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 SAN FRANCISCO. 
 
Digitized by the Internet Archive 
 
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ELEMENTS OF ARITHMETIC 
 
 FOE PKIMARY AND INTERMEDIATE CLASSES IN 
 PUBLIC AND PRIVATE SCHOOLS 
 
 BY 
 
 WILLIAM J. MILNE, Ph.D., LL.D. 
 
 President of New York State Normal College, Albany, N Y. 
 
 or xMx 
 
 UNIVERSITY 
 
 NEW YORK .;. CINCINNATI •:• CHICAGO 
 
 AMERICAN BOOK COMPANY 
 
MILNE'S MATHEMATICS 
 
 Milne's Elements of Arithmetic 
 
 Milne's Standard Arithmetic 
 
 Milne's Mental Arithmetic 
 
 Milne's Elements of Algebra 
 
 Milne's Grammar School Algebra 
 
 Milne's High School Algebra 
 Milne's Plane and Solid Geometry 
 Milne's Plane Geometry — Separate 
 
 Copyright, 1893, by American Book Company 
 
 Elmts. Ar. 
 
 E-P68 
 
PREFACE. 
 
 THIS work has been prepared to meet a demand for a book 
 that contains, within brief compass, an elementary course in 
 arithmetic. It is also designed to be an introduction to the more 
 thorough and exhaustive treatment of the science which is given in 
 the author's Standard Arithmetic. 
 
 The first part of the book contains exercises arranged in such a 
 manner that they secure an easy and natural development of the 
 ideas of numbers, and they afford sufficient practice to fix the 
 results in memory. Ready knowledge of results can be secured 
 only by frequent repetition and thorough drill, and the lessons that 
 have been prepared are believed to be admirably adapted to produce 
 expertness in computing, without sacrificing interest in the subject 
 through constant practice with abstract numbers. 
 
 When the more systematic treatment of the science is presented, 
 the pupil is led by natural, progressive, and logical steps to an 
 understanding of the definitions, principles, processes, and rules, 
 before he is required to state them ; consequently, all definitions, 
 principles, and rules are but the expressions of what he already 
 knows. It is evident, therefore, that the plan pursued in the work 
 will develop in the student the habit of investigating for himself 
 any subject which may claim his attention, and this is an extremely 
 important part of proper teaching. 
 
 The number of oral examples is large enough to supply all the 
 necessary training in " mental " arithmetic, and the frequent exer- 
 cises for review will fix in the memory all the knowledge of the 
 science that has been acquired. The treatment of the subjects has 
 been adapted to the comprehension of young pupils, and subjects 
 which are too difficult for them have been omitted ; and yet in no 
 instance has accuracy of statement or correctness of process been 
 sacrificed in what has been presented. 
 
 85i2o 3 
 
4 PREFACE. 
 
 Though the treatment of the subjects is brief and elementary, a 
 student who goes no farther in his arithmetical studies than this 
 work will take him, will have a knowledge of the science, sufficient 
 to enable him to perform, with accuracy and intelligence, most of 
 the processes required in ordinary business life. 
 
 The author desires to express his indebtedness to George I. 
 Aldrich, A.M., Superintendent of Schools of Newtonville, Mass., for 
 valuable aid in the preparation of the earlier parts of the book, and 
 for important suggestions upon the treatment of subjects through- 
 out the work. His reputation as an originator of rational and 
 scholarly methods of teaching arithmetic is a guarantee of the 
 value of his services. 
 
 "The Elements of Arithmetic " is submitted to the public with 
 the hope that it may prove to merit general favor on account of the 
 natural and rational development of the subject, and because of 
 its peculiar adaptation to the class of students for whose use it 
 was written. 
 
 WILLIAM J. MILNE. 
 
 State Normal College, 
 Albany, N.Y. 
 
SUGGESTIONS TO TEACHERS. 
 
 1. It must be borne iu mind that the pupils should first 
 be made familiar with the combinations of numbers^ before 
 any explanations or analyses are required. 
 
 2. The first part of this book is designed to give pupils 
 the knowledge of numbers; and it is suggested, therefore^ 
 that the teacher ask for results only. 
 
 3. If explanations of processes or analyses of problems 
 are required, they will divert the attention of the pupils 
 from the results, and thus hinder their progress. 
 
 4. The lessons which are given are designed to be typical 
 rather than complete, and the thoughtful teacher will devise 
 additional problems of a character similar to those given in 
 the book. 
 
 5. By going over the same lessons frequently the pupil 
 will become familiar with numbers ; but there is only one 
 way in which thorough knowledge of numbers can be ac- 
 quired, and that is by repetition of the combinations within 
 brief intervals of time. 
 
 6. The examples should at first be given in connection 
 with objects which can be conveniently obtained. From 
 these the transition is easy to concrete examples, in which 
 the splints, or counters, or buttons, or beans, stand for other 
 
 5 
 
6 , SUGGESTIONS TO TEACHERS. 
 
 units ; but too early drill upon abstract numbers is unfavor- 
 able to the develoioment of a proper comprehension of 
 arithmetic, and makes the pupils mechanical rather than 
 thoughtful. 
 
 7. It will be found advantageous to encourage the pupils 
 to form examples for themselves. Such a plan gives a 
 pleasing variety to the work of the class, and makes the 
 pupils thoughtful regarding the work. Judicious use of 
 this suggestion will aid in inspiring interest in the work. 
 
 8. The work upon the slate, or written work, should 
 come only after the oral work, and it should be given only 
 for purposes of review after the numbers have been thor- 
 oughly understood. 
 
 9. Care should be taken with the written work, that the 
 children make the figures in accordance with proper models, 
 and that everything is done neatly. 
 
 10. Be sure to require the students to answer all the 
 development exercises which precede every subject. These 
 exercises will cause the pupils to apprehend the principles 
 that are employed in the solution of the problems that 
 follow, and they will discourage the tendency on the part 
 of the children to give mechanical or formal solutions. 
 
 11. Do not fail with every lesson to review the work of 
 previous days, for it is only in this way that pupils become 
 familiar with results and processes, and success in arith- 
 metical processes is dependent to a great degree upon con- 
 stant drill. 
 
CONTENTS. 
 
 PAGE 
 
 Exercises on Numbers 1 to 100 9 
 
 Notation and Numeration o . . . 78 
 
 Notation and Numeration of United States Money .... 85 
 
 Roman Notation 87 
 
 Addition 89 
 
 Subtraction 95 
 
 Review 99 
 
 Multiplication 102 
 
 Review 114 
 
 Division 119 
 
 Review 130 
 
 Factoring 138 
 
 Tests of Divisibility 140 
 
 Cancellation . 141 
 
 Fractions o . . 145 
 
 Reduction o » . 151 
 
 Addition . , , ...... 156 
 
 Subtraction 159 
 
 Multiplication « 162 
 
 Division ........ 109 
 
 Aliquot Parts 179 
 
 Review 182 
 
 7 
 
8 
 
 CONTENTS. 
 
 Decimal Fractions o 
 
 Reduction o » , 
 Addition .... 
 Subtraction . . . 
 Multiplication . . 
 Division .... 
 Accounts and Bills 
 Review .... 
 Denominate Numbers 
 Measures of Length 
 Measures of Surface 
 Measures of Volume 
 Liquid Measures . 
 Dry Measures . . 
 Avoirdupois Weight 
 Troy Weight . . 
 Apothecaries' Weight 
 Divisions of Time , 
 Counting .... 
 Stationers' Table . 
 Reduction . . , 
 Addition . , . . 
 Subtraction . . . 
 Multiplication . . 
 Division .... 
 Practical Measurements . 
 Interest . . . o . . , 
 General Review ,00. 
 
 PAGE 
 
 188 
 191 
 193 
 194 
 195 
 198 
 201 
 203 
 205 
 205 
 206 
 207 
 208 
 209 
 209 
 210 
 210 
 211 
 213 
 213 
 214 
 217 
 218 
 219 
 220 
 221 
 226 
 229 
 
u» ^ fiA 
 
 UNIVERSITY 
 
 ELEMENTS OE^ A^RITHMETIC. 
 
 3i»4C 
 
 1. Hold up your right hand. 
 
 Hold up your left hand. 
 
 How many hands have you ? 
 
 Place your right hand behind you. 
 How many hands can I see now ? 
 Place your left hand behind you. 
 How many hands are behind you? 
 How many feet have you ? 
 How many eyes have you ? 
 How many heads have you ? 
 Name one thing you can see on my desk. 
 Show me two blocks. 
 Give Annie one block. 
 Clap your hands once. 
 Say hall twice. 
 Make a mark on your slate. 
 
 9 
 
10 ELEMENTS OF ARITHMETIC. 
 
 Make another mark. 
 
 How many marks have you made? 
 
 One mark and one mark are how many marks ? 
 
 Kub out one mark. How many marks are left? 
 
 This figure stands for one — /. 
 
 Tliis figure stands for two — 2- 
 
 Say one word. 
 
 Say two words. 
 
 Point to one book. 
 
 Point to one school-desk. 
 
 Hold up two fingers. 
 
 Tell me two boys' names. 
 
 Tell me a girl's name. 
 
 How many feet has a boy ? 
 
 How many noses has he ? 
 
 How many ears has he ? 
 
 Make as many marks as you have hands. 
 
 Make the figure that stands for two things. 
 
 Make as many marks as you have feet. 
 
 Make the figure that stands for one thing. 
 
 If I have two cents and give away one of them, how 
 many shall I have left ? 
 
 If you should liave one apple, and a girl should give 
 you one more, how many apples would you have ? 
 
 If you did not go to school on Monday and on Friday, 
 how many days were you absent that week ? 
 - Make pictures of two squares. Q []] 
 
 Make pictures of two circles. Q O 
 
 Make pictures of two triangles, j^ ^ 
 
 n and Q are how many squares? 
 
 O i^^^^d O ^^^^ 5^0 w many circles ? 
 
 A and /\ are how many triangles? 
 
THREE — 3. 
 
 11 
 
 2. How many boys are picking 
 up nuts ? 
 
 How many boys are in the tree? 
 How many boys do you see in 
 the picture? If one of the boys 
 should go home, how many would be left ? 
 Show me three pencils. 
 Make three dots on your slate. 
 Rub out one dot. How many are left? 
 Rub out another. How many have you rubbed out ? 
 How many are left? 
 
 Now put enough dots with this one dot to make three 
 dots. 
 
 One dot and two dots are how many dots? 
 James gave me two marbles, and Arthur gave me 
 one. How many marbles did they give me? 
 
 Two marbles and one marble are how many mar- 
 bles? 
 
 If I lost one of them, how many had I left? 
 Three marbles less one marble are how many marbles? 
 I see one bird on the ground, one in a bush, and 
 one flying. How many birds do I see ? 
 
12 
 
 ELEMENTS OF ARITHMETIC. 
 
 This figure stands for three — J. 
 
 Write the figures that stand for one, two, and tliree. 
 
 Tell me what tliis shows : • • and • are • • •. 
 
 |~| and [31 n are how many squares? 
 
 /\^ and A A ^^'® ^^w many triangles? 
 
 3. How many boys do you see in the picture ? 
 
 How many girls do you see ? 
 
 How many children are there ? 
 
 Three boys and one girl are how many children ? 
 
 Take four blocks. Play that they are little birds. 
 Four little birds were picking up crumbs. When one 
 flew away, how many were left? Soon afterward, two 
 more flew away. How many had then flown away? 
 How many were left ? 
 
 How many books has the largest boy in the pic- 
 ture ? 
 
 How many has the next largest? 
 
 If the largest boy should carry all the books, how 
 many would he carry ? 
 
FOUR — 4. 13 
 
 Susie and her sister found a nest with four eggs in it. 
 Her sister took two of them. How many were left for 
 Susie ? 
 
 Four eggs less two eggs are how many eggs ? 
 
 In four eggs, how many twos are there ? 
 
 How many sides has a square ? □ 
 
 How many sides has a triangle ? /\ 
 
 How many triangles can you form with four sticks ? 
 Try -It. 
 
 At a cent apiece, how many pears can you buy with, 
 four cents ? 
 
 This figure stands for four — -^. 
 
 Write the figures / , 2, 3, ^ on your slate. 
 
 How many one-cent stamps can you buy for four cents ? 
 
 When pencils cost two cents each, what will two 
 pencils cost? 
 
 James was two years old, and his sister Grace was 
 two years older. How old was Grace ? 
 
 Make two squares and write the word two beside 
 them. OO, two. 
 
 Make three circles and write the word three beside 
 them. OOO? three. 
 
 Make four triangles and write the word four beside 
 them. AAAA. four. 
 
 How many squares are D D and D D ? 
 
 Here is a shorter way of writing " Two and two are 
 four": 2 + 2 = 4. 
 
 The sign + means and^ and the sign = means are. 
 
 Read 2 + 1 = 3; 1 + 1 = 2; 3 + 1 = 4; 1 + 3 = 4. 
 
 Copy and complete : 
 
 1+3=? 2+2=? 1+1=? 1+2=? 
 
14 ELEMENTS OF ARITHMETIC. 
 
 4, How many fingers have you on one 
 hand ? 
 
 If you do not count the thumb, how 
 many are there ? 
 
 Five fingers less one finger are how 
 many fingers ? 
 
 Point to five children. 
 Make five i's on your slate. 
 How many books must I put with 
 four books to make five books ? 
 
 How many joints has your little fin- 
 ger? 
 
 How many joints has your thumb? 
 How many joints have both your little finger and 
 your thumb ? 
 
 Three things and two things are how many things ? 
 Mary has five pencils. If only two of them are sharp, 
 how many are dull ? 
 
 Alice had five yards of ribbon. She gave Sarah three 
 yards of it. How many yards had she left ? 
 
 Place five splints on your desk in this way: II II I. 
 How many twos are there in five, and hoAv many are 
 left over ? 
 
 If you had two two-cent pieces and a cent, how much 
 money would you have ? 
 
 What is another name for five cents ? 
 If you had a half-dime and spent one cent for a 
 pencil and four cents for marbles, how much money 
 would you have left ? 
 
 This figure stands for five — S'- 
 
 Write the figures /, £>^ d, ^, 6 on your slate. 
 
 How many cents are three cents and two cents? 
 
FIVE — 5. 
 
 
 Four cents and one cent? One cent and^ four cents? 
 Two cents and three cents ? 
 
 How much will be left out of five cents if I buy two 
 two-cent stamps ? 
 
 It takes five days to go to California. When we have 
 traveled three days, how many days have we to travel? 
 
 I paid three cents for a paper and had two cents left. 
 How much money had I at first ? 
 
 Write a figure that means as much as three and two. 
 Three and one. Two and one. One and one. 
 
 How many are 
 
 less four apples ? 
 
 Here is a shorter way of writing " Five less four are 
 one": 5-4 = 1. 
 
 The sign — means less. What does = mean ? 
 
 Copy and complete 
 
 : 
 
 
 5-3=? 
 
 
 3-2=? 
 
 2 + 3=? 
 
 5-5=? 
 
 
 5-4=? 
 
 4 + 1=? 
 
 4-1=? 
 
 
 5-1=? 
 
 3 + 2=? 
 
 4-2=? 
 
 
 4-3=? 
 
 2 + 2=? 
 
 
 
 O one 
 
 1. 
 
 
 
 O O two 
 
 2. 
 
 
 
 O O O three 
 
 3. 
 
 
 O 
 
 O O O four 
 
 4. 
 
 
 o o 
 
 O O O five 
 
 5. 
 
16 
 
 ELEMENTS OF ARITHMETIC. 
 
 5. Cut an apple into 
 two equal parts. 
 
 Draw a square and di- 
 vide it into two equal parts. 
 How have I divided the 
 circle placed below? 
 
 Each of these parts is 
 called a half. 
 
 How many halves are 
 there in an apple ? 
 How many halves are there in a square ? 
 Take four blocks. Place them in two equal groups. 
 How many are there in each group ? What is the name 
 of each group, or part ? 
 
 One block is what part of two blocks ? 
 James had two nuts. He gave half of them to me. 
 How many did he give me ? 
 
 What part of the nuts did he keep ? 
 
 Divide four oranges equally between Mabel 
 and Annie. What part of the four oranges 
 will each have ? 
 
 How many of the oranges will each have ? 
 * What is half of four books ? 
 Two cents is half of how many cents ? 
 Mary picked four pints of berries, but spilled 
 half of them. How many did she spill ? 
 Henry made four cents by selling newspapers, and 
 gave half of it to his sister. How much did he give to 
 her? 
 
 The wages of a boy were four dollars per week. 
 How much did he earn in half a week ? How much in 
 a week and a half ? 
 
SIX— 6. 
 
 17 
 
 6. How many bananas do you see in the bunch? 
 
 How many bananas are there in the dish? 
 
 Five bananas and one banana are how many? 
 
 Show me six fingers. 
 
 Draw a picture of a pen with six rabbits in it. 
 
 Give me one splint; now give me five more. How 
 many have I now ? 
 
 Jamie had a half-dime and a cent. To how many 
 cents was that equal ? 
 
 Henry had six pears, but he ate one. How many had 
 he left? 
 
 There were six rolls on the plate before supper, and 
 only one after supper. How many had been eaten ? 
 
 Johnnie is four years old to-day. How old will he be 
 in two more years ? 
 
 Four years and two years are how many years ? 
 
 It is two miles from Lizzie's house to school, and 
 four more miles to the station. How far does Lizzie 
 live from the station ? 
 
 EL. OF AR. — 2 
 
18 ELEMENTS OF ARITHMETIC. 
 
 I must write six letters. When I have written four 
 of them, how many more have I to write ? 
 
 Make six lines on your slate. Cross out two of them. 
 How many are not crossed out? 
 
 How many arms, hands, and feet have you ? 
 
 How many eyes, ears, and cheeks have you ? 
 
 Two cents and two cents and two cents are how 
 many cents ? 
 
 If you had six cents, how many two-cent stamps 
 could you buy with the money ? 
 
 How much must I pay for three sponges, if each one 
 costs me two cents ? 
 
 How many boots are there in three pairs of boots ? 
 
 Henry has six mittens. How many pairs of mittens 
 has he ? 
 
 Bertie found three eggs in one nest and three in 
 another. How many eggs did he find ? 
 
 Two threes are how many ? 
 
 Take six blocks. Give me half of them. How many 
 did you give me ? 
 
 Nellie had six eggs, but used half of them to make 
 a cake. How many did she use? What part of the 
 eggs was left? How many eggs were left? 
 
 In a , class of six pupils, half of the children are 
 sitting and half are standing. How many are sitting ? 
 
 Willie had six cents. He spent it for candy at a 
 cent a stick. How many sticks did he buy ? 
 
 If he had bought marbles with it at three cents 
 apiece, how many would he have bought? 
 
 This is the figure that stands for six — 6- 
 
 Write the figures f , 2, d, ^, 6, 6, on your slate. 
 
 What is the cost of three pencils, at two cents apiece ? 
 
ONE THIRD. 
 
 19 
 
 5 + 1=? 
 
 2+? = 6 
 3 + 3=? 
 ?+2 = 6 
 
 SLATE WORK. 
 6_3=? ?+5 
 
 6-D=? 
 
 6-? = 4 
 
 6 
 
 3+? = 6 
 6-1=? 
 6-? = 2 
 
 8 + 2=? 
 4-3=? 
 5-4=? 
 1 + 3=? 
 
 7. How have I divided this circle ? 
 
 Draw a line and divide it into three equal 
 parts. 
 
 Cut an api^le into 
 three equal parts. 
 
 Each of these parts 
 is called a third. 
 
 How many thirds are 
 there in a circle ? In a 
 line ? In an apple ? In 
 anything ? 
 
 Give me a third of a splint. 
 
 Take three pebbles. Divide them into three equal 
 parts. Give me a third of three pebbles. 
 
 Now take six pebbles. Divide them into three equal 
 parts. Give me a third of six pebbles. 
 
 What is a third of six horses ? 
 
 If a pie costs six cents, what will a third of a pie 
 cost? 
 
 Henry had six apples which he divided equally among 
 three friends. How many did each receive ? 
 
 School is in session six hours per day, but the little 
 children are not required to attend all the time. If they 
 have one third of the time for play, how many hours do 
 they play ? 
 
