8}$, $\frac{9}{30}$, $\frac{25}{150}$, $\frac{490}{560}$, $\frac{75}{325}$.
+
+\item Change:
+
+$a$. $\frac{1}{4}$ to $8ths$.
+
+$b$. $\frac{5}{6}$ to $12ths$.
+
+$c$. $\frac{7}{8}$ to $32ds$.
+
+$d$. $\frac{3}{5}$ to $25ths$.
+
+$e$. $\frac{2}{3}$ to $21sts$.
+
+$f$. $\frac{9}{10}$ to $50ths$.
+
+$g$. $\frac{3}{7}$ to $28ths$.
+
+$h$. $\frac{5}{9}$ to $36ths$.
+
+$i$. $\frac{6}{13}$ to $39ths$.
+
+\item How many 15ths in $\frac{3}{5}$, $\frac{2}{3}$,
+$\frac{4}{5}$, $3$, $1\frac{2}{5}$, $2\frac{2}{3}$?
+
+\item How many 12ths in $\frac{1}{6}$, $\frac{3}{4}$,
+$\frac{5}{6}$, $4$, $9$, $2\frac{1}{3}$, $7\frac{3}{6}$?
+
+\item Change to equivalent fractions:
+
+$a$. $2\frac{5}{6}$.
+
+$b$. $3\frac{3}{4}$.
+
+$c$. $12\frac{2}{3}$.
+
+$d$. $27\frac{1}{8}$.
+
+$e$. $9\frac{6}{7}$.
+
+$f$. $18\frac{2}{5}$.
+
+\item Change to equivalent entire or mixed numbers:
+
+$a$. $\frac{12}{6}$.
+
+$b$. $\frac{15}{8}$.
+
+$c$. $\frac{17}{4}$.
+
+$d$. $\frac{84}{12}$.
+
+$e$. $\frac{57}{8}$.
+
+$f$. $\frac{61}{5}$.
+
+$g$. $\frac{125}{6}$.
+
+$h$. $\frac{97}{11}$.
+
+$i$. $\frac{225}{7}$.
+
+\item Change to equivalent fractions having L.~C.~D.:
+
+$a$. $\frac{1}{2}, \frac{3}{4}, \frac{2}{3}$.
+
+$b$. $\frac{2}{3}, \frac{5}{8}, \frac{7}{12}$.
+
+$c$. $\frac{1}{4}, \frac{2}{5}, \frac{3}{10}$.
+
+$d$. $\frac{2}{3}, \frac{4}{5}, \frac{1}{6}$.
+
+$e$. $\frac{7}{8}, \frac{1}{5}, \frac{1}{2}$.
+
+$f$. $\frac{14}{15}, \frac{11}{20}, \frac{7}{9}, \frac{1}{5}$.
+
+\item Add:
+
+$a$. $\frac{1}{2}$ and $\frac{1}{3}$.
+
+$b$. $\frac{1}{4}$ and $\frac{1}{5}$.
+
+$c$. $\frac{1}{6}$ and $\frac{1}{2}$.
+
+$d$. $\frac{2}{3}$ and $\frac{3}{4}$.
+
+$e$. $\frac{2}{5}$ and $\frac{1}{6}$.
+
+$f$. $\frac{3}{7}$ and $\frac{5}{6}$.
+
+$g$. $2\frac{1}{3}$ and $3\frac{1}{5}$.
+
+$h$. $4\frac{2}{5}$ and $11\frac{5}{6}$.
+
+$i$. $14\frac{3}{4}$ and $27\frac{4}{7}$.
+
+\item Subtract:
+
+$a$. $\frac{1}{4}$ from $\frac{1}{2}$.
+
+$b$. $\frac{1}{7}$ from $\frac{1}{4}$.
+
+$c$. $\frac{1}{9}$ from $\frac{1}{6}$.
+
+$d$. $\frac{2}{5}$ from $\frac{7}{10}$.
+
+$e$. $\frac{3}{7}$ from $\frac{5}{6}$.
+
+$f$. $\frac{4}{9}$ from $\frac{1}{2}$.
+
+$g$. $2 \frac{1}{5}$ from $5 \frac{1}{2}$.
+
+$h$. $6 \frac{3}{5}$ from $15 \frac{3}{6}$.
+
+$i$. $27 \frac{7}{8}$ from $29 \frac{1}{3}$.
+
+\item Find the product of:
+
+$a$. $\frac{2}{7}$ by $3$.
+
+$b$. $\frac{3}{10}$ by $2$.
+
+$c$. $\frac{3}{8}$ by $6$.
+
+$d$. $5$ by $\frac{2}{9}$.
+
+$e$. $9$ by $\frac{7}{18}$.
+
+$f$. $4$ by $\frac{5}{6}$.
+
+$g$. $2 \frac{3}{4}$ by $5$.
+
+$h$. $12 \frac{3}{9}$ by $3$.
+
+$i$. $25$ by $2 \frac{2}{5}$.
+
+$j$. $\frac{1}{2}$ by $\frac{1}{5}$.
+
+$k$. $\frac{1}{3}$ by $\frac{4}{5}$.
+
+$l$. $\frac{5}{9}$ by $\frac{3}{7}$.
+
+$m$. $\frac{33}{54}$ by $\frac{12}{44}$.
+
+$n$. $15 \frac{2}{5}$ by $\frac{2}{3}$.
+
+$o$. $24 \frac{1}{4}$ by $\frac{5}{8}$.
+
+$p$. $36 \frac{2}{5}$ by $\frac{5}{9}$.
+
+$q$. $2 \frac{1}{3}$ by $5 \frac{1}{2}$.
+
+$r$. $3 \frac{5}{6}$ by $4 \frac{3}{5}$.
+
+\item Divide:
+
+$a$. $\frac{12}{17}$ by $3$.
+
+$b$. $\frac{10}{27}$ by $5$.
+
+$c$. $\frac{5}{7}$ by $4$.
+
+$d$. $\frac{3}{4}$ by $12$.
+
+$e$. $728 \frac{14}{15}$ by $7$.
+
+$f$. $843 \frac{1}{5}$ by $8$.
+
+$g$. $679 \frac{1}{2}$ by $6$.
+
+$h$. $\frac{1}{4}$ by $\frac{1}{8}$.
+
+$i$. $\frac{2}{3}$ by $\frac{2}{7}$.
+
+$j$. $\frac{4}{9}$ by $\frac{3}{8}$.
+
+$k$. $6$ by $\frac{2}{3}$.
+
+$l$. $9$ by $\frac{5}{6}$.
+
+$m$. $\frac{12}{25}$ by $\frac{9}{20}$.
+
+$n$. $3 \frac{1}{2}$ by $1 \frac{3}{4}$.
+
+$o$. $3 \frac{3}{5}$ by $2 \frac{2}{5}$.
+
+$p$. $48$ by $3 \frac{1}{5}$.
+
+$q$. $63$ by $4 \frac{2}{3}$.
+
+$r$. $\frac{5}{6}$ by $\frac{3}{8}$.
+
+\item Simplify:
+
+$a$. $\frac{4}{5} + \frac{1}{2} - \frac{1}{3} =$ ?
+
+$b$. $\frac{\frac{3}{4} + \frac{2}{3}}{\frac{3}{4} - \frac{2}{3}} =$ ?
+
+$c$. $\frac{5}{6} - \frac{1}{4} + \frac{4}{5} =$ ?
+
+$d$. $\frac{2 \frac{1}{3}}{4 \frac{1}{5}} =$ ?
+
+\item $16$ is $\frac{4}{7}$ of what number?
+
+\item $\frac{3}{4}$ is what part of $7$?
+
+\item $\frac{2}{5}$ is what part of $\frac{5}{6}$?
+
+\item What is $\frac{5}{8}$ of $\frac{4}{15}$?
+
+\item If $\frac{2}{3}$ of a number is $20$, what is $\frac{2}{5}$
+of the number?
+
+\item $\frac{2}{3}$ of $60$ is $\frac{4}{9}$ of what number?
+\end{enumerate}
+
+
+\chapter*{FRACTIONS.}
+\addcontentsline{toc}{chapter}{\numberline{}FRACTIONS.}
+
+
+\textbf{ 32.} In the previous exercise the different operations
+performed upon fractions in arithmetic have been reviewed. The
+principles and methods of operation are the same in algebra.
+
+\section*{REDUCTION OF FRACTIONS.}
+\addcontentsline{toc}{section}{\numberline{}REDUCTION OF
+FRACTIONS.}
+
+\textbf{ 33.} \textsc{ILLUS.} 1. Reduce $\frac{5a^2b}{10ab^2}$ to
+lowest terms.
+\[
+\frac{5a^2b \div 5ab}{10ab^2 \div 5ab} = \frac{a}{2b}
+\]
+
+\textsc{ILLUS.} 2. Reduce $\frac{a^2bx-b^3x}{a^2bx-ab^2x}$ to
+lowest terms.
+
+\begin{displaymath}
+\frac{a^2bx-b^3x}{a^2bx-ab^2x} = \frac{bx(a^2-b^2) \div bx(a-b)}{abx(a-b) \div bx(a-b)} = \frac{a+b}{a}
+\end{displaymath}
+
+
+\textit{ To reduce a fraction to its lowest terms, divide the
+terms of the fraction by their greatest common factor.}
+
+\subsubsection*{Exercise 42.}
+
+Reduce to lowest terms:
+
+\begin{enumerate}
+\item $\frac{\displaystyle 15x^3y^2z}{\displaystyle 40x^2y^2z^2}$.
+
+\item $\frac{\displaystyle 12x^5y^2z^3}{\displaystyle 30xy^3z^3}$.
+
+\item $\displaystyle \frac{x^2-7x+12}{x^3+x-20}$.
+
+\item $\displaystyle \frac{x^2+9x+18}{x^2-2x-15}$.
+
+\item $\displaystyle \frac{a^6+a^4}{a^4-1}$
+
+\item $\displaystyle \frac{x^5-x^2}{x^6-1}$
+
+\item $\displaystyle \frac{mx - ny + nx - my}{ax-2by+2bx-ay}$
+
+\item $\displaystyle \frac{ac-bd+ad-bc}{ax-2by+2ay-bx}$
+
+\item $\displaystyle \frac{ac^2d-a^3d^3}{a^3c+a^4d}$
+
+\item $\displaystyle \frac{a^2xy-x^3y^3}{ax^2-x^3y}$
+
+\item $\displaystyle \frac{a^4-b^4}{(a^2+2ab+b^2)(a^2+b^2)}$
+
+\item $\displaystyle \frac{x^8 -y^8}{(x^4+y^4)(x^4-2x^2y^2+y^4)}$
+
+\item $\displaystyle \frac{a^2x^2-16a^2}{ax^2+9ax+20a}$
+
+\item $\displaystyle \frac{a^4-14a^2-51}{a^4-2a^2-15}$
+
+\item $\displaystyle \frac{3+4x+x^2}{6+5x+x^2}$
+
+\item $\displaystyle \frac{a^2-a^2b^2}{(a-ab)^2}$
+
+\item At two-thirds of a cent apiece, what $b$ apples cost?
+
+\item James is $x$ times as old as George. If James is $a$ years
+old, how old is George?
+\end{enumerate}
+
+\textbf{34.} ILLUS. 1. Change $\frac{ac-bc-d}{c}$ to an equivalent
+entire or mixed number.
+\[
+ \frac{ac-bc-d}{c} = a - b -\frac{d}{c}
+\]
+
+\textit{To find an entire or mixed number equivalent to
+a given fraction, divide the numerator by the denominator.}
+
+
+ILLUS. 2. Change to equivalent fractions $b + \frac{a}{c}$, $b - \frac{a - x}{c}$.
+
+\[
+b + \frac{a}{c} = \frac{bc + a}{c},\quad b - \frac{a - x}{c} =
+\frac{bc - a + x}{c}
+\]
+
+\textit{ To find a fraction equivalent to a given mixed number,
+multiply the entire part by the denominator of the fraction, add,
+the numerator if the sign of the fraction be plus, subtract it if
+the sign be minus, and write the result over the denominator}.
+
+How may an entire number be changed to a fraction having a given denominator?
+
+ILLUS. 3. Change $\frac{a}{b}$, $\frac{c}{d}$, and $\frac{x}{ab}$ to equivalent fractions having a common denominator.
+\[
+\begin{array}{ccccc}
+a & \times & ad & = &a^2d\\
+\cline{1-1} \cline{5-5}
+b & \times & ad & &abd\\
+
+c & \times & ab & = &abc\\
+\cline{1-1} \cline{5-5}
+d & \times & ab & &abd\\
+
+x & \times & d & = &xd\\
+\cline{1-1} \cline{5-5}
+ab & \times & d & &abd\\
+\end{array}
+\]
+
+\textit{ To change fractions to equivalent fractions having a
+common denominator, multiply the terms of each fraction by such a
+number as will make its denominator equal to the L.C.M. of the
+given denominators.}
+
+\subsubsection*{Exercise 43.}
+
+Change to equivalent entire or mixed numbers:
+\begin{enumerate}
+\item $ \displaystyle \frac{bx - cx + m}{x}$
+
+\item $ \displaystyle \frac{mn + an + x}{n}$
+
+\item $\displaystyle \frac{x^2 + y^2 + 3}{x - y}$
+
+\item $\displaystyle \frac{a^3 + b^3 + 3}{x - y}$
+
+\item $\displaystyle \frac{6 a^3 b^2 c - 9 a^2 b^2 + 3 c}{3 a^2
+b}$
+
+\item $\displaystyle \frac{6 x^3 y^2 + 10 x^2 y^2 z - 2 m}{2 x
+y^2}$
+
+\item $\displaystyle \frac{x^3 + 2 x^2 - 2 x + 1}{x^2 - x - 1}$
+
+\item $\displaystyle \frac{2 a^3 + a^2 - 2a + 1}{a^2 + a - 2}$
+
+\item $\displaystyle \frac{x^3 + y^3}{x - y}$
+
+\item $\displaystyle \frac{a^3 - b^3}{a + b}$
+
+\item $\displaystyle \frac{8a^3}{2a - 1}$
+
+\item $\displaystyle \frac{27 x^3}{3x + 1}$
+
+\item $\displaystyle \frac{3x^3 + 8x^2 + 2}{x^2 + 2x - 1}$
+
+\item $\displaystyle \frac{2a^3 + 3a^2 + 10a - 4}{a^2 + 3a + 2}$
+
+\item $\displaystyle \frac{2a^4 + 6a^3 - 6a + 2}{a^2 +a - 1}$
+
+\item $\displaystyle \frac{3x^4 - 5x^3 + x - 1}{x^2 - x - 1}$
+
+Change to equivalent fractions:
+
+\item $\displaystyle x + y - \frac{xy}{x - y}$
+
+\item $\displaystyle a - b - \frac{2 ab}{a + b}$
+
+\item $\displaystyle -c + d + \frac{c^3 + d^3}{c^2 + cd + d^2}$
+
+\item $\displaystyle \frac{x^2y + xy^2}{x - y} + x^2 - y^2$
+
+\item $\displaystyle \frac{x+2}{3x - 1} - x - 2$
+
+\item $\displaystyle 2a - 1 - \frac{a-2}{a+3}$
+
+\item $\displaystyle a^2 - 3a + 2 - \frac{a + 3}{2a^2 - 1}$
+
+\item $\displaystyle 2 x^2 + x - 3 - \frac{x - 2}{3 x^2 + 1}$
+
+\item $\displaystyle \frac{3x + 2}{x^2 + x + 2} - 1$.
+
+\item $\displaystyle 1 - \frac{2a^2 + 3}{a^2 - 2a + 3}$.
+
+\item $\displaystyle \frac{a^2}{a^2 - a + 3} - a^2 - a - 1$.
+
+\item $\displaystyle \frac{x^4}{x^2 + x - 1} - x^2 + x - 1$.
+
+Change to equivalent fractions having a common
+denominator:
+
+\item $\displaystyle \frac{a^2}{2}, \frac{xy}{3},
+\frac{5ab^2}{4}.$
+
+\item $\displaystyle \frac{x^3}{3}, \frac{am}{2},
+\frac{3x^2y}{5}.$
+
+\item $\displaystyle \frac{7x^2}{3y}, \frac{5xy}{2a},
+\frac{x^3}{ay}.$
+
+\item $\displaystyle \frac{2a^2}{7m}, \frac{5ab}{3n},
+\frac{b^4}{mn}.$
+
+\item $\displaystyle \frac{5}{y+x}, \frac{3x}{x-y},
+\frac{2ab}{x^2-y^2}.$
+
+\item $\displaystyle \frac{2}{3+a}, \frac{5a}{a-3},
+\frac{2ax}{a^2-9}.$
+
+\item $\displaystyle \frac{b^2}{a^2-ab}, \frac{a^2b}{a^2-b^2}.$
+
+\item $\displaystyle \displaystyle \frac{m^3}{nm+n^2},
+\frac{m^2n}{m^2-n^2}.$
+
+\item $\displaystyle \frac{2}{x^4-y^4}, \frac{3}{x^2 + xy -
+2y^2}.$
+
+\item $\displaystyle \frac{5}{a^8-b^8}, \frac{2}{a^4 - 4a^2b^2 +
+3b^4}.$
+
+\item $\displaystyle \frac{a-1}{a^2-2a-15}, \frac{a+2}{a^2+a-6},
+\frac{a+3}{a^2-7a+10}.$
+
+\item $\displaystyle \frac{x+1}{x^2-x-6}, \frac{x-2}{x^2+6x+8},
+\frac{x+2}{x^2+x-12}.$
+
+\item $\displaystyle \frac{x^2+3x+2}{x^2-x-6},
+\frac{x^2-x}{x^2+5x+4}, \frac{x^2-6x+9}{x^2+x-12}.$
+
+\item If $x$ quarts of milk cost $m$ cents, what will one pint
+cost?
+
+\item Express four consecutive numbers of which $a$ is the
+largest.
+
+\item What is the next even number above $2m$?
+
+\item What is the next odd number below $4a+1$?
+
+\end{enumerate}
+
+\section*{OPERATIONS UPON FRACTIONS.}
+\addcontentsline{toc}{section}{\numberline{}OPERATIONS UPON
+FRACTIONS.}
+
+\subsection*{ADDITION AND SUBTRACTION.}
+\addcontentsline{toc}{subsection}{\numberline{}Addition and
+Subtraction.}
+
+\textbf{35.} ILLUS. 1. Find the sum of $\frac{6a^2}{9x^2}$,
+$\frac{2a^2-a}{9x^2}$, and $-\frac{3a^2+1}{9x^2}$.
+
+\begin{align*}
+\frac{6a^2}{9x^2} + \frac{2a^2-a}{9x^2} + \left(-\frac{3a^2+1}{9x^2}\right)&=\frac{6a^2 + 2a^2 - a - 3a^2 - 1}{9x^2}\\
+&= \frac{5a^2 - a - 1}{9x^2}
+\end{align*}
+
+
+ILLUS. 2. Find the sum of $\frac{x}{xy-y^2}$, $\frac{1}{x-y}$, and
+$-\frac{1}{y}$.
+
+\begin{align*}
+\frac{x}{xy-y^2} + \frac{1}{x-y} + (-\frac{1}{y})
+&= \frac{x}{y(x-y)} + \frac{y}{y(x-y)} - \frac{x-y}{y(x-y)}\\
+&= \frac{2y}{y(x-y)} = \frac{2}{x-y}
+\end{align*}
+
+\textit{To add fractions, find equivalent fractions having a
+common denominator, add their numerators, write the sum over the
+common denominator, and reduce to lowest terms.}
+
+ILLUS. 3. Subtract $\frac{a-4b}{a-2b}$ from $\frac{a-2b}{a}$.
+
+\[
+\frac{a-2b}{a}-\frac{a-4b}{a-2b}=\frac{a^2-4ab+4b^2}{a(a-2b)}-\frac{a^2-4ab}{a(a-2b)}=\frac{4b^2}{a(a-2b)}
+\]
+
+
+Make a statement of the method of subtracting one
+fraction from another.\\
+
+\subsubsection*{Exercise 44.}
+
+Find the value of:
+
+\begin{enumerate}
+\item $\displaystyle \frac{2a}{3}+\frac{a}{4}-\frac{5a}{6}$.
+
+\item $\displaystyle \frac{2x}{5}+\frac{1}{3} x-\frac{2x}{3}$.
+
+\item $\displaystyle \frac{2x}{3}-\frac{3x}{4}-\frac{4x}{5}$.
+
+\item $\displaystyle \frac{4a}{b}+\frac{3x}{2m}$.
+
+\item $\displaystyle \frac{2x}{3}-x+\frac{3x}{5}$.
+
+\item $\displaystyle \frac{m^2}{n^2}-2+\frac{n^2}{m^2}$.
+
+\item $\displaystyle \frac{x}{mn}-\frac{x+mn}{3n}+2m$.
+
+\item $\displaystyle \frac{2b+x}{3x}+\frac{5b-4x}{9x}$.
+
+\item $\displaystyle
+\frac{3m-a}{am}+\frac{b-2m}{bm}-\frac{3b-2a}{ab}$.
+
+\item $\displaystyle \frac{2a^2}{a^2-b^2}-\frac{2a}{a+b}$.
+
+\item $\displaystyle \frac{2a-3b}{a-2b}-\frac{2a-b}{a-b}$.
+
+\item $\displaystyle \frac{x+4}{x+5}-\frac{x+2}{x+3}$.
+
+\item $\displaystyle \frac{x-4y}{x-2y}-\frac{x^2-4y^2}{x^2+2xy}$.
+
+\item $\displaystyle \frac{x-7}{x+2}+\frac{x+4}{x-3}$.
+
+\item $\displaystyle
+\frac{\left(x+2a\right)^2}{x^3-8a^3}-\frac{1}{x-2a}$.
+
+\item $\displaystyle
+\frac{x^6+x^3y^3+y^6}{x^3+y^3}+\frac{x^6-x^3y^3+y^6}{x^3-y^3}$.
+
+\item $\displaystyle \frac{m - 3}{m + 2} - \frac{m - 2}{m+3} +
+\frac{1}{m - 1}$
+
+\item $\displaystyle \frac{2 (4a + b)}{15 a^2 - 15 b^2} -
+\frac{1}{5 (a+b)} - \frac{1}{3a - 3b}$
+
+\item $\displaystyle \frac{3x^3 - 3x^2 + x - 1}{3x^3 - 3x^2 - x +
+1} - \frac{3x^3 + 3x^2 - x - 1}{3x^3 + 3x^2 + x + 1}$
+
+\item $\displaystyle \frac{1}{a^2 - 7a + 12} - \frac{1}{a^2 - 5a +
+6}$
+
+\item $\displaystyle \frac{a-1}{a-2} + \frac{5 - 2a}{a^2 - 5a + 6}
++ \frac{a-2}{a-3}$
+
+\item $\displaystyle \frac{1}{a^2 + 3a + 2} + \frac{2a}{a^2 + 4a +
+3} + \frac{1}{a^2 + 5a + 6}$
+
+\item $\displaystyle \frac{x+2}{x-5} + \frac{x-3}{x+4} -
+\frac{2x^2 - 3x - 2}{x^2 - x - 20}$
+
+\item $\displaystyle \frac{m - n}{mn} + \frac{p - m}{mp} +
+\frac{n-p}{np}$
+
+\item $\displaystyle \frac{a+b}{ab} - \frac{a-c}{ac} -
+\frac{c-b}{bc}$
+
+\item $\displaystyle \frac{x^2 - yz}{(x+y)(x+z)} + \frac{x^2 -
+xz}{(y+z)(y+x)} + \frac{z^2}{(z+x)(z+y)}$
+
+\item $\displaystyle \frac{m^2 - bx}{(m+b)(m+x)} - \frac{mx -
+b^2}{(b+x)(b+m)} - \frac{mb - x^2}{(x+m)(x+b)}$
+
+\item $x$ is how many times $y$?
+
+\item If $a$ is $\frac{2}{5}$ of a number, what is $\frac{1}{5}$
+of the number?
+
+\item If $x$ is $\frac{3}{7}$ of a number, what is the number?
+
+\item Seven years ago $A$ was four times as old as $B$. If $B$ is
+$x$ years old, what is $A$'s present age?
+\end{enumerate}
+
+\textbf{ 36.} What is the effect of multiplying a number twice by
+$-1$? How many signs to a fraction?
+
+Note carefully in the following illustrations the variety of
+changes that may be made in the signs without changing the value
+of the fraction:
+
+ILLUS. 1.
+\[
+\frac{a}{b}=\frac{-a}{-b}=-\frac{-a}{b}=-\frac{a}{-b}.
+\]
+
+In case the terms of the fraction are polynomials, notice that the
+change in sign affects every term of the numerator or denominator.
+
+ILLUS. 2.
+\[
+\frac{a-b}{x-y}=\frac{b-a}{y-x}=-\frac{b-a}{x-y}=-\frac{a-b}{y-x}.
+\]
+
+ILLUS. 3.
+\[
+\frac{a-b+c}{x+y+z}=\frac{b-c-a}{-x-y-z}=-\frac{b-c-a}{x+y+z}\\
+=-\frac{a-b+c}{-x-y-z}
+\]
+
+When the denominator of the fraction is expressed in its factors,
+the variety of changes is increased.
+
+ILLUS. 4.
+\begin{align*}
+\frac{a}{(x-y)(m-n)} =& \frac{a}{(y-x)(n-m)} = \frac{-a}{(y-x)(m-n)} \\
+ =& \frac{-a}{(x-y)(n-m)} = \frac{-a}{(x-y)(m-n)} \\
+ =& \frac{a}{(y-x)(m-n)} =\frac{a}{(y-x)(n-m)} \\
+\end{align*}
+
+ILLUS. 5.