20 
 
 ELEMENTS OF ARITHMETIC. 
 
 8. Get a pint measure and a quart measure. 
 
 Which is the pint measure? Which is the quart 
 measure ? 
 
 Fill the pint measure with water. Pour it into the 
 quart measure. Does it fill it? 
 
 Fill the pint measure again and empty it into the 
 quart measure. Is the quart measure full now? How 
 many pints of water are there in it ? 
 
 How many pints of water does it take to make a 
 quart of water ? How many pints of milk make a quart 
 of milk ? How many pints of berries make a quart of 
 berries ? 
 
 I have two quarts of chestnuts. How many pint 
 bowls will it take to hold them ? 
 
 Six pints of oil will fill how many quart cans ? 
 
 What part of a quart is a pint ? 
 
 If a quart of milk costs six cents, what will a pint 
 cost? 
 
 If a pint of milk ^osts two cents, what will three 
 pints cost? 
 
 Charlie was paid a cent for every pint of cherries he 
 picked. How much did he get for picking three quarts 
 of cherries ? 
 
SEVEN — 7. 
 
 21 
 
 How many barrels are there 
 in the wagon ? 
 
 How many are there in the wagon 
 and on the ground ? 
 
 Six barrels and. one barrel are how many barrels ? 
 
 Name the days of the week. How many are there? 
 
 How many would there be if there were no Satur- 
 day? 
 
 Seven days less one day are how many days ? 
 
 How many men are busy with the apples ? 
 
 How many children are trying to helj) ? 
 
 Five men and two children are how many people ? 
 
 If two of the men should go home in the wagon, how 
 many people would be left? 
 
 If the two children should then run off to play, how 
 many would be left to work ? 
 
 How many twos are there in three ? 
 
 How many twos are there in seven ? 
 
 There were seven leaves on a twig, and five of them 
 were blown away. How many were left ? 
 
22 ELEMENTS OF ARITHMETIC. 
 
 Draw a triangle A- Draw a square Q. 
 
 How many sides has the triangle ? 
 
 How many sides has the square ? 
 
 How many sides have both figures ? 
 
 How many more sides has the square than the triangle ? 
 
 A farmer had seven tons of hay. If he sold three 
 tons, how many had he left? 
 
 If a pair of boots costs three dollars, how many pairs 
 can you buy with seven dollars ? How much will you 
 have left? 
 
 A boy bought seven pencils. He fell and broke four 
 of them. How many whole ones had he? 
 
 Show with blocks how many fours there are in seven. 
 
 How many quart measures can you fill with seven 
 pints ? What part of a quart have you left ? 
 
 What will seven peaches cost, if each one costs a cent? 
 
 This is the figure that stands for seven — /. 
 
 If I give two oranges to Lizzie and two to Annie, 
 how many do I give away ? 
 
 How many times do I give two oranges away? 
 
 Two times two oranges are how many oranges? 
 
 Two times two is written thus : 2 X 2 = 4. 
 
 What does the sign X mean? 
 
 Read and complete : 
 3x2=? 1x5=? 4x1=? 3x1=? 
 
 7x1=? 2x3=? 1x7=? 2x2=? 
 
 
 SLATE WORK. 
 
 
 6 + 1=? 
 
 7_2=? 4+? = 7 
 
 3x2=? 
 
 1 + 6=^^ 
 
 7_5=? 7-? = 3 
 
 2x2=? 
 
 7-1=? 
 
 5 + 2=? 3+? = 7 
 
 2x3=? 
 
 7-6=? 2 + 5=? 7-?=4 7x1=? 
 
EIGHT — 8. 
 
 23 
 
 10. This girl has poured seven pints of milk into the 
 dish. If she pours in one pint more, how many pints 
 will then be in the dish ? 
 
 If she empties the dish by pouring the milk into 
 quart cans, how many cans will it fill ? 
 
 After she has filled one can, how much is left in the 
 dish? 
 
 How many arms have four boys ? 
 
 How many horns have four cows ? 
 
 How many chairs must I put with six chairs to make 
 eight chairs ? 
 
 Fanny had three roses, three lilies, and two pansieSo 
 How many flowers had she ? 
 
 How many threes are found in eight? How much is 
 left? 
 
 Edith wrote eight Avords on her slate, and rubbed 
 out three of them. How many were left? 
 
 Five birds and three birds are how many birds ? 
 
 Three geese and five geese are how many geese ? 
 
 How many horses can be shod with eight horseshoes ? 
 
• • • • 
 
 24 ELEMENTS OF ARITHMETIC. 
 
 How many two-cent stamps can be bought with eight 
 cents ? 
 
 What two equal numbers make eight? 
 
 What four equal numbers make eight ? • 
 
 A gallon contains four quarts. How many gallons 
 are there in eight quarts ? 
 
 This figure stands for eight — ^. 
 
 Place eight blocks in two equal groups. How many 
 are there in each group ? 
 
 What is half of eight blocks ? 
 
 Eight sheep were in a field, and half of them got out. 
 How many were left ? 
 
 Break a splint into halves. Break each half splint 
 into two equal parts. Into how many pieces have you 
 broken the splint ? 
 
 Each piece is called a fourth^ or a quarter. 
 
 How many fourths are there in a pie ? 
 
 Draw a square on your slate. Divide it into four 
 equal parts. Do it in more than one way. 
 
 Divide four pears into four equal groups. What is a 
 fourth of four pears ? 
 
 Place eight marbles in four equal piles. What part 
 of the marbles is in each pile ? 
 
 Take a fourth of them in your hand. 
 
 A fourth of eight marbles is how many marbles ? 
 
 A boy who had eight cents spent half of what he had for 
 paper. How much did he spend ? How much had he left ? 
 
 A boy who had eight miles to go traveled one half the 
 distance by bicycle. How far did he travel by bicycle ? 
 
 A girl who earned eight dollars per month paid one 
 half of it for her necessary expenses. How much did 
 she save ? 
 
EIGHT — 8. 
 
 25 
 
 11. Find a pint measure and a gill measure. 
 Which is the smaller? 
 
 Fill the gill measure with water. Pour it into the 
 pint measure. Do this until you fill the pint measure. 
 How many gills of water fill the pint measure ? 
 How many gills are there in two pints of vinegar? 
 How many pints are there in two quarts of water ? 
 How many pints are there in three quarts of water ? 
 In eight pints of water, how many quarts are there ? 
 
 
 SLATE WORK. 
 
 
 
 2+6=? 
 
 
 8-T=? 
 
 
 8-5=? 
 
 8-? =4 
 
 
 ? x2 = 8 
 
 
 8x? = 8 
 
 2x4=? 
 
 
 ?-l = 7 
 
 
 ?+5 = 8 
 
 ^ ?+3 = 8 
 
 
 8-?=2 
 
 
 4 + 4=? 
 
 Add: 
 
 
 
 
 
 2 5 
 
 1 
 
 4 
 
 3 
 
 4 6 
 
 6 3 
 
 7 
 
 4 
 
 2 
 
 3 2 
 
 How much is : 
 
 
 
 
 
 4of4 
 
 
 iof6 
 
 
 iof4 
 
 ^of 2 
 
 
 iof3 
 
 
 iof8 
 
 iof6 
 
 
 iof 8 
 
 
 J of 2 X 2 
 
26 
 
 ELEMENTS OF ARITHMETIC. 
 
 ll^t^^^f'^ 
 
 -i'^ftt 
 
 12. David and Fred went fishing. 
 
 How many fishes has David, the boy by the fence, on 
 his string ? How many has Fred ? 
 
 How many fishes did both catch ? 
 
 How many more did David catch than Fred ? 
 
 How many postal cards can I buy with nine cents, if 
 they cost me a cent apiece ? 
 
 Emma had a bnncli of nine roses, but two of them 
 were lost. How many were left? 
 
 If you had a two-cent piece, how much more would 
 you need to make nine cents ? 
 
 Carl paid three cents for a top, three cents for a 
 pencil, and three cents for a marble. How much did 
 they all cost him ? 
 
 Three times three cents are how many cents ? 
 
 Divide nine plums equally among three girls. 
 
 What part of the plums will each receive ? 
 
 What is a third of nine pounds of sugar? 
 
 There are six boys and three girls in a class, 
 many children are there in the class ? 
 
 How 
 
NINE — 9. 27 
 
 How many more boys are there than girls ? 
 
 If there were nine eggs in a nest and three were 
 taken out, how many were there left? 
 
 Ethel had a half-dime and two two-cent pieces. How 
 much money had she ? 
 
 If she bought oranges at four cents apiece with her 
 money, how many did she buy ? 
 
 How much shall I have left out of nine dollars, after 
 paying four dollars for a pair of boots ? 
 
 How many pints are there in nine gills ? 
 
 How many quarts are there in nine pints ? 
 
 Frank wrote his name nine times on his slate. He 
 wrote it well only seven times. How many times did 
 he write it carelessly ? 
 
 How many cents are three cents and six cents more ? 
 
 If milk costs three cents a pint, how many pints can 
 I buy with nine cents ? 
 
 ^ENi) iwn) (ct>ir) 
 
 How many threes are there in nine ? 
 This figure stands for nine — ^. 
 
 Here are some shorter ways of writing the number of 
 threes in nine : 
 
 3'sin9 = 3; or 3) 9 (3 ; or 9-v-3 = 3. 
 
 Read and complete : 
 
 2'sin8=? 3'sin6=? I'sin7=? 
 
 4)8(? 6)6(? 1)5(? 9--3=? 8--8= ? 9-i-l=? 
 
28 ELEMENTS OF ARITPIMETIC. 
 
 13. Count the balls on this wire. 
 
 What two equal numbers make ten. 
 
 Two fives are how many? What is half of ten balls? 
 
 Ten cents make a dime. How many cents are there 
 in half a dime ? 
 
 Jane found a dime. She spent half a dime for a 
 doll. How much had she left ? 
 
 How many dolls, at that price, could she have bought 
 with the dime ? 
 
 How many fingers have you? 
 
 If you do not count your thumbs, how many have you ? 
 
 Ten fingers less two fingers are how many fingers ? 
 
 Place ten blocks in twos. How many twos do you 
 find in ten blocks? 
 
 At two cents apiece, what will five bananas cost ? 
 
 A boy who had ten marbles lost eight of them. How 
 many had he left ? 
 
 When Carl has read seven pages, how many pages 
 more must he read to make ten pages ? 
 
 Take four buttons away from ten buttons, and see 
 how many are left ? 
 
 Three threes and how many make ten? 
 
 How many squares can you make with ten sticks ? 
 
 These figures stand for ten — fo. 
 
 The figure on the right is called zero. What is the 
 left-hand figure called ? 
 
 Write the figures, /, 2, d, f , 6, 6, 7, 8, ^, 0, 
 
ONE HALF — ONE THIRD ^ ONE FOURTH 
 
 14. What do you call each part of this circle ? 
 
 One half is written in this wa}^ : ^. 
 
 What do you call each part of this square ? 
 
 One third is written in this way : ^. 
 
 What do you call each part of this line ? 
 
 J. X ± 
 
 1 "^ I * ! ^ I 
 
 J. 
 ^ ,...1 
 
 One fourth is written in this way : 
 
 h 
 
 Read and complete : 
 
 
 iot2=? loflO=? 
 
 Jof6=? 
 
 lof9=? lof 4=? 
 
 ^of 8=? 
 
 3 is J of? 2 is lof? 
 
 2 is 1 of? 
 
 If three bananas cost nine cents, what will each cost? 
 
 /one\ /oneN /oneA /one\ /one\ /oneN /one\/one\ /one\ 
 
 \CENT/ IcentJ Uemt/ [ciuy \cimj [cuirj IcemtJ I cent/ I cent i 
 
 What part of nine cents are three cents ? 
 
 What part of nine cents will each banana cost ? 
 
 If four caps cost eight dollars, how much will each cost ? 
 
 What part of eight dollars will each cap cost? 
 
30 
 
 ELEMENTS OF ARITHMETIC. 
 
 15. 
 
 REVIEW EXERCISES. 
 
 3 + 4=? 
 
 6-4=? 
 
 2x5=? 
 
 
 8-4 = 
 
 ? 
 
 6 + 2=? 
 
 8-2=? 
 
 3x3=? 
 
 
 9-3 = 
 
 ? 
 
 5 + 4=? 
 
 10-5=? 
 
 4x2=? 
 
 
 6-2 = 
 
 ? 
 
 2 + 7=? 
 
 5-3=? 
 
 1x7=? 
 
 
 10-5 = 
 
 ? 
 
 4 + 6=? 
 
 9-6=? 
 
 5x1=? 
 
 
 7-^l = 
 
 ? 
 
 1 + 5=? 
 
 7-1=? 
 
 3x2=? 
 
 
 10-2 = 
 
 ? 
 
 lof 10=? 
 
 7-5= ? 
 
 3+?= 5 
 
 
 5x? = 
 
 10 
 
 lof 6=? 
 
 10-7= ? 
 
 4+?= 9 
 
 
 ? x4 = 
 
 8 
 
 lof 9=? 
 
 9-?= 5 
 
 6-3= ? 
 
 
 9x1 = 
 
 ? 
 
 lof 6=? 
 
 ?-3= 5 
 
 10-? = 10 
 
 
 1 of 8 = 
 
 ? 
 
 iof 4=? 
 
 7+?=10 
 
 8-?= 4 
 
 
 iof ? = 
 
 2 
 
 iof 8=? 
 
 ?+3= 9 
 
 9-9= ? 
 
 
 iof ?= 
 
 1 
 
 2x 3=? 
 
 4-4=? 
 
 7- 4=? 
 
 
 3+? = 
 
 = 9 
 
 8+ 2=? 
 
 lof 2=? 
 
 10-10=? 
 
 
 8+? = 
 
 = 9 
 
 lof 4=? 
 
 8-4=? 
 
 2x 2=? 
 
 
 5+? = 
 
 = 9 
 
 4-f- 2=? 
 
 7x1=? 
 
 6x 1=? 
 
 
 10-? = 
 
 --2 
 
 9- 5=? 
 
 3 + 7=? 
 
 6+ 4=? 
 
 
 10-? = 
 
 = 6 
 
 10- 6=? 
 
 7 + 2=? 
 
 iof9=? 
 
 
 10-? = 
 
 = 1 
 
 Add: 2 
 
 4 1 
 
 3 
 
 2 
 
 4 
 
 
 3 
 
 1 2 
 
 3 
 
 3 
 
 1 
 
 
 1 
 
 3 
 
 1 
 
 1 
 
 2 
 
 
 2 
 
 3 4 
 
 2 
 
 4 
 
 1 
 
 
 From 9 
 
 6 8 
 
 7 
 
 8 
 
 5 
 
 
 Take 3 
 
 2 5 
 
 3 
 
 4 
 
 3 
 
 
ELEVEN TO NINETEEN. 31 
 
 16. 
 
 One ten. 
 
 •••••••• • 
 
 One ten and one are eleven ; written, //. 
 
 •••••••• •• 
 
 One ten and two are twelve ; written, 12. 
 
 • ••••• 9^ • • • 
 
 One ten and three are thirteen ; written, /3. 
 
 •••o«««« •••• 
 
 One ten and four are fourteen ; written, /^. 
 
 One ten and five are fifteen ; written, 16. 
 
 One ten and six are sixteen ; written, /6. 
 
 One ten and seven are seventeen ; written, fj. 
 
 One ten and eight are eighteen ; written, /8. 
 
 One ten and nine are nineteen ; written, /^. 
 
32 
 
 ELEMENTS OF ARITHMETIC. 
 
 iPi|#ii/ , , ,„ 
 
 .,;,/];, 
 
 
 — ■ -^-^^S^' ^"^ 
 
 17. How many frogs are there 
 in the pond? 
 If the one on the edge should jump in, how 
 many would there be in the water ? 
 
 How many ones are there in eleven frogs ? 
 
 How many tens are there in eleven frogs ? 
 
 There were eleven trees in a garden. Two of them 
 were peach trees and the rest were pear trees. How 
 many pear trees were there ? 
 
 What is the greatest number of two-cent pieces you 
 can have in eleven cents ? 
 
 A week and four days are how many days ? 
 
 How many horses can be shod all around with eleven 
 shoes ? How many shoes are left ? 
 
 Harry had three pieces of money which made eleven 
 cents. What were they? 
 
 If he should change them for two pieces of money, 
 what coins would they be ? 
 
 He took a ride upon a street car that cost him five 
 cents. How much had he left ? 
 
 There are six boys and five girls in the first clasSc 
 How many pupils are there in the class ? 
 
ELEVEN— 11. 33 
 
 Mr. Snow had eleven cows. Seven were in the 
 pasture and the others were in the barn. How many 
 were in the barn? 
 
 This figure is a pentagon. How many 
 sides has it ? 
 
 How many pentagons can you make with 
 eleven sticks ? 
 
 Make a story about nine nuts and two nuts. 
 
 Make a story about eleven dogs less ten dogs. 
 
 Two squares and a triangle have how many sides ? 
 
 Two triangles and a square have how many sides ? 
 
 There are six boys in my class to-day. When they 
 are all present, there are eleven. How many are absent? 
 
 David's father has eleven horses and Henry's has five 
 less. How many horses has Henry's father ? 
 
 A writing book and a pencil cost together eleven 
 cents. The pencil cost two cents. How much did the 
 writing book cost ? 
 
 SLATE WORK. 
 
 10+ 1=? 11-3=? 6+ 5=? 3+? = ll 
 
 11- 1-? 8 + 3=? 5+ G=? 11-?= 7 
 
 11-10=? 11-8=? 11-6=? ?+9 = ll 
 
 11^ 1_? 3 + 8=? 11- 5=? 5 6 11 
 
 11^10=? 11-3=? 11-f- 5=? 11 2 9 
 
 9+ 2=? 11-8=? 11- 6=? 11-?= 5 
 
 2+9=? T + 4=? 11-11=? ?-8=3 
 
 11- 2=? 11-4=? 11- 0=? ?+T = ll 
 
 11- 9=? 11-7=? llx 1=? G+? = ll 
 
 11h- 2=? 4 + 7=? 1x11=? 3 + 7= ? 
 
 11- 9=? 4)11(? 11-11=? 10-?= 6 
 
 EL. OF AR. 3 
 
34 
 
 ELEMENTS OF ARITHMETIC. 
 
 ^^\Vf />V)^]T 18. How many eggs are there in 
 /l ' ^- 'this nest? 
 
 AVhat other name is given to twelve eggs? 
 If you should buy a dozen oranges, how many 
 would you get ? 
 
 If they should cost you a cent apiece, how much would 
 you pay for them ? 
 
 If you should put half of them in a dish, how many 
 would there be in the dish? How many others would 
 there be ? 
 
 What is another name for six oranges, or six of 
 anything ? 
 
 There are six working days in a week. How many 
 working days are there in two Aveeks ? 
 
 A baby had twelve spools to ]Dlay with, and one rolled 
 away. How many were left ? 
 
 If it takes ten yards of lace to trim a dress and two 
 yards to trim an apron, how much will be needed to 
 trim both ? 
 
 I have six two-cent pieces. How much money have I? 
 
 How many two-cent stamps can I buy with a dime 
 and two cents ? 
 
TWELVE — 12. 35 
 
 James bought a dozen bananas. He ate three, gave 
 three to his brother, and three to his sistero How many- 
 had he left? 
 
 How many threes are there in twelve ? 
 
 Place twelve beans in groups of three beans each. 
 
 Into how many groups have you separated the twelve 
 beans ? What part of the whole is each group ? 
 
 What is a fourth of twelve miles ? 
 
 How many months are there in a year ? Name them. 
 
 If Willie is out of school during June, July, August, 
 and September, how many months in the year does he 
 attend school ? 
 
 Three horses need how many shoes? 
 
 What part of twelve shoes does each horse need ? 
 
 If the ground is covered with snow during a third of 
 the year, how. many months is it covered? How many 
 months in the year is the ground bare ? 
 
 Five plants in the room are in blossom, and seven are 
 not. How many plants are there in the room ? 
 
 Take a foot rule. Measure a foot on your desk. 
 Measure the width of a table. See hoAv your rule is 
 divided. What is each of the larger parts of a foot 
 called? How many inches are there in a foot? 
 