+
+\begin{align*}
+\frac{a-b}{(x-y)(c-d)}&= \frac{b-a}{(y-x)(c-d)} = \frac{a-b}{(y-x)(d-c)}\\
+&= -\frac{a-b}{(y-x)(c-d)} = \textrm{etc.}
+\end{align*}
+
+What is the effect on the value of a fraction of changing the
+signs of two factors of either denominator or numerator? Why? If
+the signs of only one factor of the denominator are changed, what
+must be done to keep the value of the fraction the same?
+
+Write three equivalent fractions for each of the following by
+means of a change in the signs:
+
+\begin{enumerate}
+\item $\displaystyle \frac{xy}{mn}$
+
+\item $\displaystyle \frac{3ab}{2x^3}$
+
+\item $\displaystyle \frac{x+y}{a-b}$
+
+\item $\displaystyle \frac{1-x}{3c-a^2}$
+
+\item $\displaystyle \frac{x-y-x}{c+d-a}$
+
+\item $\displaystyle \frac{3a-x+y^2}{m^2+2c^2d-z}$
+
+Write six equivalent fractions for each of the following:
+
+\item $\displaystyle \frac{x}{(a-b)(m-n)}$
+
+\item $\displaystyle \frac{3a^2bc}{(2x-y)(a-z)}$
+
+\item $\displaystyle \frac{a-m}{(c-d)(x-y)}$
+
+\item $\displaystyle \frac{2c+d}{(a-b+c)(x-z)}$
+
+Write as many equivalent fractions as possible for
+each of the following:
+
+\item $\displaystyle \frac{c}{(a-b)(x-y)(m-n)}$
+
+\item $\displaystyle \frac{a}{(c-d)(m-x)(y-b)}$
+
+\item $\displaystyle \frac{a^2x}{(2m-n)(c-x)(d-y^2)}$
+
+\item $\displaystyle \frac{(x-y)(2a-b)}{(m-x)(a-c)(d-y)}$
+\end{enumerate}
+
+ILLUS. 1. Find the value of $\frac{1}{x+1}-\frac{1}{1-x}-\frac{x}{x^2-1}$.
+
+\[
+\begin{array}{rl}\displaystyle
+\frac{1}{x+1}-\frac{1}{1-x}-\frac{x}{x^2-1}
+&\displaystyle = \frac{1}{x+1}+\frac{1}{x-1}-\frac{x}{x^2-1}\\
+&\displaystyle = \frac{x-1}{x^2-1}+\frac{x+1}{x^2-1}-\frac{x}{x^2-1}\\
+&\displaystyle = \frac{x}{x^2-1}
+\end{array}
+\]
+
+ILLUS. 2. Find the value of $\frac{1}{(a-b)(b-c)} + \frac{1}{(b-a)(a-c)} + \frac{1}{(c-a)(c-b)}$.
+
+\[
+\begin{array}{c}
+\displaystyle \frac{1}{(a-b)(b-c)} + \frac{1}{(b-a)(a-c)} + \frac{1}{(c-a)(c-b)}\\
+\displaystyle = \frac{1}{(a-b)(b-c)} - \frac{1}{(a-b)(a-c)} + \frac{1}{(a-c)(b-c)}\\
+\displaystyle = \frac{a-c}{(a-b)(b-c)(a-c)} - \frac{b-c}{(a-b)(b-c)(a-c)} + \frac{a-b}{(a-b)(b-c)(a-c)}\\
+\displaystyle = \frac{2a-2b}{(a-b)(b-c)(a-c)}\\
+\displaystyle = \frac{2}{(b-c)(a-c)}
+\end{array}
+\]
+
+\subsubsection*{Exercise 45.}
+
+Find the value of:
+\begin{enumerate}
+\item $\displaystyle \frac{2x}{x^2-4} + \frac{1}{2-x} +
+\frac{1}{2+x}$.
+
+\item $\displaystyle \frac{3a}{a^2-9} + \frac{1}{3-a} -
+\frac{1}{3+a}$.
+
+\item $\displaystyle m^2 - \frac{am^2}{a-m} - \frac{am^2}{m+a} -
+\frac{2a^2m^2}{m^2-a^2}$.
+
+\item $\displaystyle
+\frac{20a-4}{4a^2-1}+\frac{3}{1-2a}-\frac{6}{1-2a}$.
+
+\item $\displaystyle \frac{1}{x-y} + \frac{3y^3}{y^3-x^3} -
+\frac{1}{3(1+a)}$.
+
+\item $\displaystyle
+\frac{a+3}{6(a^2-1)}+\frac{1}{2(1-a)}+\frac{1}{3(1+a)}$.
+
+\item $\displaystyle
+\frac{1}{5(3b-y^2)}-\frac{1}{2(3b+y^2)}-\frac{11b-7y^2}{10(y^4-9b^2)}$.
+
+\item $\displaystyle
+\frac{3}{(1-x)(3-x)}+\frac{1}{(2-x)(x-3)}-\frac{1}{(x-1)(x-2)}$.
+
+\item $\displaystyle
+\frac{a}{(a-b)(b-c)}+\frac{1}{c-a}-\frac{a-b}{(c-a)(c-b)}$.
+
+\item $\displaystyle
+\frac{1}{(z-a)(z-y)}+\frac{1}{(a-z)(a-y)}+\frac{1}{(y-z)(y-a)}$.
+
+\item $\displaystyle
+\frac{1}{(1-x)(1-y)}-\frac{x^2}{(x-1)(y-x)}-\frac{y^2}{(y-1)(x-y)}$.
+
+\item $\displaystyle a-\frac{a^2}{a+1}-\frac{a}{1-a}$.
+
+\item $\displaystyle 1-a+a^2-a^3+\frac{a}{1+a}$.
+
+\item $\displaystyle
+\frac{1}{x^2-7x+12}-\frac{2}{x^2-6x+8}+\frac{2}{x^2-5x+6}$.
+
+\item What will $a$ pounds of rice cost if $b$ pounds costs 43\\
+cents?
+
+\item $x$ is $\frac{4}{9}$ of what number?
+
+\item $m$ is $\frac{5}{x}$ of what number?
+
+\item $y$ is $\frac{a}{b}$ of what number?
+\end{enumerate}
+
+\subsection*{MULTIPLICATION AND DIVISION.}
+\addcontentsline{toc}{subsection}{\numberline{}Multiplication and
+Division.}
+
+\textbf{ 37.} ILLUS. $\displaystyle \frac{x^3}{y} \times c =
+\frac{x^3c}{y}$, $\displaystyle \frac{4x^y}{5a^2bc^3}\times ab =
+\frac{4x^2y}{5ac^3}$
+
+\textit{ A fraction is multiplied by multiplying the numerator or
+dividing the denominator.}\footnotemark
+
+\subsubsection*{Exercise 46.}
+
+Multiply:
+
+\begin{enumerate}
+\item $\displaystyle \frac{a}{b^2}$ by $x$.
+
+\item $\displaystyle \frac{3a^2bc}{7x^2y^3}$ by $y^2$.
+
+\item $\displaystyle \frac{1}{2c^2d}$ by $a^2d$.
+
+\item $\displaystyle \frac{4abc^2}{9x^2y}$ by $3xy$.
+
+\item $\displaystyle \frac{3mn}{2x^2y^2}$ by $6x^2y$.
+
+\item $\displaystyle \frac{x-y}{2mn}$ by $3x$.
+
+\item $\displaystyle \frac{2a-b}{3x^2-xy}$ by $x$.
+
+\item $\displaystyle \frac{x^2}{x^3-y^3}$ by $x-y$.
+
+\item $\displaystyle \frac{3x+3y}{x-y}$ by $x^2-y^2$.
+
+\item $\displaystyle \frac{a^2-4a-21}{a^2-3a-10}$ by $a-5$.
+
+\item $\displaystyle x+y-\frac{4xy}{x+y}$ by $x+y$.
+
+\item $\displaystyle a-b+\frac{4ab}{a-b}$ by $a-b$.
+
+\item $\displaystyle \frac{1}{x-y+z}+\frac{2z}{(x-y)^2-z^2}$ by
+$x^2-xy-xz$.
+
+\item $\displaystyle \frac{1}{a-b+c}-\frac{2b}{(a+c)^2-b^2}$ by
+$ac+bc+c^2$.
+
+\footnotetext{In arithmetic, which of these two methods did you
+find would apply to all examples? When either method may he
+applied to any given example, which is preferable, and why?}
+
+\item How many units of the value of $\frac{1}{5}$ are there in 2?
+
+\item How many units of the value of $\frac{1}{x}$ are there in 5?
+
+\item How many sixths are there in $4a$?\\
+\end{enumerate}
+
+\textbf{ 38.} ILLUS. $\displaystyle \frac{xy^3}{2c}\div y^3=\frac{x}{3c}, \frac{5abc}{3mn^2}$\\
+
+\textbf{ \textit{ A fraction is divided by dividing the numerator
+or multiplying the denominator.\footnotemark}}
+
+\subsubsection*{Exercise 47.}
+
+Divide:
+
+\begin{enumerate}
+\item $\displaystyle \frac{a^2x}{b}$ by $ax$.
+
+\item $\displaystyle \frac{3m^2n}{2xy}$ by $2x$.
+
+\item $\displaystyle \frac{18cd^5}{5x}$ by $6cd^3$.
+
+\item $\displaystyle \frac{6x^2y}{m}$ by $9mn$.
+
+\item $\displaystyle \frac{a}{c+d}$ by $3x$.
+
+\item $\displaystyle \frac{3x^3-3xy}{2m^2n}$ by $3x$.
+
+\item $\displaystyle \frac{2ab-2b^4}{5x^2yz}$ by $2ab$.
+
+\item $\displaystyle \frac{5(a^2-b^2)}{xy}$ by $a-b$.
+
+\item $\displaystyle \frac{m+n}{2m^2-2mn}$ by $m^2-n^2$.
+
+\item $\displaystyle \frac{x^2+5x-14}{x^2-7x+12}$ by $x-2$.
+
+\item $\displaystyle ax+bx+a+b+\frac{a+b}{3x}$ by $a+b$.
+
+\item Multiply $\displaystyle \frac{x^2-5x+6}{x^2+3x-4}$ by $x-1$.
+
+\footnotetext{See previous foot-note.}
+
+\item Divide $\displaystyle \frac {5 b c^2}{6 a x^3} $ by $10 a b
+c$.
+
+\item Multiply $\displaystyle 1 + \frac{3ac}{4 x^2 y} $ by $ 2 a
+c $.
+
+\item Divide $\displaystyle \frac{5 x^2 y - 5 y^3}{2 x^2 z} $ by
+$ 4 x + 4 y $.
+
+\item Divide $\displaystyle \frac{4 x^2 y}{3 m n^2} $ by $ 6 a x
+y $.
+
+\item A can do a piece of-work in $2 \frac{1}{3}$ days, and B in
+$2\frac{1}{2}$ days. How much of the work can they both together
+do in one day?
+
+\item If C can do a piece of work in $m$ days, and D works half as
+fast as G, how much of the work can D do in one day?
+\end{enumerate}
+
+\textbf{ 39.} ILLUS. 1. $\displaystyle \frac{x}{y} \times
+\frac{a}{4} = \frac{a x}{4 y} $
+
+ILLUS. 2. $\displaystyle \frac{7 x^5 y^4}{5 a^3 z^2} \times
+\frac{20a^2 z^3}{42 x^4y^3} = \frac{2 xyz}{3 a}$
+
+ILLUS. 3. $\displaystyle \frac{ab-ax-2 b+2 x}{b^3-x^3} \times
+\frac{b^2 x+b x^2+x^3}{a-2} $
+\begin{center} $\displaystyle = \frac{(a-2) (b-x) x (b^2+b x+x^2)}{(b-x) (b^2+b x+x^2) (a-2)} = x$\\ \end{center}
+
+\textit{ To multiply one fraction by another, multiply the
+numerators together for the numerator of the product and the
+denominators for its denominator, and reduce to lowest terms.}
+
+\subsubsection*{Exercise 48.}
+
+Simplify:
+
+\begin{enumerate}
+
+\item $\displaystyle \frac{7a^2b^2c}{18x^2y^2z} \times
+\frac{9xyz^2}{28abc^2}$
+
+\item $\displaystyle \frac{9a^2b^2}{8nx^3y^2} \times
+\frac{5x^2y^2}{2am^2n} \times \frac{24m^2n^2x}{90ab^2}$
+
+\item $\displaystyle \frac{x^2 - 16y^2}{xy - 4y^2} \times
+\frac{2y}{x + 4y}$
+
+\item$\displaystyle \frac{3a - b}{2b} \times \frac{18ab +
+6b^2}{9a^2 - b^2}$
+
+\item$\displaystyle \frac{abd + cd^2}{a^2d - abc} \times \frac{acd
+- bc^2}{ab^2 + bcd}$
+
+\item$\displaystyle \frac{2ax - 4y}{mn^2 + any} \times \frac{mny +
+ay^2}{a^2x - 2ay}$
+
+\item$\displaystyle \frac{x^2 - x - 6}{x^2 + x - 2} \times
+\frac{x^2 + 3x - 4}{x^2 - 2x - 3}$
+
+\item$\displaystyle \frac{a^2 + 2a - 3}{a^2 + a - 2} \times
+\frac{a^2 + 7a + 10}{a + 3}$
+
+\item$\displaystyle \frac{x^4 - 64x}{3x^2 - 2x} \times \frac{9x^2
+- 4}{x^2 - 4x}$
+
+\item$\displaystyle \frac{m^2 - 25}{m^2 + 2m - 15} \times
+\frac{m^4 - 27m}{m^2 - 5m}$
+
+\item$\displaystyle \frac{a^3 - a^2y + ay^2}{x - 3} \times
+\frac{ax + xy - 3a -3y}{a^3 + y^3}$
+
+\item$\displaystyle \frac{x^3 + x^2 - x - 1}{x^2 + 6x + 9} \times
+\frac{x^2 + 2x - 3}{2x^3 + 4x^2 + 2x}$
+
+\item$\displaystyle \frac{a^3 - a^2 - a + 1}{a^2 + 4a + 4} \times
+\frac{a^2 + 3a + 2}{3a^3 - 6a^2 + 3a}$
+
+\item$\displaystyle (m + \frac{mn}{m - n})(n - \frac{mn}{m + n})$
+
+\item A grocer purchased $y$ pounds of tea for \$25. Another
+grocer purchased 4 pounds less for the same money. What was the
+price per pound which each paid?
+
+\item What is $\displaystyle \frac{a}{3}$ of $\displaystyle
+\frac{2}{c}$?
+
+\item Two numbers differ by 28, and one is eight-ninths of the
+other. What are the numbers?
+\end{enumerate}
+
+\textbf{ 40.} ILLUS. $\displaystyle \frac{b}{c}\div
+\frac{x}{y}=\frac{b}{c}\times\frac{y}{x}=\frac{by}{cx}$
+
+\textit{ To divide one fraction by another, invert the divisor and
+proceed as in multiplication}.
+
+\subsubsection*{Exercise 49.}
+
+
+
+Simplify:
+\begin{enumerate}
+\item $\displaystyle \frac{18a^6b^2c}{35x^3y^3z}\div
+\frac{3a^4b}{5x^2yz}$.
+
+\item $\displaystyle \frac{7axy^2}{8m^2n}\div
+\frac{2xz^3}{3ab^2m}$.
+
+\item $\displaystyle \frac{4x^2y^3z^4}{3a^2b^3c^4}\div
+\frac{3x^4y^3z^2}{4a^4b^3c^2}$.
+
+\item $\displaystyle \frac{2am^3n^2}{3x^3yz^2}\div
+\frac{3a^3mn^2}{2x^3yz^2}$.
+
+\item $\displaystyle \frac{x^2-7x+10}{x^2}\div
+\frac{x^2-4}{x^2+5x}$.
+
+\item $\displaystyle \frac{x-6}{x-3}\div \frac{x^2-3x}{x^2-5x}$.
+
+\item $\displaystyle \frac{x-2}{x-7}\div
+\frac{x^2-9x+20}{x^2-10x+21}\div \frac{x^2-4x+3}{x^2-5x+4}$.
+
+\item $\displaystyle \frac{a^2-7a}{a-8}\div
+\frac{a^2+10a+24}{a^2-14a+48}\div \frac{a^2-7a+6}{a^2+3a-4}$.
+
+\item $\displaystyle \frac{a+b}{(a-b)^2}div
+\frac{a^2+b^2}{a^2-b^2}\times\frac{a^4-b^4}{(a+b)^3}$.
+
+\item $\displaystyle
+\frac{x^2-y^2}{x^2+y^2}\times\frac{x^4-y^4}{(x-y)^3}\div
+\frac{(x+y)^2}{x-y}$.
+
+\item $\displaystyle \frac{a^2-2a-8}{a^2+2a-15}\div
+\frac{a^2+5a+6}{a^2-4a+3}\div \frac{a^2-5a+4}{a^2+8a+15}$.
+
+\item $\displaystyle \frac{x^2-9x+20}{x^2-5x-14}\div
+\frac{x^2-7x+12}{x^2-x-42}\times\frac{x^2-x-6}{x^2+11x+30}$.
+
+\item $\displaystyle \frac{a^2 - a - 2}{a^2 + 2a - 8} \times
+\frac{a^2 + 5a}{ab + b} \div \frac{a^2 - 25}{a^2 - a - 20}$.
+
+\item $\displaystyle \frac{m - 2}{2m - 14} \times \frac{m^2 - 5m -
+14}{2m^2 + 4m} \times \frac{4m^2}{m^2 - 4} \div \frac{2mn -
+14n}{m^2 - 5m - 14}$.
+
+\item $ \displaystyle \left(\frac{a+1}{a-1} + 1\right)
+\left(\frac{a-1}{a+1} + 1\right)
+ \div
+ \left(\frac{a+1}{a-1} - 1\right) \left(1 - \frac{a-1}{a+1}\right)$.
+
+\item $\displaystyle \left(1 + \frac{x}{1-x}\right) \left(1 -
+\frac{x}{1+x}\right) \div \left(1 + \frac{1+x}{1-x}\right) \left(1
+- \frac{1-x}{1+x}\right)$.
+
+\item How much times 7 is 21?
+
+\item How much times $\frac{3}{14}$ is $\frac{6}{7}$?
+
+\item How much times $\displaystyle \frac{c}{d}$ is $\displaystyle
+\frac{x}{4}$?
+
+\item How much times $\displaystyle \frac{5bz}{7mx^3}$ is
+$\displaystyle \frac{3xy^2z}{4amn^2}$?
+
+\end{enumerate}
+
+What may be done to a fraction without changing its value? How
+multiply a fraction? When is a fraction in its lowest terms? How
+reduce a fraction to lowest terms? How divide one fraction by
+another? How add fractions? How divide a fraction? What is the
+effect of increasing the denominator of a fraction? What is the
+effect of dividing the numerator of a fraction? What is the effect
+of subtracting from the denominator of a fraction? How multiply
+two or more fractions together? What is the effect of adding to
+the numerator of a fraction? What is the effect of multiplying a
+fraction by its denominator? What changes in the signs of a
+fraction can be made without changing the value of the fraction?
+How change an entire number to a fraction with a given
+denominator?
+
+\subsection*{INVOLUTION, EVOLUTION, AND FACTORING.}
+\addcontentsline{toc}{subsection}{\numberline{}Involution,
+Evolution and Factoring.}
+
+\textbf{ 41.} ILLUS. 1. $\displaystyle \left(\frac{a}{b}\right)^2
+= \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2}$.
+
+What is the square of $\frac{3x}{2y}$? What is the cube of $\frac{x}{y}$? of\\
+$\frac{2a^2b}{xy}$? What is the square root of $\frac{64x^2}{9y^4}$? What is the\\
+cube root of $\frac{8a^3x^6}{27b^9y^3}$? How find any power of a fraction?\\
+How find any root of a fraction?\\
+
+ILLUS. 2. $\displaystyle \left(\frac{m}{n} + \frac{1}{x}\right)\left(\frac{m}{n} - \frac{1}{x}\right) = \frac{m^2}{n^2} - \frac{1}{x^2}$.\\
+
+
+ILLUS. 3. $\displaystyle \left(\frac{x}{y} - 3\right)\left(\frac{x}{y} - 6\right) = \frac{x^2}{y^2} - \frac{9x}{y} + 18$.\\
+
+ILLUS. 4. Factor $ \frac{b^4}{a^4} - \frac{y^4}{x^4}$.\\
+\begin{center}
+$\displaystyle \frac{b^4}{a^4} - \frac{y^4}{x^4} =
+\left(\frac{b^2}{a^2} + \frac{y^2}{x^2}\right) \left(\frac{b}{a} +
+\frac{y}{x}\right) \left( \frac{b}{a} - \frac{y}{x}\right)$.
+\end{center}
+
+ILLUS. 5. Factor $x^4 + x^3 + \frac{1}{4}$.
+\begin{center}
+$\displaystyle x^4 + x^2 + \frac{1}{4} = \left(x^2 +
+\frac{1}{2}\right)^2$.
+\end{center}
+
+\subsubsection*{Exercise 50.}
+
+Find by inspection the values of each of the following:
+\begin{enumerate}
+
+\item $\displaystyle \left(\frac{2x^2}{ab^3}\right)^2$.
+
+\item $\displaystyle \left(\frac{a^2b^3}{4y}\right)^3$.
+
+\item $\displaystyle \left(-\frac{2xy}{3a^2m^3}\right)^5)$.
+
+\item $\displaystyle \left(\frac{x(a-b)}{3a^2b}\right)^4$.
+
+\item $\displaystyle \left(-
+\frac{5x^2y(a+b^2)^2}{2ab^3(x^2-y)^3}\right)^3$.
+
+\item $\displaystyle \left(-
+\frac{3am^4(2a+3b)^3}{4x^2y^2(m-n)^2}\right)^3$.
+
+\item $\displaystyle \left( -\frac{ 3a^5xy^2z^3 }{ 2bc^4d^4 }
+\right)^4$.
+
+\item $\displaystyle \left( -\frac{ 4x^7y^2z^3 }{ 3ab^6d^3 }
+\right)^4$.
+
+\item $\displaystyle \sqrt[3]{ \frac{ 8a^3b^6 }{ 27m^6n^9 } }$.
+
+\item $\displaystyle \sqrt{ \frac{ 36m^2n^8 }{ 121a^4b^6 } }$.
+
+\item $\displaystyle \sqrt[4]{ \frac{ 256x^4y^8 }{ 81a^4b^{12} }
+}$.
+
+\item $\displaystyle \sqrt[5]{ - \frac{32x^5y^10}{ 243m^10n^15 }
+}$.
+
+\item $\displaystyle \sqrt[3]{ -\frac{ 27x^6(a-b)^9 }{ 64a^6b^{12}
+} }$.
+
+\item $\displaystyle \sqrt{ \frac{ 4x^2 + 12xy + 9y^2 }{ 16x^4y^8
+} }$.
+
+\item $\displaystyle \sqrt{ \frac{ 8x^2(x+y)^5 }{ 50y^4(x+y) } }$.
+
+\item $\displaystyle \sqrt[3]{ -\frac{ 81a^3(a-b)^7 }{ 24b^9(a-b)
+} }$.
+
+\item $\displaystyle \left( \frac{x}{y} + \frac{a}{b} \right)
+ \left( \frac{x}{y} - \frac{a}{b} \right)$.
+
+\item$\displaystyle \left( \frac{a}{b} - \frac{c}{d} \right)
+ \left( \frac{a}{b} + \frac{c}{d} \right)$.
+
+\item $\displaystyle 3ab\left(\frac{2x}{ab}+ \frac{x^2}{2mn}-
+\frac{ax}{3mb}\right)$.
+
+\item $\displaystyle \frac{x^2}{yz}
+ \left( \frac{2x}{3} - \frac{3y^2z}{2x} + \frac{x^2y}{m^2z} \right)$.
+
+\item $\displaystyle \left( \frac{a^4}{b^4} + \frac{c^4}{d^4}
+\right)
+ \left( \frac{a^2}{b^2} + \frac{c^2}{d^2} \right)
+ \left( \frac{ a }{ b } + \frac{ c }{ d } \right)
+ \left( \frac{ a }{ b } - \frac{ c }{ d } \right)$.
+
+\item $\displaystyle \left( \frac{x^2}{16} + \frac{9y^2}{25m^1}
+\right)
+ \left( \frac{x}{4} + \frac{3y}{5m} \right)
+ \left( \frac{x}{4} - \frac{3y}{5m} \right)$.
+
+\item $\displaystyle \left( \frac{x}{y} + 1 \right)
+ \left( \frac{x^2}{y^2} - \frac{x}{y} + 1 \right)$.
+
+\item $\displaystyle \left( \frac{a}{b} - 1 \right)
+ \left( \frac{a^2}{b^2} + \frac{a}{b} + 1 \right)$.
+
+\item $\displaystyle \left( \frac{x}{y} - \frac{3a^2}{2b}
+\right)$.
+
+\item $\displaystyle \left( \frac{2y}{c^2} + \frac{9}{2d}
+\right)^2$.
+
+\item $\displaystyle \left( \frac{a}{b} + 2 \right) \left(
+\frac{a}{b} - 5 \right)$.
+
+\item $\displaystyle \left( \frac{x}{y} - \frac{a}{b} \right)^3$.