 If I had a piece of string a foot long and cut off four 
 inches, how many inches would be left ? 
 
 What number of inches must I add to seven inches 
 to make a foot ? 
 
 How many inches long is a strap that is just one half 
 of a foot in length ? 
 
 Six cows have how man}^ horns? 
 
 Each of three windows has four panes of glass. How 
 many panes of glass are there in the windows ? 
 
36 ELEMENTS OF ARITHMETIC, 
 
 SLATE WORK. 
 
 11+ 1=? 
 
 5 + 3 = 
 
 =, •? 
 
 
 12-4=? 
 
 12-6=? 
 
 12- ? = 11 
 
 9+? = 
 
 = 12 
 
 
 3x4=? 
 
 ?-7 = 5 
 
 ? +11 = 12 
 
 3 + 9 = 
 
 = 2 
 
 ~i 
 
 Lof 12=? 
 
 ll-? = 8 
 
 12-11=? 
 
 ? -9 = 
 
 = 3 
 
 
 7+?=12 
 
 ?-6 = Q 
 
 ?+ 2 = 12 
 
 4x3 = 
 
 = ? 
 
 
 12-7=? 
 
 4 + 6=? 
 
 2 + 10=? 
 
 12-3 = 
 
 = ? 
 
 
 ?-5 = 7 
 
 11-7=? 
 
 12- ? = 10 
 
 iof 12 = 
 
 = ? 
 
 
 5 + 7=? 
 
 9-6=? 
 
 ?-10 = 2 
 
 8+? = 
 
 = 12 
 
 
 6-\-6=? 
 
 3x3=? 
 
 6x 2=? 
 
 12-8 = 
 
 ~ ? 
 
 
 12-? = 6 
 
 ?+4 = 12 
 
 12- 2=? 
 
 4 + 8 = 
 
 = ? 
 
 
 Lof 12=? 
 
 8 + 3=? 
 
 12- 5=? 
 
 12-? = 
 
 = 8 
 
 
 ^ 2x? = 12 
 
 10-7=? 
 
 8+ 4=? 
 
 ?x3= 
 
 = 12 
 
 
 12-7=? 
 
 ? +9 = 12 
 
 12- ?=7 
 
 5+? = 
 
 = 12 
 
 
 12-8=? 
 
 ? -8 = 4 
 
 8+ ? = 12 
 
 12-5 = 
 
 = ? 
 
 
 12-4=? 
 
 ? x6 = 12 
 
 12- 3=? 
 
 4x?= 
 
 = 12 
 
 
 ?+5 = 12 
 
 3+? = 12 
 
 12- ? = 2 
 
 7 + 5 = 
 
 - 9 
 
 
 ?+6 = 12 
 
 ?-7 = 5 
 
 Add: 
 
 
 
 
 
 
 4 1 
 3 5 
 
 2 4 
 
 3 
 4 
 4 
 
 
 1 
 
 2 
 3 
 
 2 
 3 
 4 
 
 3 6 
 
 4 1 
 3 4 
 
 From 11 
 
 9 
 
 
 10 
 
 11 
 
 10 11 
 
 Take 8 
 
 5 
 
 
 7 
 
 5 
 
 2 7 
 
 Six 2's = ? 
 
 Two 
 
 6's = 
 
 :? 
 
 Two 3's=: 
 
 = ? 2)6(? 
 
 Three 4's = ? 
 
 Four 
 
 3's = 
 
 :? 
 
 Three 2's = 
 
 =? 4)12(? 
 
 Five 2^s = ? 
 
 Three 3's = 
 
 :? 
 
 Four 2's=: 
 
 =? ?)12(2 
 
THIRTEEN — 13. 
 
 87 
 
 19. Count the number of stripes in the flag. 
 
 What colors are the stripes ? How many red ones are 
 there ? How many Avhite ones ? 
 
 If one of the red ones should be torn off, how many 
 stripes would be left ? 
 
 Byron placed eleven blocks in one group and two in 
 another. How many blocks were there in both groups? 
 
 If he should put one of the eleven blocks with the 
 other group, how many would there be in both groups? 
 How many in each group ? 
 
 What is the largest piece of money that can be used 
 in paying thirteen cents ? 
 
 How much more is needed to make thirteen cents ? 
 
 I had thirteen cents, and I spent four cents for stampSc 
 How many cents had I left ? 
 
 Four two-cent stamps and a five-cent stamp cost how 
 much ? 
 
 I walked six miles yesterday and seven miles to-day. 
 How many miles did I walk in all ? 
 
38 
 
 ELEMENTS OF ARITHMETICo 
 
 
 
 
 
 SLATE WORK. 
 
 Supply the Signs o 
 
 7 + 6 = 
 
 ? 
 
 
 10 + 
 
 3=? 
 
 13-?= 6 
 
 3 4 12 
 
 13-6 = 
 
 '? 
 
 
 13- 
 
 3=? 
 
 13-?= 2 
 
 7 2 9 
 
 13-7 = 
 
 ? 
 
 
 13- 
 
 10=? 
 
 8+? = 13 
 
 11 5 6 
 
 6 + 7 = 
 
 ? 
 
 
 3 + 10=? 
 
 3+? = 13 
 
 8 2 4 
 
 8 + 5 = 
 
 ? 
 
 
 11 + 
 
 2= ? 
 
 13-?= 4 
 
 4 2 8 
 
 13-8 = 
 
 ? 
 
 
 13- 
 
 2= ? 
 
 13-? = 10 
 
 4 2 6 
 
 13-5 = 
 
 ? 
 
 
 13- 
 
 11=? 
 
 13-?= 8 
 
 ? + ? = 13 
 
 5 + 8 = 
 
 ? 
 
 
 2 + 11=? 
 
 9+? = 13 
 
 ?-5= 5 
 
 9 + 4 = 
 
 ? 
 
 
 13x 
 
 1=? 
 
 2+? = 13 
 
 ?-?= 9 
 
 13-4 = 
 
 ? 
 
 
 Ix 
 
 13=? 
 
 5+? = 13 
 
 ? X ?= 9 
 
 13-9 = 
 
 ? 
 
 
 13- 
 
 1=? 
 
 13-? = 12 
 
 ? — ?= 2 
 
 4 + 9 = 
 
 ? 
 
 
 13- 
 
 13=? 
 
 13-?= 7 
 
 ? + ? = 12 
 
 Acid: 
 
 
 
 
 
 
 
 2 
 
 
 8 
 
 
 8 
 
 5 4 
 
 6 • 2 
 
 5 
 
 
 4 
 
 
 2 
 
 6 2 
 
 4 5 
 
 6 
 
 
 5 
 
 
 3 
 
 1 6 
 
 3 6 
 
 5 
 
 
 6 
 
 
 7 
 
 5 6 
 
 5 4 
 
 5 
 
 
 3 
 
 
 4 
 
 5 5 
 
 8 4 
 
 2 
 
 
 2 
 
 
 2 
 
 1 2 
 
 4 
 
 From 
 
 
 13 
 
 
 13 
 
 13 12 
 
 13 12 
 
 Take 
 
 
 8 
 
 
 2 
 
 6 10 
 
 9 7 
 
 From 
 
 
 12 
 
 
 13 
 
 13 13 
 
 12 13 
 
 Take 
 
 
 5 
 
 
 7 
 
 4 5 
 
 9 3 
 
FOURTEEN — 14. 
 2 3 
 
 39 
 
 • • •• •••• •• 
 
 20. How many dots are there in each of these figures? 
 
 What two numbers malve fourteen in the first figure ? 
 In the second ? In the tliird ? In the f ourtli ? 
 
 How many days are there in two weeks ? 
 
 How many more days are there in two wrecks than in 
 one week? 
 
 At seven cents apiece, how many tops can I buy with 
 fourteen cents ? 
 
 Fourteen geese went down to the pond. Seven of 
 them soon came back. How many remained at the pond ? 
 
 With your blocks, count by twos to fourteen. 
 
 Begin with one block, and count by twos to thirteen. 
 
 I have fourteen cents in two-cent pieces. Hoav many 
 tv.^o-cent pieces have I ? 
 
 If you had a dime, how many more cents would it 
 take to make fourteen cents ? 
 
 How many cents do you need to put with a half-dime 
 to make fourteen cents ? 
 
 How many pints are there in seven quarts ? 
 
 How many quarts are there in fourteen pints ? 
 
 A foot and two inches are how many inches ? 
 
 Kobert sharpened fourteen j^ei^cils for me. The 
 points of two of them w^ere broken. How many Avere 
 left with good points ? 
 
 I have two three-cent pieces. How many more cents 
 will it take to make fourteen cents ? 
 
 Marie is 8 years old to-day. In how many years will 
 she be 14 ? 
 
40 ELEMENTS OF ARITHMETIC. 
 
 
 
 SLATE 
 
 WORK. 
 
 
 2x7=? 
 
 7 + 
 
 ? = 14 
 
 Supply the Signs. 
 5 9 14 
 
 ? + ?=14 
 
 6 + 8=? 
 
 ? _ 
 
 10= 4 
 
 14 8 6 
 
 ?-? = 10 
 
 14-9=? 
 
 ? + 
 
 5 = 14 
 
 12 2 14 
 
 ? X ?=14 
 
 iofl4 = 
 
 ? 14- 
 
 ? = 11 
 
 14 4 10 
 
 ?-?= 7 
 
 7)14(? 
 
 14- 
 
 ?= 2 
 
 14 7 7 
 
 ? + ? = 12 
 
 7x.2=? 
 
 5x 
 
 2= ? 
 
 12-6= ? 
 
 11-?= ? 
 
 2)14(? 
 
 1)14(? 
 14)14(? 
 14x1=? 
 
 ? X 
 
 4x 
 
 '3x 
 3x 
 
 2 = 12 
 
 2= ? 
 ?= 6 
 3= ? 
 
 12-4= ? 
 12-3= ? 
 12 — 2= ? 
 12-1= ? 
 
 12—?= ? 
 13-?= ? 
 14-?= ? 
 10-?= ? 
 
 Add: 
 
 
 
 
 
 2 5 
 
 3 4 
 
 7 5 
 
 3 
 
 7 
 4 
 
 4 
 3 
 
 7 
 
 1 6 
 
 3 2 
 8 • 6 
 
 2 3 
 7 4 
 5 7 
 
 From 
 Take 
 
 14 
 
 8 
 
 13 
 5 
 
 14 12 
 
 10 7 
 
 14 13 
 
 9 8 
 
 21. Take your foot rule and draw a line on the board 
 just as long as the rule. How long is the line ? 
 
 Add to this line another of the same length, and 
 then another. How long is the whole line now ? 
 
 Take the yard-stick and measure the line with it. 
 How does it compare in length with a yard ? 
 
 How many feet are there in a yard of anything ? 
 
 Measure off a yard on the edge of the table. 
 
 What is always measured by the yard ? 
 
 How many feet are there in two yards of string ? 
 
 A room twelve feet long will need how many yards 
 of carpet to extend the length of the room ? 
 
FIFTEEN — 15. 41 
 
 22. How many rows of squares 
 are there in this figure ? 
 
 How many squares are there in 
 each row? 
 
 How many squares are there in 
 the three rows ? 
 
 Tell from the picture how many threes make fifteen ? 
 How many fives ? 
 
 I have a dime and a half-dime. For how many cents 
 can I exchange them ? 
 
 Look at the figure given above and tell me what a 
 third of fifteen squares is. 
 
 In fifteen feet, how many yards are there ? 
 
 There were fifteen bananas in a bunch. After ten 
 were sold how many were left ? 
 
 What three pieces of money make fifteen cents ? 
 
 If a pound of beef costs ten cents, what will a pound 
 and a half cost ? 
 
 Count by threes to fifteen. 
 
 Write on your slate in a vertical line enough threes 
 to make fifteen. How many did you write ? 
 
 Five triangles have how many sides ? 
 
 A dozen and three oranges are how many oranges? 
 
 Johnny is fifteen months old. Hoav many months 
 more than a year is that ? 
 
 I bought three oranges for fifteen cents. How much 
 did I pay for each orange ? 
 
 Three times what number equals fifteen ? 
 
 Five times what number equals fifteen ? 
 
 A boy ran on three errands, receiving five cents for 
 each. How much did he earn ? 
 
 How many eggs are a dozen and a quarter eggs? 
 
42 
 
 ELEMENTS OF ARITHMETIC. 
 
 23. Make a figure like the one on the previous page 
 without the two inside horizontal lines. 
 
 Into how many equal parts is 
 this figure divided? 
 
 Each of these parts is called a 
 
 fifth. 
 
 How many fifths are there in 
 the figure which you drew ? 
 
 How many fifths are there in any figure ? Li any line ? 
 
 If a pie were divided into five equal parts and you 
 were given one of the parts, what part of the pie would 
 you receive ? 
 
 Now you may draw the two inside horizontal lines 
 as shown in the previous figure. How many squares 
 have you now made ? 
 
 How many squares are there in a fifth of the figure ? 
 
 What is a fifth of fifteen sheep ? 
 
 Separate ten blocks into five equal parts. What is a 
 fifth of ten blocks ? 
 
 This is the way we write a fifth : \. 
 
 
 SLATE WORK. 
 
 
 10+ 5=? 
 
 15- 3=? 
 
 T+? = 15 
 
 15-?= 5 
 
 15- 5=? 
 
 5x 3=? 
 
 9+? = 15 
 
 15-?= 3 
 
 15-10=? 
 
 iofl5=? 
 
 15-? = 11 
 
 15-?= 1 
 
 5 + 10=? 
 
 15- 3=? 
 
 15-?= 5 
 
 15-? = 15 
 
 3x 5=? 
 
 15-12=? 
 
 13+? = 15 
 
 3x ? = 15 
 
 iofl5=? 
 
 12+ 3=? 
 
 15-?= 8 
 
 5x '.^ = 15 
 
 15-f 5=? 
 
 3 + 12=? 
 
 8+? = 15 
 
 Ix ? = 15 
 
SIXTEEN — 16. 
 
 43 
 
 24. Count the plates in these two piles. 
 
 How many are there in each pile ? 
 
 How many are there in both piles ? 
 
 If these plates were in two equal piles, how many 
 would there be in each ? 
 
 Take sixteen pieces of paper and imagine that they 
 are plates. Place them in two equal piles. What is 
 one half of sixteen plates ? 
 
 There were sixteen children in a class. One stormy 
 day one half of them were absent. How many were 
 present ? 
 
 How many shoes do four horses need ? 
 
 How many oxen can be shod with sixteen shoes, if 
 each ox requires eight shoes ? 
 
 With your blocks count by twos to sixteen. 
 
 Write on your slates in a vertical line a sufficient 
 number of twos to make sixteen. How many twos are 
 there in sixteen ? 
 
 Begin with sixteen and count backward by twos. 
 
 There are four quarts in a gallon. Prove it. 
 
44 ELEMENTS OF ARITHMETIC. 
 
 How many quarts are there in two gallons? 
 How many quarts are there in three gallons ? 
 In sixteen quarts, how many gallons are there ? 
 What part of a gallon is a quart ? 
 How many pints are there in a gallon ? 
 
 
 SLATE 
 
 WORK. 
 
 
 4x4= ? 
 
 iofl6= ? 
 
 16- 9 = ? 
 
 3 + ? = 16 
 
 8x2= ? 
 
 I of 16= ? 
 
 16- 4 = ? 
 
 . ?-8= 8 
 
 2x? = lG 
 
 8+? = 16 
 
 16-10 = ? 
 
 ? + 5 = 16 
 
 16-^2= ? 
 
 16-7= ? 
 
 16-12 = ? 
 
 ?-h9 = 16 
 
 16-4= ? 
 
 ? + 4 = 16 
 
 16- 5 = ? 
 
 16-?= 5 
 
 16h-8= ? 10 + 6= ? 16- 6 = ? 16-?= 3 
 
 Add: 
 
 3 
 
 4 
 
 
 8 
 
 5 
 
 
 6 
 
 4 
 
 4 
 
 5 
 
 
 2 
 
 6 
 
 
 3 
 
 2 
 
 2 
 
 3 
 
 
 2 
 
 3 
 
 
 2 
 
 3 
 
 7 
 
 3 
 
 
 4 
 
 2 
 
 
 4 
 
 4 
 
 Subtract : 
 
 
 
 
 
 
 
 
 16 
 
 16 
 
 
 15 
 
 15 
 
 
 16 
 
 16 
 
 10 
 
 15 
 
 
 8 
 
 4 
 
 
 9 
 
 5 
 
 Answer instantly : 
 
 
 
 
 
 
 7+ 8 
 
 
 15- 
 
 -5 
 
 lof 16 
 
 
 3x 3 
 
 9+ 4 
 
 
 16- 
 
 ^1 
 
 lof 
 
 8 
 
 
 4x 4 
 
 8- 5 
 
 
 16- 
 
 -8 
 
 lof 
 
 4 
 
 
 2x 2 
 
 16- 9 
 
 
 16- 
 
 -4 
 
 lof 
 
 2 
 
 
 3+ 8 
 
 14-11 
 
 
 16- 
 
 -2 
 
 1 of 14 
 
 
 8+ 8 
 
 12+ 4 
 
 
 14- 
 
 ^2 
 
 lof 
 
 6 
 
 
 5+ 8 
 
 3x 4 10-2 lof 12 11- 7 
 
 8x 2 12-2 16-12 13- 6 
 
ONE EIGHTH. 45 
 
 25. Into how many parts is this line divided? How 
 do these parts compare in size ? 
 
 Each of these equal parts is called an eighth. 
 
 How many eighths are there in this square ? 
 
 Point to an eighth of this square. 
 
 An eighth is written in this way — -J. 
 
 Separate eight blocks into eight equal parts. 
 
 What is an eighth of eight blocks ? 
 
 If you should divide sixteen pears equally 
 
 among eight boys, what part of them would each boy 
 get? 
 
 How many would each boy get ? 
 
 What is an eighth of sixteen horses ? 
 
 A farmer had sixteen tons of hay. If he should sell 
 an eighth of it, how many tons would he have left ? 
 
 Draw a square. Divide it into halves. Now divide 
 your square into four equal parts. 
 
 What is each part called? 
 
 How many fourths are there in the square ? 
 
 How many fourths are there in half of the square ? 
 
 What, then, is another name for half of the square ? 
 
 What other name might we give to two fourths of a 
 pie? 
 
 Now divide this square into eight equal parts. What 
 is each part called ? 
 
 How many eighths are there in the square ? 
 
 How many eighths are there in half of the square ? 
 
 How many eighths are there in a fourth of the square ? 
 
 Tell me all the names you know for a half of anything. 
 
 Give me one other name for a fourth of anything. 
 
46 
 
 ELEMENTS OF ARITHMETIC. 
 
 ••• •• •••• 
 
 •• •• •••• 
 
 • • • • e • 
 
 • • • • • • 
 
 26. Copy these figures on your slate. 
 
 Fill in the blank half of each with enough dots to 
 make seventeen in each one. 
 
 Name three pieces of money that will make seventeen 
 cents o 
 
 There were seventeen pine trees in a field. If seven 
 of them were chopped down, how many were left stand- 
 ing? 
 
 In front of my house there are nine maple trees on 
 one side of the street and eight on the other. How 
 many are there on both sides ? 
 
 It is seventeen miles from one station to the next. 
 When the train has gone eleven miles from one station, 
 how many miles has it to go before it reaches the other ? 
 
 A house has six windows in one side, seven in another, 
 and two in each end. How many windows are there in 
 the house? 
 
 A man earned seventeen dollars a week and spent 
 twelve dollars of it. How much had he left ? 
 
 Mr. White sold his turkeys for seventeen dollars. 
 How many sheep at five dollars apiece could he buy 
 with the money ? How much would he have left ? 
 
 A newsboy who bought seventeen papers sold all but 
 five of them. How many did he sell ? 
 
 Two boys start from the same place. One walks eight 
 miles in one direction, the other walks nine miles in the 
 other direction. How far are they apart then ? 
 
SEVENTEEN — 17. 
 
 47 
 
 
 SLATE WORK. 
 
 
 
 17- 2=? 
 
 12-f-?-17 
 
 17^ 5=? 
 
 
 15 = 17-? 
 
 IT- 8 = ? 
 
 17-?= 7 
 
 17- 
 
 -10 = ? 
 
 
 17 = 10-f? 
 
 17-f- 2=? 
 
 17-? = 13 
 
 17- 
 
 - 8 = ? 
 
 
 17 = 2x8-h? 
 
 17-^ 3 = ? 
 
 ? + 8 = 17 
 
 17- 
 
 - 6 = ? 
 
 
 17 = 3x5-f? 
 
 17-f- 4 = ? 
 
 6-f-? = 17 
 
 17- 
 
 - 9 = ? 
 
 
 17 = 7x2-f? 
 
 7-^10 = ? 
 
 17-? = 10 
 
 17- 
 
 - 7=? 
 
 
 17-4x3-f? 
 
 Add: 
 
 
 
 
 
 2 
 
 5 4 
 
 3 6 
 
 9 
 
 4 
 
 3 
 
 4 8 
 
 7 
 
 2 
 
 1 
 
 3 
 
 Add: 
 
 12 
 
 5 
 
 15 
 
 2 
 
 7 
 10 
 
 6 
 11 
 
 Answer instantly : 
 
 9-1- 4 
 
 7-f 8 
 6-fll 
 
 4-f- 8 
 
 5-f 9 
 
 9h- 8 
 
 6+ 4 
 
 12- 5 
 15- 8 
 17- 9 
 16-12 
 11- 5 
 
 13- 8 
 
 14- 7 
 
 7x2 
 4x3 
 3x3 
 6x2 
 4x4 
 2x5 
 
 5 
 12 
 
 4 
 13 
 
 9 
 
 8 
 
 9 
 
 6 
 
 7 
 
 5 
 
 8 
 
 6 
 
 5 
 
 4 
 
 8 
 
 6 
 
 5 
 
 8 
 
 Oat of 
 
 17 
 
 14 
 
 15 
 
 16 
 
 17 
 
 17 
 
 Take 
 
 8 
 
 8 
 
 10 
 
 7 
 
 16 
 
 3 
 
 1)16( 
 2)16( 
 4)16( 
 8)16( 
 16)16( 
 5)15( 
 7)14( 
 
48 
 
 ELEMENTS OF ARITHMETIC. 
 
 :^^^==-^ 
 
 h-#2- j»r^^:^ — " '^^^^mr^ 
 
 
 °^======:^— — --— — ^^ 1\ 
 
 1 
 
 27. How many birds are resting on these wires? 
 
 How many are there in each group ? 
 
 If one from the larger group should join the smaller 
 group, how many would there be in each group then ? 
 
 There are eighteen windows in a house. If ten of 
 Miem are in the sides of the house, how many are there 
 in the ends ? 
 
 Mary's rose-bush had eighteen rose-buds on it, and 
 she picked twelve of them. How many rose-buds were 
 left on the bush? 
 
 Lizzie made three button-holes each day for a week, 
 beginning on Monday. How many had she made by 
 Saturday night? 
 
 At three cents apiece, how many peaches can I buy 
 with eighteen cents? 
 
 Fred learned six verses of a poem each day for three 
 days. How many verses did he learn ? 
 
 What part of the eighteen verses did he learn each 
 day? 
 
 A dozen and a half eggs are how many eggs ? 
 
 If lace is worth a cent an inch, what will eighteen 
 inches of it cost? 
 
ONE SIXTH. 
 
 49 
 
 28. Into how many parts is this figure divided ? 
 
 Each part is called a sixth. 
 
 How many sixths are there in it? 
 
 How many sixths are there in a pie ? In a melon *? 
 
 A sixth is written in this way — -J . 
 
 Make a picture of a pie and divide it into sixths. How 
 many sixths are there in half a pie ? In one third ? 
 
 If you cut a card that is six inches long into sixths, 
 how long will each piece be ? 
 
 What is a sixth of twelve words ? 
 
 John's overcoat cost eighteen dollars. His hat cost 
 one sixth as much. What did his hat cost ? 
 
 Draw a line. Divide it into halves, and then into 
 sixths. 
 
 Draw another line of the same length. Divide it into 
 thirds, and then into sixths. 
 
 How many sixths are there in half of the line ? 
 
 How many sixths are there in a third of the line ? 
 
 One half is how many fourths ? How many sixths ? 
 How many eighths ? 
 
 Express with figures one sixth, two sixths, five sixths, 
 three sixths. 
 
 Write all the expressions that you know which mean 
 the same as one half. 
 
 Write all the expressions that you know which mean 
 the same as one fourth. 
 
 How many sixths are 1 sixth and 3 sixths ? 
 How many eighths are 3 eighths and 4 eighths ? 
 
 EL, OF AR. — ^4 
 
50 ELEMENTS OF ARITHMETIC. 
 
 
 
 SLAT! 
 
 : WORK. 
 
 
 
 
 18-2 = ? 
 
 6X?: 
 
 = 18 
 
 
 18- 6. 
 
 = ? 
 
 
 iof 16 = ? 
 
 18^3 = ? 
 
 3X?: 
 
 = 18 
 
 
 18- 9. 
 
 = ? 
 
 
 lof 16 = ? 
 
 18-6 = ? 
 
 9X?: 
 
 = 18 
 
 
 18-12 = 
 
 = 9 
 
 
 iof 16 = ? 
 
 18-9 = ? 
 
 2X?: 
 
 = 18 
 
 
 18-15 = 
 
 = 9 
 
 
 iof 15 = ? 
 
 -J- of 18=? 
 
 9+?: 
 
 = 18 
 
 
 18- 7 = 
 
 _ 9 
 
 
 iof 12 = ? 
 
 iof 18 = ? 
 
 10+?: 
 
 = 18 
 
 
 18- 3 = 
 
 =3 ? 
 
 
 Iof 6 = ? 
 
 iof 18 = ? 
 
 8+? = 
 
 = 18 
 
 
 18- 5 = 
 
 33 ? 
 
 
 Iof 9 = ? 
 
 12 + ? = 18 
 
 9 + 8 
 
 = 2 
 
 
 iof? = 
 
 = 2 
 
 
 11-7=? 
 
 18-?= 2 
 
 14-6 
 
 = 9 
 
 
 iof? = 
 
 = 6 
 
 
 8 + 4 = ? 
 
 6 + ? = 18 
 
 6 + 9: 
 
 -7 
 
 
 lof? = 
 
 = 4 
 
 
 14-7 = ? 
 
 18-? = 14 
 
 8 + 5: 
 
 =2 
 
 
 lof? = 
 
 = 3 
 
 
 6x3 = ? 
 
 18-? = 10 
 
 b + 6. 
 
 = 9 
 
 
 lof? = 
 
 = 5 
 
 
 4 + 9 = ? 
 
 7+?=18 
 
 9 + 7: 
 
 = ? 
 
 
 4of?= 
 
 = 1 
 
 
 1 of 15 = ? 
 
 Add: 
 
 
 
 
 
 
 
 
 4 5 
 
 2 
 
 
 3 
 
 6 
 
 
 7 
 
 6 
 
 3 2 
 
 9 
 
 
 7 
 
 4 
 
 
 1 
 
 3 
 
 6 T 
 
 2 
 
 
 2 
 
 5 
 
 
 4 
 
 4 
 
 5 4 
 
 4 
 
 
 4 
 
 3 
 
 
 5 
 
 4 
 
 3 4 
 
 5 
 
 
 4 
 
 5 
 
 
 5 
 
 6 
 
 6 6 
 
 8 
 
 
 1 
 
 1 
 
 
 2 
 
 2 
 
 7 2 
 
 2 
 
 
 3 
 
 8 
 
 
 7 
 
 5 
 
 2 3 
 
 1 
 
 
 6 
 
 4 
 
 
 2 
 
 1 
 
 Subtract : 
 
 
 
 
 
 
 
 
 18 
 
 18 
 
 18 
 
 
 18 
 
 18 
 
 
 18 
 
 17 
 
 10 
 
 14 
 
 
 9 
 
 7 
 
 
 5 
 
ISmSTETEEN — 19. 
 
 51 
 
 29. How many doves do 
 you see in this picture ? 
 
 What two numbers make 
 nineteen ? 
 
 Take nineteen blocks. I 
 will name one part of nine- 
 teen, and you may find out 
 the other part with your 
 blocks: 12; 6; 8; 15; 5; 
 16; 7; 9; 11; 4; 13; 10; 
 14; 3; 17. 
 
 Two weeks and five days 
 are how many days ? 
 
 Alice is nineteen years old. 
 How old was she ten years 
 ago? 
 
 There were nineteen cars at 
 the station. After an engine 
 had drawn away twelve of 
 them, how many were left? 
 
 Out of nineteen pupils, five 
 were absent. How many 
 were present? 
 
 In a house there are nine- 
 teen rooms. If eight are up- 
 stairs, how many are down- 
 stairs ? 
 
 How many pairs of boots at two dollars a pair can I 
 buy with nineteen dollars ? How much shall I have left ? 
 How many triangles can you make with nineteen 
 sticks ? How many sticks will be left ? 
 How many are nineteen less ten? 
 
52 ELEMENTS OF ARITHMETIC 
 
 How many gallon cans can be filled with nineteen 
 quarts of milk ? How many quarts will be left ? 
 
 How many five-cent stamps can you buy with nine- 
 teen cents ? How many cents will be left ? 
 
 
 
 SLATE 
 
 WORK. 
 
 
 
 10 + 9 = ? 
 
 19- 
 
 -9 = ? 
 
 13 + ? = 19 
 
 
 19 = 3x5+? 
 
 12 + 7=? 
 
 19- 
 
 -4 = ? 
 
 19-? = 10 
 
 
 19 = 4x4 + ? 
 
 19-8 = ? 
 
 19- 
 
 -7=? 
 
 11 + ? = 19 
 
 
 19 = 5x2+? 
 
 19-6 = ? 
 
 19- 
 
 -5 = ? 
 
 19-? = 14 
 
 
 19 = 3x6+? 
 
 14 + 5 = ? 
 
 19- 
 
 -8 = ? 
 
 8 + ? = 19 
 
 
 19 = 7x2+? 
 
 19-7 = ? 
 
 19- 
 
 -6 = ? 
 
 19-?= 9 
 
 
 19 = 2x9 + ? 
 
 Answer ins 
 
 jtantly : 
 
 
 
 
 3x6 
 
 
 8 + 9 
 
 2x8 
 
 
 lofl2 
 
 15-^5 
 
 
 5 + 9 
 
 2x5 
 
 
 lof 8 
 
 13-6 
 
 
 6 + 9 
 
 2x3 
 
 
 -1- of 18 
 
 lofl8 
 
 
 4 + 9 
 
 2x9 
 
 
 i-ofl6 
 
 7 + 8 
 
 
 11 + 9 
 
 2x7 
 
 
 lof 10 
 
 5x3 
 
 
 15-9 
 
 2x4 
 
 
 iofl6 
 
 14-7 
 
 
 19-9 
 
 2x6 
 
 
 iof 8 
 
 17-9 
 
 
 14-9 
 
 .3x5 
 
 
 i of 12 
 
 iofl2 
 
 
 18-9 
 
 3x2 
 
 
 4 of 8 
 
 8 + 6 
 
 
 12-9 
 
 3x4 
 
 
 iofl6 
 
 Add: 
 
 
 
 
 
 
 2 
 
 6 
 
 3 
 
 4 
 
 3 
 
 5 
 
 3 
 
 4 
 
 5 
 
 4 
 
 3 
 
 T 
 
 5 
 
 2 
 
 6 
 
 4 
 
 3 
 
 3 
 
 4 
 
 3 
 
 4 
 
 4 
 
 3 
 
 2 
 
 5 
 
 4 
 
 1 
 
 3 
 
 6 
 
 2 
 
REVIEW EXERCISES. 63 
 
 30. REVIEW EXERCISES. 
 
 1 + 5=? 
 
 11 + 5=? 
 
 9-4 = ? 
 
 19-4=? 
 
 3 + 6 = ? 
 
 13 + 6 = ? 
 
 9-8 = ? 
 
 19-8=? 
 
 5 + 4-? 
 
 15 + 4 = ? 
 
 7-5=? 
 
 17-5=? 
 
 2 + 6 = ? 
 
 12 + 6 = ? 
 
 7-3=? 
 
 17-3 = ? 
 
 4 + 4 = ? 
 
 14 + 4 = ? 
 
 5-4=? 
 
 15-4=? 
 
 6 + 2 = ? 
 
 16 + 2 = ? 
 
 5-2 = ? 
 
 15-2 = ? 
 
 3 + 4 = *^ 
 
 13 + 4 = ? 
 
 8-7 = ? 
 
 18-7 = ? 
 
 7 + 2 = ? 
 
 17 + 2 = ? 
 
 8-5 = ? 
 
 18-5 = ? 
 
 4 + 2 = ? 
 
 14 + 2=? 
 
 6-4 = ? 
 
 16-4 = ? 
 
 5 + 3 = ^ 
 
 15 + 3 = ? 
 
 6-1 = ? 
 
 16-1=? 
 
 2 + 7 = ? 
 
 12 + 7 = ? 
 
 4-3 = ? 
 
 14-3 = ? 
 
 3 + 5=? 
 
 13 + 5 = ? 
 
 4-2 = ? 
 
 14-2 = ? 
 
 4 + 3 = ? 
 
 14 + 3 = ? 
 
 8-4 = ? 
 
 18-4 = ? 
 
 5 + 1 = ? 
 
 15 + 1 = ? 
 
 8-6 = ? 
 
 18-6 = ? 
 
 1 + 8 = ? 
 
 11 + 8 = ? 
 
 7-6 = ? 
 
 17-6 = ? 
 
 2 + 4 = ? 
 
 12 + 4 = ? 
 
 7-4 = ? 
 
 17-4=? 
 
 3 + 2 = ? 
 
 13 + 2 = ? 
 
 7-7 = ? 
 
 17-7=? 
 
 4 + 1 = ? 
 
 14 + 1 = ? 
 
 9-7 = ? 
 
 19-7=? 
 
 1 + 3=? 
 
 11 + 3 = ? 
 
 9-3 = ? 
 
 19-3=? 
 
 4 + 5 = ? 
 
 14 + 5 = ? 
 
 G-B = ? 
 
 16-5 = ? 
 
 6 + 3 = ? 
 
 16 + 3 = ? 
 
 6-2 = ? 
 
 16-2 = ? 
 
 1 + 7 = ? 
 
 11 + 7 = ? 
 
 3-2 = ? 
 
 13-2 = ? 
 
 1 + 4 = ? 
 
 11 + 4 = ? 
 
 4-1 = ? 
 
 14-1 = ? 
 
 8 + 3 = ? 
 
 11 + 6 = ? 
 
 9-6=? 
 
 13-5 = ? 
 
 6 + 1=? 
 
 13 + 3 = ? 
 
 7-2=? 
 
 14-4 = ? 
 
 8 + 5 = ? 
 
 12 + 5 = '^ 
 
 8-3 = ? 
 
 15-6 = ? 
 
 7 + 7 = ? 
 
 15 + 2 = ? 
 
 9-5 = ? 
 
 18-9 = ? 
 
54 ELEMENTS OF ARITHMETIC, 
 
 31. Take ten splints and tie them together. Take 
 ten more, and make another bundle. How many tens 
 have you? 
 
 Two tens are called tiventy. They are written ^0, 
 
 Take another bundle of ten. How many tens have 
 you now ? 
 
 Three tens, called thirty^ are written 30 ; four tens, 
 forty ^ are written ^0 ; five tens, fifty ^ are written 60 ; 
 six tens, %ixty^ are written 60 ; seven tens, seventy^ are 
 written /O ; eight tens, eighty^ are written SO ; nine 
 tens, ninety^ are written ^0. 
 
 Count by tens to ninety. 
 
 Write the numbers from ten to ninety on your slate. 
 
 Begin at 90 and count back by tens to 0. 
 
 Put away all but two bundles of tens. Ten and ten 
 are how many ? 
 
 How many fives are there in 10 ? How many in 20 ? 
 
 How many twos are there in 10? How many in 20? 
 
 Count by 2's to 20, writing the numbers on your slate, 
 thus: 0, 2, ^, etc. 
 
 Count by 4's in the same way. Count by 5's in the 
 same way. 
 
 A boy had 20 quarts of berries and sold one half of 
 them. How many did he sell? How many had he left? 
 
 There were 15 birds on a tree. How many more 
 must come to make 20 ? 
 
 How many dimes can you get for 20 cents ? 
 
 How many half dimes can you get for 20 cents ? 
 
TWENTY — 20. 55 
 
 There are two pints in a quart. How many quarts 
 are there in 20 pints? 
 
 Four quarts make a gallon. How many gallons are 
 there in 20 quarts ? 
 
 
 
 SLATE 
 
 WORK. 
 
 
 10- 
 
 -7=? 
 
 10-6 = ? 
 
 lof 10 = ? 
 
 1x2 = ? 
 
 20- 
 
 -7 = ? 
 
 20-6 = ? 
 
 lof 20 = ? 
 
 2x2 = ? 
 
 10- 
 
 -4 = ? 
 
 10-5 = ? 
 
 -iof 10 = ? 
 
 3x2 = ? 
 
 20- 
 
 -4 = ? 
 
 20-5 = ? 
 
 lof20 = ^ 
 
 4x2 = ? 
 
 10- 
 
 -2 = ? 
 
 20-5 = ? 
 
 10- 1 = ? 
 
 5x2 = ? 
 
 20- 
 
 -2 = ? 
 
 20-2 = ? 
 
 20- 1 = ? 
 
 6x2=? 
 
 10- 
 
 -8 = ? 
 
 1x5=? 
 
 20- 4 = ? 
 
 7x2 = ? 
 
 20- 
 
 -8 = ? 
 
 2 5's = ? 
 
 20-10 = ? 
 
 8x2 = ? 
 
 10- 
 
 -3=? 
 
 3 5\s = ? 
 
 10-f- 4 = ? 
 
 9x2 = ? 
 
 20- 
 
 ~3 = ? 
 
 4 5's = ? 
 
 10-10 = ? 
 
 10x2 = ? 
 
 11_4+ 6 = ? 20-7 + 6 = ? 13-9 + 4 = ? 20-8-4 = ? 
 
 11_8+ 5 = ? 20-2-9 = ? 20-9-4 = ? 16-7-8 = ? 
 
 17_8+ 6 = ? 20-7 + 6 = ? 11-6 + 8 = ? 19-2-8 = ? 
 
 12-6+ 7 = ? 12-7 + 6 = ? 20-5-9 = ? 20-7-5 = ? 
 
 15-6+ 7 = ? 14-7 + 9 = '? 20-9-8 = '? 19-8-7 = ? 
 
 20-6+ 5 = ? 20-9 + 4 = ? 13-5 + 9 = ? 18-8-6=? 
 
 11_1+ 8 = ? 20-6-8 = ? 18-9 + 3 = ? 15-9-3=? 
 
 17_9 + 10 = ? 19-2 + 2 = ? 20-8-7 = ? 20-3-8 = ? 
 
 14_8+ 7 = ? 13-5 + 8=? 17-9 + 3 = ? 18-8-6 = ? 
 
 13_94_11:=? 20-9-2 = ? 20-4-8 = ? 13-7-2 = ? 
 
 15_6 + 10 = ? 18-8 + 6 = ? 20-7-3 = ? 19-8 + 2 = ? 
 
 19_5+ 6 = ^ 20-6+4 = -:> 20-8-6 = ? 18-8-5 = ? 
 
 20-8- 3 = ? 19-7-2 = ? 15 + 4-7 = ? 12 + 8-5=? 
 
56 
 
 ELEMENTS OF ARITHMETIC. 
 
 TENS AND ONES. 
 
 □ □□iniaaai aniLlLuLQQQDLaL rn ni ni n run m 
 
 32. Look at the first picture. How many piles of 
 ten blocks do you see ? How many other blocks ? 
 
 This number is called twenty-one^ and it is written ^/, 
 
 How many tens do you see in the second picture? 
 How many ones ? 
 
 This number is called thirty-three^ and it is written 33, 
 
 How many tens are there in the next picture ? How 
 many ones ? 
 