+
+Factor:
+
+\item $\displaystyle \frac{x^4}{a^2} - \frac{b^2}{y^4}$
+
+\item $\displaystyle \frac{a^2}{b^4} - \frac{x^8}{y^2}$
+
+\item $\frac{a^4}{m^4} - \frac{b^4}{x^4}$
+
+\item $\displaystyle \frac{8a^3}{y^3} + \frac{b^3}{c^3}$
+
+\item $\displaystyle \frac{x^3}{27a^3} - \frac{y^3}{c^3}$
+
+\item $\displaystyle \frac{81a^4}{b^4} - \frac{x^4y^4}{625z^4}$
+
+\item $\displaystyle \frac{m^2}{y^2} + \frac{2m}{y} - 15$
+
+\item $\displaystyle \frac{x^2}{a^2} - \frac{2x}{a} - 8$
+
+\item $\displaystyle \frac{y^2}{4} + 2 + \frac{4}{y^2}$
+
+\item $\displaystyle \frac{x^2}{a^2} + \frac{a^2}{x^2} - 2$
+
+\item A dog can take 2 leaps of $a$ feet each in a second. How
+many feet can he go in 9 seconds?
+
+\item How many weeks would it take to build a stone wall if
+$\frac{1}{a}$ of it can be built in one day?
+\end{enumerate}
+
+\section*{COMPLEX FRACTIONS.}
+\addcontentsline{toc}{section}{\numberline{}COMPLEX FRACTIONS.}
+
+\textbf{ 42} ILLUS.$ \displaystyle \frac{a + \frac{b}{c}}{d}$ ,
+$\displaystyle \frac{a}{d - \frac{x}{y}}$, $\displaystyle \frac{a
++ \frac{b}{c}}{d - \frac{x}{y}}$, $\displaystyle
+\frac{\frac{a}{x}}{\frac{b}{c}}$
+
+A \textbf{ complex fraction} is one which has a fraction in one or
+both of its terms.
+
+What is a simple fraction? What is the effect of multiplying a
+simple fraction by its denominator or by any multiple of its
+denominator? By what must the numerator of the complex fraction
+$\displaystyle \frac{ a + \frac{ax}{b} }{ \frac{c}{d} +
+\frac{x}{ab} } $be multiplied to make the numerator an entire
+number? By what multiply the denominator to make it an entire
+number? If both numerator and denominator are multiplied, under
+what conditions will the value of the fraction remain the same?
+
+ILLUS. 1. Simplify $\displaystyle \frac{ a + \frac{ax}{b} }{
+\frac{c}{d} + \frac{x}{ab} }$.
+\[
+\begin{array}{ccccc}
+ a &+& \frac{ax}{b} & \times & abd \\
+ \cline{1-3}
+ \frac{c}{d} &+& \frac{x}{ab} & \times & abd
+\end{array}
+ = \frac{ a^2bd + a^2dx }{ abc + dx }
+\]
+
+ILLUS. 2. Simplify $\displaystyle \frac{ \frac{1}{1-x} -
+\frac{1}{1+x} }
+ { \frac{1}{1-x} + \frac{1}{1+x} }$.
+\[
+\begin{array}{ccccc}
+ \frac{1}{1-x} &-& \frac{1}{1+x} & \times & (1-x)(1+x) \\
+ \cline{1-3}
+ \frac{1}{1-x} &+& \frac{1}{1+x} & \times & (1-x)(1+x)
+\end{array}
+ = \frac{1 + x - 1 + x}{1 + x + 1 - x} = \frac{2x}{2} = x
+\]
+
+\textit{To simplify a complex fraction, multiply each term of the
+complex fraction by the L.C.M. of the denominators of the
+fractions in the terms, and reduce.}
+
+\subsubsection*{Exercise 51.}
+
+Simplify:
+
+\begin{enumerate}
+\item $\displaystyle \frac{1+\frac{a}{b}}{\frac{x}{b}-1}$.
+
+\item $\displaystyle \frac{x+\frac{a}{y}}{m-\frac{b}{y}}$.
+
+\item $\displaystyle \frac{x}{a+\frac{b}{c}}$.
+
+\item $\displaystyle \frac{x-\frac{1}{x^2}}{1-\frac{1}{x}}$.
+
+\item $\displaystyle \frac{\frac{a}{b}-\frac{b}{a}}{a-b}$.
+
+\item $\displaystyle \frac{1+m^3}{1+\frac{1}{m}}$.
+
+\item $\displaystyle
+\frac{\frac{x}{y}-\frac{z}{x}}{\frac{a}{y}-\frac{b}{x}}$.
+
+\item $\displaystyle \frac{\frac{ab}{c}-3d}{3c-\frac{ab}{d}}$.
+
+\item $\displaystyle \frac{1+\frac{1}{a-1}}{1-\frac{1}{a+1}}$.
+
+\item $\frac{1+a+a^2}{1+\frac{1}{a}+\frac{1}{a^2}}$.
+
+\item $\displaystyle \frac{x-3-\frac{2}{x-4}}{x-1+\frac{2}{x-4}}$.
+
+\item $\displaystyle
+\frac{\frac{a}{a+b}+\frac{a}{a-b}}{\frac{2a}{a^2-b^2}}$.
+
+\item $\displaystyle
+\frac{\frac{a+1}{a-1}+\frac{a-1}{a+1}}{\frac{a+1}{a-1}-\frac{a-1}{a+1}}$.
+
+\item $\displaystyle
+\frac{\frac{a^3+b^3}{a^2-b^2}}{\frac{a^2-ab+b^2}{a-b}}$.
+
+\item $\displaystyle 1-\frac{1}{1+\frac{2}{a-2}}$.
+
+\item $\displaystyle
+\frac{\frac{a+2b}{b}-\frac{a}{a+b}}{\frac{a+2b}{a+b}+\frac{a}{b}}$.
+
+\item How many pounds of pepper can be bought for $y$ dollars, if
+$x$ pounds cost $2x$ dimes?
+
+\item How many inches in $y$ feet? How many yards?
+
+\item A man bought $x$ pounds of beef at $x$ cents a pound,
+and handed the butcher a $y$-dollar bill. How many cents
+change should he receive?
+
+\item $x$ times $I$ is how many times $m$?
+\end{enumerate}
+
+\subsubsection*{Exercise 52. (Review.)}
+
+\begin{enumerate}
+\item Divide $x^3 - 19x + 6x^7 + 20 + 8x^2 + 16x^4-x^6-11x^5$
+by $x^2 + 4-3x+2x^3$.
+
+\item Find two numbers whose sum is 100 and whose
+difference is 10.
+
+\item Find the G.C.F. of $x^4-1$, $x^5+ x^4-x^3-x^2 + x + 1$, and
+$x^2 - x - 2$.
+
+\item Factor $x^{10} - 14x^5y + 49y^2$, $a^9-b^9$, $3a^2-3a-216$.
+
+\item Reduce $\displaystyle
+\frac{(2a+2b)(a^2-b^2)}{(a^2+2ab+b^2)(a-b)}$ to lowest terms.
+
+\item Factor $\displaystyle
+\frac{16x^4}{a^4b^4}-\frac{c^4}{81y^4}$.
+
+\item If $x$ and $y$ stand for the digits of a number of two
+places, what will represent the number?
+
+\item Find the L.C.M. of $a^2-5a + 6$, $a^2-16$, $a^2-9$, $a^2- 7a
++ 12$, and $a^2 - 4$.
+
+\item If one picture costs $a$ cents, how many can be bought
+for $x$ dollars?
+
+Simplify:
+
+\item $\displaystyle \frac{x - 3}{x - 2} + \frac{2 (1 - x)}{x^2 -
+6x + 8} - \frac{x - 1}{x - 4}$.
+
+\item $\displaystyle \frac{1}{(2 - x)(3 - x)} - \frac{2}{(x - 1)(x
+- 3)} + \frac{1}{(x - 1)(x - 2)}$.
+
+\item $\displaystyle \frac{x^2 + xy}{x - y} \times \frac{x^3 -
+3x^2y + 3xy^2 - y^3}{x^2 - y^2} \div \frac{2xy - 2y^2}{3}$.
+
+\item $\displaystyle \left( \frac{1}{m} + \frac{1}{n} \right) (a +
+b) - \left( \frac{a + b}{m} \right) - \left( \frac{a - b}{n}
+\right)$.
+
+\item $\displaystyle \left( x - \frac{1}{1 + \frac{2}{x - 1}}
+\right) \div \frac{x + \frac{1}{x}}{\frac{1}{x} + 1}$.
+
+\item $\displaystyle \frac{\frac{a}{b} + \frac{b}{a} -
+1}{\frac{a^2}{b^2} + \frac{a}{b} + 1} \times \frac{1 +
+\frac{b}{a}}{a - b} \div \frac{1 + \frac{b^3}{a^3}}{\frac{a^2}{b}
+- \frac{b^2}{a}}$.
+
+\item Prove that $3(a + b)(a + c)(b + c) = (a + b + c)^3 - (a^3 +
+b^3 + c^3)$.
+
+\item How many sevenths in $6xy$?
+
+\item Divide $\displaystyle 2x^5 - \frac{x^4}{12} +
+2\frac{3}{4}x^3 - 2x^2 - x$ by $3x^2 + x$.
+\end{enumerate}
+
+\chapter*{EQUATIONS}
+\addcontentsline{toc}{chapter}{\numberline{}EQUATIONS.}
+\addcontentsline{toc}{section}{\numberline{}SIMPLE.}
+
+
+\begin{tabular}{llc}
+\textbf{43}.& ILLUS. 1.& $27 + 10x = 13x + 23$.\\
+&ILLUS. 2. & $\displaystyle \frac{a^2}{x} + \frac{b}{2} =
+\frac{4b^2}{x} +
+\frac{a}{4}$.\\
+\end{tabular}
+
+An \textbf{equation} is an expression of the equality of two
+numbers. The parts of the expression separated by the
+sign of equality are called the \textbf{members} of the equation.
+
+The last letters of the alphabet are used to represent
+unknown numbers, and known numbers are represented by
+figures or by the first letters of the alphabet.
+
+\begin{tabular}{lcr}
+\textbf{44}. &ILLUS. 1. & $5x+20 = 105$. \\
+&& $5x=85$. \\
+&ILLUS. 2. & $3x-18=42$. \\
+&& $3x=60$.\\
+&ILLUS. 3. & $\frac{x}{12}=2$. \\
+&& $x=24$.\\
+&ILLUS. 4. &$7x=49$.\\
+&& $x=7$. \\
+\end{tabular}
+
+\textbf{\textit{Any changes may be made in an equation which do
+not destroy the equality of the members.}}
+
+Name some of the ways in which such changes may
+be made.
+
+\begin{tabular}{lcl}
+\textbf{45}. &ILLUS. 1. &$x + b = a$. Subtracting $b$ from each
+member, $x = a - b$.\\
+&ILLUS. 2. &$x-b=c$. Adding $b$ to each member, $x = c + b$.
+\end{tabular}
+
+In each of these illustrations $b$ has been transposed
+(changed over) from the first to the second member,
+and in each case its sign has been changed. Hence,
+
+\textbf{\textit{Any term may be transposed from one member of an
+equation to the other provided its sign be changed.}}
+
+\textbf{46.} ILLUS. $\displaystyle \frac{ab + x}{b^2} - \frac{b^2
+- x}{a^2b} = \frac{x - b}{a^2} - \frac{ab - x}{b^2}$.
+
+Multiply both members by $a^2b^2$,
+\begin{equation*}
+a^3b + a^2x - b^3 + bx = b^2 x - b^3 - a^3b + a^2x.
+\end{equation*}
+\textit{To clear an equation of fractions, multiply each member of
+the equation by the L.C.M. of the denominators of the fractional
+terms.}
+
+\textbf{47.} ILLUS. 1. Solve \begin{eqnarray*}
+x + 4 + 2(x - 1) & = & 3x + 4 - (5x - 8) \\
+x + 5 + 2x - 2 & = & 3x + 4 - 5x - 8 \\
+\text{Transposing, } x + 2x - 3x + 5x & = & 4 + 8 - 4 + 2 \\
+5x & = & 10 \\
+x & = & 2 \\
+\end{eqnarray*}
+
+ILLUS. 2. Solve $\displaystyle \frac{5x + 3}{8} - \frac{3 - 4x}{3}
++ \frac{x}{2} = \frac{31}{2} - \frac{9 - 5x}{6}$. Multiply by 24,
+\begin{eqnarray*}
+3(5x + 3) - 8(3 - 4x) + 12x & = & 372 - 4(9 - 5x) \\
+15x + 9 - 24 + 32x + 12x & = & 372 - 36 + 20x \\
+\text{Transposing, } 15x + 32x + 12x - 20x & = & 372 - 36 - 9 + 24 \\
+39x & = & 351 \\
+x & = & 9 \\
+\end{eqnarray*}
+
+
+\textit{ To solve an equation, clear of fractions if necessary,
+transpose the terms containing the unknown number to one member
+and the known terms to the other, unite the terms, and divide both
+members by the coefficient of the unknown number}.
+
+\subsubsection*{Exercise 53.}
+
+Solve:
+
+\begin{enumerate}
+\item $22-6x=34-12x$.
+\item $5x-4=10x+ 11$.
+\item $23-8x=80-11x$.
+\item $5x-21=7x + 5$.
+\item $18x-43=17-6x$.
+\item $18-8x=12x- 87$.
+\item $9x-(2x-5)=4x+(13 + x)$.
+\item $15x-2(5x-4)-39 = 0$.
+\item $12x-18x+17=8x+3$.
+\item $21x-57=6x-14x+30$.
+\item $5x-27-11x+16=98-40x-41$.
+\item $14(x-2)+3(x+1)= 2(x-5)$.
+\item $6(23-x)-3x=3(4x-27)$.
+\item $3(x-1)-2(x-3)+(x-2)-5=0$.
+\item $(x+5)(x-3)=(x+2)(x-5)$.
+\item $(x+4)(x+7)=(x+2)(x+11)$.
+\item $(x-1)(x+4)(x-2)=x(x-2)(x+2)$.
+\item $(x-5)(x+3)-(x-7)(x-2)-2(x-1)=-12$.
+\item $(x+2)^{2}-(x-1)^{2}=5(2x+3)$.
+\item $(2x+3)(x+3)-14=(2x+1)(x+1)$.
+\item $(x+1)^{2}+(x-5)^{2}=2(x+5)^{2}$.
+
+\item $7x - 15 + 4x - 6 = 4x - 9 - 9x$.
+
+\item Divide the number 105 into three parts, such that the second
+shall be 5 more than the first, and the third three times the
+second.
+
+\item A man had a certain amount of money; he earned four times as
+much the next week, and found \$30. If he then had seven times as
+much as at first, how much had he at first?
+
+\item How many fourths are there in $7x$?
+
+\item How long will it take a man to build $x$ yards of wall if he
+builds $z$ feet a day?
+
+\item $\displaystyle 6x - \frac{x + 4}{3} = \frac{x}{3} + 28$.
+
+\item $\displaystyle \frac{2x - 1}{5} - \frac{x + 12}{3} -
+4\frac{4}{5} = \frac{2x}{3}$.
+
+\item $\displaystyle x - \frac{x}{4} + 25 = \frac{x}{3} +
+\frac{x}{2} + 21$.
+
+\item $\displaystyle \frac{x - 3}{3} + \frac{5}{21} = -\frac{x -
+8}{7}$.
+
+\item $\displaystyle \frac{8 - 5x}{12} + \frac{5x - 6}{4} -
+\frac{7x + 5}{6} = 0$.
+
+\item $\displaystyle \frac{3 + 5x}{4} + \frac{x + 2}{2} =
+1\frac{5}{9}$.
+
+\item $\displaystyle \frac{2x - 1}{3} - \frac{13}{42} = \frac{5x -
+4}{6}$.
+
+\item $\displaystyle \frac{1 - 11x}{7} - \frac{7x}{13} =
+\frac{2}{13} - \frac{8x - 15}{3}$.
+
+\item $\displaystyle \frac{2}{x}-\frac{3}{2x} =
+\frac{7}{24}-\frac{2}{3x}$.
+
+\item $\displaystyle \frac{1}{4}+\frac{1}{3x} =
+\frac{11}{36}+\frac{1}{6x}$.
+
+\item $\displaystyle
+\frac{3}{4x}-\frac{2}{x}+\frac{x-2}{2x}+7\frac{5}{12} =
+5+\frac{4}{6x}$.
+
+\item $\displaystyle \frac{2x+3}{5x}-\frac{3}{x}+4 =
+\frac{1}{2x}+\frac{3}{4x}+2\frac{23}{40}$.
+
+\item $\displaystyle \frac{x+9}{11}-\frac{2-x}{5} =
+\frac{x+5}{7}$.
+
+\item $\displaystyle \frac{x+4}{5}-\frac{4-x}{7} = \frac{x+1}{3}$.
+
+\item $\displaystyle \frac{2}{5}(x-6)-\frac{3}{16}(x-1) =
+\frac{5}{12}(4-x)-\frac{5}{48}$.
+
+\item $\displaystyle \frac{4+x}{4}-\frac{1-x}{7}-\frac{1}{5}(8-x)
+= \frac{x-23}{5}+7$.
+
+\item How long will it take a man to walk $x$ miles if he walks 15
+miles in $b$ hours?
+
+\item What is the interest on $m$ dollars for one year at 5 per
+cent?
+
+\item What are the two numbers whose sum is 57, and whose
+difference is 25?
+
+\item What is the interest on $b$ dollars for $y$ years at 4 per
+cent?
+
+\item $(c+a)x+(c-b)x = c^2$.
+
+\item $(a-b)x+(a+b)x = a$.
+
+\item $2x+a(x-2) = a+6$.
+
+\item $b(2x-a)-a^2 = 2x(a+b)-3ab$.
+
+
+\item $\displaystyle a^2+c^2=\frac{cx}{a}+\frac{ax}{c}$.
+
+\item $b^4-x^2+2bx=(b^2+x)(b^2-x)$.
+
+\item $x^2+4a^2+a^4=\left(x+a^2\right)^2$.
+
+\item $b^2(x-b)+a^2(x-a)=abx$.
+
+\item $\displaystyle \frac{2x+5}{5x+3}=\frac{2x-4}{5x-6}$.
+
+\item $\displaystyle \frac{6x+5}{2x-3}=\frac{3x-4}{x+1}$.
+
+\item $\displaystyle \frac{2+9x}{3(6x+7)}=\frac{2x+9}{35+4x}$.
+
+\item $\displaystyle \frac{3}{2-5x}-\frac{4}{1-3x}=0$.
+
+\item $\displaystyle \frac{2(3x+4)}{1+2x}-1=\frac{2(19+x)}{x+12}$.
+
+\item $\displaystyle 1-\frac{x}{x+2}=\frac{4}{x+6}$.
+
+\item $\displaystyle
+\frac{x^2}{1-x^2}+\frac{x+1}{x-1}=\frac{3}{x+1}$.
+
+\item $\displaystyle \frac{2+3x}{1-x}+5=\frac{2x-4}{x+2}$.
+
+\item $\displaystyle
+\frac{5}{6+2x}+\frac{2}{x+1}=\frac{2\frac{1}{2}}{2+2x}+\frac{4}{3+x}$.
+
+\item $\displaystyle
+\frac{2(1-2x)}{1-3x}+\frac{1}{6}=\frac{1-3x}{1-2x}$.
+
+\item $\displaystyle
+\frac{x-8}{x-6}-\frac{x}{x-2}=\frac{x-9}{x-7}-\frac{x+1}{x-1}$.
+
+\item $\displaystyle \frac{x+5}{x+8} -
+\frac{x+6}{x+9}=\frac{x+2}{x+5}-\frac{x+3}{x+6}$.
+
+\item Find three consecutive numbers whose sum is 81. \item A's
+age is double B's, B's is three times C's, and C is $y$ years old.
+What is A's age?
+
+\item How many men will be required to do in $a$ hours what $x$
+men do in 6 hours?
+
+\item Find the sum of three consecutive odd numbers of which the
+middle one is $4x + 1$.
+\end{enumerate}
+
+\subsubsection*{Exercise 54}.
+\begin{enumerate}
+\item In a school of 836 pupils there is one boy to every
+three girls. How many are there of each?
+
+\item Divide 253 into three parts, so that the first part shall
+be four times the second, and the second twice the third.
+
+\item The sum of the ages of two brothers is 44 years, and
+one of them is 12 years older than the other. Find their
+ages.
+
+\item Find two numbers whose sum is 158, and whose difference
+is 86.
+
+\item Henry and Susan picked 16 quarts of berries.
+Henry picked 4 quarts less than three times as many as
+Susan. How many quarts did each pick?
+
+\item Divide 127 into three parts, such that the second
+shall be 5 more than the first, and the third four times
+the second.
+
+\item Twice a certain number added to four times the
+double of that number is 90. What is the number?
+
+\item I bought some five-cent stamps, and twice as many
+two-cent stamps, paying for the whole 81 cents. How
+many stamps of each kind did I buy?
+
+\item Three barns contain 58 tons of hay. In the first
+barn there are 3 tons more than in the second, and 7 less
+than in the third. How many tons in each barn?
+
+\item If I add 18 to a certain number, five times this
+second number will equal eleven times the original number.
+What is the original number?
+
+\item In a mixture of 48 pounds of coffee there is one-third
+as much Mocha as Java. How much is there of
+each?
+
+\item The half and fifth of a number are together equal to
+56. What is the number?
+
+\item What number increased by one-third and one-fourth
+of itself, and 7 more, equals 45?
+
+\item What number is doubled by adding to it three-eighths
+of itself, one-third of itself, and 14?
+
+\item A grocer sold 27 pounds of sugar, tea, and meal.
+Of meal he sold 3 pounds more than of tea, and of sugar
+6 pounds more than of meal. How many pounds of each
+did he sell?
+
+\item A son is two-sevenths as old as his father. If the
+sum of their ages is 45 years, how old is each?
+
+\item Two men invest \$2990 in business, one putting in
+four-ninths as much as the other. How much does each
+invest?
+
+\item In an election 47,519 votes were cast for three candidates.
+One candidate received 2061 votes less, and the
+other 1546 votes less, than the winning candidate. How
+many votes did each receive?
+
+\item John had twice as many stamps as Ralph, but after
+he had bought 65, and Ralph had lost 16, they found that
+they had together 688. How many had each at first?
+
+\item Find three consecutive numbers whose sum is 192.
+
+\item If 17 be added to the sum of two numbers whose
+difference is 12, the result will be 61. What are the
+numbers?
+
+\item Divide 120 into two parts such that five times one
+part may be equal to three times the other.
+
+\item Mr. Johnson is twice as old as his son; 12 years
+ago he was three times as old. What is the age of each?
+
+\item Henry is six times as old as his sister, but in 3 years
+from now he will be only three times as old. How old is
+each?
+
+\item Samuel is 16 years older than James; 4 years ago
+he was three times as old. How old is each?
+
+\item Martha is 5 years old and her father is 30. In
+how many years will her father be twice as old as Martha?
+
+\item George is three times as old as Amelia; in 6 years
+his age will be twice hers. What is the age of each?
+
+\item Esther is three-fourths as old as Edward; 20 years
+ago she was half as old. What is the age of each?
+
+\item Mary is 4 years old and Flora is 9. In how many
+will Mary be two-thirds as old as Flora?
+
+\item Harry is 9 years older than his little brother; in
+6 years he will be twice as old. How old is each?
+
+\item Divide $2x^4+27xy^3-81y^4$ by $x+3y$.
+
+\item Prove $\left(a^2+ab+b^2\right)^2-\left(a^2-ab+b^2\right)^2=4ab(a^2+b^2)$.
+
+\item Find the value of $\displaystyle
+\frac{x-y}{y}+\frac{2x}{x-y}-\frac{x^3+x^2y}{x^2y-y^3}$.
+
+\item Solve $\displaystyle \frac{5x-7}{2}-3x=\frac{2x+7}{3}-14$.
+
+\item Mr. Ames has \$132, and Mr. Jones \$43. How
+much must Mr. A. give to Mr. J. so that Mr. J. may have
+three-fourths as much as Mr. A.?
+
+\item A has \$101, and B has \$35; each loses a certain
+sum, and then A has four times as much as B. What was
+the sum lost by each?
+
+\item A certain sum of money was divided among A, B, and C; A and B
+received \$75, A and C \$108, and B and C \$89. How much did each
+receive?
+
+\textit{ Suggestion}. Let $x$ equal what A received.
+
+\item Mary and Jane have the same amount of money.
+If Mary should give Jane 40 cents, she would have one-third
+as much as Jane. What amount of money has
+each?
+
+\item An ulster and a suit of clothes cost \$43; the ulster and a
+hat cost \$27; the suit of clothes and the hat cost \$34. How much
+did each cost?
+
+\item John, Henry, and Arthur picked berries, and sold them; John
+and Henry received \$4.22, John and Arthur \$3.05, Henry and
+Arthur \$3.67. How much did each receive for his berries?
+
+\item A can do a piece of work in 4 days, and B can do it
+in 6 days. In what time can they do it working together?
+
+\textit{ Suggestion}. Let $x$ equal the required time. Then find
+what part of the work each can do in one day.
+
+\item Mr. Brown can build a stone wall in 10 days, and Mr.
+Mansfield in 12 days. How long would it take them to do
+it working together?
+
+\item Mr. Richards and his son can hoe a field of corn in
+9 hours, but it takes Mr. Richards alone 15 hours. How
+long would it take the son to hoe the field?
+
+\item A can do a piece of work in 4 hours, B can do it in
+6 hours, and C in 3 hours. How long would it take them
+working together?
+
+\item A can mow a field in 6 hours, B in 8 hours, and with
+the help of C they can do it in 2 hours. How long would
+it take C working alone?
+
+\item A tank can be emptied by two pipes in 5 hours and
+7 hours respectively. In what time can it be emptied
+by the two pipes together?
+
+\item A cistern can be filled by two pipes in 4 hours and
+6 hours respectively, and can be emptied by a third in 15
+hours. In what time could the cistern be filled if all three
+pipes were running?