 This number is cdXl^di forty-jive^ and it is written ^5. 
 
 How many tens are there in the last picture ? How 
 many ones ? 
 
 This number is called jifty-ttvo^ and is written 6^. 
 
 We have learned to read tens by calling a number 
 expressing 3 tens thirty^ four tens/or^?/, etc. ; and when 
 a number expresses tens and ones, we read it by naming 
 the tens first and then the ones^ without any words 
 between them. 
 
 We can tell whether a figure stands for ones or for 
 tens by the place it occupies. If it represents ones^ it 
 has the first place to the rights if te7is the next to the left. 
 
 Read the following numbers by telling how many tens 
 and how many ones there are. 
 
 Thus : 25 is 2 tens and 5 ones, or twenty-five. 
 37 14 83 96 12 20 21 38 82 29 30 26 
 99 31 24 76 33 58 16 23 25 67 52 95 
 68 77 53 74 83 76 39 26 34 51 90 50 
 
TWENTY-ONE — 21. 57 
 
 Begin with one and add two each time until you reach 
 twenty-one. 
 
 Count by threes to 21. 
 
 Three feet make 1 yard. If John has 21 feet of 
 string for his kite, how many yards has he ? 
 
 How many weeks are there in 21 days ? 
 
 If Kate makes 7 loaves of cake every day for 3 days, 
 and sells on one day 8 loaves and on another day 6, how 
 many has she left? 
 
 What is i of 21 ? What is | of 21 ? 
 
 If Fred buys 15 papers of one kind and 6 of another, 
 and sells 8 of them, how many has he left ? 
 
 How much must Jamie pay for 3 balls, if their prices 
 are 6 cents, 9 cents, and 4 cents? How many cents 
 will he have left, if he had 21 cents at first ? 
 
 Harold worked 10 examples one day, 7 the next day, 
 and 4 the next. How many did he work in the three days ? 
 If all but four were correct, how many were correct ? 
 
 If I have 21 nuts, to how many children can I give 7 
 each ? 
 
 If I rode north 6 miles, then east 11 miles, then south 
 6 miles, how far was I from the starting-place ? 
 
 If I traveled from home 11 miles on Monda}^ 7 miles 
 farther on Tuesday, and 3 miles on Wednesday ; and 
 on my way back on Thursday walked 10 miles and on 
 Friday 6 miles, how many miles was I then from home ? 
 
 9 + 2 = ? 
 
 49 + 2 = ? 
 
 21-7 = ? 
 
 3x3 = ? 
 
 7 + 4 = ? 
 
 37 + 4 = ? 
 
 iof21=^ 
 
 4x3=? 
 
 8 + 3 = ? 
 
 78 + 3 = ? 
 
 |of 21 = ^ 
 
 5x3 = ? 
 
 19 + 2=? 
 
 79 + 2 = ? 
 
 lof 21 = ^ 
 
 6x3=? 
 
 17 + 4 = ? 
 
 47 + 4 = ? 
 
 f of 21 = ? 
 
 7x3 = ^ 
 
 18 + 3 = ? 
 
 58 + 3=? 
 
 1 of 21 = ? 
 
 1x7 = ? 
 
58 ELEMENTS OF ARITHMETIC. 
 
 RAPID WORK. 
 
 8-1-3 + 5-7-3 + 10-8 + 6-9 + 1 + 8 = ? 
 
 13-6 + 7-5-2 + 7 + 3-10-6 + 8 + 7-8 + 9=? 
 
 11 + 8-7-6 + 13-10-7 + 6 + 8-9-3 + 8 + 9 = ? 
 
 21-2-7-4-5+6+8-9+4+9-6+7-4=? 
 
 13 + 5 + 2-17 + 6 + 5-7 + 9-11 + 15-13 + 9-8 + 4 = ? 
 
 15-6 + 7-9 + 12-11 + 6-9-2 + 16-13 + 15-10 = ? 
 
 7 + 8-9 + 12-11 + 6-9-2 + 16-13 + 15-18 + 1 = ? 
 
 33. Twenty and two are how many ? 
 
 Count by twos to twenty-two. 
 
 What is one half of twenty-two ? 
 
 How many figures are required to write twenty-two ? 
 
 For what does the left hand figure stand ? 
 
 For what does the right hand figure stand ? 
 
 Donald has 3 pansies, 4 pinks, 5 violets, 6 roses, and 
 4 lilies. How many flowers has he ? 
 
 There are 22 children in Fred's class. If 11 stand in 
 a row, how many rows are there ? 
 
 22-2-2-2, etc., to 18 + ?=22 20 + ? = 22 
 
 22-3-3-3, etc., to 1 13 + ? = 22 17+? = 22 
 
 22-4-4-4, etc., to 2 15+? = 22 14 + ?=22 
 
 22-5-5-5, etc., to 2 19 + ? = 22 16+?=22 
 
TWENTY-TWO — 22. 59 
 
 How many two-cent postage stamps can I buy for 22 
 cents ? For 20 cents ? For 10 cents ? For 18 cents ? 
 
 1 17 
 
 2 15 This is the way Emma adds a column and tests it to 
 
 3 12 see if it is correct. 
 
 2^ ^\ She found the sum to be 18. She then proved her 
 
 o q result correct by taking in succession each number 
 
 n from 18, thus: 1 from 18 leaves 17 ; 2 from 17 leaves 
 ^ ^ 15 ; 3 from 15 leaves 12 ; 1 from 12 leaves 11 ; 2 from 
 
 1 ^11 leaves 9, etc. After taking away the last number, 
 
 2 3 was left. Test your results in the same way in the 
 
 3 following examples. 
 
 18 
 
 Add and test : 
 
 8231 30 20 3321 
 
 2332 20 30 4323 
 
 2 2 2 3 10 30 3 3 2 3 
 
 2 1 3 2 30 20 4 3 2 3 
 3312 20 10 3323 
 2233 10 20 4323 
 2223 30 30 3323 
 1233 20 30 4323 
 
 3 4 1 3 10 20 3 3 2 3 
 
 The sign $ stands for dollars. 
 
 14 IT $5 6 8 5 9 4 3 2 1 
 42534237136 
 47565426442 
 42533633858 
 42021311243 
 12121111141 
 
 22-i-2 = ? 22^11 = ? 1 of 22 = ? J- of 22 = ? 
 ^\of22 = ? -j\of22 = ? j^of22 = ? ^^^ of 22 = ? 
 
60 ELEMENTS OF ARITHMETIC. 
 
 34. 20 boys were snow-balling and 3 more joined 
 tliem. How many were there then ? 
 
 If each boy threw 1 snow-ball, how many snow-balls 
 were thrown? 
 
 If all but 3 hit the mark, how many snow-balls hit the 
 mark ? 
 
 9+ 4 = ? 
 
 10 + 4 = ? 
 
 29 + 4 = ? 
 
 39 + 4=? 
 
 6+ T = ? 
 
 16 + 7 = ? 
 
 26 + 7=? 
 
 36 + 7 = ? 
 
 8+ .5 = ? 
 
 18 + 5 = ? 
 
 48 + 5=? 
 
 78 + 5=? 
 
 3 + 10 = ? 
 
 19 + 4 = ? 
 
 49 + 4 = ? 
 
 72 + 9 = ? 
 
 Write the numbers from 20 to 30; from 30 to 40; 
 from 19 to 10; from 50 to 60; from 50 to 40; from 80 
 
 to 90. 
 
 What is the largest number you can write, using but 
 one figure ? What is the smallest ? 
 
 What is the smallest number you can write, using 
 only two figures? 
 
 What is the largest number that can be written with 
 two figures ? 
 
 Add first by lines and then by columns : 
 2+5+3+4+1+2+3+3=? 
 6+4+3+3+1+3+2+3=? 
 5+5+5+5+2+2+2+2=? 
 4+4+4+4+4+1+1+0=? 
 1+2+4+3+4+3+4+3=? 
 
 Add, and test your results 
 
 as 
 
 before 
 
 ;: 
 
 
 
 
 2 3 3 
 
 2 
 
 4 
 
 4 
 
 6 
 
 Q 
 
 2 
 
 3 
 
 2 
 
 4 
 
 2 2 4 
 
 9 
 
 3 
 
 2 
 
 7 
 
 9 
 
 4 
 
 4 
 
 6 
 
 4 
 
 8 13 
 
 1 
 
 2 
 
 5 
 
 2 
 
 5 
 
 6 
 
 3 
 
 3 
 
 4 
 
 4 9 5 
 
 4 
 
 1 
 
 3 
 
 3 
 
 3 
 
 7 
 
 4 
 
 2 
 
 4 
 
 7 6 5 
 
 6 
 
 If 
 7 
 
 4 
 
 5 
 
 3 
 
 4 
 
 3 
 
 6 
 
 6 
 

 '/^^' 
 
 TWENTY-FOUR — 24. " ^-^Gl 
 
 If 23 passengers travel by coach and 7 are seated on 
 top, liow many are there inside ? 
 
 Morton earned 15 cents and his uncle gave him 8 
 cents more. He then bought 3 oranges at 3 cents apiece. 
 How many cents had he left ? 
 
 If 3'OU have 13 cents, how many more cents do you 
 need to buy a book that costs 20 cents ? 
 
 How many more cents do you need to buy one that 
 costs 23 cents? 
 
 Twenty-three children belong to Miss C's class, but 
 only 19 are present. How many are absent ? 
 
 There. were 6 persons in a car, when 7 more were 
 taken in; soon 5 left and 9 got on. How many were 
 there then in the car ? 
 
 If the car could seat 23 people, how many more could 
 be seated ? 
 
 If a lady goes to Boston with 23 dollars and spends 
 16 dollars, how many dollars has she left ? 
 
 35. Count by 2's to 24. 
 
 Count by 3's to 24, then back to 0. 
 
 Count by 4's to 24, then back to 0. 
 
 How many two-cent postage stamps can I buy for 24 
 cents ? 
 
 How many yards are there in 24 feet ? 
 
 How many gallons are there in 24 quarts ? 
 
 Twelve things make a dozen. How many dozen but- 
 tons has Ethel, if she has 24 buttons ? 
 
 Mr. B. had 24 eggs. He put them into boxes holding 
 6 eggs each. How many boxes did he fill ? 
 
 If the boxes had held 12 eggs, how many boxes would 
 he have used? 
 
62 
 
 ELEMENTS OF ARITHMETIC. 
 
 Write all the pairs of numbers, which, added together, 
 
 make 24. 
 
 24-2-3-4-4-3-6=? 
 
 24-2-2-2, etc., to 0. 
 
 24-3-3-3, etc., to 0. 
 
 Subtract each number less than 10 from 24 as many 
 times as you can, and write the results on your slate. 
 
 1x4 = ? 
 
 3x3 = ? 
 
 12- 
 
 -6 = ? 
 
 24- 
 
 - 4 = ? 
 
 2x4 = ? 
 
 4x3 = ? 
 
 12- 
 
 -2 = '^ 
 
 24- 
 
 - 8 = ? 
 
 3x4 = ? 
 
 5x3 = ? 
 
 9- 
 
 -3 = ? 
 
 24- 
 
 -12 = ? 
 
 4x4 = ? 
 
 6x3 = ? 
 
 18- 
 
 -3 = ? 
 
 24- 
 
 _ 2 = ? 
 
 5x4 = ? 
 
 7x3 = ? 
 
 12- 
 
 -4 = ? 
 
 1 of 24 = ? 
 
 6x4 = ? 
 
 8x3 = ? 
 
 16- 
 
 -4 = ? 
 
 iof24=? 
 
 1x6 = ? 
 
 1x8 = ? 
 
 24- 
 
 -3 = ? 
 
 fof 24 = ? 
 
 2x6 = ? 
 
 2x8 = ? 
 
 21- 
 
 =-3 = ? 
 
 4 of 24 = ? 
 
 3x6 = ? 
 
 3x8 = ? 
 
 8- 
 
 -4 = ? 
 
 fof 24 = ? 
 
 4x6 = ? 
 
 24-3 = ? 
 
 8- 
 
 -2 = ? 
 
 1 of 24 = ? 
 
 1x3 = ? 
 
 18-2 = ? 
 
 15- 
 
 -5 = ? 
 
 fof 24=? 
 
 2x3=? 
 
 24-6 = ? 
 
 12- 
 
 -3 = ? 
 
 iof24 = ? 
 
 Twelve inches make a foot. How many feet are 
 there in 24 inches ? 
 
 Harold has 2 dimes and 4 cents. How much money 
 has he ? How many bananas, at 6 cents each, can he 
 buy with his money ? 
 
 If he should prefer to buy oranges at 3 cents, how 
 many could he buy ? 
 
 If he should give half his money to his brother, how 
 much would each have ? 
 
 David walked 24 miles in 8 hours. How far did he 
 walk in one hour? 
 
 KachBl had 24 postal cards. She used 6 at one time, 
 
TWENTY-FIVE — 25. 63 
 
 5 at another time, and 7 at another time. How many 
 had she left ? 
 
 Daisy had 2 dozen dolls' cups. She piled them up, 
 placing half a dozen in a pile. How many piles had she ? 
 
 Harold, Reuben, James, and Burton have 6 cents each, 
 and each one buys a peach for 2 cents. How much 
 more money do they need to buy a ball for 20 cents? 
 
 A piece of cloth contained 24 yards ; 1 of it was sold. 
 How many yards were sold ? How many yards remained 
 unsold ? 
 
 Thornton bought 2 dozen postal cards. How much 
 did he pay for them ? If he gave the clerk 25 cents, how 
 much change should he receive ? 
 
 Mr. B, worked at his shop from 7 o'clock in the morn- 
 ing until 12 o'clock noon, and from 1 to 4 o'clock in 
 the afternoon, and at home he worked from 7 to 9 
 o'clock. How many hours did he work ? 
 
 36. Lucy has two dimes and a half-dime. To how 
 many cents are they equal ? 
 
 She spent all but 4 cents for 3 paper dolls. How 
 many cents did she spend ? What was the cost of each 
 doll? 
 
 If Lulu paid 25 cents for a slate and sponge, and the 
 sponge alone cost 6 cents, what was the price of the 
 slate ? 
 
 A milkman who had 25 quarts of milk put it in gal- 
 lon (4 quart) cans. How many cans did he fill, and 
 how much milk had he left ? 
 
 How many people could he supply with 1 quart a 
 day ? With 5 quarts ? With 3 quarts, and how much 
 would be left ? 
 
64 ELEMENTS OF ARITHMETIC. 
 
 Harold is 9 years old, and Arthur is 4 years younger. 
 In how many years will Arthur be 25 years old ? How 
 many years before Harold will be 25 ? 
 
 Daisy had 25 acorns, which she placed in 5 equal 
 piles. How many did she put in each pile ? 
 
 What is 1 of 25 cents? -J- of 25 children? -J- of 25 
 hens ? I" of 25 trees ? 
 
 Lena had a quarter of a dollar (25 cents), and spent 
 8 cents. How much had she left ? 
 
 The cook used 2 dozen eggs for the Christmas dinner. 
 If she had 25 eggs at first, how many had she left ? 
 
 How many half-dimes can you get in change for a 
 quarter of a dollar ? 
 
 24 + 1 = ? 25-4-4-4, etc. 
 
 25-2-2, etc. 25-5-5-5, etc. 
 
 25-3-3-3, etc. 25-6-6-6, etc. 
 
 1x5 = ? 6x 5 = ? 25-f- 1 = ? 20+? = 25 ?-fl6 = 25 
 
 2x5 = ? Tx 5 = ? iof25 = ? 15+? = 25 ? + 14 = 25 
 
 3x5 = ? 8x 5 = ? |of25 = ? 23 + ? = 25 ? + 21 = 25 
 
 4x5 = ? 25- 5 = ? f of25 = ? 5 + ? = 25 ? + ll = 25 
 
 5x5 = ? 25-10 = ? |of25 = '> 7-|-?=25 ?-M8 = 25 
 
 37. Two tens and six are how many? 
 
 If I have 26 pictures and give away one half of them, 
 how many do I give away? How many have I left? 
 2x? = 26 26-13 = ? 
 
 Write all the pairs of numbers you can whose sum 
 equals 26. 
 
 Agnes bought 26 buttons at the store. The buttons 
 were on cards, having 12 buttons each. How many 
 cards did she buy, and how many buttons besides ? 
 
TWENTY-SIX — 26. ^^ 
 
 In our school there are 23 pupils. On Saturday 5 
 boys and 3 girls went skating, 6 girls and 4 boys went 
 coasting, and the rest went to Boston. How many pupils 
 went to Boston ? 
 
 James solved 7 examples on Monday, 8 on Tuesday, 
 and 6 on Wednesday. How many did he solve in all ? 
 
 A pole stands 7 feet in the mud, 6 feet in the water, 
 and 9 feet above water. How long is the pole ? 
 
 I counted the birds on 4 telegraph wires. On the first 
 wire there were 6 birds, on the second, 5, on the third, 
 6, and on the fourth, 6. How many birds did I count? 
 
 A boy had 6 copper cents, 2 half-dimes, and a dime. 
 How much money had he ? 
 
 Ethel has five half-dimes and a cent. How much 
 money has she ? 
 
 How many Christmas cards, at 4 cents each, can Ethel 
 buy with her money, and how many cents will she have 
 left? 
 
 Frank is 26 years old, and Philip is 19. What is the 
 difference in their ages ? 
 
 A newsboy buys 16 Heralds and 10 Transcripts. He 
 sells 12 Heralds and 7 Transcripts. How many papers 
 has he yet on hand ? 
 
 A man had 26 gallons of vinegar. He first sold one 
 half of it, and later he sold 4 gallons more. How many 
 gallons did he sell ? How many had he left ? 
 
 A boy had 12 slate pencils and 11 lead pencils ; he 
 gave away half the slate pencils. How many pencils 
 had he left? 
 
 Count by 2's to 26. 
 
 Write the table of 2's to 26, thus : 
 
 1x2=? 2x2 = ? 3x2 = ? etc. 
 
 EL. OF AR. — 5 
 
ee 
 
 ELEMENTS OF AKITHMETIC. 
 
 38. If there are 10 marbles in each of the bags in 
 
 the picture, how many mar- 
 bles are there in all? 
 
 Two tens and 7 ones are 
 how many? 
 
 Two dimes and 7 cents are 
 how many cents ? 
 
 Fred had 26 apples in his 
 basket, and 1 in his hand. 
 How many had he ? 
 To how many children can he give 3 apples each? 
 9 each ? 5 each, and how many will be left ? 
 How many yards are there in 27 feet? 
 If I divide 27 pieces of candy equally among 3 chil= 
 dren, how many pieces will each have ? 
 
 What is one third of 27 ? Two thirds of 27 ? 
 A grocer having 27 barrels of flour, put them in 9 
 equal piles. How many barrels were there in each pile ? 
 What is one ninth of 27 melons ? Two ninths ? 
 If I have f of 27 dollars, how many dollars have I ? 
 A man having 27 cows sold | of them. How many 
 did he sell ? How many had he left ? 
 
 A fruit dealer found that ^ of his 27 baskets of peaches 
 
 was spoiled. How many were spoiled? 
 baskets of good peaches had he left ? 
 
 4x3 = ? 
 1x3 = ? 
 6x3 = ? 
 2x3 = ? 
 8x3 = ? 
 9x3 = ? 
 
 3x3 = ? 
 7x3 = ? 
 5x3 = ? 
 1x9 = ? 
 2x9 = ? 
 3x9 = ? 
 
 4x7 = ? 
 27-^3 = ? 
 
 27-^9 = ? 
 
 18-3 = ? 
 
 12-3 = ? 
 
 3)3( 
 
 6-3 = ? 
 24-3 = ? 
 
 24-8 = ? 
 
 18-3 = ? 
 3)9( 
 7)21( 
 
 How many 
 
 lof 27 = ? 
 iof 21 = ? 
 lof 15 = ? 
 
 n 
 
 J of 27 
 
 18^2 = ? 
 18-9 = ? 
 
TWENTY-SEVEN — 27. 67 
 
 Add by lines and columns : 
 
 3+3+3+3+3+3+3+3=^ 4+8+4+7+4+1=? 
 
 4+2+5+7+1+6+0+2=? 5+4+5+3+4+6=? 
 
 8+4+1+9+0+2+2+2=? 5+3+1+7+5+4=? 
 
 6+6+3+4+4+2+1+5=? 7+2+8+6+3+3=? 
 
 7+8+2+5+4+5+4+1=? 6+4+2+4+4+7=? 
 
 27 + 1 = ? 
 
 Count by 2's to 28. 
 
 Count by 4's to 28. 
 
 Count by 7's to 28. 
 
 How many tens and how many ones are there in 28 ? 
 
 If Nellie puts 28 peaches on four plates, putting the 
 same number on each plate, how many will there be on 
 each plate ? 
 
 If she gives away the peaches on one plate, what part 
 of the whole will she give away ? What is ^ of 28 ? 
 
 Max bought 7 pairs of shoes for 28 dollars. What 
 was the cost of 1 pair ? 
 
 What is I of 28? fof28? f of 28? 
 
 How many five-cent pieces and how many cents will 
 make 28 cents ? 
 
 Fred was sick for 28 days. How many weeks was 
 that? 
 
 Mro Brown had 28 pecks of potatoes, which he sold 
 by the bushel (4 pecks) o How many bushels did he 
 sell? 
 
 28 pints equal how many quarts ? 
 28 quarts equal how many gallons ? 
 
 William spent one half of his 28 cents for a slate and 
 5 cents for a lead penciL How much did he spend? 
 How much had he left ? 
 
68 ELEMENTS OF ARITHMETIC. 
 
 28-5-9-7-3-4=? 
 28-3-6-2-2-1=? 
 28-=9-2-8-l-7 = ? 
 28-4-4-4-4-4 = ? 
 
 28-T-T-T-T=? 
 
 Add and prove by subtracting as on page 59. 
 
 4 
 
 5 
 
 6 
 
 2 
 
 1 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 3 
 
 2 
 
 4 
 
 4 
 
 6 
 
 2 
 
 8 
 
 4 
 
 3 
 
 3 
 
 4 
 
 4 
 
 9 
 
 4 
 
 9 
 
 4 
 
 4 
 
 4 
 
 7 
 
 4 
 
 
 
 7 
 
 5 
 
 3 
 
 4 
 
 2 
 
 1 
 
 3 
 
 4 
 
 6 
 
 4 
 
 4 
 
 28 + 1 = '? 
 
 Beginning with 1 count by 2's to 29, 
 Beginning with 1 count by 4's to 29. 
 Beginning with 1 count by 7's to 29, 
 
 39. After giving 9 cents for some candy, Anita had 
 2 dimes remaining« How many cents did she have at 
 first? 
 
 At 9 cents a spool, how much will 3 spools of silk 
 cost ? 
 
 One day Alice worked 4 hours at 7 cents an hour, 
 and gave 10 cents for car fare. How many cents had 
 she left? 
 
 Henry had 29 cents, but spent 10 cents for a top. 
 How much had he left? How much did he lack of 
 having a quarter of a dollar? 
 
 A boy who had 9 cents went on 3 errands, earning 5 
 cents for each. How much money did he have then ? 
 
 An organ grinder received at one house 10 cents, at 
 
TWENTY-NINE — 29. 69 
 
 another 10 cents, and at another 9 cents. How much 
 did he receive from them all ? 
 
 Henry caught 15 fishes and Oren 14. How many did 
 both catch ? 
 
 After spending 19 cents for fishing tackle, a boy had 
 10 cents left. How much had he at first ? 
 
 Mary went to school 18 days in January and 11 in 
 February. How many days did she attend in those 
 months ? 
 
 29-5-4-3-6=? 
 29-7-3-4-2=? 
 29-8-8-8-3 = ? 
 29-5-6-4-9=? 
 29-8-3-9-9=? 
 
 40. Morton paid 3 dimes for a picture-book. What 
 did it cost ? 
 
 Lulu had 3 dimes, and spent one half her money. How 
 many cents did she spend ? 
 
 Susan's clothes-line is 10 yards long. How many 
 feet long is it ? 
 
 How many half-dimes can I get in exchange for 30 
 cents ? 
 
 How many oranges, at 5 cents apiece, can be bought 
 for 3 dimes ? 
 
 Lena bought half a dozen bananas at 5 cents each. 
 What did they cost ? 
 
 Lucy has 25 cents and a half-dime. How many quarts 
 of cherries, at 6 cents a quart, can she buy with the 
 money ? 
 
 Clarence lives 6 blocks from the schoolhouse. How 
 many blocks does he walk to and from school each day, 
 if he goes home to dinner ? 
 
70 ELEMENTS OF ARITHMETIC. 
 
 Some girls wanted to buy a game that cost 30 cents ; 
 3 gave 2 cents each ; 4 gave 3 cents each ; 2 gave 4 
 cents each. How many girls contributed? How much 
 more do they need to pay for the game ? 
 
 Leila bought 3 pounds of sugar at 7 cents a pound, 
 and gave the storekeeper 3 dimes. What change should 
 she receive ? 
 
 10 + ? = 30 18 + ? = 30 14 + ? = 30 
 
 17 + ? = 30 15-f? = 30 12 + ? = 30 
 
 20 + ?-30 19 + ? = 30 ll + ? = 30 
 
 25 + ? = 30 16 + ? = 30 30-8 = ? 
 
 From 30 take 2 as many times as you can, naming 
 results ; as, 28, 26, etc. 
 
 In the same way subtract 3, 5, 6, 7, 4, 8. 
 
 Anson had 30 rabbits, but 10 of them got away. How 
 many had he left ? 
 
 Mary had 2 dimes and a five-cent piece. After pay- 
 ing 10 cents for a composition book and 5 cents for a 
 pencil, how much had she left ? 
 
 Hannah attended school every school day for 30 days, 
 but her sister Sarah was absent 8 days during this time. 
 How many days did Sarah attend ? 
 
 A month is commonly regarded as 30 days. How 
 many days is half a month ? 
 
 The fare from Andes to Ray field is 15 cents. A 
 pupil's ticket from Andes to Rayfield and back is only 
 18 cents. How much cheaper is the pupil's ticket than 
 the regular fare ? 
 
 A boy who had 3 dimes paid 25 cents for a ticket to 
 a museum. How much money had he left? 
 
REVIEW EXERCISES. 71 
 
 REVIEW EXERCISES. 
 41. Answers to be given instantljo 
 
 iof 10 = ? 
 
 16-9 = ? 
 
 20-5 = ? 
 
 19-7=? 
 
 lof 20 = ? 
 
 15-3 = ? 
 
 24-4 = ? 
 
 3 3's=? 
 
 7x2 = ? 
 
 15-8 = ? 
 
 30-10 = ? 
 
 12-4 = ? 
 
 30-5 = ? 
 
 25-8 = ? 
 
 6 + 7=? 
 
 18-6 = ? 
 
 9 + 5 = ? 
 
 3 10's = ? 
 
 11-9 = ? 
 
 2x8 = ? 
 
 iof2T = ? 
 
 4)28( 
 
 4 + 7=? 
 
 12 2's = ? 
 
 |of27 = ^ 
 
 28-7 = ? 
 
 8-3 = ? 
 
 iof 30=? 
 
 30-3 = ? 
 
 13-9 = ? 
 
 18-3 = ? 
 
 8 2's=? 
 
 28^7 = ^ 
 
 ii-2=:> 
 
 3x7=? 
 
 21-3 = ? 
 
 .iof28 = ^ 
 
 9 + 9 = ? 
 
 13-7 = ? 
 
 7x2 = ? 
 
 13-8 = ? 
 
 19 + 9 = ? 
 
 9 + 8 = ? 
 
 6x5 = ? 
 
 3x9 = ^ 
 
 11-6 = ? 
 
 19+-8 = ? 
 
 4x6 = ? 
 
 9x3 = ? 
 
 5 + 7 = ? 
 
 5x5=? 
 
 24-3 = ? 
 
 12-5 = ? 
 
 17-8 = ? 
 
 iof25='> 
 
 lof 16 = ? 
 
 lof24 = ^ 
 
 3x6 = ^ 
 
 lVof20 = '? 
 
 iofl8 = ? 
 
 f of 24 = ? 
 
 9x3 = ? 
 
 {of 21 = ? 
 
 J^of20 = ? 
 
 24-12 = ? 
 
 7x3 = ? 
 
 12 + 6 = ? 
 
 4x8 = ? 
 
 24-8 = ? 
 
 18-3 = ? 
 
 24-6 = ? 
 
 8x2=? 
 
 9-5 = ? 
 
 8 3's = ? 
 
 3x5 = ? 
 
 22-2 = ? 
 
 3x4 = ? 
 
 16-5 = ? 
 
 lof21=? 
 
 7x3=? 
 
 3x8 = ? 
 
 12-8 = ? 
 
 f of 21=? 
 
 21-7 = ? 
 
 lof 16 = '^ 
 
 7 + 8 = ? 
 
 13-4 = ? 
 
 iof 16=? 
 
 1 of 16 = ? 
 
 4 3's = ? 
 
 iof 24 = ? 
 
 f of 16 = ? 
 
 2x12 = ? 
 
 14-9 = ? 
 
 5 3's = ? 
 
 6 3's=? 
 
 9-5 = '^ 
 
 19-3 = ? 
 
 15-8 = ? 
 
 8 + 7=? 
 
72 ELEMENTS OF ARITHMETIC. 
 
 19-5 = ? 
 
 25-5 = ? 
 
 4 4's = ? 
 
 2 + 9 = ? 
 
 29-5 = ? 
 
 7 4's = ? 
 
 6 4's = ? 
 
 11 2\s = ? 
 
 8)24( 
 
 12-9 = ? 
 
 3 6's = ? 
 
 4x7 = ? 
 
 11 2's = ? 
 
 17-8 = ? 
 
 lof24 = ? 
 
 iof24 = ? 
 
 2x9 = ? 
 
 30-5 = ? 
 
 5 + 9 = ? 
 
 f of 24 = ? 
 
 3x9 = ? 
 
 23-7 = ? 
 
 3 + 5 = ? 
 
 18-2 = ? 
 
 9)27( 
 
 3x6 = ? 
 
 13 + 5 = ? 
 
 -1 of 30 = ? 
 
 18-9 = ? 
 
 19-8 = ? 
 
 28-f-4 = ? 
 
 2x9 = ? 
 
 18-9 = ? 
 
 29-8 = ? 
 
 8 3's = ? 
 
 22 — 11 = ? 
 
 5 6's = ? 
 
 20-4 = ? 
 
 2 8's = ? 
 
 iof 9 = ? 
 
 NUMBERS FROM 30 TO 100. 
 
 42. Count by 4's to 32. 
 Write the table of 4's, thus ; 
 
 1x4 = ? 2x4 = ? 3x4=? etc. 
 
 There are 4 quarts in a gallon. How many quarts 
 are there in 2 gallons ? In 4 gallons ? In 7 gallons ? 
 In 8 gallons ? 
 
 There were 4 sleds on Morton's hill last Saturday. 
 If each one held 8 children, how many children were 
 coasting on the hill ? 
 
 1x8 = ? 2x8 = ? 3x8 = ? 4x8=-:^ 
 
 Bertie had 32 cents. If he gave Helen ^ of his 
 money, how many cents did he give her ? 
 
 Mr. Young had 32 sheep, but he sold | of them. 
 How many did he sell? How many had he left? 
 
 John had 30 cents ; he spent one half of it for a slate, 
 and 2 cents for a sponge. How much had he left? 
 Then, if his mother gave him 10 cents, how much had he? 
 
NUMBERS FEOM 30 TO 100. 73 
 
 How many pecks (8 quarts) are there in 32 quarts ? 
 How many 3-cent pieces can I get for 3 dimes and 3 
 cents ? 
 
 Write the table of 3's, thus : 
 
 Ix 3 = ? 2x 3 = ? 3x 3 = ?etc. 
 
 1x11 = ? 2x11 = ? 3x11 = ? 
 
 Julia had 33 cents, and Olive J- as many. How 
 many cents had Olive? 
 
 33-3 = ? 33-11 = ? iof33 = ? J^ of 33=? 
 
 A Avalk measured 11 yards in length. Since a yard 
 is 3 feet long, how many feet long was the walk ? 
 
 I spent 33 cents for balls at 11 cents apiece. How 
 many balls did I buy ? 
 
 Count by 2's to 34, and back from 34 to 0. 
 
 17 + 17=? 2x17 = ? 34-2 = ? 
 
 Two men start from the same place and travel in 
 opposite directions. One travels at the rate of 3 miles 
 an hour, the other at the rate of 2 miles an hour. How 
 far apart are they at the end of 2 hours? 5 hours? 
 7 hours ? 8 hours ? 6 hours ? 
 
 Count by 5's to 60. Count by 7's to 70. Count by 
 lO's to 70. 
 
 Donna worked 7 examples each day for 5 days. How 
 many did she work ? 
 
 Mrs. H. bought from the milkman 35 quarts of milk 
 in one week. What was the average number of quarts 
 she bought per day ? 
 
 How many days are there in 2 weeks ? In 5 weeks ? 
 In 4 weeks ? In 3 weeks ? In 6 weeks ? In 8 weeks ? 
 In 7 weeks ? 
 
74 ELEMENTS OF ARITHMETIC. 
 
 If 5 men can do a piece of work in 7 days, how long 
 will it take 1 man to do it ? 
 
 How many weeks are there in 35 days ? In 14 days ? 
 In 28 days? In 42 days? 
 
 -iof35 = ? iof35 = ? 35-7-? 
 
 I of 35 = ? fof35 = ? 35-5 = ? 
 
 Two men start from the same place and travel in 
 opposite directions. One travels 9 miles an hour, the 
 other travels 7 miles an hour. How far apart will they 
 be in 2 hours ? 
 
 Suppose the men traveled in the same direction, how 
 far apart would they be in 2 hours ? In 7 hours ? In 
 11 hours? In 12 hours? 
 
 Harry earned 7 half-dimes while Lucy earned 3 dimes 
 and a five-cent piece. Which earned the larger sum of 
 mone}^ ? 
 
 At 7 cents a yard, what is the cost of 4 yards of 
 cambric? Of 3 yards? Of 5 yards? Of 7 yards? 
 Of 6 yards ? 
 
 How many weeks was Lulu away at the seashore, if 
 she was gone 21 days ? 14 days ? 28 days ? 42 days ? 
 63 days ? 49 days ? b^ days ? 
 
 1x6=? 2x6=? 3x6=? 4x6=? 5x6=? ^xQ>=l 
 7x6 = ? 8x6 = ? 9x6 = ? 10x6 = ? 11x6 = ? 12x6=? 
 
 Emily is 18 years old, and her uncle is twice as old. 
 How old is he ? 
 
 What is i- of 1 36 ? J of 36 men ? -| of 36 bushels ? 
 
 How many times is 4 contained in 12? In 24? In 
 36? In 20? In 16? In 8? In 28 ? In 32 ? 
 
 There are 6 working days in a week. How many 
 working days are there in 2 weeks ? In 6 weeks ? In 3 
 
NUMBERS FROM 30 TO 100. 75 
 
 weeks ? In 5 
 
 weeks ? In 4 weeks ? In 
 
 7 weeks? 
 
 In 
 
 10 weeks? I] 
 
 :l 8 
 
 weeks ? 
 
 In 9 weeks? 
 
 In 11 weeks? 
 
 In 12 weeks? 
 
 
 
 
 
 
 12- 6 = ? 
 
 24- 
 
 -6 = ? 
 
 30-6=? 
 
 36-6=? 
 
 
 24- 6 = ^ 
 
 m- 
 
 -6 = ? 
 
 60^6 = ? 
 
 72-6 = ? 
 
 
 6x 6-'^x8 
 
 6: 
 
 k6=?x9 
 
 6x6=?xl2 
 
 5x6 = ? 
 
 x3 
 
 2xl0 = ?x4 
 
 4; 
 
 ><4 = 2x? 
 
 6x2=?x4 
 
 4x4 = ? 
 
 x8 
 
 48- 6 = ? 
 
 18- 
 
 f-6 = ? 
 
 42-6 = ? 
 
 
 
 How many yards of cloth, at $3 a yard, will pay for 
 5 barrels of flour at $ 6 per barrel ? 
 
 Willard sold 12 barrels of apples at $8 per barrel, and 
 spent half the money for books at 1 2 each. How many 
 books did he buy? 
 
 Otto sold 6 quarts of berries at 6 cents a quart, and 
 spent the money for 9 bananas. How much did he pay 
 apiece for the bananas ? 
 
 There are 9 children in the drawing class, and 4 times 
 as many in the arithmetic class. How many are there 
 in the arithmetic class ? 
 
 At 4 cents a yard, what is the cost of 2 yards of 
 ribbon? Of 4 yards? Of 6 yards? Of 9 yards? Of 
 10 yards ? Of 8 yards ? Of 8 yards ? 
 
 How many weeks' work are there in 36 working 
 days? In 18? In 24? In 30? In 12? In 42? In 
 60? In 72? 
 
 Count by 2's to 38. 
 9 + 9 = ? 19 + 9 = ? 29 + 9 = ? etc. 19 + 19 = ? etc. 
 
 Count by 3's to 39. Count by 4's to 48. 
 
 At 4 cents apiece, what must I pay for 5 oranges? 
 For 3? For 8? For 6? For 10? For 4? For 7? 
 For 9 ? For 11 ? For 1 dozen ? 
 
76 ELEMENTS OF ARITHMETIC. 
 
 Bert rode his pony 8 miles a day for 5 days. How 
 far did he ride ? 
 
 How many eggs, at 4 cents each, will pay for 5 yards 
 of ribbon at 8 cents a yard ? 
 
 If a stage goes 8 miles an hour, how many hours will 
 it take to travel 24 miles ? 40 miles ? 16 miles ? 48 
 miles ? 32 miles ? 56 miles ? 72 miles ? 64 miles ? 
 
 What is I of 40? i of 40? f of 40? ^ of 40? 
 JoOf40? 3-3^ of 40? iof40? |of40? |of40? 
 
 How many live-cent loaves of bread can Florence buy 
 with 25 cents ? With 35 cents ? With 40 cents ? With 
 50 cents? 
 
 How many gallons are there in 12 quarts? In 28 
 quarts ? In 36 quarts ? In 40 quarts ? In 48 quarts ? 
 In 16 quarts? 
 
 How many quarts are there in 16 pints ? In 24 pints ? 
 In 20 pints ? In 8 pints ? In 40 pints ? In 38 pints ? 
 
 How many pecks are there in 3 bushels? In 8 
 bushels? In 10 bushels? In 6 bushels? In 4 bushels? 
 In 9 bushels ? In 7 bushels ? 
 
 If one pencil costs 5 cents, what is the cost of 3 pen- 
 cils? Of 5 pencils? Of 4 pencils? Of 2 pencils? 
 Of 6 pencils? Of 7 pencils? 
 
 At 7 cents a quart, what is the cost of 4 quarts of 
 blueberries ? Of 6 quarts ? Of 3 quarts ? Of 2 quarts ? 
 Of 7 quarts ? 
 
 Count by 7's to 70. Count by 8's to 80. 
 
 Lucy bought 6 paper dolls at 7 cents each. She gave 
 the clerk 50 cents. What change should she receive? 
 6x7=2x? 
 
 Donna has 9 half-dimes in her pocket book. To how 
 many cents are they equal ? 
 
NUMBERS FROM 30 TO 100, 77 
 
 How many toy books, at 6 cents each, can she buy, and 
 how many cents will she have left? 
 
 If Clarence sleeps 8 hours at night, how many hours 
 does he sleep in 3 nights ? In 5 nights ? In 7 nights ? 
 In 4 nights ? In 6 nights ? 
 
 Helen had 3 five-cent pieces, and Nannie had 3 dimes. 
 How many sheets of drawing paper, at 9 cents each, can 
 they buy if they put their money together ? 
 
 Count by 5's to 60. Count by 9's to 90. 
 
 How many 5's are there in 25 ? In 40 ? In 15 ? In 
 45? In 30? In 50? In 35? In 60? In 20? In 55? 
 
 How many 9's are there in 18? In 45? In 27? In 
 36-? In 54? etc. 
 