+
+\item A and B together can do a piece of work in 8 days,
+A and C together in 10 days, and A by himself in 12 days.
+In what time can B and C do it? In what time can A, B,
+and C together do it?
+
+\item John and Henry can together paint a fence in 2
+hours, John and Lewis together in 4 hours, and John by
+himself in 6 hours. In what time can the three together
+do the painting?
+
+\item C can do a piece of work in $a$ days, and D can do the
+same work in $b$ days. In how many days can they do it
+working together?
+
+\item James and Thomas can do a piece of work in $d$ days
+and James alone can do it in $c$ days. How long would it
+take Thomas alone?
+
+\item Solve $\displaystyle
+\frac{3+x}{3-x}-\frac{1+x}{1-x}-\frac{2+x}{2-x}=1$.
+
+\item Find the value of $\displaystyle
+\frac{x^2}{(x-y)(x-z)}+\frac{y^2}{(y-x)(y-z)}+\frac{z^2}{(z-x)(z-y)}$.
+
+\item Expand $(x^2-2)(x^2+2)(x^2+3)(x^2-3)$.
+
+\item Factor $9a^2+12ab+4b^2$, $12+7x+x^2$, $ac-2bc-3ad+6bd$.
+
+\textit{ Illustrative Example}. At what time between 1 o'clock and
+2 o'clock are the hands of a clock (1) together? (2) at right
+angles? (3) opposite to each other?
+
+How far does the hour hand move while the minute
+hand goes around the whole circle? How far while the
+minute hand goes half around? What part of the distance
+that the minute hand moves in a given time does the hour
+hand move in the same time?
+
+\begin{figure}[htbp]
+\centering \includegraphics[scale=0.75]{fig1.eps}\\
+\caption{}
+\end{figure}
+\begin{figure}[htbp]
+\centering \includegraphics[scale=0.75]{fig2.eps}\\
+\caption{}
+\end{figure}
+\begin{figure}[htbp]
+\centering \includegraphics[scale=0.75]{fig3.eps}\\
+\caption{}
+\end{figure}
+
+(1) Let $AM$ and $AH$ in all the figures denote the positions
+of the minute and hour hands at 1 o'clock, and $AX$
+(Fig. 1) the position of both hands when together.
+
+\[
+\begin{array}{rrcl}
+\text{Let}\qquad & x &=& \text{number of minute spaces in arc } MX.\\
+ &MX &=& MH + HX. \\
+ & x &=& 5 + \frac{x}{12}. \text{ Solution gives } x=5\frac{5}{11}.
+\end{array}
+\]
+
+Hence, the time is $5\frac{5}{11}$ minutes past 1 o'clock.
+
+(2) Let $AX$ and $AB$ (Fig. 2) denote the positions of
+the minute and hour hands when at right angles.
+\[
+\begin{array}{rrcl}
+\text{Let } & x &=& \text{ number of minute spaces in arc } MBX. \\
+ & MBX &=& MH + HB + BX. \\
+&x &=&5 + \frac{x}{12} + 15.\text{ Solution gives } x=21\frac{9}{11}.
+\end{array}
+\]
+
+Hence, the time is $21\frac{9}{11}$ minutes past 1 o'clock.
+
+(3) Let $AX$ and $AB$ (Fig. 3) denote the positions of
+the minute and hour hands when opposite.
+
+
+\begin{tabular}{rrcl}
+Let & $x$ &=& number of minute spaces in arc $MBX$. \\
+ &$MBX$ &=& $MH +HB + BX.$ \\
+ & $x$ &=& $5+\frac{x}{12}+30$. Solution gives $x=38\frac{2}{11}$.
+\end{tabular}
+
+
+Hence, the time is $38\frac{2}{11}$ minutes past 1 o'clock.
+
+\item At what time are the hands of a clock together
+between 2 and 3? Between 5 and 6? Between 9 and 10?
+
+\item At what time are the hands of a clock at right
+angles between 2 and 3? Between 4 and 5? Between 7
+and 8?
+
+\item At what time are the hands of a clock opposite each
+other between 3 and 4? Between 8 and 9? Between 12
+and 1?
+
+\item At what times between 4 and 5 o'clock are the hands
+of a watch ten minutes apart?
+
+\item At what time between 8 and 9 o'clock are the
+hands of a watch 25 minutes apart?
+
+\item At what time between 5 and 6 o'clock is the minute
+hand three minutes ahead of the hour hand?
+
+\item It was between 12 and 1 o'clock; but a man, mistaking
+the hour hand for the minute hand, thought that
+it was 55 minutes later than it really was. What time
+was it?
+
+\item At what time between 11 and 12 o'clock are the
+hands two minutes apart?
+
+\textit{Illustrative Example}. A courier who travels at the rate
+of 6 miles an hour is followed, 5 hours later, by another
+who travels at the rate of $8\frac{1}{2}$ miles an hour. In how many
+hours will the second overtake the first?
+\begin{tabular}{rrcl}
+Let & $x$ &=& number of hours the second is traveling. \\
+ & $x + 5$ &=& number of hours the first is traveling. \\
+ & $8\frac{1}{2}x$ &=& distance the second travels. \\
+ & $6(x+5)$ &=& distance the first travels. \\
+ & $8\frac{1}{2}x$ &=& $6 (x + 5)$. Solution gives $x=12$. \\
+\multicolumn{4}{l}{He will overtake the first in 12 hours.}
+\end{tabular}
+
+\item A messenger who travels at the rate of 10 miles an
+hour is followed, 4 hours later, by another who travels at
+the rate of 12 miles an hour. How long will it take the
+second to overtake the first?
+
+\item A courier who travels at the rate of 19 miles in 4 hours
+is followed, 8 hours later, by another who travels at the rate
+of 19 miles in 3 hours. In what time will the second
+overtake the first? How far will the first have gone
+before he is overtaken?
+
+\item A train going at the rate of 20 miles an hour is
+followed, on a parallel track, 4 hours later, by an express
+train. The express overtakes the first train in $5\frac{1}{3}$ hours.
+What is the rate of the express train?
+
+\item A messenger started for Washington at the rate of
+$6\frac{1}{2}$ miles an hour. Six hours later a second messenger
+followed and in $4\frac{7}{8}$ hours overtook the first just as he was
+entering the city. At what rate did the second messenger
+go? How far was it to Washington?
+
+\item How far could a man ride at the rate of 8 miles an
+hour so as to walk back at the rate of 4 miles an hour and
+be gone only 9 hours?
+
+\item Two persons start at 10 A.M. from towns A and B,
+$55\frac{1}{2}$ miles apart. The one starting from A walks at the
+rate of $4\frac{1}{4}$ miles an hour, but stops 2 hours on the way;
+the other walks at the rate of $3\frac{3}{4}$ miles an hour without
+stopping. When will they meet? How far will each
+have traveled?
+
+\textit{ Suggestion}. Let x equal the number of hours.
+
+\item A boy who runs at the rate of $12\frac{1}{2}$ yards per second,
+starts 16 yards behind another whose rate is 11 yards per
+second. How soon will the first boy be 8 yards ahead of
+the second?
+
+\item A rectangle whose length is 4 ft. more than its
+width would have its area increased 56 sq. ft. if its length
+and width were each made 2 ft. more. What are its
+dimensions?
+
+\item The length of a room is double its width. If the
+length were 3 ft. less and the width 3 ft. more, the area
+would be increased 27 sq. ft. Find the dimensions of the
+room.
+
+\item A floor is two-thirds as wide as it is long. If the
+width were 2 ft. more and the length 4 ft. less, the area
+would be diminished 22 sq. ft. What are its dimensions?
+
+\item A rectangle has its length and width respectively
+4 ft. longer and 2 ft. shorter than the side of an equivalent
+square. Find its area.
+
+\item An enclosed garden is 24 ft. greater in length than in
+width. 684 sq. ft. is used for a walk 3 ft. wide extending
+around the garden inside the fence. How long is the
+garden?
+
+\item Factor $\displaystyle \frac{x^2}{4}-\frac{y^2z^4}{9m^6}$,
+$x^6-27y^3$, $a^{16}-b^{16}$, $2c-4c^3+2c^5$.
+
+\item Extract the square root of $12x^4-24x+9+x^6-22x^3-4x^5+28x^2$.
+
+\item What must be subtracted from the sum of
+$4x^3+3x^2y-y^3$, $4x^2y-3x^3$, $7x^2y+9y^3-2x^2y$, to leave the remainder
+$2x^3-3x^2y+y^3$?
+
+\item Find the G.C.F. of $x\left(x+1\right)^2$, $x^2(x^2-1)$, and
+$2x^3-2x^2-4x$.
+
+\item From one end of a line I cut off 5 feet less than one-fifth
+of it, and from the other end 4 feet more than one-fourth of it,
+and then there remained 34 feet. How long was the line?
+
+\item A can do twice as much work as B, B can do twice
+as much as C, and together they can complete a piece of
+work in 4 days. In what time can each alone complete the
+work.
+
+\item Separate 57 into two parts, such that one divided by
+the other may give 5 as a quotient, with 3 as a remainder.
+
+\item Divide 92 into two parts, such that one divided by
+the other may give 4 as a quotient, with 2 as a remainder.
+
+\item Fourteen persons engaged a yacht, but before sailing,
+four of the company withdrew, by which the expense
+of each was increased \$4. What was paid for the yacht?
+
+\item Find two consecutive numbers such that a fifth of
+the larger shall equal the difference between a third and an
+eighth of the smaller.
+
+\item A is 24 years older than B, and A's age is as much
+above 50 as B's is below 40. What is the age of each?
+
+\item Find the number, whose double added to 16 will
+be as much above 70 as the number itself is below 60.
+
+\item A hare takes 5 leaps to a dog's 4, but 3 of the dog's
+leaps are equal to 4 of the hare's; the hare has a start of
+20 leaps. How many leaps will the hare take before he is
+caught?
+
+\textit{ Suggestion}. Let $5x$ equal the number of leaps the hare
+will take, and let $m$ equal the length of one leap.
+
+\item A greyhound takes 3 leaps to a hare's 5, but 2 of
+the greyhound's leaps are equal to 4 of the hare's. If the
+hare has a start of 48 leaps, how soon will the greyhound
+overtake him?
+
+\item A hare has 40 leaps the start of a dog. When will
+he be caught if 5 of his leaps are equal to 4 of the dog's,
+and if he takes 7 leaps while the dog takes 6?
+\end{enumerate}
+
+
+\section*{SIMULTANEOUS EQUATIONS.}
+\addcontentsline{toc}{section}{\numberline{}SIMULTANEOUS.}
+
+\[
+\textbf{48. } \textrm{ILLUS. } \left. \begin{array}{cc} x + y = 8, & x=3\\
+4x-y=7, & y=5 \end{array} \right\} \textrm{ in both equations. }
+\]
+
+\textbf{Simultaneous equations} are equations in which the
+same unknown numbers have the same value.
+
+One equation containing more than one unknown
+number cannot be solved. There must be as many
+simultaneous equations as there are unknown numbers.
+
+\[
+ \textrm{ILLUS. 1. Solve} \left\{ \begin{array}{c} x + 3y = 17,\\
+2x+y=9. \end{array} \right.
+\]
+
+Multiply the first equation by 2;
+
+\[
+ \left. \begin{array}{cc} \textrm{then} & 2x + 6y = 34, \\
+\textrm{but} & \underline{2x+y=9}\\
+\textrm{ Subtracting, } & 5y=25\\
+& y=5. \end{array} \right.
+\]
+
+To find the value of $x$, substitute the value of $y$ in the
+second equation:
+
+\[
+2x + 5 = 9, 2x = 4, x = 2.
+\]
+
+\[
+ Ans. \left\{ \begin{array}{l} x = 2,\\
+y=5 \end{array} \right.
+\]
+
+\[
+ \textrm{ILLUS. 2. Solve} \left\{ \begin{array}{c} 3x + 4y = 12,\\
+5x - 6y=1. \end{array} \right.
+\]
+
+Multiply the first equation by 3, and the second equation
+by 2,
+
+\[ \begin{array}{l rrr l}
+ & 9x & {}+ 12y = & 36 \\
+ & 10x & {}- 12y = & 2 \\
+\cline{2-4} \text{Adding, } & 19x & & 38 & \therefore x
+= 2.
+\end{array}
+\]
+
+Substituting, $6 + 4y = 12$, $4y = 6$, $y = 1\frac{1}{2}$.
+
+\textit{Multiply one or both of the equations by such a number
+that one of the unknown numbers shall have like coefficients. If
+the signs of the terms having like coefficients are alike,
+subtract one equation from the other; if unlike, add the
+equations}.
+
+\subsubsection*{Exercise 55.}
+Solve: \begin{enumerate} \item
+$
+ \left\{ \begin{array}{rcrcl}
+ x &+& y &=& 4, \\
+3x &-& 2y &=& 7 .
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+ x &-& y &=& 2, \\
+2x &+& 5y &=& 18 .
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+5x &+& 2y &=& 47, \\
+2x &-& y &=& 8 .
+\end{array} \right.
+$
+\item
+$
+ \left\{ \begin{array}{rcrcl}
+4x &-& 3y &=& 10, \\
+6x &+& 4y &=& 49 .
+\end{array} \right.
+$
+\item
+$
+ \left\{ \begin{array}{rcrcl}
+ 8x &-& 2y &=& 6, \\
+ 10x &+& 7y &=& 36 .
+\end{array} \right.
+$
+\item
+$
+ \left\{ \begin{array}{rcrcl}
+ 2x &-& 5y &=& -11, \\
+ 3x &+& y &=& 9 .
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+ 7x &-& 3y &=& 41, \\
+ 2x &+& y &=& 12 .
+\end{array} \right.
+$ \item
+$
+ \left\{ \begin{array}{rcrcl}
+ 2x &+& 9y &=& -5, \\
+11x &+& 15y &=& 7 .
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+ 4y &-& 2x &=& 4, \\
+10y &+& 3x &=& -8 .
+\end{array} \right.
+$
+\item
+$
+ \left\{ \begin{array}{rcrcl}
+3x &-& 5y &=& 15, \\
+5x &+& 3y &=& 8 .
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+3y &-& 2x &=& 3, \\
+4y &-& 6x &=& 2\frac{1}{3} .
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+3x &+ & 2y &=& 11,\\
+ 7x &-& 5y &=& 190.
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+\frac{1}{2}x &+& \frac{1}{3}y &=& 11,\\
+ 8x &+& \frac{2}{5}y &=& 102.
+\end{array} \right.
+$
+\item
+$
+\left\{ \begin{array}{rcrcl}
+ 5x &+& 2y &=& 66,\\
+\frac{x}{3} &+& \frac{3y}{4} &=& 15\frac{1}{2}.
+\end{array} \right.
+$
+
+\item $\left\{ \begin{matrix}\frac{3x}{5} - \frac{2y}{7} = 35, \\ x + 2y = -63.\end{matrix}\right.$
+
+\item $ \left\{ \begin{matrix}x - \frac{3y}{5} = 6,
+\\ \frac{2x}{3} + 7y = 189.\end{matrix}\right.$
+
+\item $\left\{ \begin{matrix}\frac{x + 2y}{3x - y} = 1, \\
+\frac{4y - x}{3 + x - 2y} = 2\frac{1}{2}.\end{matrix}\right.$
+
+\item $\left\{ \begin{matrix}\frac{x + 2y}{x - 2} = -5\frac{2}{3},
+\\ \frac{2y - 4x}{3 - y} = -6.\end{matrix}\right.$
+
+\item $\left\{ \begin{matrix}y - \frac{2y + x}{3} = \frac{2x + y}{4} - 8\frac{3}{4}, \\ \frac{3x + y}{2} - \frac{y}{3} = \frac{109}{10} + \frac{4y - x}{5}.\end{matrix}\right.$
+
+\item $\left\{ \begin{matrix}x + y = a, \\ x - y = b.\end{matrix}\right.$
+
+\item $\left\{ \begin{matrix}\frac{3x - 19}{2} + 4 = \frac{3y + x}{3} + \frac{5x - 3}{2}, \\ \frac{4x + 5y}{16} + \frac{2x + y}{2} = \frac{9x - 7}{8} + \frac{3y + 9}{4}.\end{matrix}\right.$
+
+\item $\left\{ \begin{matrix}\frac{1}{5}(3x - 2y) + \frac{1}{3}(5x - 3y) = x, \\ \frac{4x - 3y}{2} + \frac{2}{3}x - y = 1 + y.\end{matrix}\right.$
+
+\item If 1 is added to the numerator of a fraction, its
+value is $\frac{1}{8}$; but if 4 is added to its denominator, its value
+is $\frac{1}{4}$. What is the fraction?
+
+\textit{ Suggestion}. Letting $x$ equal the numerator, and $y$ the
+denominator, form two equations.
+
+\item If 2 is subtracted from both numerator and denominator
+of a certain fraction, its value is $\frac{3}{5}$; and if 1 is
+added to both numerator and denominator, its value is $\frac{2}{3}$.
+What is the fraction?
+
+\item If 2 is added to both numerator and denominator of
+a certain fraction, its value is $\frac{2}{3}$; but if 3 is subtracted
+from both numerator and denominator, its value is $\frac{1}{2}$.
+What is the fraction?
+
+\item If 3 be subtracted from the numerator of a certain
+fraction, and 3 be added to the denominator, its value will
+be $\frac{1}{2}$; but if 5 be added to the numerator, and 5 be subtracted
+from its denominator, its value will be 2. What is
+the fraction?
+
+\item The sum of two numbers divided by 2 is 43, and
+their difference divided by 2 is 19. What are the numbers?
+
+\item The sum of two numbers divided by 3 gives as a
+quotient 30, and their difference divided by 9 gives 4.
+What are the numbers?
+
+\item Five years ago the age of a father was four times
+that of his son; five years hence the age of the father will
+be $2\frac{1}{3}$ times that of the son. What are their ages?
+
+\item Seven years ago John was one-half as old as Henry,
+but five years hence he will be three-quarters as old.
+How old is each?
+
+\item $A$ and $B$ own herds of cows. If $A$ should sell 6
+cows, and $B$ should buy 6, they would have the same number;
+if $B$ should sell 4 cows to $A$, he would have only
+half as many as A. How many cows are there in each
+herd?
+
+\item The cost of 5 pounds of tea and 7 pounds of coffee is
+\$4.94; the cost of 3 pounds of tea and 6 pounds of coffee is
+\$3.54. What is the cost of the tea and coffee per pound?
+
+\item What is the price of corn and oats when 4 bushels of corn
+with 6 bushels of oats cost \$4.66, and 5 bushels of corn with 9
+bushels of oats cost \$6.38?
+
+\item A merchant mixes tea which cost him 87 cents a pound with
+tea which cost him 29 cents a pound. The cost of the mixture is
+\$17.98. He sells the mixture at 55 cents a pound and gains
+\$2.92. How many pounds of each did he put into the mixture?
+\end{enumerate}
+
+
+
+\subsection*{QUADRATIC EQUATIONS.}
+\addcontentsline{toc}{section}{\numberline{}QUADRATIC.}
+
+\begin{tabular}{llc} \textbf{ 49}. &ILLUS. 1.& $ax^2 = b$, $7x^2-10 =
+5+2x^3$.\\
+& ILLUS. 2. &$x^2+8x = 20$, $ax^2+bx-c = bx^2+d$.
+\end{tabular}
+
+A \textbf{ quadratic equation} is an equation in which, the
+highest power of the unknown number is a square. It is called an
+equation of the second degree.
+
+If it contains only the second power of the unknown number (Illus.
+1), it is called a \textbf{ pure quadratic equation}. If it
+contains both the first and second powers of the unknown number
+(Illus. 2), it is called an \textbf{ affected quadratic equation}.
+
+\textbf{ 50}. ILLUS. Solve $x^2-\frac{x^2-10}{3} =
+35-\frac{x^2+50}{5}$.
+\begin{align}
+15x^2-5x^2+50 &= 525-3x^2-150.\\
+ 13x^2 &= 325.\\
+ x^2 &= 25.\\
+ x &= {\pm}5.
+\end{align}
+
+\textit{ To solve a pure quadratic equation, reduce to the form
+$x^2 = a$ and take the square root of each member}.
+
+\subsubsection*{Exercise 56}.
+
+Solve:
+\begin{enumerate}
+\item $5x^2-12 = 33$.
+
+\item $3x^2+4 = 16$.
+
+\item $4x^2+ll = 136-x^2$.
+
+\item $5(3x^2-1) = 11(x^2+1)$.
+
+\item $\displaystyle \frac{2}{5x^2}-\frac{1}{3x^2} =
+\frac{4}{15}$.
+
+\item $\displaystyle \frac{x^2-1}{6}+\frac{1}{4} =
+\frac{x^2+1}{8}$.
+
+\item $(x+3)^2 = 6x+58$.
+
+\item $\displaystyle 6x+2+\frac{16}{x} =
+\frac{15+40x}{4}-1\frac{3}{4}$.
+
+\item $\displaystyle
+\frac{x-3}{x-1}+\frac{x+1}{x+3}+\frac{8}{x^2+2x-3} = 0$.
+
+\item $\displaystyle \frac{x+1}{x-1}+\frac{2(x-3)}{x-2} =
+\frac{16-9x}{x^2-3x+2}$.
+
+\item $\displaystyle
+\frac{1}{(2-x)(3-x)}-\frac{2}{(1-x)(x-3)}+\frac{1}{(x-1)(x-2)} =
+\frac{1}{2-x}+\frac{1}{(x-1)(2-x)(x-3)}$.
+
+\item $\displaystyle
+\frac{1}{6x+6}-\frac{1}{2x+2}+\frac{10}{3-3x^2} =
+\frac{x}{3(1-x)}$.
+
+\item A father is 30 years old, and his son is two years
+old. In how many years will the father be three times as
+old as his son?
+
+\item Divide the number 112 into two parts such that the
+smaller divided by their difference will give as a quotient
+3.
+
+\item The numerator of a fraction is 4 less than the denominator;
+if 30 be added to the denominator, or if 10 be subtracted
+from the numerator, the resulting fractions will
+be equal. What is the original fraction?
+
+\end{enumerate}
+\textbf{ 51.} ILLUS. Solve
+$\frac{x-1}{x-2}-\frac{x-3}{x-4}=-\frac{2}{3}$.
+
+\begin{eqnarray*}
+3x^2-15x+12-3x^2+15x-18 & = & -2x^2+12x-16. \\
+2x^2-12x & = & -10. \\
+x^2-6x & = & -5. \\
+x^2-6x-5 & = & 0. \\
+(x-5)(x-1) & = & 0. \\
+\end{eqnarray*}
+
+This equation will be satisfied if either factor is equal
+to zero. Placing each factor in turn equal to zero, and
+solving,
+
+\begin{tabular}{rclrcl}
+x-5 & = & 0, & x-1 & = & 0, \\
+x & = & 5; & x & = & 1. \\
+\end{tabular}
+
+\textit{ Ans.} $x=5$ or $1$.
+
+\textit{ To solve an affected quadratic equation, reduce the
+equation to the form $x^2 + bx + c = O$, factor the first member,
+place each factor in turn equal to zero, and solve the simple
+equations thus formed.}
+
+\subsubsection*{Exercise 57.}
+
+Solve:
+
+\begin{enumerate}
+\item $x^{2}+3x=18$.
+\item $x^{2}+5x=14$.
+\item $x(x-1)=72$.
+\item $x^{2}=10x-21$.
+\item $23x=120+x^{2}$.
+\item $187=x^{2}+6x$.
+\item $x^{2}-2bx=-b^{2}$.
+\item $x^{2}=4ax-3a^{2}$.
+\item $x^{2}+(a-1)x=a$.
+\item $adx-acx^{2}=bcx-bd$.
+\item $(x+3)(x-3)=8(x+3)$.
+\item $(x+2)(x-5)=4(x-4)$.
+
+\item $\displaystyle \frac{x}{5}+\frac{2}{x}=1\frac{2}{5}$.
+
+\item $\displaystyle \frac{x}{3}-2=\frac{x^{2}}{12}-\frac{x}{2}$.
+
+\item $\displaystyle \frac{8}{x-2}-3+\frac{x+1}{7}=0$.
+
+\item $\displaystyle \frac{x}{x+1}-2\frac{1}{6}+\frac{x+1}{x}=0$.
+
+\item $\displaystyle x+4=3x-\frac{24}{x-1}$.
+
+\item $\displaystyle
+\frac{x-1}{x+1}+\frac{x+3}{x-3}=\frac{2(x+2)}{x-2}$.
+
+\item $\displaystyle
+\frac{x-1}{x+2}-\frac{3x^{2}+2}{x^{2}-4}=\frac{3x}{2-x}$.
+
+\item $\displaystyle
+\frac{2x}{3-x}+\frac{2x(x-3)}{x^2-9}=\frac{x-3}{x+3}$.
+
+\item At what time between 4 and 5 o'clock are the hands of a
+clock opposite each other?
+
+\item John, having three times as much money as Lewis, gave Lewis
+\$2, and then had twice as much as Lewis. How much had each at
+first?
+
+\item A fish is 3 feet long; its head is equal in length to the
+tail, and its body is five times the length of the head and tail
+together. What is the length of the head?
+
+\item In how many days can A, B, and C build a boat
+if they work together, provided A alone can build it in
+24 days, B in 18 days, and C in 30 days?
+
+\end{enumerate}
+
+The above method of solving affected quadratic equations is the
+simplest of three methods commonly used, and will not solve all
+possible cases; the method given for solving simultaneous
+equations is only one of three known methods; the cases in
+factoring are less than half of those usually taken. In fact, we
+have made only a beginning in the subject of algebra; much more
+lies ahead along the lines which we have been following. \textit{
+Can you grasp more clearly the conditions given in any problem
+presented to you, and see more definitely just what is required,
+than when you began this study? Do you possess greater ability to
+think out problems? Has the use of letters to represent numbers
+made you think more exactly what is to be done, and what the
+operations mean?} If so, your knowledge of numbers is broader, and
+you already know that
+
+\textbf{ \textit{ Algebra is the knowledge which has for its
+object general truths about numbers.}}
+
+\subsubsection*{Exercise 58. (General Review.)}
+\begin{enumerate}
+\item When $a=l$, $b = 3$, $c = 5$, and $d = O$, what is the
+value of
+
+$\displaystyle
+\frac{4a+b^2+b^2c^2+ad}{b^2+c^2+d^2}-\frac{1+a^2c^2}{a^2+c^2+d^2}
++\frac{a^2+b^2+d^2}{1+a^2b^2+bd}-\frac{a^2+2ab+b^2}{b^2-2bc+c^2}$?
+
+\item Prove that $\displaystyle
+(x^{2}+xy+y^{2})(x^{2}-xy+y^2)=\frac{x^{6}-y^{6}}{x^{2}-y^{2}}$
+
+\item Solve $(x+5)^{2}-(x+1)^{2}-16x=(x-1)^{2}-(x-5)^{2}$.
+
+\item A tank is filled by two pipes, $A$ and $B$, running
+together, in 12 hours, and by the pipe $B$ alone in 20 hours.
+In what time will the pipe $A$ alone fill it?
+
+\item Find the G.C.F. of $x^{3}+1-x-x^{2}$, $x^{3}+x-1-x^{2}$,
+$x^{4}-1$, and $x^{2}-4x+3$.
+
+\item Divide $a^{5}+a^{4}b-a^{3}b^{2}+a^{3}-2ab^{2}+b^{3}$ by $a^{3}-b+a$.
+
+\item Find the square root of $5y^{4}+1+12y^{5}-2y-2y^{3}+4y^{6}+7y^{2}$.
+
+\item Expand
+\[
+(x+1)(x+2)-(2x+1)(2x+3)+(x-4)(x-9)+(x-5)^{2}.
+\]
+
+\item Solve $\displaystyle
+\frac{x+2}{x-2}+\frac{x-2}{x+2}=\frac{5}{2}$.
+
+\item Simplify
+
+\[
+\left(\frac{1}{x-1}-\frac{3}{(x+3)(x-1)}\right)\div\left(\frac{1}{x+3}+\frac{1}{(x-1)(x+3)}\right).
+\]
+
+\subsection*{II.}
+
+\item Add $2(a-c)^{3}-10x^{3}y-7(a-c)$, $6(a-c)-2(a-c)^{3}-10x^{3}y$,
+$3(a-c)-(a-c)^{3}+2x^{3}y$, $2(a-c)+x^{3}y-(a-c)^{3}$,
+$4(a-c)+5(a-c)^3+2x^{3}y$, $3(a-c)-2x^{3}y-6(a-c)^3$.
+
+\item Solve $bx-b^{2}=3b^{2}-4bx$.
+
+\item Factor $x^{6}+2x^{3}-3$, $ax^{2}-ay^{2}+by^{2}-bx^{2}$, $27x^3+(y+z)^{3}$.
+
+\item Find the fraction which becomes equal to one when
+six is added to the numerator, and equal to one-third when
+four is added to the denominator.
+
+\item Simplify $\displaystyle \frac{ \frac{a^2}{b^3} + \frac{1}{a}
+}
+ { \frac{a}{b^3} - \frac{1}{b} + \frac{1}{ab} }$.
+
+\item Solve $\displaystyle
+\frac{7x+9}{4}=\left(x-\frac{2x-1}{9}\right)+7$.
+
+\item Six years ago John was five times as old as Sarah.
+If he is twice as old as Sarah now, what are their ages?
+
+\item Multiply together $\frac{1-x^2}{1+y}$, $\frac{1-y^2}{x+x^2}$, and $1+\frac{x}{1-x}$.
+
+\item Simplify
+$a^2 - (b^2-c^2) - \{b^2 - (c^2-a^2)\} + \{c^2 - (b^2-a^2)\}$.
+
+\item $x$ times $y$ is how many times $a$?
+
+
+\subsection*{III.}
+
+\item Add $2x + y - 2a + 55\frac{1}{2}b$, $24b-y + 2x + a$, $3a -
+2y - 4x - 81b$, and subtract the result from $2y + 3a +
+\frac{1}{2}b + 3x$.
+
+\item Divide $\frac{11}{8}a^2-\frac{5}{4}a^3-\frac{1}{2}a+a^4$ by $a^2-\frac{1}{2}a$.
+
+\item A can do a piece of work in 3 days which B can do
+in 5 days. In what time can they do it working together?
+
+\item Simplify $\displaystyle \frac{a-b}{a^2-ab+b^2}
+ + \frac{ab}{a^3+b^3} + \frac{1}{a+b}$.
+
+\item Factor $x^2-9x-52$, $1-a^{16}$, $(a^2 + b^2)^2 + 2(a^4-b^4) + (a^2-b^2)^2$.
+
+\item Solve $\displaystyle \frac{3x}{4} - \frac{x-10}{2} = x -6
+-\frac{x-4}{2}$.
+
+\item The sum of the ages of a man and his son is 100
+years; one-tenth of the product of their ages exceeds the
+father's age by 180. How old are they?
+
+\item Solve $x = 9 - \frac{y}{2}$, $y = 11 + \frac{x}{3}$.
+
+\item From what must $3x^4 - 2x^3 + x - 6$ be subtracted
+to produce unity?
+
+\item Find the following roots: $\sqrt{5.5225}$, $\sqrt[3]{32.768}$.
+
+\subsection*{IV.}
+
+\item Find the value of $\displaystyle \frac{4x^3+2y^3}{ab} +
+\frac{2y^3 + 4z^3}{z^3+y^2} - \frac{b^3-z^2b}{a^2 b}$, if $x=1$,
+$y = 2$, $z = 0$, $a = 4$, and $b = 5$.
+
+\item Solve $\displaystyle \frac{4}{x-6} - \frac{3}{x-9} =
+\frac{1}{x-3}$.
+
+\item Find three consecutive numbers whose sum is 78.
+
+\item Find the G.C.F. of $2a^3 - 12a - 2a^2$, $a^4 - 4a^2$ and
+$4a^3b + 16ab + 16a^2b$.
+
+\item Divide $\displaystyle \frac{x^4 - y^4}{x^2 - 2xy + y^2}$ by
+$\frac{x^2 + xy}{x-y}$.
+
+\item A fraction becomes $\frac{3}{4}$ by the addition of three to the
+numerator and one to the denominator. If one is subtracted
+from the numerator and three from the denominator, it becomes
+$\frac{1}{2}$. What is the fraction?
+
+\item Expand
+$\left(\frac{3a^2b\left(m+n\right)^2}{4xy^3}\left(a-b\right)^3\right)^3$,
+$\sqrt{\frac{50x^4\left(a+b\right)^7}{32y^6z^2(a+b)}}$.
+
+\item If a certain number is multiplied by itself, the result is
+$9x^4 - 4x + 10x^2 + 1 - 12x^3$. Find the number.
+
+\item Simplify $\displaystyle \frac{ax-x^2}{\left(a+x\right)^2}
+\times \frac{a^2+ax}{\left(a-x\right)^2} \div
+\frac{2ax}{a^2-x^2}$.
+
+\item Solve $18x-20y = 3$, $\displaystyle
+\frac{4y-2}{3}-\frac{5x}{2}= 0$.
+
+\subsection*{V.}
+
+\item Factor $x^4+5x^2+6$, $x^2-14x+49$, $x^2-\left(y+z\right)^2$.
+
+\item Add $xy-\frac{9}{8}x-\frac{7}{12}(x^2-y^2)-5x^2y^2$, $\frac{5}{8}x-xy+9x^2y^2
++\frac{2}{3}(x^2-y^2)$, $\frac{1}{9}x^2y^2-xy+\frac{1}{4}x+\frac{3}{4}(x^2-y^2)$, $2xy+\frac{1}{4}x
+-\frac{5}{6}(x^2-y^2)-4x^2y^2$.
+
+\item At what times between 7 and 8 o'clock are the
+hands of a clock six minutes apart?
+
+\item Simplify $\displaystyle \frac{x^2-5x+6}{x^2-2x+1} \times
+\frac{x^2-4x+3}{x^2-4x+4} \div \frac{x^2-6x+9}{x^2-3x+2}$.
+
+\item Solve $\displaystyle \frac{x+2}{b+2}=2-\frac{x+1}{b+1}$.
+
+\item Factor $\displaystyle \frac{x^6}{y^6}-\frac{a^2b^4}{c^2}$,
+$\displaystyle \frac{x^2}{y^2}-\frac{5x}{y}-14$, $\displaystyle
+\frac{x^2}{y^2}-2+\frac{y^2}{x^2}$.
+
+\item A, who works only two-thirds as fast as B, can
+build a stone wall in 12 days. In what time could A
+and B together build the wall?
+
+\item Solve $\displaystyle \frac{x+y}{2}-\frac{x-y}{3}=8$,
+$\frac{x+y}{3}+\frac{x-y}{4}=11$.
+
+\item Expand $\left(1+2x\right)^3$, $\left(2x^2-3a^2b^3\right)^4$.
+
+\item Reduce $\displaystyle
+\frac{(a^4+2a^2b^2+b^4)(a^4+b^4)}{a^8-b^8}$ to lowest terms.
+
+\subsection*{VI.}
+
+\item $y$ is how much greater than $x$?
+
+\item Subtract $3x^3+4x^2y-7xy^2+10y^3$ from $4x^3-2x^2y
++4xy^2+4y^3$ and find the value of the remainder when
+$x=2$ and $y=1$.
+
+\item The length and width of a rectangle are respectively
+5 feet longer and 4 feet.shorter than the side of an equivalent
+square. What is its area?
+
+\item Find the L.C.M. of $a^2-3-2a$, $a^2-1$, and $2a^2
+-6a + 4$.
+
+\item Simplify $\displaystyle
+\frac{\frac{b}{4a}-1+\frac{a}{b}}{\frac{b}{2a}-\frac{2a}{b}}$.
+
+\item Solve $\displaystyle \frac{x-1}{3}+\frac{3}{x-1}=2$.
+
+\item Factor $a^4b+8ac^3bm^6$, $4c^3x^2+cy^2+4c^2xy$, $x^6-1$.
+
+\item Multiply $\displaystyle
+1-\frac{1}{2}x-\frac{1}{3}x^2+\frac{1}{4}x^3$ by
+$1-\frac{1}{3}x^2-\frac{1}{4}x^3-\frac{1}{2}x$.
+
+\item Find the cube root of $6x^4+7x^3+3x^5+6x^2+x^6+1+3x$.
+
+\item Divide $12x^2y^2- 4y^4 - 6x^3y + x^4$ by $x^2+2y^2-3xy$.
+
+\subsection*{VII.}
+
+\item Add $\frac{1}{10}a^3-\frac{4}{5}a^4-\frac{1}{5}a^2+\frac{3}{10}a$, $\frac{1}{4}a^2-\frac{4}{5}a-\frac{5}{7}a^4-\frac{1}{8}a^3$,
+$\frac{5}{7}a^4+\frac{1}{8}a^3+\frac{3}{4}a^2+\frac{2}{5}a$, $\frac{4}{5}a^4+\frac{1}{5}a^2+\frac{2}{5}a^3+\frac{1}{10}a$.
+
+\item Solve $x(a-x)+x(b-x)=2(x-a)(b-x)$.
+
+\item Factor $x^4-22x^2-75$, $16-x^8$, $\left(a+b\right)^2-\left(a-b\right)^2$.
+
+\item A piece of work can be finished by 3 men in 8 days,
+or by 5 women in 6 days, or by 6 boys in 6 days. In what
+time can 2 men, 3 women, and 3 boys do the work?
+
+\item Solve $\displaystyle
+\frac{3x+19}{2}-\left(\frac{x+1}{6}+3\right)=\frac{5x+2}{3}-\left(3-\frac{3x-1}{2}\right)$.
+
+\item Expand $\displaystyle
+\left(\frac{a}{b}-\frac{c}{d}\right)^3$, $\displaystyle
+\left(\frac{c}{d}+1\right)\left(\frac{c^2}{d^2}-\frac{c}{d}+1\right)$.
+
+\item What number is that, the sum of whose third and
+fourth parts is less by two than the square of its sixth part?
+
+\item Solve $\displaystyle \frac{x}{5}-\frac{y}{7}=1$,
+$\frac{2x}{3}-\frac{y}{2}=3$.
+
+\item Divide $m$ by $1+y$ to four terms.
+
+\item If $x$ is $\frac{3}{5}$ of a number, what is the number?
+
+
+\subsection*{VIII.}
+
+\item The head of a fish is 6 inches long, the tail is
+as long as the head and half the body, and the body is
+as long as the head and tail. What is the length of the
+fish?
+
+\item Add $4a - 5x - 15y$, $a + 18x + 8y$, $4a - 7x+11y$,
+$a+3x+5y$, and multiply the result by the difference between $11a
++ 7y$ and $10a +6y - x$.
+
+\item Divide $2x^2+\frac{9}{2}x^4+\frac{8}{9}$ by $2x+3x^2+\frac{4}{3}$.
+
+\item How many numbers each equal to $1-2x+x^2$ must
+be added together to equal $5x^6-6x^5+1$?
+
+\item Factor $a^3+5a^2-4a-20$, $x^6-y^6$, $2x^5-8x^3y^2+6xy^4$.
+
+\item A courier who travels at the rate of 5 miles an hour
+is followed, 4 hours later, by another who travels at the rate
+of 15 miles in 2 hours. In how many hours will the
+second overtake the first?
+
+\item Divide $\displaystyle \frac{1}{1-x}-\frac{1}{1+x}$ by
+$\displaystyle \frac{1}{1-x}+\frac{1}{1+x}$.
+
+\item Solve $3x-4y=-6$, $10x+2y=26$.
+
+\item $3xy-3a^2+4b^2-5cd+4xy-6a^2-7b^2+7cd+3xy
+-6a^2+6b^2-3cd-5xy+7a^2-6b^2+4cd+4xy+7a^2
+-7b^2+4cd-6xy-6a^2+3b^2-7cd+7a^2=?$
+
+\item Simplify
+$3x-5-\{2(4-x)-3(x-2)\}+\{3-(5+2x)-2\}$.
+\end{enumerate}
+
+\chapter*{ANSWERS TO A FIRST BOOK IN ALGEBRA.}
+
+
+\subsection*{Exercise 1.}
+
+\begin{enumerate} \item 43; 86.
+
+\item Carriage, \$375; horse, \$125.
+
+\item C, \$31; J, \$155.
+
+\item 8; 56.
+
+\item 8 miles.
+
+\item Needles, 8; thread, 64.
+
+\item 224 girls; 448 boys.
+
+\item 25; 275.
+
+\item H, 6 qts.; J, 18 qts.
+
+\item Lot, \$720; house, \$3600.
+
+\item Mr. A, 72; son, 24.
+
+\item 50 A.; 300 A.
+
+\item Dict., \$7.20; rhet, \$.90.
+
+\item 112; 4144.
+
+\item Aleck, 56; Arthur, 8.
+
+\item Mother, 28; daughter, 4.
+
+\item J, 15 yrs.; M, 5 yrs.
+
+\end{enumerate}
+
+\subsection*{Exercise 2.}
+\begin{enumerate}
+
+ \item Necktie, \$.75; hat, \$3; boots, \$3.75.
+
+\item 30; 45; 15 miles.
+
+\item James, 15; sister, 5; brother, 10.
+
+\item Pig, \$10; cow, \$30; horse, \$50.
+
+\item A, 35; B, 15; C, 5.
+
+\item 12; 48.
+
+\item 8 men; 40 women.
+
+\item Henry, \$200; John, \$400; James, \$800.
+
+\item 4500 ft.; 13,500 ft.; 27,000 ft.
+
+\item 15; 45; 60 pigeons.
+
+\item 165; 33; 11.
+
+\item A, \$44; B, \$11; C, \$55.
+
+\item Calf, \$8; cow, \$16; horse, \$48.
+
+\item 150; 450 gal.
+
+\item Cow, \$30; lamb, \$5.
+
+\item Tea, 90; coffee, 30.
+
+\item Mrs. C, \$25,000; Henry, \$5000.
+\end{enumerate}
+
+\subsection*{Exercise 3.}
+
+\begin{enumerate}
+
+\item 14 boys; 21 girls.
+
+\item 14 yrs.; 29 yrs.
+
+\item 492; 587 votes.
+
+\item 22; 48.
+
+\item J, 79; H, 64.
+
+\item Flour, 27 bbls.; meal, 30 bbls.
+
+\item 23 Hol.; 40 Jer.
+
+\item \$18; \$26.
+
+\item 40; 59.
+
+\item 16; 19; 21.
+
+\item 21; 17; 24.
+
+\item \$10,000; \$11,500; \$12,700.
+
+\item 21; 38; 6.
+
+\item 51; 28; 16 sheep.
+
+\item A, 253; B, 350; C, 470 votes.
+
+\item 17; 12; 24 A.
+
+\item 36; 20; 55.
+
+\item \$50,000; \$44,000; \$24,000.
+
+\end{enumerate}
+
+\subsection*{Exercise 4.}
+
+\begin{enumerate}
+\item C, 34; H, 15.
+
+\item 26 pear; 7 apple.
+
+\item J, 16 qts.; M, 7 qts.
+
+\item 65.
+
+\item 18.
+
+\item 24.
+
+\item 11.
+
+\item Tea, \$8.76; coffee, \$1.63.
+
+\item 15; 33 rooms.
+
+\item 5; 6; 12.
+
+\item 17; 20; 100.
+
+\item \$5000; \$3000; \$10,000.
+
+\item \$50; \$68; \$204.
+
+\item A, \$5000; B, \$10,500; C, \$31,500.
+
+\item 8000; 24,250; 48,500ft.
+
+\item Daughter, \$25,000; son, \$40,000; widow, \$160,000.
+
+\item Father, 14 qts.; older son, 7 qts.; younger son, 4 qts.
+
+\item H, 200 stamps; J, 185 stamps; T, 189 stamps.
+
+\end{enumerate}
+
+\subsection*{Exercise 5.}
+
+\begin{enumerate}
+
+\item Blue, 5 yds.; white, 15 yds.
+
+\item 3.
+
+\item Walked 2 hrs.; rode 8 hrs.
+
+\item Book, \$2; lamp, \$4.
+
+\item 12.
+
+\item 12 twos; 24 fives.
+
+\item Tea, 67; coffee, 32.
+
+\item Crackers,18 gingersnaps, 25
+
+\item Lamp, \$1; vase, \$1.50.
+
+\item House, \$4500; barn, \$3300.
+
+\item 12,000; 13,500 ft.
+
+\item 29 gal.; 24 gal.
+
+\item Johnson, \$6000; May, \$1500.
+
+\item 27; 10; 42.
+
+\item 3; 17; 51.
+
+\item 3 bbls.; 9 boxes.
+
+\item 18; 90; 180.
+
+\item 84; 132.
+
+\end{enumerate}
+
+\subsection*{Exercise 6.}
+\begin{enumerate}
+\item 4.
+
+\item 7.
+
+\item 12 yrs.
+
+\item \$6.25.
+
+\item \$8.
+
+\item 8 sheep.
+
+\item 121; 605.
+
+\item 142; 994.
+
+\item \$500; \$1450; \$2900.
+
+\item W, 6 yrs.; J, 9 yrs.
+
+\item 25; 15 marbles.
+
+\item 16.
+
+\item Oranges, 35 cents; apples, 20 cents.
+
+\item 9; 15.
+
+\item 30 yrs.; 32 yrs.
+
+\item Cow, \$30; horse,\$45.
+
+\item 5.
+
+\item Boots, \$5; clothes, \$18.
+\end{enumerate}
+
+\subsection*{Exercise 7.}
+\begin{enumerate}
+\item 14 yrs.; 56 yrs.
+
+\item Corn, 60; wheat, 300.
+
+\item \$3000; \$9000.
+
+\item 70 miles; 35 miles.
+
+\item 45; 720.
+
+\item 189.
+
+\item J, 3 yrs.; M, 15 yrs.
+
+\item 96.
+
+\item \$40,000.
+
+\item 24 marbles.
+
+\item \$30,000.
+
+\item 300 oranges.
+
+\item 90.
+
+\item 60,000ft.
+
+\item 70ft.
+
+\item 72 sq. rds.
+
+\item A, \$22,500; B, \$7500.
+
+\item 30.
+\end{enumerate}
+\subsection*{Exercise 8.}
+\begin{enumerate}
+\item 4.
+
+\item 45 marbles.
+
+\item 12; 24; 6 cows.
+
+\item 16.
+
+\item 45.
+
+\item 6048.
+
+\item \$3000.
+
+\item 24.
+
+\item 14.
+
+\item 48,000 ft.
+
+\item 30.
+
+\item 18; 9; 63.
+
+\item 56; 21; 7.
+
+\item 16; 4; 56; 36.
+
+\item Coffee, 18 lbs.; tea, 20 lbs.; cocoa, 24 lbs.
+
+\item \$2500; \$5000; \$7500.
+
+\item J, 9 cents; P, 81 cents.
+
+\item \$12,000.
+\end{enumerate}
+
+\subsection*{Exercise 9.}
+
+\begin{enumerate}
+ \item 35; 21.
+
+\item 24; 18.
+
+\item 42; 30 miles.
+
+\item 30; 54 yrs.
+
+\item J, 10 boxes; H, 16 boxes.
+
+\item 33 tons; $27\frac{1}{2}$ tons.
+
+\item John, 28yrs.; James, 32yrs.
+
+\item \$1000; \$625.
+
+\item 240 girls; 180 boys.
+
+\item 150 lemons.
+
+\item 21,000 ft.; 6000 ft.
+
+\item 20; 12; 10; l0 miles.
+
+\item 126 cu. yds.
+
+\item M, 390; H, 130.
+
+\item 39; 41; 32; 27.
+
+\item 3205; 2591; 1309.
+
+\item 20 miles; 4 miles; 48 miles.
+\end{enumerate}
+
+\subsection*{Exercise 10.}
+
+\begin{enumerate}
+
+\item $x + 9$.
+
+\item $a + p$.
+
+\item $8b$.
+
+\item $x + y$.
+
+\item $c + 5$.
+
+\item $dx$.
+
+\item $m + l + v + c$ dols.
+
+\item $x + y + z$ yrs.
+
+\item $bm$.
+
+\item $d + 1$.
+
+\item $y + z + s$ cts.
+
+\item $m + 1$.
+
+\item $yx$.
+
+\item $x + 40 + a$.
+
+\item $28$; $46$.
+\end{enumerate}
+
+\subsection*{Exercise 11.}
+
+\begin{enumerate}
+\item $a - b$ or $b - a$.
+
+\item $b - 10$.
+
+\item $a + b - c$.
+
+\item $a -2$, $a - 1$, $a$, $a + 1$, $a + 2$.
+
+\item $a - b$ dols.
+
+\item $c - 8$.
+
+\item $x - 3$, $x - 6$, $x - 9$.
+
+\item $c - b$ dols.
+
+\item $x - 5$.
+
+\item $x$, $x + 9$, or $x$, $x - 9$.
+
+\item $x - 75$ dols.
+
+\item $m + x$ dols.
+
+\item $c - f$ cts.
+
+\item $b - e$ dols.
+
+\item $l + 4 + m - x$ dols.
+
+\item $c - a - b$.
+
+\item $429; 636$ votes.
+
+\item $m + x - y + b - z$ dols.
+
+\item $80 - c$ dols.
+
+\item $x, 60 - x$.
+\end{enumerate}
+
+\subsection*{Exercise 12.}
+
+\begin{enumerate}
+
+ \item $2x$.
+
+\item $xyz$.
+
+\item $100x$ cts.
+
+\item $abc$.
+
+\item $ad$ cts.
+
+\item $mb$ miles.
+
+\item $ax$ hills.
+
+\item $x^3$.
+
+\item $a^9$.
+
+\item $d^a$.
+
+\item $3m^3 + a^2$.
+
+\item $x^{2m} + x^m$.
+
+\item $\frac{6}{100}mx$ dols., or $6mx$ cts.
+
+\item $3c - 8$ boys; $4c - 8$ boys and girls.
+
+\item $9x$ days.
+
+\item $3x$ thirds.
+
+\item $5b$ fifths.
+
+\item $m - x + 2a$ dols.
+
+\item $12a - 39$.
+
+\end{enumerate}
+
+\subsection*{Exercise 13.}
+
+\begin{enumerate}
+\item $\frac{5a}{3c}$.
+
+\item $\frac{y}{100}$ dols.
+
+\item $\frac{x}{a}$ books.
+
+\item $\frac{m}{y}$ days.
+
+\item $\frac{x}{b}$ dols.
+
+\item $\frac{a + b}{c}$.
+
+\item $a + \frac{b}{c}$.
+
+\item $a + \frac{a}{2}$, or $\frac{3}{2}a$.
+
+\item $300x$.
+
+\item $18b - 3x$ dols.
+
+\item A, $\displaystyle \frac{1}{x}$; B, $\displaystyle
+\frac{1}{y}$; C, $\displaystyle \frac{1}{z}$; all, $\displaystyle
+\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$.
+
+\item $a^2$ sq. ft.
+
+\item $100a + 10b + 25c$ cts.
+
+\item $\displaystyle \frac{x}{y}$.
+
+\item $\displaystyle \frac{n}{m}$ chestnuts.