 If a boy can buy 9 marbles for a cent, how many 
 marbles can he buy for 3 cents ? For 2 cents ? For 5 
 cents ? For 4 cents ? For 6 cents ? For 7 cents ? 
 
 Bert 's pony can travel 8 miles an hour. How much 
 farther can he travel in 6 hours than Harold's pony, 
 which can travel only 6 miles an hour? 
 
 (9x 5)-(5x6) = ? ( 4xl0)-(4x8)=? 
 
 (4x 9)-(3x4)=? ( 5xl0)-(6x6) = ? 
 
 (7x 7)-(6x7)-? ( 7x 4)-(9x3) = ? 
 
 (6x 5)-(3x5) = ? ( 8x 6)-(5x4)=? 
 
 (4xl0)-(5x7) = ? ( 5x 5)-(3x5) = ? 
 
 (4x 4) + (5x2)=? (12x 3)-(4x9)=? 
 
 (6x 7)-(3x7) = ? ( 6x 4) + (3x5)=? 
 
 (5x 8)-(3x3)=? ( 3x 6) + (3x4)=? 
 
 (3xll) + (2x5) = ? ( 8x 4)-(4x7) = 
 
 How many lO's are there in 30 ? In 50 ? In 90 ? In 
 100? 
 
 What is I of 50 cents ? Of 40 cents ? Of 30 cents ? 
 
NOTATION AND NUMERATION. 
 
 43. We have learned that ten ones make one ten. ISTow 
 take ten tens and tie them together. Ten tens are called 
 one hundred. 
 
 In writing hundreds, the figure representing hundreds is 
 placed in the third place from the right. 
 Thus, one liundred is written 100. 
 
 44. Write with figures : 
 
 Two hundred. Three hundred. Nine hundred. 
 
 Seven hundred. Four hundred. Six hundred. 
 
 Five hundred. Eight hundred. One hundred. 
 
 45. When numbers are expressed by three figures, the 
 figure at the right represents ones or units, the second figure 
 tens, and the third hundreds « 
 
 46. In reading numbers expressed by three figures, the 
 
 tens are read after the hundreds without the word and. 
 
 Thus, 478 is read four hundred seventy-eight, instead of four hun- 
 dred and seventy-eight. 205 is read two hundred five, not two 
 hundred and five. 
 
 78 
 
NOTATION AND NUMERATION. 79 
 
 EXERCISES^ 
 
 47. What number is represented in the picture on page 78 ? 
 Show with objects, and tell what you understand by 
 
 303 121 763 286 431 295 333 107 113 103 
 400 200 101 111 321 117 115 240 276 189 
 
 Write the following numbers, placing units under units, 
 tens under tens, etc. .: 
 Three hundred forty-six. Sixteen. 
 
 Five hundred eighteen. Six hundred eightj^-fiveo 
 
 Ninety-two. Forty-seven. 
 
 One hundred seventy-five. Seven hundred sixteen. 
 Two hundred eighty. Two hundred two. 
 
 Seven hundred eleven. Four hundred eleven. 
 
 Four hundred forty-four. Eight hundred. 
 
 Three hundred five. Five hundred twenty-nine. 
 
 Eight hundred seventeen. Nine hundred seventy-three. 
 Six hundred nine. Seven hundred sixty-four. 
 
 Read the numbers you have written. 
 
 How many figures are needed to express hundreds ? 
 
 48. The method of expressing numbers by figures or 
 letters is called Notation. 
 
 • 49. The method of expressing numbers by figures is 
 called the Arabic Notation. 
 
 It is called the Arabic Notation because it was first introduced into 
 Europe by the Arabs. 
 
 50. The method of reading numbers is called Numeration. 
 
 EXERCISES. 
 
 51. 1. Write in columns of ten numbers each all the 
 numbers from one hundred to one hundred forty-nine. 
 
 2. From two hundred fifty -one to three hundred. 
 
 3. What number is next greater than 549^ 328, 191, 400, 
 589, 240, 379, 599, 109, 765 ? 
 
80 ELEMENTS OF ARITHMETIC. 
 
 4. Make the following numbers larger by one hundred, 
 and write them : 17, 89, 30, 4, 52, 305. 
 
 5. What number comes next before 100, 700, 201, 870, 310 ? 
 
 6. How many hundreds are there in each of the folio w= 
 ing numbers ? How many tens ? How many units ? 
 
 468 453 754 902 999 837 964 815 
 125 602 882 176 731 395 618 274 
 
 52. What is the smallest number you can express by 
 three figures ? The largest ? 
 999 If we add 1 to 999, we have nine units and 1, 
 1 making ten units or one ten. Adding the one ten and 
 -jr^QQ the nine tens, w^e have ten tens or one hundred. Add- 
 ing the one hundred to the nine hundreds, we have 
 ten hundreds, which we call by the new name one thousand. 
 7000 is read, seven thousand. 1720 is read, one thousand ^even 
 hundred twenty. 
 
 EXERCISES. 
 
 Eead the following : 
 
 1. 2. 
 
 3. 4. 
 
 5. 6. 7. 
 
 2000 3000 
 
 5000 7000 
 
 4000 8000 9000 
 
 3791 4809 
 
 7002 9100 
 
 4700 2083 1005 
 
 4263 5074 
 
 2380 1234 
 
 5678 7985 9090 
 
 8. What place, counting from 
 
 the right, do thousands 
 
 occupy ? 
 
 
 
 9. What is the smallest whole number that can be ex- 
 pressed by four figures ? What is the greatest ? 
 
 10. How many units, tens, and hundreds are expressed 
 by each of the following numbers ? 
 
 Write and read : 
 
 16. 17. 
 
 5055 9100 
 
 7643 1001 
 
 5010 1110 
 
 11. 
 
 12. 
 
 13. 
 
 14. 
 
 15. 
 
 6709 
 
 7891 
 
 7000 
 
 7777 
 
 1098 
 
 7253 
 
 8005 
 
 7700 
 
 4028 
 
 7005 
 
 1389 
 
 6666 
 
 7770 
 
 2003 
 
 4809 
 
KOTATION AND NUMERATION. 81 
 
 18. Read the numbers in 11, 12, 13, and 14, reading the 
 thousands and hundreds together as hundreds. 
 
 Thus, 6709, or sixty-seven hundred nme. 
 
 19. In 8888, for what does each of the S's stand ? 
 
 20. What effect would it have upon the value of 7 in 71, 
 if a cipher were placed to the right of the 1 ? 
 
 53. When we reach a thousand, we begin to count the 
 thousands as we did the units, thus : 1 thousand, 2 thousand, 
 etc., up to 999 thousand, and when we have a thousand 
 thousand, ^ e call the number one million. 
 
 A thousand million we call a billion. 
 A thousand billion we call a trillion. 
 
 54. For convenience, when more than three figures are 
 employed to express any whole number, they are divided 
 into groups called Periods, the first of which, counting from 
 the right, is used to denote any number from 1 to 999 units; 
 the second, from 1 to 999 thousands; the third, from 1 to 
 999 millions, etc. 
 
 For convenience in reading, the periods are usually sepa- 
 rated from each other by commas. 
 
 Table. 
 
 EL. OF AR. — 
 
82 ELEMENTS OF ARITHMETIC. 
 
 The first number in the cable is read : seven hundred forty-three 
 trillion, nine hundred eighty-six billion, Jive hundred forty-three 
 million, two hundred six thousand, seven hundred ?iineteen. 
 
 Eead the foregoing numbers, looking at tlie headings 
 until you become familiar with the names of the periods. 
 
 55. Wlieyi tue read numbers, we begin at the right and sep- 
 arate the numbers into periods of three figures each. Then, 
 beginning at the left, we read each period as if it stood alone, 
 adding its name. 
 
 Each period, except the highest or left-hand period, must contain 
 three figures. 
 
 EXERCISES. 
 
 66. Write and read : 
 
 1. 2. 3. 4. 
 
 700,000 903,744 300,300 17,019 
 
 965,237 200,080 37,961 586,240 
 
 83,016 147,906 145,808 770,300 
 
 728,000 400,004 307,029 562,999 
 
 5. 6. 7. 
 
 1,789,645 769,834,521 211,467,431 . 
 
 29,783,240 888,971,646 462,293,070 
 
 8. Erase the figure at the right hand of each number 
 in 5, 6, and 7, arrange the remaining figures in periods 
 anew, by placing commas where they belong, and then read 
 the numbers. 
 
 9. Continue thus with the given numbers, erasing the 
 figures one by one, pointing off into periods correctly, and 
 reading the numbers. 
 
 10. Make the following numbers larger by one thousand. 
 By three thousand. By seven thousand. By nine thousand. 
 
 763 265 110 703 
 
NOTATION AND NUMERATION. 83 
 
 11. What number is next greater than 1000? 1699? 
 2019? 5082? 7200? 1239? 7379? 6299? 9999? 
 7676 ? 8109 ? 4099 ? 
 
 12. What number comes next before 3000 ? 6000? 5400? 
 2040? 2010? 5790? 1000? 2010? 9801? 
 
 13. How many hundreds and tens are there in 34 tens? 
 In 86 tens ? In 37 tens ? In 42 tens ? In 128 tens ? In 
 564 tens ? 
 
 14. How many hundreds are there in 386 units ? In 864 
 units ? In 130 units ? In 175 units ? In 679 units ? In 
 534 tens ? In 643 tens ? In 161 tens ? In 879 tens ? 
 
 15. How many thousands and hundreds are there in 75 
 hundreds ? In 51 hundreds ? In 42 hundreds ? In 78 hun- 
 dreds ? How many in 250 tens ? In 175 tens ? In 549 
 tens ? In 543 tens ? How many in 9875 units ? 
 
 16. Make the following numbers larger, first by ten thou- 
 sand ; then by fifty thousand ; then by seventy thousand : 
 834 227 386 700 652 200 507 993 
 
 17. What number is next greater than 15,999 ? 150,109? 
 199,999? 777,779? 976,190? 275,300? 789,999? 
 
 18. What number is next less than 400,001 ? 600,000 ? 
 289,223? 500,000? 943,000? 727,800? 482,176? 
 100,000? 400,680? 400,999? 700,010? 300,000? 
 
 WRITTEN EXERCISES. 
 57. Write in figures : 
 
 1. Six thousand, eight hundred fort}^ 
 
 2. Eight thousand, two hundred fifty-seven. 
 
 3. Nineteen thousand, five hundred twenty-seven. 
 
 4. Eighty-four thousand, six hundred thirty-one. 
 6. One hundred thousand, seven hundred ten. 
 
 6. Fifty-eight thousand, six hundred one. 
 
 7. Two hundred thousand, four hundred seventy-five. 
 
 8. Four hundred eighteen thousand, two hundred seven. 
 
84 ELEMENTS OF ARITHMETIC. 
 
 9. Five hundred seven thousand, thirty. 
 
 10. Seven hundred fifteen thousand, nine hundred twenty-, 
 four. 
 
 11. Thirty-two million, four hundred fifteen thousand^ 
 two hundred thirty -three. 
 
 12. One hundred six million, eight hundred nine thou- 
 sand, seven hundred fifty. 
 
 13. Two hundred sixty-five million, fifteen thousand, one 
 hundred seventy-nine. 
 
 14. Three hundred sixty -five million, forty-nine. 
 
 15. Nine million, nine thousand, nine. 
 
 16. Eighty million, eighty thousand, eighty. 
 
 17. Six hundred million, six hundred thousand, six hun> 
 dred. 
 
 18. Thirty-six million, eighteen thousand. 
 
 19. Twenty-four million, eight thousand, eight. 
 
 20. Two hundred million, two thousand, two. 
 
 21. Thirty-nine billion, sixty million, sixty thousand, 
 sixty. 
 
 22. Fifty-two billion, one million, one thousand, one. 
 
 23. Sixty billion, sixty million, sixty thousand, sixty. 
 
 24. Eight hundred million, eighty-four thousand. 
 
 25. Seventy-seven billion, seventy-seven million, seventyN 
 seven thousand, seventy-seven. 
 
 26. Four hundred twenty-six million, four hundred thou- 
 sand, twenty-five. 
 
 27. One hundred seventy million, seven. 
 
 28. Sixteen hundred seventy-eight. 
 
 29. One trillion, seventeen million. 
 
 30. Four hundred fifty-six million, ten. 
 
 31. Five hundred nine million, four hundred thousand, 
 twenty-five. 
 
 32. Nineteen hundred two. 
 
 33. One billion, six hundred thousand, four. 
 
NOTATION AND NUMERATION. 85 
 
 NOTATION AND NUMERATION OF UNITED 
 STATES MONEY. 
 
 58. Write the names of all the United States coins you 
 have seen, and tell of what each is made. What paper 
 money have you seen? 
 
 In the currency of the United States, 
 
 10 mills make 1 cent. 
 100 cents make 1 dollar. 
 
 59. The Sign of Dollars is ^. It is written before the 
 number. 
 
 Thus, $27 is read, twenty-seven dollars. 
 
 60. In writing cejits and mills, a period called the Decimal 
 Point is placed before the number of cents. 
 
 61. Cents occupy the Jirst tico places at the right of the 
 decimal point, and 7nills the third place. 
 
 Thus, .$0.35 is read, thirty-five cents; $2,875 is read, two dollars 
 eighty-seven cents five mills. 
 
 Mills are not coined. In business five mills or more are called one 
 cent, and less than five are disregarded. 
 
 62. If the number of cents is less than ten, a cipher must 
 be written in the first place at the right of the decimal point. 
 
 Thus, five cents is written $.05; three cents, $.03; seven dollars 
 two cents five mills is written $ 7.025. 
 
 EXERCISES. 
 
 63. Read the following : 
 
 1. 2. 3. 4. 5. 
 
 ^1.00 ^1.75 ^1.65 33.185 $0.25. 
 
 $4.87 $18.05 $15.43 $70.10 $1.05. 
 
 $3.04 $24.16 $100.50 $9,125 $ .40. 
 
 $7. $2,357 $10. $0,375 $7.20. 
 
86 ELEMENTS OF ARITHMETIC. 
 
 Write the following in columns^ keeping decimal points 
 in a vertical line : 
 
 1. Eighteen dollars twenty-four cents. 
 
 2. Twenty-six dollars thirty-six cents. 
 
 3. Fifty-one dollars five cents five mills. 
 
 4. Thirty-eight dollars sixteen cents three mills. 
 
 5. Thirty-nine dollars nine cents. 
 
 6. Thirty dollars five cents. 
 
 7. One hundred dollars ten cents four mills. 
 
 8. Six dollars six cents. 
 
 9. Write one cent, two cents, three cents, etc., to fifty 
 cents. 
 
 10. How many cents are there in f 1? In ^ 2 ? In $3 ? 
 Inf4? In $8? In $5? In $9? In $7? In ^6? 
 
 11. Eead as cents: $2; ^2.02; $3; $3.05; $3.75; 
 $4.17; $9.08; $11.11. 
 
 12. Write the following as cents, using the character ^ 
 for cents : 
 
 $1.25 $1.10 $3.75 $7.45 $9.67 $8.80 
 $7.05 $9.81 $4.13 $4.18 ^5M $6.72 
 
 13. Eead the following as dollars and cents : 
 Thus, 1728j^ = 17 hundred 28 cents, or 17 dollars 28 cents. 
 
 2375^ 300^ 1015^ 1005^ 680p 2793^ 
 
 1390^ 970^ 700)^ 57345^ 3250^ 1809;^ 
 
 14. Write the same, using the sign of dollars. 
 
 15. How many mills are there in 2^? In 5^? In 7^? 
 In 9^ ? In 10^ ? In 70^ ? In 30^ ? In 90^ ? In 15^ ? 
 
 16. How many mills are there in lOOj^ ? In 300^? In 
 $1? In $2? In $9? In $1.20? In $1.50? 
 
 17. Eead the following as dollars, cents, and mills : 
 2765 mills 5639 mills 2793 mills 
 
 235 mills 9050 mills 4505 mills 
 
 18» Write the above, using the dollar sign. 
 
NOTATION AND NUMERATION. 87 
 
 ROMAN NOTATION. 
 
 64. The method of expressing numbers by means of 
 letters is called Roman Notation. 
 
 It is called Roman Notation because it was originally used by the 
 ancient Romans. 
 
 65. In this system, seven capital letters are used to 
 express numbers, viz. : 
 
 Letters: I, V, X, L, C, D, M. 
 
 Values: 1, 5, 10, 50, 100, 500, 1000. 
 
 By combining these letters according to certain principles, 
 any number can be expressed. 
 
 66. Principles. — 1. Wien a letter is repeated, its value 
 is repeated. 
 
 Thus, X represents ten ; XX, twenty ; XXX, thirty ; CC, two hun- 
 dred ; MM, two thousand. 
 
 2. When a letter is jjlaced before another of greater value, 
 
 its value is to be taken from that of the greater. 
 
 Thus, I represents one, V, five, and X, ten. But IV represents 
 four ; IX, nine ; XL, forty ; XC, ninety. 
 
 3. When a letter is placed aftei' another of greater value, 
 
 their values are to be united. 
 
 Thus, XV represents fifteen ; LXX, seventy ; LXXX, eighty ; MC, 
 one thousand, one hundred. 
 
 4. When a letter is placed, between tivo letters, each of greater 
 
 value than itself, its vcdue is to be taken from the smn of the 
 
 other two. 
 
 Thus, XIV represents fourteen ; XIX, nineteen ; CIV, one hun- 
 dred four. 
 
 5. A bar placed over a letter increases its vcdue a thousand 
 fold. 
 
 Th us, V represents five ; V. five thousand ; LXX represents seventy ; 
 LXX, seventy thousand. 
 
88 
 
 ELEMENTS OF ARITHMETIC. 
 
 Table. 
 
 1= 1 
 
 XIII = 13 
 
 LX: 
 
 = 60 
 
 11= 2 
 
 XIV = 14 
 
 LXX: 
 
 = 70 
 
 111= 3 
 
 XV = 15 
 
 XC: 
 
 = 90 
 
 TV= 4 
 
 XVI = 16 
 
 C: 
 
 = 100 
 
 V= 5 
 
 XIX = 19 
 
 cc 
 
 = 200 
 
 VI = 6 
 
 XX = 20 
 
 cccc 
 
 = 400 
 
 VII = 7 
 
 XXI = 21 
 
 CD 
 
 = 400 
 
 VIII = 8 
 
 XXIX = 29 
 
 D 
 
 = 500 
 
 IX = 9 
 
 XXX = 30 
 
 DCC 
 
 = 700 
 
 X = 10 
 
 XXXIV = 34 
 
 M 
 
 = 1000 
 
 XI = 11 
 
 XL = 40 
 
 MMM 
 
 = 3000 
 
 XII = 12 
 
 L = 50 
 EXERCISES. 
 
 MDCCCXCII 
 
 = 1892 
 
 Eead the following : 
 
 
 
 XX 
 
 XIX CDXX 
 
 XLIV XXV 
 
 XXXV LXIV 
 
 XL 
 XC 
 
 XCIV 
 
 LXX 
 
 XIX 
 
 XCIX 
 
 VIII 
 
 MDLIV 
 
 XXI 
 
 LXIX CCXXIV 
 
 LX 
 
 MDCCC 
 
 XLIX 
 
 XXXVI CCCLIX 
 
 DCXL 
 
 MMDC 
 
 MDCCCX 
 
 CCXXVI DLXXI 
 
 MDXL 
 
 DCCL 
 
 cxcv 
 
 CCXLIV IMMMD 
 
 IVDXL 
 
 CCXCV 
 
 Express the following by Eoman Notation : 
 
 
 23 61 
 
 84 35 312 51 
 
 7 1010 
 
 1900 
 
 34 19 59 47 419 493 1800 8000 
 15 36 62 86 226 499 1492 9000 
 27 43 97 214 384 278 1607 1721 
 

 of 
 
 ADDITION. 
 
 67. 1. How many blocks are 5 blocks, 3 blocks, and 7 
 blocks ? 
 