+
+\item 12; 18 apples. \end{enumerate}
+
+\subsection*{Exercise 14.} \begin{enumerate}
+\setcounter{enumi}{9}
+
+\item 11.
+
+\item 7.
+
+\item 21.
+
+\item 78.
+
+\item 46.
+
+\item $-74$.
+
+\item $-1\frac{1}{3}$.
+
+\item $-4\frac{1}{4}$.
+
+\item 5.
+
+\item 6 apples; 12 pears.
+
+\item 36 years.
+
+\end{enumerate}
+
+\subsection*{Exercise 15.}
+
+\begin{enumerate}
+\item $24x$.
+
+\item $25ab$.
+
+\item $-18ax^3$.
+
+\item $-42x$.
+
+\item $10a^2$.
+
+\item $-10abc^2$.
+
+\item $6ab-x^2$.
+
+\item $10ax - 4bc$.
+
+\item $-16a^2$.
+
+\item $8a^4b + 3ab - x^5$.
+
+\item $\frac{3}{2}a$.
+
+\item $-\frac{7}{12}b$.
+
+\item $m + d + c - x$ cts.
+
+\item $a - x - 5 + y$ miles.
+
+\item $5a + 4b + 5c$.
+
+\item $x + y - z$.
+
+\item $-3z - a$.
+
+\item $2x^3 + 4x^2 - 2x + 17$.
+
+\item $a^3 + b^3 + c^3$.
+
+\item $2a^m + 1$.
+
+\item $2a^2b^2c$.
+
+\item $23x^3 - 20x^2 + 27x + 6$.
+
+\item $x^5y + 12x^4y^2 - 16x^3y^3 - 8xy^5$.
+
+\item $5x + 3y + z - a - 3b$.
+
+\item $a^3 + b^3 + c^3 - 3abc$.
+
+\item $mb + c$ men.
+
+\item $x - 10$ cows; $z + 19$ horses.
+
+\item 22 girls; 30 boys. \end{enumerate}
+
+\subsection*{Exercise 16.}
+
+\begin{enumerate}
+\item $2a^3$.
+
+\item $12a^2b$.
+
+\item $-9xy^3$.
+
+\item $4x^my$.
+
+\item $8x^2 - 3ax$.
+
+\item $5xy + 7by$.
+
+\item $2a^m$.
+
+\item $9 ax$.
+
+\item $-3a - b + 14c$.
+
+\item $4x - y + 2z$.
+
+\item $8x^4 - 2x^3 + 4x^2 - 15x + 14$.
+
+\item $20a^2b^2 + 16a^2b$.
+
+\item $4x^3 - 2$.
+
+\item $2x^m - x^{2m} - x^{3m}$
+
+\item $2a^{2n} - 18a^nx^n - 9x^{2n}$.
+
+\item $\frac{4}{3}a^2 - \frac{7}{2}a - \frac{1}{2}$.
+
+\item $-2x^4y - 3x^3y^2 + 5xy^4 - y^5$.
+
+\item $x - y + a$.
+
+\item $-3a^2$.
+
+\item $8x^3 - 2x$.
+
+\item $27y^3 - 3z^3 - 6x^3 + 4yz^2 - 11z^2x$.
+
+\item $4x^2 - 16x + 64$.
+
+\item $-4a^2 + 6b^2 - 8bc + 6ab$.
+
+\item $2x^4 - 3x^2 + 2x - 4$.
+
+\item $5a^3 + 2a + 2$.
+
+\item $-11a^2b + 4ab^2 - 12a^2b^2 - b^3$.
+
+\item $b - a$.
+
+\item $x - 3$.
+
+\item $40 - y$ yrs.
+
+\item $\frac{23}{a}$ hrs.
+\end{enumerate}
+\subsection*{Exercise 17.} \begin{enumerate}
+
+\item $2x + a + b + c - d$.
+
+\item $a + c$.
+
+\item $2a^2b-a^3-2b^3-ab^2$.
+
+\item $3xy - x^2 - 3y^2$.
+
+\item $4b - 4c$.
+
+\item $-2y$.
+
+\item $-6b + 4c$.
+
+\item $-b$.
+
+\setcounter{enumi}{16}
+
+\item $5(x - y)$.
+
+\item $150 - 7(x + y)$.
+
+\item $x + 8$ yrs.
+
+\item $3(x - 35)$ dols.
+\end{enumerate}
+
+\subsection*{Exercise 18.} \begin{enumerate}
+
+\item $35 cx$.
+
+\item $-51 acxy$.
+
+\item $21ax^4y^3$.
+
+\item $10a^3b^3c^4$.
+
+\item $18acx^3y^3$.
+
+\item $30a^3b^2c^4$.
+
+\item $-x^3y^3z^3$.
+
+\item $-a^4b^5c^2$.
+
+\item $-\frac{2}{9}a^2cx^6y^4$.
+
+\item $-\frac{3}{20}a^3b^5c^4$.
+
+\item $\frac{100}{ab}$.
+
+\item $100x$.
+
+\item $100a + 10b + c$.
+
+\item $x + 7$ or $x - 7$.
+\end{enumerate}
+
+\subsection*{Exercise 19.} \begin{enumerate}
+
+\item $x^4y^2 + x^3y^3 + x^2y^4$.
+
+\item $a^4b - a^3b^2 +a^2b^3$.
+
+\item $-2a^4b + 6a^3b^2 - 2ab^4$.
+
+\item $24x^4y^2 + 108x^3y^3 + 81xy^5$.
+
+\item $a^5b^2 - \frac{6}{25}a^4b^3 - \frac{2}{5}a^3b^4$.
+
+\item $x^3 + y^3$.
+
+\item $x^5 - 4x^4 + 5x^3 - 3x^2 + 2x - 1$.
+
+\item $x^5 + x^4 - 4x^3 + x^2 + x$.
+
+\item $x^2y^2 - 2xy^2n + y^2n^2 - m^2n^2 + 2xm^2n - x^2m^2$.
+
+\item $x^7 + x^6 + 2x^5 + x^2 + x + 2$.
+
+\item $a^6 + b^6$.
+
+\item $x^3 - 3xyz + y^3 + z^3$.
+
+\item $x^7 - y^7$.
+
+\item $x^8 - 8x^4a^4 + 16a^8$.
+
+\item $a^6 + 2a^3y^3 - 9a^4y^4 + y^6$.
+
+\item $x^6 + x^5 + 2x^4 - 11x^3 - 17x^2 - 34x - 12$.
+
+\item $6x^6 - 17x^5 - 12x^4 - 14x^3 + x^2 + 12x + 4$.
+
+\item $5(x + y)$; $4(x - y)$.
+
+\item $\frac{12}{35}$ of the field.
+
+\item $\frac{1}{a} + \frac{1}{b}$.
+\end{enumerate}
+
+\subsection*{Exercise 20.} \begin{enumerate}
+
+\item $x^2 + 9x + 14$.
+
+\item $x^2 + 7x + 6$.
+
+\item $x^2 - 7x + 12$.
+
+\item $x^2 - 7x + 10$.
+
+\item $x^2 + 3x - 10$.
+
+\item $x^2 + 4x - 21$.
+
+\item $x^2 - x - 42$.
+
+\item $x^2 - x - 30$.
+
+\item $x^2 - 13x + 22$.
+
+\item $x^2 - 14x + 13$.
+
+\item $y^2 - 2y - 63$.
+
+\item $x^2 + 20x + 51$.
+
+\item $y^2 - 13y - 30$.
+
+\item $y^2 + 18y + 32$.
+
+\item $a^4 + 2a^2 - 35$.
+
+\item $a^2 - 81$.
+\item $m^4 - 18m^2 + 32$.
+\item $b^6 + 2b^3 - 120$.
+\item $x^2 - \frac{3}{4}x + \frac{1}{8}$.
+\item $y^2 + \frac{1}{2}y + \frac{1}{18}$.
+\item $m^2 + \frac{1}{3}m - \frac{2}{9}$.
+\item $a^2 + \frac{1}{5}a - \frac{6}{25}$.
+\item $x^2 - \frac{7}{6}x + \frac{1}{3}$.
+\item $y^2 + \frac{19}{20}y + \frac{3}{20}$.
+\item $21 - 10x + x^2$.
+\item $15 -8x + x^2$.
+\item $42 - x - x^2$.
+\item $33 + 8x - x^2$.
+\item $x^2 - 9$.
+\item $y^2 - 25$.
+\item $21$.
+\item $12$ cows.
+\end{enumerate}
+
+
+\subsection*{Exercise 21.}
+
+\begin{enumerate}
+
+\item $a^4b^2$.
+\item $x^3y^6$.
+\item $a^8b^2$.
+\item $-x^9y^6$.
+\item $27 a^6y^3$.
+\item $49 a^2b^4c^6$.
+\item $x^5y^5z^{10}$.
+\item $m^8n^4d^4$.
+\item $-125 x^9y^{12}z^3$.
+\item $121 c^{10}d^{24}x^8$.
+\item $\frac{1}{4} x^4a^2m^6$.
+\item $\frac{1}{9} a^2b^6c^2$.
+\item $225 c^{12}d^2x^4$.
+\item $-729 x^3y^{15}z^6$.
+\item $a^{36}b^8c^{16}d^8$.
+\item $-x^{40}y^5z^{15}m^{10}n^5$.
+\item $\frac{4}{9} a^4b^2c^8$.
+\item $\frac{25}{36} m^2n^4x^6$.
+\item $8 b$ days.
+\item $10 a$ mills; $\frac{a}{100}$ dols.
+
+\end{enumerate}
+
+\subsection*{Exercise 22.}
+
+\begin{enumerate}
+
+\item $z^3 + 3z^2x + 3zx^2 + x^3$.
+\item $a^4 + 4a^3y + 6a^2y^2 + 4ay^3 + y^4$.
+\item $x^4 - 4x^3a + 6x^2a^2 - 4xa^3 + a^4$.
+\item $a^3 - 3a^2m + 3am^2 - m^3$.
+\item $m^2 + 2am + a^2$.
+\item $x^2 - 2xy + y^2$.
+\item $x^6 + 3x^4y^2 + 3x^2y^4 + y^6$.
+\item $m^6 - 2m^3y^2 + y^4$.
+\item $c^8 - 4c^6d^2 + 6c^4d^4 - 4c^2d^6 + d^8$.
+\item $y^6 + 3y^4z^4 + 3y^2z^8 + z^{12}$.
+\item $x^4y^2 + 2x^2yz + z^2$.
+
+\item $a^8b^4 - 4a^6b^3c + 6a^4b^2c^2-4a^2bc^3 + c^4$.
+
+\item $a^6 - 3a^4b^3c + 3a^2b^6c^2-b^9c^3$.
+\item $x^4y^2 - 2x^2ymn^3 + m^2n^6$.
+\item $x^3 + 3x^2 + 3x + 1$.
+\item $m^2 - 2m + 1$.
+\item $b^8 - 4b^6 + 6b^4 - 4b^2 + 1$.
+\item $y^9 + 3y^6 + 3y^3 + 1$.
+\item $a^2b^2 - 4ab + 4$.
+\item $x^4y^2 - 6x^2y + 9$.
+\item $1 - 4x + 6x^2 - 4x^3 + x^4$.
+\item $1 - 3y^2 + 3y^4 - y^6$.
+\item $4x^2 + 12xy^2 + 9y^4$.
+\item $27a^3b^3 - 27a^2b^2x^2y + 9abx^4y^2 - x^6y^3$.
+\item $256m^4n^{12} - 768m^3n^9a^2b + 864m^2n^6a^4b^2
+ - 432mn^3a^6b^3 + 81a^8b^4$.
+\item $\frac{1}{4}x^2 - xy + y^2$.
+\item $1 - x^2 + \frac{1}{3}x^4 - \frac{1}{27}x^6$.
+\item $x^8 - 12x^6 + 54x^4 - 108x^2 + 81$.
+\item $20a^2 - d$ horses.
+\item $100x - a^2$ cts.
+\item $a(25 - x)$ cts.
+\item $3$.
+\item $-228$.
+\end{enumerate}
+
+\subsection*{Exercise 23.} \begin{enumerate}
+
+\item $14x - 7$.
+
+\item $b^8 - 2b^4 + 1$.
+
+\item $10x^2 + 7y^2$.
+
+\item \$160; \$80; \$60.
+
+\item 13; 21.
+
+\item $2a^3 + 4a^2 + 10$.
+
+\item $\frac{1}{3} a^2 - \frac{4}{3} ab + \frac{1}{2} b^2$.
+
+\item $48a^7 b^6 c^7$.
+
+\item $\frac{16}{81}x^4 y^8 z^{12}$.
+
+\item $2x^2 - 8x + 26$.
+
+\item $8a^6 b^3 - 36a^4 b^2 xy + 54a^2 bx^2 y^2 - 27x^3 y^3$.
+
+\item $x^3 - 3x^2 + 2y - 6$.
+
+\item $x^6 - 3x^4 y^2 + 3x^2 y^4 - y^6$.
+
+\item $x^4 - 1$.
+
+\item $x^4 - y^4$.
+
+\item $176\frac{1}{2}$ lbs.; $140\frac{1}{2}$ lbs.
+
+\item watch, \$200; chain, \$150.
+
+\item $2x + 4$.
+
+\item $8\,ay$.
+
+\item $1 + \frac{2}{3} b + \frac{1}{9} b^2 - \frac{1}{4} a^2$.
+\end{enumerate}
+
+
+\subsection*{Exercise 24.} \begin{enumerate}
+
+\item $5xy$.
+
+\item $13ab$.
+
+\item $3a^2$.
+
+\item $5x^2 y^3$.
+
+\item $-17x$.
+
+\item $-11x^3 y$.
+
+\item $4xz^2$.
+
+\item $9a^2c^2$.
+
+\item $2a^2 b^3$.
+
+\item $-3x^2y$.
+
+\item $-5x^3 y^3$.
+
+\item $-5a^2 c^4$.
+
+\item $\frac{4}{5} x^3y$.
+
+\item $-3a^3 m^3$.
+
+\item $8m^3 x^4$.
+
+\item $-6x^2 y^2 z^3$.
+
+\item $2(x + y)^2 z^2$.
+
+\item $5(a - b)^2 x$.
+
+\item $\frac{1}{2} x^3 y^3 z^2$.
+
+\item $-\frac{1}{3} a^2 b^3$.
+
+\item $x^9 y^6 - 9x^7 y^5 + 27x^5 y^4 - 27 x^3 y^3$.
+
+\item $\frac{b}{2a}$ miles.
+
+\item $\frac{8y}{x}$ days.
+
+\item $\frac{ab}{c}$.
+\end{enumerate}
+
+\subsection*{Exercise 25.}
+\begin{enumerate}
+
+\item $3ab^2 - 7b + 15a^3 x$.
+
+\item $5x^2 y + 3y - 9xy^3$.
+
+\item $8x^3 y^5 - 4x^2 y^2 - 2y$.
+
+\item $13a^2 b - 9ab^2 + 7b$.
+
+\item $-\frac{6}{7} a^2x^2 + \frac{3}{2} ax^3$.
+
+\item $-\frac{1}{3} x^2 + 2y^2$.
+
+\item $-4y^3 z^3 + 3x^2 y^3 z^4 - xy$.
+
+\item $20ac - 31a^2 b^4 c^2$.
+
+\item $x^6 - \frac{5}{2} x^3 + \frac{1}{2} x^4 - 4x - \frac {3}{2}
+x^2$.
+
+\item $8 + \frac{32}{3} y^4 - 16y^3$.
+
+\item $\frac{2}{3} a - \frac{1}{6} b - c$.
+
+\item $3x - 2y - 4$.
+
+\item $xy$ men.
+
+\item $\frac{60x}{a}$ minutes.
+
+\item $\frac{100b}{x}$ apples.
+\end{enumerate}
+
+\subsection*{Exercise 26.}
+\begin{enumerate}
+
+\item $x - 7$.
+
+\item $x - 3$.
+
+\item $x^2 + 5$.
+
+\item $y^2 - 6$.
+
+\item $x^2 - 5x - 3$.
+
+\item $a^2 + 2a - 4$.
+
+\item $x^2 + xy + y^2$.
+
+\item $a^2 - ab + b^2$.
+
+\item $8a^3 + 12a^2 b + 18ab^2 + 27b^3$.
+
+
+\item $27x^5 + 9x^4y + 3x^2y^2 + y^3$.
+
+\item $x^3 + x^2y + 3xy^2 + 4y^3$.
+
+\item $a^3 + 4a^2b - 3ab^2 - 2b^3$.
+
+\item $x^4 + 2x^2 - 3x + 1.$
+
+\item $x^3 - 3x^2 + x - 1$.
+
+\item $a^2 - a - 1$.
+
+\item $x^4 - x^3 - x^2$.
+
+\item $a^8 + a^5 + a^2$.
+
+\item $a^10 - a^8 + a^6 - a^4$.
+
+\item $x^2 + 2xy + 2y^2$.
+
+\item $2a^2 - 6ab + 9b^2$.
+
+\item $\frac{1}{2} x^2 + xy - \frac{1}{3} y^2$.
+
+\item $\frac{1}{3} x^2 - \frac{1}{2} xy + \frac {2}{3} y^2$.
+
+\item $\frac{1}{2}\left(x - y\right)^3 - \left(x - y\right)^2
+- \frac{1}{4}(x - y)$.
+
+\item $2x^2 - 3y$.
+
+\item $4x^3 - 4x^2 - 6x + 6$.
+
+\item $2a^3 + a^2b - 2ab^2 - b^3$.
+
+\item $a^4 + a^2b^2 + b^4$.
+
+\item $x^4 + x^2y^2 + y^4$.
+
+\item $3a^3 + 2b^2$.
+
+\item $1 + 2x - 2x^2 + 2x^3 - \text{etc}$.
+
+\item $1 - a - a^2 - a^3 - \text{etc}$.
+
+\item $2 + \frac{3}{2} a + \frac{3}{4} a^2 + \frac{3}{8} a^3 +
+\text{etc}$.
+
+\item $3 - \frac{4}{3} x + \frac{4}{9} x^2 - \frac{4}{27} x^3
++ \text{etc}$.
+
+\item $\displaystyle \frac{x}{5}$ hrs.
+
+\item $\displaystyle \frac{x + my + bc}{n}$ dols.
+
+\item $\displaystyle \frac{am + bp}{m + p}$ cts.
+
+\item $y - 11$ yrs.
+\end{enumerate}
+
+
+\subsection*{Exercise 27.}
+\begin{enumerate}
+
+\item $4ab^3$.
+
+\item $3xy^2$.
+
+\item $- 2x^2y$.
+
+\item $- 2a^2b^3$.
+
+\item $3ab^2$.
+
+\item $3xy^3$.
+
+\item $2x^2y$.
+
+\item $3x^4y^3$.
+
+\item $\frac{2}{3} m^3y^2$.
+
+\item $\frac(2)(3) a^3b^2$.
+
+\item $- \frac{3}{4} x^3y^4$.
+
+\item $- \frac{2}{3} ab^3$.
+
+\item $x^2(a - b)$.
+
+\item $a^3(x^2 + y^2)$.
+
+\item $2ab^3\left(x^2 - y\right)^2$.
+
+\item $4x^2y\left(m^3 + y\right)^3$.
+
+\item $\frac{3}{2}a^2b^3$.
+
+\item $\frac{1}{2}x^2y$.
+
+\item $10a^3b^3c^4$.
+
+\item $4y$.
+
+\item $15x^3y^3z^5$.
+
+\item $5b$.
+
+\item $\displaystyle \frac{a}{x + y}$ hrs.; $\displaystyle
+\frac{ax}{x + y}$ miles.; $\displaystyle \frac{ay}{x + y}$ miles.
+
+\item $26$.
+
+\item $2(m - 6)$; $2m - 6$.
+\end{enumerate}
+
+
+\subsection*{Exercise 28.}
+\begin{enumerate}
+
+\item $2x - 3y$.
+
+\item $x^2 + 5xy^3$.
+
+\item $4abc^2 - 7xy^2z$.
+
+\item $\frac{1}{2} x - y^2z$.
+
+\item $ab^3 + \frac{1}{3}c^4$.
+
+\item $x^2 - 2x - 1$.
+
+\item $x^2 + 3x + 4$.
+
+\item $2x^2 - x + 2$.
+
+\item $3x^2 + x - 1$.
+
+\item $x^3 + x^2 - x + 1$.
+
+\item $x^3 + 2x^2 + x - 4$.
+
+\item $2x^3 - x^2 - 3x + 1$.
+
+\item $90 - x$.
+
+\item $10x + y$.
+\end{enumerate}
+
+
+\subsection*{Exercise 29.}
+
+
+\begin{enumerate}
+\item $3x-y$.
+\item $5x^2-1$.
+\item $3b^2+4a$.
+\item $x^2-2y^2$.
+\item $1+3x$.
+\item $1-7m$.
+\item $4x^2-\frac{1}{2}$.
+\item $3x^3+\frac{1}{3}$.
+\item $2a^2-a+1$.
+\item $x^3-x^2+x$.
+\item $x^2-x-1$.
+\item $\frac{1}{2}a^2+2a-1$.
+\item $4x$ in.
+\item $27x^3$.
+\item $4y$ ft.
+\item $a-b$ miles north; $a+b$ miles.
+\end{enumerate}
+
+\subsection*{Exercise 30.}
+
+
+\begin{enumerate}
+\item $45$.
+\item $97$.
+\item $143$.
+\item $951$.
+\item $8.4$.
+\item $.95$.
+\item $308$.
+\item $.0028$.
+\item $3.9$.
+\item $73$.
+\item $62.3$.
+\item $83.9$.
+\item $3.28$.
+\item $50.5$.
+\item $5.898-$.
+\item $2.646-$.
+\item $.501-$.
+\item $33$ pieces.
+\item $74$ men.
+\item $104.9+$ in.
+\item $92$ trees.
+\end{enumerate}
+
+
+\subsection*{Exercise 31.}
+
+\begin{enumerate}
+\item $5(a^2-5)$
+\item $16(1+4xy)$.
+\item $2a(1-a)$.
+\item $15a^2(1-15a^2)$.
+\item $x^2(x-1)$.
+\item $a^2(3+a^3)$.
+\item $a(a-b^2)$.
+\item $a(a+b)$.
+\item $2a^3(3+a+2a^2)$.
+\item $7x(1-x^2+2x^3)$.
+\item $x(3x^2-x+1)$.
+\item $a(a^2-ay+y^2)$.
+\item $(x+y)(3a+5mb-9d^2x)$.
+\item $5(a-b)(1-3xy-a^2b)$.
+\item $4xy(x^2-3xy-2y^2)$.
+\item $2axy^5(3x^2-2xy+y^2-ay^4)$.
+\item $17x^2y(3x^3-2x^2y+y^3)$.
+\item $3ab(2ab-a^2b^2c-3b^2c+c^2)$.
+\item $3ax(x^6-8+3x^4-x^3-3x^5)$.
+\item $27a^5b^2c^3(a^3-3a^2b+3ab^2-b^3-c^3)$.
+\item $m+d+c-x$ cts.
+\item $12$ beads.
+\end{enumerate}
+
+\subsection*{Exercise 32.}
+
+\begin{enumerate}
+\item $(a+b)(x+y)$.
+\item $(x+b)(x+a)$.
+\item $(a-b)(x^2+y^2)$.
+\item $(x+5)(x-a)$.
+\item $(a-x)(x+b)$.
+\item $(x-4y)(x+my)$.
+\item $(x^3+2)(2x-1)$.
+\item $(m-n)(x-a)$.
+\item $(x^2+1)(x+1)$.
+
+\item $(y^2 + 1)(y - 1)$.
+\item $(x^4 - x^2 + 1)(x + 1)$.
+\item $(ax + by + c)(a + b)$.
+\item $(a - b - c)(x - y)$.
+\item $(3a - 2b)(x + y)$.
+\item $(2a + 3b - c)(x - y)$.
+\item $3a(2x + y)(m - n)$.
+\item $250 + yA$.; $30 - x$ horses.
+\item $12$.
+\item $70$.
+\end{enumerate}
+
+\subsection*{Exercise 33.}
+
+\begin{enumerate}
+
+\item $(x+y)(x-y)$.
+
+\item $(m + n)(m - n)$.
+
+\item $(ab^2 + cd)(ab^2 - cd)$.
+
+\item $(mp^2 - x^3y^2)(mp^2 + x^3y^2)$.
+
+\item $(a^3bx^2 + m^2c^4y^{10})(a^3bx^2 - m^2c^4y^{10})$.
+
+\item $(xy^2z^2 + cdm^2)(x^2y^2z^2 - cdm^2)$.
+
+\item $(x^2y +a^3y^2)(x^2y - a^3y^2)$.
+
+\item $(g^3c^3 - x^3z^4)(g^3c^3 + x^3z^4)$.
+
+\item $(2a - 3x)(2a + 3x)$.
+
+\item $(4m - 3n)(4m + 3n)$.
+
+\item $(9xy^2 - 5bd)(9xy^2 - 5bd)$.
+
+\item $(27m^2cx^5 + 100y^2)(27m^2cx^5 - 100y^2)$.
+
+\item $(11m + 8x)(11m - 8x)$.
+
+\item $(x^2 + y^2)(x + y)(x - y)$.
+
+\item $(m^4 + a^4)(m^2 + a^2)(m + a)(m - a)$.
+
+\item $(a^4b^4 + 1)(a^2b^2 + 1)(ab + 1)(ab - 1)$.
+
+\item $(x^8 + b^8)(x^4 + b^4)(x^2 + b^2)(x + b)(x - b)$.
+
+\item $(4a^2 + 1)(2a + 1)(2a - 1)$.
+
+\item $a(a + x)(a - x)$.
+
+\item $5b^2(b + a)(b - a)$.
+
+\item $(a - b)(x + y)(x - y)$.
+
+\item $5a(a + y)(1 + a)(1 - a)$.
+
+\item $(m + y - a)(m - y)$.
+
+\item $(a - x + 1)(a + x)$.
+
+\item $x - 3y$.
+
+\item $3x + 11$ yrs.
+
+\end{enumerate}
+
+\subsection*{Exercise 34.}
+
+\begin{enumerate}
+
+\item $(x + y)(x^2 - xy + y^2)$.
+\item $(c + d)(c^2 - cd + d^2)$.
+\item $(a + bc)(a^2 - abc + b^2c^2)$.
+\item $(ax + y)(a^2x^2 - axy + y^2)$.
+\item $(2abc^2 + m^2)(4a^2b^2c^4 - 2 abc^2m^2 + m^4)$.
+\item $(x^2y^3 + 6a)(x^4y^6 - 6ax^2y^3 + 36a^2)$.
+\item $(a^2 + b^2)(a^4 - a^2b^2 + b^4)$.
+\item $(4x^2+y^2)(16x^4 - 4x^2y^2 + y^4)$.
+\item $(x + 2)(x^2 - 2x + 4)$.
+\item $(3 + ab^2)(9 -3ab^2 + a^2b^4)$.
+\item $(y + 1)(y^2 - y + 1)$.
+\item $(1 + bc^3)(1 - bc^3 + b^2c^5)$.
+\item $(\frac{1}{2}a^2b + c^3)(\frac{1}{4}a^4b^2
+ - \frac{1}{2}a^2bc^3 + c^5)$.
+
+\item $(\frac{1}{4}x + 1)(\frac{1}{2}x^2 - \frac{1}{4}x + 1)$.
+
+\item $(m + n + 2)\{(m + n)^2 - 2(m + n) + 4\}$.
+
+\item $(1 - x - y)\{1 - (x - y) + (x - y)^2\}$.
+
+\item $2a^2(xy^2 + a^2b^3) (x^2y^4 - a^2b^3xy^2 + a^4b^6)$.
+
+\item $(m - n)(x + y)$.
+
+\item $(x - y)(a + b)(a^2 - ab + b^2)$.
+
+\item $(a + b)(a^2 - ab + b^2)(a - b)(a^2 + ab + b^2)$.
+
+\item $(27x^3 + 4y^3)(27x^3 - 4y^3)$.
+
+\item $(1 + x^4)(1 + x^2)(1 + x)(1 - x)$.
+
+\item $3(d-26)$; $3d - 104$ dols.
+
+\item $10y$.
+
+\item $x - 8$.
+
+\end{enumerate}
+
+\subsection*{Exercise 35.}
+\begin{enumerate}
+
+
+\item $(x - a)(x^2 +ax + a^2)$.
+
+\item $(c - b)(c^2 + cb + b^2)$.
+
+\item $(a - xy)(a^2 + axy + x^2y^2)$.
+
+\item $(mn - c)(m^2n^2 + mnc + c^2)$.
+
+\item $(3m - 2x)(9m^2 + 6mx + 4x^2)$.
+
+\item $8(x - 2y)(x^2 + 2xy + 4y^2)$.
+
+\item $(4ax^2b - 5m^3cy^2)(16a^2x^4b^2 + 20abcm^3x^2y^2 + 25m^6c^2y^4)$.
+
+\item $27(bc^2y - 2a^3m^2x^2)(b^2c^4y^2 + 2a^3bc^2m^2x^2y + 4a^2m^4x^4)$.
+
+\item $(2x^3y - 5)(4x^6y^2 + 10x^3y + 25)$.
+
+\item $(3 - 4mx^2y)(9+12mx^2y + 16m^2x^4y^2)$.
+
+\item $(a-b^2)(a^2 + ab^2 + b^4)$.
+
+\item $(m^2 - x)(m^4 + m^2x + x^2)$.
+
+\item $(x-1)(x^2 + x + 1)$.
+
+\item $(1-y)(1 + y + y^2)$
+
+\item $(\frac{1}{3}xy^2 - b^3)(\frac{1}{9}x^3y^4 + \frac{1}{3}xy^2b^3 + b^6)$.
+
+\item $(2-\frac{1}{4}m^2n)(4+\frac{1}{2}m^2n + \frac{1}{16}m^4n^2)$.
+
+\item $\{1 - (a + b)\} \{1+ (a+b) + (a+b)^2\}$.
+
+\item $\{xy - (x-y^2)\} \{x^2y^2 + xy(x - y^2) + (x - y^2)^2\}$.
+
+\item $x(x^2 - y)(x^4 + x^2y + y^2)$.
+
+\item $(a + m)(b - c)(b^2 + bc + c^2)$.
+
+\item $(x + y)(x^2 - xy + y^2)(x - y)(x^2 + xy + y^2)$.
+
+\item $(x^5 + x^2 - 1)(x - 1)$.
+
+\item $(a - b)(a + b + a^2 + ab + b^2)$.
+
+\item $(x + y)(mx^2 - mxy + my^2 - 1)$.
+
+\item $\frac{75}{xy}$ hrs.
+
+\item $100x + 10x + y$.
+\end{enumerate}
+
+\subsection*{Exercise 36.}
+
+\begin{enumerate}
+
+\item $(a + x)^2$.
+
+\item $(c - d)^2$.
+
+\item $(2a + y)^2$.
+
+\item $(a + 2b)^2$.
+
+\item $(x - 3c)^2$.
+
+\item $(4x - y)^2$.
+
+\item $(a + 1)^2$.
+
+\item $(3 - x)^2$.
+
+\item $(x - 5)^2$.
+
+\item $(2y - 3)^2$.
+
+\item $(3x + 4)^2$.
+
+\item $(3a - 1)^2(3a + 1)^2$.
+
+\item $\left(3c+11d\right)^2$.
+\item $\left(2y-9\right)^2$.
+\item $\left(x^2-3y\right)^2$.
+\item $\left(3-2x^2\right)^2$.
+\item $\left(4xy^2-3a^8\right)^2$.
+\item $\left(5b+3c^2y\right)^2$.
+\item $\left(7a+2x^2y\right)^2$.
+\item $\left(\frac{1}{3}x^2-y^2\right)^2$.
+\item $\left(\frac{1}{2}a+2b\right)^2$.
+\item $\left(\frac{2}{3}x^3y-z^4\right)^2$.
+\item $x^3\left(x^3-2y^4\right)^2$.
+\item $2y\left(3a+2y^2\right)^2$.
+\item $3ax^3\left(ax-5b\right)^2$.
+\item $\left(x+y-a^2\right)^2$.
+\item $\left\{(m^2-n^2)-(m^2+n^2)\right\}^2$ or $\left(-2n^2\right)^2$.
+\item $(a+b)(a+b+6)$.
+\item $(x-y)(x-y-3)$.
+\item $\left(x-y\right)^2\left(x^2+xy+y^2\right)^2$.
+\item $y^2$.
+\item $9y^2$.
+\item $\pm 2cd$.
+\item $1$.
+\item $9$.
+\item $4y^4$.
+\item $25y^2$.
+\item $\pm 60ab^3c$.
+\item $\pm 70a^2b^3cd$.
+\item $16a^2$ or $28a^2$.
+\item $ay$ or $353ay$.
+\item $4a^2b^2$ or $16a^2b^2$.
+\item $x^2$ or $-3x^2$.
+\item $a$ or $-3a$.
+\item $b-a$; $6(b-a)$; $b+a$; $3(b+a)$ miles.
+\item $\frac{8}{15}$ of the cistern.
+\item $\frac{b-9}{3}$; $\frac{b-9}{3}+9$ years.
+\end{enumerate}
+
+\subsection*{Exercise 37.}
+
+\begin{enumerate}
+\item $(a+2)(a+1)$.
+\item $(x+6)(x+3)$.
+\item $(x-3)(x-2)$.
+\item $(a-5)(a-2)$.
+\item $(y-8)(y-2)$.
+\item $(c-3)(c+2)$.
+\item $(x+5)(x-1)$.
+\item $(x+6)(x-1)$.
+\item $(y+13)(y-5)$.
+\item $(a-11)(a+7)$.
+\item $(x-9)(x+7)$.
+\item $(a+15)(a-5)$.
+\item $(a-11)(a-13)$.
+\item $(a^3-13)(a^3+9)$.
+\item $(x^3+11)(x^3-7)$.
+\item $(6+x)(5+x)$.
+\item $(7+a)(3+a)$.
+\item $(7-x)(5-x)$, or $(x-7)(x-5)$.
+\item $(9-x)(4-x)$.
+\item $(c+3d)(c-d)$.
+\item $(a+5x)(a+3x)$.
+\item $(x-5y)(x+4y)$.
+\item $(x^2+4y)(x^2-3y)$.
+\item $(x^2-5y^2)(x+3y)(x-3y)$.
+\item $x^3(x-12)(x-11)$.
+\item $a^4(a-7)(a-5)$.
+\item $3a(x+3)(x-3)(x+2)(x-2)$.
+\item $3a\left(a-2b\right)^2$.
+\item $(c-a)(c+a)(cd-1)(c^2d^2+cd+1)$.
+\item $(a+b)(x-3)(x-2)$.
+\item $\frac{1}{a}+\frac{1}{b}$.
+\item $\frac{7x}{2}$ days.
+\end{enumerate}
+
+
+\subsection*{Exercise 38.}
+
+\begin{enumerate}
+
+\item $9a^2 - 9a - 4$.
+
+\item $3y^3 + 2y^2 - 2y - 1$.
+
+\item $x^4 - y^4$.
+
+\item $a^3 - 5a^2 + 26a - 2$.
+
+\item $x^6y^3 - 12x^4y^2z^3a^4 + 48x^2yz^8a^8 - 64z^9a^{12}$.
+
+\item $\left(x-10\right)\left(x-1\right)$.
+
+\item $\left(a + b + c + d\right)\left(a + b - c - d\right)$.
+
+\item $\left(2x^2 - 3\right)\left(3x + 2\right)$.
+
+\item $x^4\left(3x^2 - 11\right)^2$.
+
+\item $\left(2c^2 - d^3\right)\left(4c^4 + 2c^2d^3 + d^6\right)$.
+
+\item $\left(1 + 4x\right)\left(1 - 4x + 16x^2\right)$.
+
+\item $\left(a+1\right)\left(a^2 - a + 1\right)
+\left(a - 1\right)\left(a^2 + a + 1\right)$.
+
+\item $\left(x+2\right)\left(x+1\right)\left(x-1\right)$.
+
+\item $\left(x-y\right)\left(x^4 + x^3y + x^2y^2 + xy^3 + y^4\right)$.
+
+\item $\left(9+x\right)\left(3+x\right)$.
+
+\item $m^2 + 2m - 4$.
+
+\item $-3m^2 + 8mn + 3n^2$.
+
+\item $52 - 25x - 52x^2$.
+
+\item $2x^3$.
+
+\item 18; 19; 20.
+
+\item 60.
+
+\item 16.
+\end{enumerate}
+
+\subsection*{Exercise 39.}
+
+
+\begin{enumerate}
+
+\item $3ab^2$.
+
+\item $5xy$.
+
+\item $2xy^3\left(x+y\right)$.
+
+\item $3a^2b\left(a-b\right)$.
+
+\item $m-n$.
+
+\item $9x^4-4$.
+
+\item $x-5$.
+
+\item $x+3$.
+
+\item $a+2$.
+
+\item $y\left(y-1\right)$.
+
+\item $x-y$.
+
+\item $3a\left(c^2-a^2\right)$.
+
+\item $x^2\left(2x^3-y^2\right)$.
+
+\item $\displaystyle \frac{ax+by}{a+b}$ cts.
+
+\item $\displaystyle \frac{5}{100}x$ dols.
+
+\item 5; 11.
+
+\end{enumerate}
+
+
+\subsection*{Exercise 40.}
+
+
+\begin{enumerate}
+\item $\left(a^2-4\right)\left(a+5\right)$.
+
+\item $\left(x^3+1\right)\left(x+1\right)$.
+
+\item $\left(x^2-9\right)\left(x-5\right)$.
+
+\item $\left(x^6-1\right)\left(x^2-3\right)$.
+
+\item $2\left(x^2-1\right)\left(x-3\right)\left(x+2\right)$.
+
+\item $\left(a+1\right)^3\left(m-2\right)$.
+
+\item $\left(a^2-4\right)\left(a^2-25\right)\left(a^2-9\right)$.
+
+\item $\left(x-1\right)^3\left(y+3\right)$.
+
+\item $\left(x^2-1\right)\left(x^2-9\right)\left(x^2-16\right)$.
+
+\item $2x^3\left(x^3+1\right)\left(x-1\right)$.
+
+\item $\left(1-a^4\right)^2$.
+
+\item $6axz\left(a^2 - x^2\right)$.
+
+\item $\left(m^2-n^2\right)\left(a^2-10a+21\right)$.
+
+\item $6bx\left(1-b^3\right)\left(1+b\right)$.
+
+\item $5ab\left(a+x\right)\left(a-x\right)^2\left(a-x-1\right)$.
+
+\item $c + y$ degrees.
+
+\item $b - y$, or $y - b$ degrees.
+
+\item $\frac{2}{x}$.
+
+\item $x^3 + 12$.
+
+\item \$\,306; \$\,1836.
+\end{enumerate}
+
+\subsection*{Exercise 42.}
+
+
+\begin{enumerate}
+\item $\displaystyle \frac{3x}{8z}$.
+
+\item $\displaystyle \frac{2x^4}{5y}$.
+
+\item $\displaystyle \frac{x-3}{x+5}$.
+
+\item $\displaystyle \frac{x+6}{x-5}$.
+
+\item $\displaystyle \frac{a^4}{a^2-1}$.
+
+\item $\displaystyle \frac{x^2}{x^3+1}$.
+
+\item $\displaystyle \frac{m+n}{a+2b}$.
+
+\item $\displaystyle \frac{c+d}{x+2y}$.
+
+\item $\displaystyle \frac{d(c-ad)}{a^2}$.
+
+\item $\displaystyle \frac{y(a+xy)}{x}$.
+
+\item $\displaystyle \frac{a-b}{a+b}$.
+
+\item $\displaystyle \frac{x^2+y^2}{x^2-y^2}$.
+
+ \item $\displaystyle \frac{a(x - 4)}{x + 5}$.
+
+ \item $\displaystyle \frac{a^2 - 17}{a^2 - 5}$.
+
+ \item $\displaystyle \frac{1 + x}{2 + x}$.
+
+ \item $\displaystyle \frac{1 + b}{1 - b}$.
+
+\item $\displaystyle \frac{2}{3}b$, or $\displaystyle
+\frac{2b}{3}$ cts.
+
+\item $\displaystyle \frac{a}{x}$ years.
+\end{enumerate}
+
+\subsection*{Exercise 43.}
+
+\begin{enumerate}
+\item $\displaystyle b-c+\frac{m}{x}$.
+
+\item $\displaystyle m+a+\frac{x}{n}$.
+
+\item $\displaystyle x+y+\frac{3}{x-y}$.
+
+\item $\displaystyle a^2-ab+b^2-\frac{2}{a+b}$.
+
+\item $\displaystyle 2abc-3b+\frac{c}{a^2b}$.
+
+\item $\displaystyle 3x^2+5xz-\frac{m}{xy^2}$.
+
+\item $\displaystyle x+3+\frac{2x+4}{x^2-x-1}$.
+
+\item $\displaystyle 2a-1+\frac{3a-1}{a^2+a-2}$.
+
+\item $\displaystyle x^2+xy+y^2+\frac{2y^3}{x-y}$.
+
+\item $\displaystyle a^2-ab+b^2-\frac{2b^3}{a+b}$.
+
+\item $\displaystyle 4a^2+2a+1+\frac{1}{2a-1}$.
+
+\item $\displaystyle 9x^2-3x+1-\frac{1}{3x+1}$.
+
+\item $\displaystyle 3x+2-\frac{x-4}{x^2+2x-1}$.
+
+\item $\displaystyle 2a-3+\frac{15a+2}{a^2+3a+2}$.
+
+\item $ 2a^2+4a-2$.
+
+\item $3x^2-2x+1$.
+
+\item $\displaystyle \frac{x^2-y^2-xy}{x-y}$.
+
+\item $\displaystyle \frac{a^2-2ab-b^2}{a+b}$.
+
+\item $\displaystyle \frac{2d^3}{c^2+cd+d^2}$.
+
+\item $\displaystyle \frac{x^3+y^3}{x-y}$.
+
+\item $\displaystyle \frac{4-4x-3x^2}{3x-1}$.
+
+\item $\displaystyle \frac{2a^2+4a-1}{a+3}$.
+
+\item $\displaystyle \frac{2a^4-6a^3+3a^2+2a-5}{2a^2-1}$.
+
+\item $\displaystyle \frac{6x^4+3x^3-7x^2-1}{3x^2+1}$.
+
+\item $\displaystyle \frac{4x-x^2}{x^2-x+2}$.
+
+\item $\displaystyle -\frac{a^2+2a}{a^2-2a+3}$.
+
+\item $\displaystyle -\frac{a^4+2a^2+2a+3}{a^2-a+3}$.
+
+\item $\displaystyle \frac{x^2-2x+1}{x^2+x-1}$.
+
+\setcounter{enumi}{41}
+
+\item $\displaystyle \frac{m}{2x}$ ct.
+
+\item $a-3$, $a-2$, $a-1$, $a$.
+
+\item $2m+2$.
+
+\item $4a-1$.
+\end{enumerate}
+
+
+\subsection*{Exercise 44.}
+\begin{enumerate}
+
+\item $\displaystyle \frac{a}{12}$.
+
+\item $\displaystyle \frac{x}{15}$.
+
+\item $\displaystyle \frac{43 x}{60}$.
+
+\item $\displaystyle \frac{4 a m + 3 b x}{2 b m}$.
+
+\item $\displaystyle \frac{4 x}{15}$.
+
+\item $\displaystyle \frac{m^{4} - 2 m^{2} n^{2} + n^{4}}{m^{2}
+n^{2}}$.
+
+\item $\displaystyle \frac{3 x - m x + 5 m^{2} n}{3 m n}$.
+
+\item $\displaystyle \frac{11 b - x}{9 x}$.
+
+\item 0.
+
+\item $\displaystyle \frac{2 a b}{a^{2} - b^{2}}$.
+
+\item $\displaystyle \frac{b^{2}}{(a - 2 b)(a - b)}$.
+
+\item $\displaystyle \frac{2}{(x + 5)(x + 3)}$.
+
+\item $\displaystyle - \frac{4 y^{2}}{x (x - 2 y)}$.
+
+\item $\displaystyle \frac{2 x^{2} - 4 x + 29}{(x + 2)(x - 3)}$.
+
+\item $\displaystyle \frac{2 a x}{x^{3} - 8 a^{3}}$.
+
+\item $\displaystyle \frac{2 x^{9}}{x^{6} - y^{6}}$.
+
+\item $\displaystyle \frac{m^{2} + 11}{(m - 1)(m + 2)(m + 3)}$.
+
+\item 0.
+
+\item $\displaystyle \frac{2(9 x^{4} + 1)}{9 x^{4} - 1}$.
+
+\item $\displaystyle \frac{2}{(a - 2)(a - 3)(a - 4)}$.
+
+\item 2.
+
+\item $\displaystyle \frac{2}{a + 3}$.
+
+\item $\displaystyle \frac{x + 25}{x^{2} - x - 20}$.
+
+\item 0.
+
+\item $\displaystyle \frac{2}{a}$.
+
+\item $\displaystyle \frac{x^{2} + x z - y z}{(x + z)(y + z)}$.
+
+\item 0.
+
+\item $\displaystyle \frac{x}{y}$.
+
+\item $\displaystyle \frac{a}{2}$.
+
+\item $\displaystyle \frac{7 x}{3}$.
+
+\item $4 x - 21$.
+
+\end{enumerate}
+
+
+\subsection*{Exercise 45.}
+
+\begin{enumerate}
+\item $\displaystyle \frac{2}{x + 2}$.
+
+\item $\displaystyle \frac{a}{a^{2} - 9}$.
+
+\item $m^{2}$.
+
+\item $\displaystyle \frac{1}{2 a + 1}$.
+
+\item $\displaystyle \frac{3 y}{x^{2} + x y + y^{2}}$.
+
+\item $\displaystyle - \frac{1}{3(a^{2} - 1)}$.
+
+\item $\displaystyle - \frac{b}{5(y^{4} - 9 b^{2})}$.
+
+\item $\displaystyle \frac{1}{(1 - x)(3 - x)}$.
+
+\item $\displaystyle \frac{b}{(b - c)(a - b)}$.
+
+\item 0.
+
+\item 1.
+
+\item $\displaystyle \frac{2 a^{2}}{a^{2} - 1}$.
+
+\item $\displaystyle \frac{1 + a - a^{4}}{1 + a}$.
+
+\item $\displaystyle \frac{1}{(x - 2)(x - 3)}$.
+
+\item $\displaystyle \frac{43 a}{b}$.
+
+\item $\displaystyle \frac{9 x}{4}$.
+
+\item $\displaystyle \frac{m x}{5}$.
+
+\item $\displaystyle \frac{b y}{a}$.
+\end{enumerate}
+
+\subsection*{Exercise 46.}
+
+\begin{enumerate}
+\item $\displaystyle \frac{ax}{b^2}$. \item $\displaystyle
+\frac{3a^2 bc}{7x^2 y}$. \item $\displaystyle \frac{a^2}{2c^2}$.
+\item $\displaystyle \frac{4abc^2}{3x}$. \item $\displaystyle
+\frac{9mn}{y}$. \item $\displaystyle \frac{3x(x-y)}{2mn}$. \item
+$\displaystyle \frac{2a-b}{3x-y}$. \item $\displaystyle
+\frac{x^2}{x^2+xy+y^2}$. \item $3\left(x+y\right)^2$. \item
+$\displaystyle \frac{a^2-4a-21}{a+2}$. \item $x^2-2xy+y^2$. \item
+$a^2+2ab+b^2$. \item $x$. \item $c$. \item $10$. \item $5x$. \item
+$24a$.
+\end{enumerate}
+
+\subsection*{Exercise 47.}
+
+\begin{enumerate}
+\item $\displaystyle \frac{a}{b}$.
+
+\item $\displaystyle \frac{3m^2 n}{4x^2 y}$.
+
+\item $\displaystyle \frac{3d^2}{5x}$.
+
+\item $\displaystyle \frac{2x^2 y}{3m^2 n}$.
+
+\item $\displaystyle \frac{a}{3x(c+d)}$.
+
+\item $\displaystyle \frac{x^2-y}{2m^2n}$.
+
+\item $\displaystyle \frac{a-b^3}{5ax^2yz}$.
+
+\item $\displaystyle \frac{5(a+b)}{xy}$.
+
+\item $\displaystyle \frac{1}{2m\left(m-n\right)^2}$.
+
+\item $\displaystyle \frac{x+7}{x^2-7x+12}$.
+
+\item $\displaystyle x+1+\frac{1}{3x}$.
+
+\item $\displaystyle \frac{x^2-5x+6}{x+4}$.
+
+\item $\displaystyle \frac{c}{12a^2 x^3}$.
+
+\item $\displaystyle 2ac+\frac{3a^2 c^2}{2x^2 y}$.
+
+\item $\displaystyle \frac{5y(x-y)}{8x^2 z}$.
+
+\item $\displaystyle \frac{2x}{9amn^2}$.
+
+\item $\displaystyle \frac{29}{35}$.
+
+\item $\displaystyle \frac{1}{2m}$.
+\end{enumerate}
+
+\subsection*{Exercise 48.}
+
+\begin{enumerate}
+\item $\displaystyle \frac{abz}{8cxy}$. \item $ \frac{2}{3}$.
+\item $2$. \item $3$. \item $\displaystyle \frac{cd}{ab}$. \item
+$\displaystyle \frac{2y}{an}$. \item $\displaystyle
+\frac{x+4}{x+1}$. \item $a+5$. \item $\displaystyle
+\frac{(x^2+4x+16)(3x+2)}{x}$. \item $m^2+3m+9$. \item $a$. \item
+$\displaystyle \frac{\left(x-1\right)^2}{2x(x+3)}$. \item
+$\displaystyle \frac{\left(a+1\right)^2}{3a(a+2)}$. \item
+$\displaystyle \frac{m^2 n^2}{m^2-n^2}$. \item $\displaystyle
+\frac{25}{y}$; $\displaystyle \frac{25}{y-4}$ dols. \item
+$\displaystyle \frac{2a}{3c}$. \item $252$; $224$.
+\end{enumerate}
+
+\subsection*{Exercise 49.}
+
+\begin{enumerate}
+\item $\displaystyle \frac{6a^2bc}{7xy^2}$. \item $\displaystyle
+\frac{21ab^2y^2}{16mnz^3}$. \item $\displaystyle
+\frac{16a^2z^2}{9c^2x^2}$. \item $\displaystyle
+\frac{4m^2y^2}{9a^2x^2}$. \item $\displaystyle
+\frac{x^2-25}{x^2+2x}$. \item $\displaystyle
+\frac{(x-5)(x-6)}{(x-3)(x-3)}$. \item $\displaystyle
+\frac{x-2}{x-5}$. \item $\displaystyle \frac{a(a-7)}{a+6}$. \item
+$1$. \item $1$. \item $1$. \item $\displaystyle \frac{x-5}{x+5}$.
+\item $\displaystyle \frac{a}{b}$. \item $\displaystyle
+\frac{m}{2n}$. \item $a^2$. \item $\displaystyle \frac{1}{4x}$.
+\item $3$. \item $4$. \item $\displaystyle \frac{dx}{4c}$. \item
+$\displaystyle \frac{21x^4y^2}{20abn^2}$.
+\end{enumerate}
+
+\subsection*{Exercise 50.}
+
+\begin{enumerate}
+\item $\displaystyle \frac{4x^4}{a^2b^6}$.
+
+\item $\displaystyle \frac{a^3b^9}{64y^3}$
+
+\item $\displaystyle -\frac{32x^5y^5}{243a^{10}m^{15}}$.
+
+\item $\displaystyle \frac{x^4 \left(a-b \right)^4}{81a^8b^4}$.
+
+\item $\displaystyle -\frac{125x^5y^3\left(a+b^2\right)^6}{8a^3b^9
+\left(x^2-y \right)^9}$.
+
+\item $\displaystyle -\frac{27a^3m^{12} \left( 2a+3b \right)
+^9}{64x^6y^9 \left( m-n \right) ^6}$.
+
+\item $\displaystyle
+\frac{81a^{20}x^4y^8z^{12}}{16b^4c^{16}d^{16}}$.
+
+\item $\displaystyle
+\frac{256x^{28}y^8z^{12}}{81a^4b^{24}d^{12}}$.
+
+\item $\displaystyle \frac{2ab^2}{3m^2n^3}$.
+
+\item $\displaystyle \frac{6mn^4}{11a^2b^3}$.
+
+\item $\displaystyle \frac{4xy^2}{3ab^3}$.
+
+\item $\displaystyle -\frac{2xy^2}{3m^2n^3}$.
+
+\item $\displaystyle -\frac{3x^2\left(a-b\right)^3}{4a^2b^4}$.
+
+\item $\displaystyle \frac{2x+3y}{4x^2y^4}$. \item $\displaystyle
+\frac{2x\left(x+y\right)^2}{5y^2}$. \item $\displaystyle
+-\frac{3a \left(a-b \right)^2}{2b^3}$. \item $\displaystyle
+\frac{x^2}{y^2}-\frac{a^2}{b^2}$. \item $\displaystyle
+\frac{a^2}{b^2}-\frac{c^2}{d^2}$. \item
+$6x+\frac{3abx^2y}{2mn}-\frac{a^2x}{m}$. \item $\displaystyle
+\frac{2x^3}{3yz}-\frac{3xy}{2}+\frac{x^4}{m^2z^2}$. \item
+$\displaystyle \frac{a^8}{b^8}-\frac{c^8}{d^8}$. \item
+$\displaystyle \frac{x^4}{256} - \frac{81y^4}{625m^4}$. \item
+$\displaystyle \frac{x^3}{y^3}+1$. \item $\displaystyle
+\frac{a^3}{b^3}-1$. \item $\displaystyle
+\frac{x^2}{y^2}-\frac{3a^2x}{by}+\frac{9a^4}{4b^2}$. \item
+$\displaystyle \frac{4y^2}{c^4}+\frac{18y}{c^2d}+\frac{81}{4d^2}$.
+\item $\displaystyle \frac{a^2}{b^2}-\frac{3a}{b}-10$.
+
+\item $\displaystyle \frac{x^3}{y^3}-\frac{3ax^2}{by^2}+
+ \frac{3a^2x}{b^2y}-\frac{a^3}{b^3}$.
+
+\item $\displaystyle
+\left(\frac{x^2}{a}-\frac{b}{y^2}\right)\left(\frac{x^2}{a}+
+ \frac{b}{y^2}\right)$.
+
+\item $\displaystyle
+\left(\frac{a}{b^2}-\frac{x^4}{y}\right)\left(\frac{a}{b^2}+
+ \frac{x^4}{y}\right)$
+
+\item $\displaystyle \left(\frac{a^2}{m^2}+\frac{b^2}{x^2}\right)
+ \left(\frac{a}{m}+\frac{b}{x}\right)
+ \left(\frac{a}{m}-\frac{b}{x}\right)$.
+
+\item $\displaystyle \left(\frac{2a}{y}+\frac{b}{c}\right)
+ \left(\frac{4a^2}{y^2}-\frac{2ab}{cy}+\frac{b^2}{c^2}\right)$.
+
+\item $\displaystyle
+\left(\frac{x}{3a}-\frac{y}{c}\right)\left(\frac{x^2}{9a^2}+
+ \frac{xy}{3ac}+\frac{y^2}{c^2}\right)$.
+
+\item $\displaystyle
+\left(\frac{9a^2}{b^2}+\frac{x^2y^2}{25z^2}\right)
+ \left(\frac{3a}{b}+\frac{xy}{5z}\right)
+ \left(\frac{3a}{b}-\frac{xy}{5z}\right)$.
+
+\item $\displaystyle
+\left(\frac{m}{y}-5\right)\left(\frac{m}{y}+3\right)$.
+
+\item $\displaystyle
+\left(\frac{x}{a}-4\right)\left(\frac{x}{a}+2\right)$.
+
+\item $\displaystyle \left(\frac{y}{2}+\frac{2}{y}\right)^2$.
+
+\item $\displaystyle \left(\frac{x}{a}-\frac{a}{x}\right)^2$.
+
+\item $18a$ ft.
+
+\item $\displaystyle \frac{a}{6}$ weeks.
+\end{enumerate}
+
+\subsection*{Exercise 51}
+\begin{enumerate}
+
+\item $\displaystyle \frac{b+a}{x-b}$.
+
+\item $\displaystyle \frac{xy+a}{my-b}$.
+
+\item $\displaystyle \frac{cx}{ac+b}$.
+
+\item $\displaystyle \frac{x^2+x+1}{x}$.
+
+\item $\displaystyle \frac{a+b}{ab}$.
+
+\item $m(m^2-m+1)$.
+
+\item $\displaystyle \frac{x^2-yz}{ax-by}$.
+
+\item $\displaystyle -\frac{d}{c}$.
+
+\item $\displaystyle \frac{a+1}{a-1}$.
+
+\item $a^2$.
+
+\item $\displaystyle \frac{x-5}{x-3}$.
+
+\item $a$.
+
+\item $\displaystyle \frac{a^2+1}{2a}$.
+
+\item $1$.
+
+\item $\displaystyle \frac{2}{a}$.
+
+\item $l$.
+
+\item $5y$ lbs.
+
+\item $12y$ in.; $\displaystyle \frac{y}{3}$ yds.
+
+\item $100y-x^2$ cts.
+
+\item $\displaystyle \frac{lx}{m}$.
+\end{enumerate}
+
+\subsection*{Exercise 52.} \begin{enumerate}
+
+\item $3x^4-2x^3-x+5$.
+
+\item $45$; $55$.
+
+\item $x+1$.
+
+\item $(x^5-7y)^2$, $(a-b)(a^2+ab+b^2)(a^6+a^3b^3+b^6)$,
+ $3(a-9)(a+8)$.
+
+\item $2$.
+
+\item $\displaystyle
+\left(\frac{4x^2}{a^2b^2}+\frac{c^2}{9y^2}\right)
+ \left(\frac{2x}{ab}+\frac{c}{3y}\right)
+ \left(\frac{2x}{ab}-\frac{c}{3y}\right)$.
+
+\item $10x+y$.
+
+\item $(a^2-16)(a^2-9)(a^2-4)$.
+
+\item $\displaystyle \frac{100x}{a}$.
+
+\item $\displaystyle -\frac{6}{x-4}$.
+
+\item $0$.
+
+\item $\displaystyle \frac{3x}{2y}$.
+
+\item $\displaystyle \frac{2b}{n}$.
+
+\item $1$.
+
+\item $1$.
+
+\setcounter{enumi}{16}
+
+\item $42xy$.
+
+\item $\displaystyle \frac{2x^3}{3} - \frac{x^2}{4} + x - 1$.
+\end{enumerate}
+
+\subsection*{Exercise 53.}
+
+\begin{enumerate}
+
+\item $x = 2$.
+
+\item $x = -3$.
+
+\item $x = 19$.
+
+\item $x = -13$.
+
+\item $x = 2\frac{1}{2}$.
+
+\item $x = 5\frac{1}{4}$.
+
+\item $x = 4$.
+
+\item $x = 6\frac{1}{5}$.
+
+\item $x = 1$.
+
+\item $x = 3$.
+
+\item $x = 2$.
+
+\item $x = 1$.
+
+\item $x = 10\frac{3}{7}$.
+
+\item $x = 2$.
+
+\item $x = 1$.
+
+\item $x = 3$.
+
+\item $x = 4$.
+
+\item $x = 3$.
+
+\item $x = -3$.
+
+\item $x = 1$.
+
+\item $x =\frac{6}{7}$.
+
+\item $x = \frac{3}{4}$.
+
+\item $17$; $22$; $66$.
+
+\item $\$\,15$.
+
+\item $28x$ fourths.
+
+\item $\displaystyle \frac{3x}{z}$ days.
+
+\item $x = 5\frac{1}{2}$.
+
+\item $x = -15$.
+
+\item $x = 48$.
+
+\item $x = 4$.
+
+\item $x = -5$.
+
+\item $x = -\frac{1}{9}$.
+
+\item $x = \frac{1}{7}$.
+
+\item $x = 9$.
+
+\item $x = 4$.
+
+\item $x = 3$.
+
+\item $x = 1$.
+
+\item $x = 2$.
+
+\item $x = 2$.
+
+\item $x =11$.
+
+\item $x = 6$.
+
+\item $x = 8$.
+
+\item $\displaystyle \frac{bx}{15}$ hrs.
+
+\item $\displaystyle \frac{5m}{100}$ dols.
+
+\item $16$; $41$.
+
+\item $\displaystyle \frac{4by}{100}$ dols.
+
+\item $\displaystyle x = \frac{c^2}{ a - b + 2c }$.
+
+\item $x = \frac{1}{2}$.
+
+\item $x = 3$.
+
+\item $\displaystyle x = \frac{2b - a}{2}$.
+
+\item $\displaystyle x = \frac{ac(a^2 + c^2)}{a + c}$.
+
+\item $x = 0$.
+
+\item $x = 2$.
+
+\item $x = a + b$.
+
+\item $x = \frac{2}{3}$.
+
+\item $x = \frac{1}{4}$.
+
+\item $x = 1$.
+
+\item $x = \frac{5}{11}$.
+
+\item $x = 2$.
+
+\item $x = 2$.
+
+\item $x = 4$.
+
+\item $x = 6$.
+
+\item $x = 1$.
+
+\item $x = \frac{7}{17}$.
+
+\item $x = 4$.
+
+\item $x = -7$.
+
+\item $26$; $27$; $28$.
+
+\item $6y$ years.
+
+\item $\displaystyle \frac{6x}{a}$ men.
+
+\item $12x + 3$.
+\end{enumerate}
+
+\subsection*{Exercise 54.}
+
+\begin{enumerate}
+
+\item $209$ boys; $627$ girls.
+
+\item $184$; $46$; $23$.
+
+\item $16$; $28$ yrs
+
+\item $36$; $122$.
+
+\item S., $5$; H., $11$ qts.
+
+\item $17$; $22$; $88$.
+
+\item $9$.
+
+\item $9$ fives; $18$ twos.
+
+\item $18$; $15$; $25$ tons.
+
+\item $15$.
+
+\item $12$ M.; $36$ J.
+
+\item $80$.
+
+\item $24$.
+
+\item $48$.
+
+\item t., 5 lbs.; m., 8 lbs.; s., 14 lbs.
+
+\item 35; 10 yrs.
+
+\item \$2070, \$920.
+
+\item 17,042; 14,981; 15,496.
+
+\item R., 213; J., 426.
+
+\item 63; 64; 65.
+
+\item 16; 28.
+
+\item 45; 75.
+
+\item 24; 48 yrs.
+
+\item 2; 12 yrs.
+
+\item J., 12 yrs.; S., 28 yrs.
+
+\item 20 yrs.
+
+\item A., 6 yrs.; G., 18 yrs.
+
+\item Ed., 40 yrs.; Es., 30 yrs.
+
+\item 6 yrs.
+
+\item 3; 12 yrs.
+
+\item $2 x^3 - 6 x^2y + 18xy^2 - 27 y^3$.
+
+\setcounter{enumi}{32}
+
+\item $\displaystyle \frac{y}{x - y}$.
+
+\item $x = 7$.
+
+\item \$32.
+
+\item \$13.
+
+\item A, \$47; B, \$28; C, \$61.
+
+\item 80 cts.
+
+\item u., \$18; cl., \$25; h., \$9.
+
+\item J., \$ 1.80; H., \$2.42; A., \$1.25
+
+\item $2\frac{2}{5}$ days.
+
+\item $5\frac{5}{11}$ days.
+
+\item $22\frac{1}{2}$ hrs.
+
+\item $1\frac{1}{3}$ hrs.
+
+\item $4\frac{4}{5}$ hrs.
+
+\item $2\frac{11}{12}$ hrs.
+
+\item $2\frac{6}{7}$ hrs.
+
+\item $17\frac{1}{7}$; $7\frac{1}{17}$ days.
+
+\item $1\frac{5}{7}$ hrs.
+
+\item $\displaystyle \frac{ab}{a + b}$ days.
+
+\item $\displaystyle \frac{cd}{c - d}$ days.
+
+\item $x = 1\frac{1}{2}$.
+
+\item 1.
+
+\item $x^8 - 13x^4 + 36$.
+
+\item $(3a + 2b)^2$; $(3 + x)(4 + x)$; $(a - 2b)(c - 3d)$.
+
+\item \begin{tabular}{ll}(1)& $10\frac{10}{11}$ m. \\
+ (2)& $27\frac{3}{11}$ m. \\
+ (3) &$49\frac{1}{11}$ m.
+ \end{tabular}
+
+\item \begin{tabular}{lll} (1) &$27\frac{3}{11}$ m. \\
+ (2)& $5\frac{5}{11}$ m.; &$38\frac{2}{11}$ m. \\
+ (3)& $21\frac{9}{11}$ m.; &$54\frac{6}{11}$ m.
+ \end{tabular}
+
+\item \begin{tabular}{ll}(1)& $49\frac{1}{11}$ m. \\
+ (2) & $10\frac{10}{11}$ m. \\
+ (3) & $32\frac{8}{11}$ m.
+ \end{tabular}
+
+\item $10\frac{10}{11}$ m., $32\frac{8}{11}$ m.
+
+\item $16\frac{4}{11}$ m.
+
+\item $30\frac{6}{11}$ m.
+
+\item $5\frac{5}{11}$ m. past 12M.
+
+\item $57\frac{9}{11}$ m.
+
+\item 20 hrs.
+
+\item 24 hrs.; 152 hrs
+
+\item 35 miles.
+
+\item $14\frac{1}{2}$ miles; $70\frac{11}{16}$ miles.
+
+\item 24 miles.
+
+\item 6 P.M.; 30 miles from B; $25\frac{1}{2}$ miles from A.
+
+\item 16 sec.
+
+\item 11 ft. by 15 ft.
+
+\item 12 ft. by 24 ft.
+
+\item 14 ft. by 21 ft.
+
+\item 16 sq. ft.
+
+\item 48 ft. by 72 ft.
+
+\item
+\begin{tabular}{l}
+ $\displaystyle \left( \frac{x}{2} + \frac{yz^2}{3m^3} \right)
+ \left( \frac{x}{2} - \frac{yz^2}{3m^3} \right) $; \\
+ $(x^2 - 3y)(x^4 + 3x^2y + 9 y^2) $; \\
+ $(a^8 + b^8)(a^4 + b^4)$ etc.; \\
+ $2c(c^2 - 1)^2$.
+\end{tabular}
+
+\item $x^3 -2x^2 + 4x -3$.
+
+\item $7y^3 + 15x^2y - x^3$.
+
+\item $x(x + 1)$.
+
+\item 60 ft.
+
+\item A, 7 days; B, 14 days; C, 28 days.
+
+\item 9; 48.
+
+\item 18; 74.
+
+\item \$140.
+
+\item 24; 25.
+
+\item A, 57 yrs.; B 33 yrs.
+
+\item 38.
+
+\item 300 leaps.
+
+\item With 144 leaps.
+
+\item After 560 leaps.
+\end{enumerate}
+
+\subsection*{Exercise 55.}
+
+\begin{enumerate}
+\item $x=3$, $y=1$.
+
+\item $x=4$, $y=2$.
+
+\item $x=7$, $y=6$.
+
+\item $x=5\frac{1}{2}$, $y=4$.
+
+\item $x=1\frac{1}{2}$, $y=3$.
+
+\item $x=2$, $y=3$.
+
+\item $x=5$, $y=-2$.
+
+\item $x=2$, $y=-1$.
+
+\item $x=-2\frac{1}{4}$, $y=-\frac{1}{8}$.
+
+\item $x=2\frac{1}{2}$, $y=-1\frac{1}{2}$.
+
+\item $x=\frac{1}{2}$, $y=1\frac{1}{3}$.
+
+\item $x=15$, $y=-17$.
+
+\item $x=12$, $y=15$.
+
+\item $x=6$, $y=18$.
+
+\item $x=35$, $y=-49$.
+
+\item $x=21$, $y=25$.
+
+\item $x=3$, $y=2$.
+
+\item $x=\frac{1}{2}$, $y=4$.
+
+\item $x=12$, $y=15$.
+
+\item $x=\frac{a+b}{2}$, $y=\frac{a-b}{2}$.
+
+\item $x=39$, $y=-56$.
+
+\item $x=-2$, $y=-1\frac{17}{21}$.
+
+\item $ \frac{7}{24}$.
+
+\item $ \frac{11}{17}$.
+
+\item $\frac{8}{13}$.
+
+\item $\frac{11}{13}$.
+
+\item $24$; $62$.
+
+\item $27$; $63$.
+
+\item $13$; $37$.
+
+\item J., $13$ yrs.; H., $19$ yrs.
+
+\item A, $36$ cows; B, $24$ cows.
+
+\item tea, $54$; coffee, $32$.
+
+\item corn, $61$; oats, $37$.
+
+\item $12$ lbs of $87$ kind; $26$ lbs of $29$ kind.
+\end{enumerate}
+
+\subsection*{Exercise 56.}
+
+\begin{enumerate}
+
+\item $x=\pm 3$.
+
+\item $x=\pm 2$.
+
+\item $x=\pm 5$.
+
+\item $x=\pm 2$.
+
+\item $x=\pm \frac{1}{2}$.
+
+\item $x=\pm 1$.
+
+\item $x=\pm 7$.
+
+\item $x=\pm 2$.
+
+\item $x=\pm 1$.
+
+\item $x=\pm 2$.
+
+\item $x=\pm 2$.
+
+\item $x=\pm 3$.
+
+\item $12$ yrs.
+
+\item $48$; $64$.
+
+\item $\frac{17}{21}$
+\end{enumerate}
+
+\subsection*{Exercise 57.}
+
+
+\begin{enumerate}
+\item $x = 3 \text{ or } -6$.
+
+\item $x = 2 \text{ or } -7$.
+
+\item $x = 9 \text{ or } -8$.
+
+\item $x = 7 \text{ or } 3$.
+
+\item $x = 8 \text{ or } 15$.
+
+\item $x = 11 \text{ or } -17$.
+
+\item $x = b \text{ or } b$.
+
+\item $x = a \text{ or } 3a$.
+
+\item $x = 1 \text{ or } -a$.
+
+\item $x = \frac{d}{c} \text{ or } -\frac{b}{a}$.
+
+\item $x = 11 \text{ or } -3$.
+
+\item $x = 6 \text{ or } 1$.
+
+\item $x = 5 \text{ or } 2$.
+
+\item $x = 6 \text{ or } 4$.
+
+\item $x = 6 \text{ or } 16$.
+
+\item $x = 2 \text{ or } -3$.
+
+\item $x = 5 \text{ or } -2$.
+
+\item $x = 0 \text{ or } 5$.
+
+\item $x = 0 \text{ or } -3$.
+
+\item $x = 3 \text{ or } 3$.
+
+\item $54\frac{6}{11}$ m.
+
+\item J., \$18; L., \$6.
+
+\item $3$ in.
+
+\item $7\frac{31}{47}$ days.
+\end{enumerate}
+
+\subsection*{Exercise 58.}
+
+
+\begin{enumerate}
+\item $3$. \setcounter{enumi}{2} \item $x=3$. \item $30$ hrs.
+\item $x-1$. \item $a^2+ab-b^2$. \item $1-y+3y^2+2y^3$. \item
+$60-x^2-28x$. \item $x=\pm 6$. \item $1$. \item
+$11(a-c)-17x^3y-3\left(a-c\right)^3$. \item $x=\frac{4b}{5}$.
+\item $(x^3+3)(x-1)(x^2+x+1)$; $(a+b)(x-y)(x+y)$;
+$(3x+y+z)\{9x^2-3x(y+z)+\left(y+z\right)^2\}$. \item
+$\frac{5}{11}$. \item $a+b$. \item $x=5$. \item J., $16$; S., $8$.
+\item $\displaystyle \frac{1-y}{x}$. \item $a^2-3b^2+3c^2$. \item
+$\displaystyle \frac{xy}{a}$. \item $a+3x+4y+2b$. \item
+$a^2-\frac{3}{4}a+1$. \item $1\frac{7}{8}$ days. \item
+$\displaystyle \frac{2a^2}{a^3+b^3}$. \item $(x-13)(x+4)$;
+$(1+a^8)(1+a^4)(1+a^2)$ etc.; $\left(a^2+b^2+a^2-b^2\right)^2$; or
+$\left(2a^2\right)^2$. \item $x=36$. \item $60$; $40$ yrs. \item
+$x=3$, $y=12$. \item $3x^4-2x^3+x-5$. \item $2.35$, $3.2$. \item
+$3\frac{7}{16}$. \item $x=0$. \item $25$; $26$; $27$. \item
+$a(a+2)$. \item $\displaystyle \frac{x^2+y^2}{x}$.
+
+\item $ \frac{3}{7}$.
+
+\item $\displaystyle
+-\frac{27a^6b^3\left(m+n\right)^6}{64x^3y^9\left(a-b\right)^9}$;
+$\displaystyle \pm \frac{5x^2\left(a+b\right)^3}{4y^3z}$.
+
+\item $3x^2-2x+1$. \item $\frac{1}{2}$. \item $x=-\frac{2}{3}$,
+$y=-\frac{3}{4}$. \item $(x^2+3)(x^2+2)$; $\left(x-7\right)^2$;
+$(x+y+z)(x-y-z)$. \item $\frac{1}{9}x^2y^2+xy$. \item
+$31\frac{7}{11}$ m.; $44\frac{8}{11}$ m. \item $1$.
+
+\item $x=b$.
+
+\item $\displaystyle\left( \frac{x^3}{y^3} + \frac{ab^2}{c}
+\right) \left( \frac{x^3}{y^3} - \frac{ab^2}{c} \right)$;
+$\displaystyle \left( \frac{x}{y} - 7 \right) \left( \frac{x}{y} +
+2 \right)$; $\displaystyle \left( \frac{x}{y} - \frac{y}{x}
+\right)^2$.
+
+\item $4\frac{4}{5}$ days. \item $x=18$, $y=6$. \item
+$1+6x+12x^2+8x^3$;
+$16x^8-96x^6a^2b^3+216x^4a^4b^6-216x^2a^5b^9+81a^8b^{12}$. \item
+$\displaystyle \frac{a^2+b^2}{a^2-b^2}$. \item $y-x$. \item $0$.
+\item $400$ sq. ft. \item $2(a^2-1)(a-3)(a-2)$.
+
+\item $\displaystyle -\frac{2a-b}{2(2a+b)}$.
+
+\item $x=4$ or $4$. \item $ab(a+2cm^2)(a^2-2acm^2+c^2m^4)$;
+$c\left(2cx+y\right)^2$; $(x+1)(x^2-x+1)(x-1)(x^2+x+1)$. \item
+$1-x-\frac{5}{12}x^2+\frac{1}{3}x^3+\frac{1}{9}x^4-\frac{1}{16}x^5$.
+\item $x^2+x+1$. \item
+$x^2-3xy+y^2+\frac{9xy^3-6y^4}{x^2+2y^2-3xy}$. \item
+$a^2+\frac{1}{2}a^3$.
+
+\item $\displaystyle x=\frac{2ab}{a+b}$.
+
+\item $(x+5)(x+5)(x^2+3)$; $(4+x^4)(2+x^2)(2-x^2)$; $(2a)(2b)$.
+
+\item $3\frac{3}{4}$ days.
+
+\item $x=5$.
+
+\item $\displaystyle
+\frac{a^3}{b^3}-\frac{3a^2c}{b^2d}+\frac{3ac^2}{bd^2}-\frac{c^3}{d^3}
+$; $\displaystyle \frac{c^3}{d^3}+1$. \item $24$ or $-3$. \item
+$x=15$, $y=14$. \item $m-my+my^2-my^3+$ etc. \item $\displaystyle
+\frac{5x}{3}$. \item $4$ ft. \item
+$10a^2+19ax-19ay+9x^2+18xy+9y^2$. \item
+$\frac{3}{2}x^2-x+\frac{2}{3}$. \item $5x^4+4x^3+3x^2+2x+1$. \item
+$(a+2)(a-2)(a+5) $; $(x+y)(x^2-xy+y^2)(x-y)(x^2+xy+y^2) $;
+$2x(x^2-3y^2)(x+y)(x-y)$. \item $8$ hrs. \item $x$. \item $x=2$,
+$y=3$. \item $3xy-7b^2$. \item $6x-23$.
+\end{enumerate}
+
+
+\newpage
+\small
+\pagenumbering{gobble}
+\begin{verbatim}
+
+End of Project Gutenberg's A First Book in Algebra, by Wallace C. Boyden
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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..d8ff5d2 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #13309 (https://www.gutenberg.org/ebooks/13309) |