 2. How many splints are 3 splints, 2 splints, and 8 
 splints ? 
 
 3. How many oranges are 7 oranges, 2 oranges, and 5 
 oranges ? 
 
 4. How many are 5 and 6 and 7 ? 
 6. How many are 7 and 8 and 6 ? 
 
 6. How many are 5 and 9 and 2 and 6 ? 
 
 7. How many are 3 and 7 and 4 and 8 ? 
 
 8. How many are 4 and 8 and 5 and 9 ? 
 
 9. What have you been doing with the numbers given 
 above ? 
 
 68. The process of finding a number that is equal to two 
 or more given numbers is called Addition. 
 
 69. The result obtained by adding is the Sum, or Amount. 
 
 70. The Sign of Addition is a small upright cross (+). 
 
 It is read j)lus, and is placed between the numbers to be 
 
 added. 
 
 Thus, 4 + 2 is read 4 plus 2, and means that 4 and 2 are to be 
 added. 
 
 71. The Sign of Equality is two short parallel horizontal 
 lines ( = ). It is read equals, or is equal to. 
 
 Thus, 3 + 2 = 5 is read 3 plus 2 equals 5. 
 
90 ELEMENTS OF ARITHMETIC. 
 
 WRITTEN EXERCISES. 
 72. 1 . Add 13, 12, 24, 30, and 14. 
 
 Explanation. — In adding, the numbers are written in 
 columns, units under units, tens under tens, and it is more 
 12 convenient to begin with the units' column. Adding, we find 
 24 there are 13 units. 13 means 1 ten and 3 units ; therefore the 3 is 
 OA written in the units' column, and the 1 ten is added to the tens. 
 Adding the tens, we find that there are 9 tens. Therefore the 
 ^ result is 93. 
 
 QQ In adding, do not say 4 and 4 are eight and 2 are ten. 
 
 etc. , simply name results, thus : 4, 8, 10, 13. 
 
 13 
 
 Find the sum 
 
 of 
 
 
 
 
 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 72 
 
 73 
 
 84 
 
 57 
 
 ^27 
 
 $21 
 
 $96 
 
 $20 
 
 85 
 
 97 
 
 17 
 
 46 
 
 72 
 
 53 
 
 81 
 
 38 
 
 90 
 
 16 
 
 52 
 
 93 
 
 18 
 
 38 
 
 45 
 
 47 
 
 36 
 
 68 
 
 78 
 
 16 
 
 19 
 
 73 
 
 78 
 
 93 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 14. 
 
 15. 
 
 16. 
 
 17. 
 
 39 
 
 54 
 
 89 
 
 2S 
 
 59 
 
 2^ 
 
 45 
 
 68 
 
 29 
 
 90 
 
 24 
 
 11 
 
 83 
 
 27 
 
 17 
 
 39 
 
 72 
 
 71 
 
 74 
 
 50 
 
 17 
 
 48 
 
 18 
 
 74 
 
 45 
 
 29 
 
 93 
 
 33 
 
 26 
 
 16 
 
 13 
 
 m 
 
 13 
 
 36 
 
 35 
 
 ^Q 
 
 28 
 
 36 
 
 14 
 
 23 
 
 18. 
 
 19. 
 
 20. 
 
 21. 
 
 22. 
 
 23. 
 
 24. 
 
 25. 
 
 45 
 
 62 
 
 42 
 
 Q5 
 
 43 
 
 Q>^ 
 
 bQ 
 
 42 
 
 58 
 
 87 
 
 24 
 
 89 
 
 69 
 
 42 
 
 49 
 
 46 
 
 34 
 
 49 
 
 85 
 
 38 
 
 22 
 
 25 
 
 17 
 
 34 
 
 22 
 
 36 
 
 48 
 
 76 
 
 ^^ 
 
 34 
 
 21 
 
 5S 
 
 67 
 
 55 
 
 34 
 
 45 
 
 74 
 
 28 
 
 36 
 
 91 
 
ADDITION. 91 
 
 26. Two birds fly in opposite directions, one 19 miles, 
 and the other 27 miles. How far apart are they ? 
 
 27. Max earned 24 dollars, and John earned 9 dollars 
 more than Max. How many dollars did both earn ? 
 
 28. Baby is 4 years old ; mamma is seven times as old 
 as baby ; grandma is twice as old as mamma. What is the 
 sum of their ages ? 
 
 29. In a store window I saw 38 books, 25 toys, 17 slates, 
 and 20 globes. How many articles did I see ? 
 
 30. In one class in our school there are 11 boys and 9 
 girls ; in another, 17 boys and 18 girls ; in another, 10 boys 
 and 14 girls. How many boys are there in all ? How 
 many girls ? How many children ? 
 
 31. A fisherman caught 46 trout, 75 bass, and 13 blue- 
 fish. How many fishes did he catch ? 
 
 32. Fred weighs 83 pounds, and Nathan weighs 97 pounds. 
 How much do they both weigh ? 
 
 33. In one book there are 98 pages, in another 89, and 
 in a third 75. How many pages are there in all the books ? 
 
 34. A certain tract of land was divided into 4 farms; 
 one containing 13 acres, another 37 acres, the third 70 acres, 
 and the fourth 48 acres. How many acres did the original 
 tract contain ? 
 
 35. The distance from Alburn to Grade is 6 miles, from 
 Grade to Newton 8 miles, from Newton to Arnold 9 miles, 
 and from Arnold to Houston 8 miles. How far is it from 
 Alburn to Houston ? 
 
 36. Elsie solved 18 examples on Monday, 14 on Tues- 
 day, and 12 on Wednesday. How many examples did she 
 solve in all ? 
 
 37. If I buy a collar for 25 cents, some gloves for 37 
 cents, some ribbon for 8 cents, and a paper of needles for 9 
 cents, what do they all cost ? 
 
 38. A farmer has in one bin 28 bushels of oats, in another 
 
92 ELEMENTS OF ARITHMETIC. 
 
 36, in another 23. How many bushels has he in the three 
 bins ? 
 
 39. If I sell 8 chickens to one man, 18 to another, and 
 have 24 left, how many had I at first ? 
 
 40. Mr. Atwater bought a cow for 36 dollars, a sheep 
 for 8 dollars, and a calf for 13 dollars. How much did he 
 pay for all ? 
 
 41. Susie has 38 cents, and her brother has 17 cents more 
 than she has. How many cents have they both ? 
 
 42. Morton's geography cost 75 cents, his reader 39 
 cents, and his slate 18 cents. How much did they all cost ? 
 
 43. Mr. Barnes has 3 fields. In one he put 36 cows, in 
 another 42 cows, and in the third 27 cows. How many 
 cows has he in the three fields ? 
 
 44. A fisherman sold 4 fishes for the following sums : the 
 first for 37 cents, the second for 44 cents, the third for 54 
 cents, the fourth for 48 cents. What did he receive for 
 all? 
 
 45. On a Christmas tree I counted 12 dolls, 25 balls, 28 
 flags, 5 whistles, 3 drums, 25 candy -bags, and 14 -books. 
 How many things did I count ? 
 
 46. Thornton has 68 marbles, Oliver has 10 less than 
 Thornton. How many have both boys together ? 
 
 47. One stalk of a plant has 43 leaves, a second stalk 
 has 36 leaves, and a third has 18 leaves. How many leaves 
 have they all ? 
 
 48. A baker made at one time 37 loaves, at another 56, 
 at another 28. How many loaves did he make in all ? 
 
 49. A milkman sold 72 quarts of milk on Monday, 56 on 
 Tuesday, 85 on Wednesday, and 49 on Thursday. How 
 many quarts did he sell in the four days ? 
 
 50. A butcher sold 24 pounds of veal, 37 pounds of porkj 
 38 pounds of chickens, and 49 pounds of beef. How many 
 pounds of meat did he sell ? 
 
ADDITION, 93 
 
 51. A mail has 45 sheep, 28 cows, and 15 calves. How 
 many animals has he ? 
 
 62. If I spend 8 dollars a week for board, 3 dollars for 
 car fare, and have 14 dollars left of my week's earnings, 
 how much did I earn ? 
 
 53. Wilfred paid 28 dollars for a desk. He paid 5 dol- 
 lars to have it polished, and then sold it for 7 dollars 
 more than it cost him. How much dil he get for the 
 desk? 
 
 54. Commencing with 28, find the sum of 28 and the 
 succeeding numbers less than 36 ? 
 
 55. A cloth merchant sold 5 rolls of cloth, containing 
 respectively 46, 52, 63, 29, and 76 yards. How many yards 
 did he sell ? 
 
 56. A saddle cost 14 dollars, a wagon 60 dollars more 
 than the saddle, and a horse as much as the saddle and 
 wagon together How much did the horse cost? How 
 much did all cost ? 
 
 57. Harold earned 87 cents. If he previously had 36 
 cents, and then found 7 cents, how many cents had he in 
 all? 
 
 58. Howard wrote 38 lines of German on Monday, 27 on 
 Tuesday, 15 on Wednesday, 17 on Thursday, and 26 on 
 Friday. How many lines did he write during the week ? 
 
 59. A man sold three lots. For one he received $ 695, 
 for another $ 734, and for the third $ 392. How much did 
 he receive for all ? 
 
 60. A owns a farm of 518 acres, B one of 425 acres, and 
 C one of 392 acres. How many acres do they all own ? 
 
 61. In a city there were four schools, the first containing 
 396 pupils, the second 683 pupils, the third 718 pupils, 
 and the fourth as many as the second and third together. 
 How many pupils attended the schools ? How many at- 
 tended the fourth school ? 
 
94 ELEMENTS OF ARITHMETIC. 
 
 Add the following : 
 
 
 
 
 
 
 
 
 62. 
 
 63. 
 
 64. 
 
 
 65. 
 
 
 66. 
 
 
 67. 
 
 68. 
 
 68 
 
 54 
 
 37 
 
 
 81 
 
 
 39 
 
 
 55 
 
 38 
 
 45 
 
 32 
 
 93 
 
 
 39 
 
 
 23 
 
 
 38 
 
 43 
 
 93 
 
 81 
 
 86 
 
 
 44 
 
 
 87 
 
 
 37 
 
 71 
 
 87 
 
 36 
 
 94 
 
 
 87 
 
 
 65 
 
 
 25 
 
 64 
 
 94 
 
 57 
 
 83 
 
 
 39 
 
 
 34 
 
 
 61 
 
 83 
 
 86 
 
 91 
 
 27 
 
 
 86 
 
 
 56 
 
 
 39 
 
 73 
 
 83 
 
 87 
 
 55 
 
 
 54 
 
 
 83 
 
 
 24 
 
 98 
 
 37 
 
 32 
 
 66 
 
 
 37 
 
 
 74 
 
 
 86 
 
 72 
 
 69. 
 
 70. 
 
 
 71. 
 
 
 72. 
 
 
 73. 
 
 74. 
 
 452 
 
 374 
 
 
 418 
 
 
 519 
 
 
 735 
 
 875 
 
 378 
 
 516 
 
 
 372 
 
 
 327 
 
 
 872 
 
 912 
 
 684 
 
 834 
 
 
 384 
 
 
 684 
 
 
 659 
 
 386 
 
 359 
 
 912 
 
 
 596 
 
 
 519 
 
 
 738 
 
 596 
 
 871 
 
 816 
 
 
 387 
 
 
 329 
 
 
 486 
 
 837 
 
 964 ., 
 
 325 
 
 
 426 
 
 
 548 
 
 
 684 
 
 542 
 
 832 
 
 426 
 
 
 385 
 
 
 819 
 
 
 396 
 
 816 
 
 576 
 
 845 
 
 
 713 
 
 
 384 
 
 
 845 
 
 347 
 
 75. 
 
 76. 
 
 
 
 77. 
 
 
 
 78. 
 
 
 79. 
 
 3785 
 
 7351 
 
 
 
 7638 
 
 
 
 6835 
 
 
 381 
 
 4651 
 
 4286 
 
 
 
 1842 
 
 
 
 7867 
 
 
 3468 
 
 3278 
 
 3875 
 
 
 
 3451 
 
 
 
 392 
 
 
 193 
 
 5964 
 
 1894 
 
 
 
 4265 
 
 
 
 8437 
 
 
 8200 
 
 8137 
 
 5768 
 
 
 
 7169 
 
 
 
 24 
 
 
 57 
 
 8462 
 
 3681 
 
 
 
 8376 
 
 
 
 6174 
 
 
 93 
 
 5738 
 
 5437 
 
 
 
 1452 
 
 
 
 5762 
 
 
 3956 
 
 9124 
 
 8796 
 
 
 
 3814 
 
 
 
 8600 
 
 
 4184 
 
 6783 
 
 5106 
 
 
 
 6842 
 
 
 
 439 
 
 
 324 
 
 3854 
 
 7340 
 
 
 
 6937 
 
 
 
 3764 
 
 
 47 
 
 4265 
 
 4165 
 
 
 
 5843 
 
 
 
 158 
 
 
 596 
 
 7384 
 
 9354 
 
 
 
 7165 
 
 
 
 99 
 
 
 1938 
 
SUBTRACTION. 
 
 73. 1. How many cents has George if he lost 5 cents of 
 the 9 cents which he had ? 
 
 2. Ten horses were drawing wagons, all but 5 of which 
 were gray. How many gray horses were there ? 
 
 3. Bertha had 13 cents and spent 6 cents. How much 
 had she left ? 
 
 4. How many are 10 less 3 ? 15 less 8 ? 17 less 7 ? 
 
 5. How many are 16 less 7 ? 19 less 9 ? 13 less 5 ? 
 
 6. How many are 12 less 8 ? 14 less 5 ? 15 less 7 ? 
 
 7. How many are 13 less 6 ? 17 less 9 ? 18 less 9 ? 
 
 8. What have you been doing with the above numbers ? 
 
 74. The process of finding the difference between two 
 numbers, or of taking part of a number from it and finding 
 how many are left, is called Subtraction. 
 
 75. The number from which another is taken is called 
 the Minuend. 
 
 76. The number to be subtracted is the Subtrahend. 
 
 77. The number remaining after one number is taken 
 from another is called the Remainder. 
 
 The remainder is the difference between the minuend and 
 the subtrahend. 
 
 78. The Sign of Subtraction is a short horizontal line 
 ( — ). It is called Minus. 
 
 Thus, 8 — 3 is read, 8 minus 3, and means that 3 is to be subtracted 
 from 8. 
 
 95 
 
ELEMENTS OF ARITHMETICo 
 
 WRITTEN EXERCISES. 
 
 79. 1. From 869 subtract 423. 
 
 Explanation. — For convenience, the less 
 number is written under the greater, units under 
 units, tens under tens, etc. Beginning at the right, 
 each order of units in the subtrahend is subtracted 
 separately from the same order in the minuend. 
 3 units are 6 units, which are written under units. 
 
 Minuend 869 
 Subtrahend 423 
 
 Remainder 446 
 
 Thus, 9 units - 
 
 6 tens — 2 tens are 4 tens, which are written under the tens. 
 
 8 hundreds — 4 hundreds are 4 hundreds, which are written under 
 the hundreds. Hence the remainder is 446. 
 
 Proof. — 446, the remainder, plus 423, the subtrahend, equals 869, 
 the minuend. Hence the result is correct. 
 
 Copy, subtract, and prove : 
 
 2. 
 
 3. 
 
 4. 5. 
 
 6. 
 
 7. 
 
 8. 
 
 888 
 
 999 
 
 564 899 
 
 329 
 
 957 
 
 864 
 
 356 
 
 276 
 
 213 486 
 
 115 
 
 516 
 
 231 
 
 9. 
 
 10. 
 
 11. 12. 
 
 13. 
 
 14. 
 
 15. 
 
 435 
 
 768 
 
 975 543 
 
 879 
 
 768 
 
 599 
 
 234 
 
 253 
 
 432 133 
 
 217 
 
 642 
 
 * 143 
 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 From 
 
 876 
 
 789 
 
 968 
 
 999 
 
 888 
 
 Take 
 
 242 
 
 176 
 
 723 
 
 275 
 
 234 
 
 21. Susie had two bundles of 10 splints each, and 6 more. 
 How many splints had she ? She gave her teacher 8 of the 
 splints. How many had she left ? 
 
 She found the 6 ones were not enough to give 
 her teacher, so she took one bundle of the tens, 
 untied it, and put it with her 6 ones, making 16 
 ones besides the 1 ten which she had not untied. 
 She took 8 away from the 16 and found she had 8 
 left, and also the 1 ten, or 18 left in all. 
 
 Do just as Susie did. 
 
 116 
 
 %^ 
 8 
 
 18, Remainder. 
 
SUBTRACTION. 97 
 
 22. Take 1 bundle of one hundred splintS; 4 bundles of 
 10 splints each, and 5 single splints. How many have you 
 in all? Give your teacher 27 of the splints. Compare 
 what you did with the following solution: 
 
 3 15 
 
 14p 
 
 2 7 
 
 118, Remainder. 
 
 23. Annie had 2 bundles of one hundred splints eachc 
 She gave Fred 78. How many had she left ? 
 
 Annie took 1 bundle of one hundred, untied it, and found 
 there were 10 tens in it. She left 9 of the tens a moment and 
 untied 1 bundle of 10, finding 10 ones in it. Of these she 
 gave 8 to Fred, and had 2 left. Then taking up the 9 tens, 
 she gave 7 tens to Fred, and had 2 tens left, and besides 
 these the 1 bundle of one hundred. She had, therefore, 
 122 left. 
 
 Eind the difference between the following, and supply the 
 numbers omitted in minuend, subtrahend, or remainder : 
 
 24. 
 
 25. 
 
 
 26. 
 
 
 27. 
 
 28. 
 
 29. 
 
 30. 
 
 31. 
 
 34 
 
 86 
 
 
 92 
 
 
 41 
 
 70 
 
 83 
 
 44 
 
 96 
 
 17 
 
 69 
 
 
 55 
 
 
 18 
 
 29 
 
 45^ 
 
 28_ 
 
 73 
 
 32. 
 
 33. 
 
 
 34. 
 
 
 35. 
 
 36. 
 
 37. 
 
 38. 
 
 39. 
 
 99 
 
 72 
 
 
 77 
 
 
 77 
 
 91 
 
 85 
 
 88 
 
 65 
 
 45 
 
 48 
 
 
 43 
 
 
 29 
 
 88 
 
 17 
 
 33 
 
 28. 
 
 40. 
 
 41. 
 
 
 42. 
 
 
 43. 
 
 44. 
 
 45. 
 
 46. 
 
 47. 
 
 871 
 
 761 
 
 
 240 
 
 
 166 
 
 752 
 
 631 
 
 382 
 
 546 
 
 257 
 
 347 
 
 
 18 
 
 
 139 
 
 239 
 
 408 
 
 272 
 
 338 
 
 
 
 48. 
 
 
 49. 
 
 50. 
 
 51. 
 
 52. 
 
 53„ 
 
 54. 
 
 Minuend 
 
 ^ 
 
 300 
 
 1 ] 
 
 204 
 
 500 
 
 607 
 
 908 
 
 550 
 
 600 
 
 Subtrahend, 
 
 171 
 
 
 78 
 
 183 
 
 179 
 
 127 
 
 258 
 
 379 
 
 Remainder, 
 
 
 
 
 
 
 
 
 
 
 EL. 
 
 or 
 
 AK.- 
 
 -7 
 
 
 
 
 
 
98 
 
 ELEMENTS OF ARITHMETICo 
 
 
 65. 56. 
 
 57. 58. 
 
 59. 60. 
 
 61. 
 
 Minuend, 
 
 635 591 
 
 624 350 
 
 870 980 
 
 527 
 
 Subtrahend, 
 
 469 268 
 
 156 123 
 
 456 379 
 
 318 
 
 Remainder, 
 
 
 
 
 
 
 62. 63. 
 
 64. 65, 
 
 66. 
 
 67. 
 
 Minuend, 
 
 ? 769 
 
 ? 780 800 
 
 ? 
 
 Subtrahend, 
 
 17 ? 
 
 238 ? 
 
 9 
 
 502 
 
 Remainder, 
 
 136 144 
 
 196 123 191 
 
 139 
 
 
 68. 
 
 69. 
 
 70. 
 
 
 Minuend, 
 
 $541 
 
 $905 
 
 768 children. 
 
 
 Subtrahend, 
 
 $•492 
 
 $184 
 
 599 
 
 
 Remainder, 
 
 $ 
 
 $ 
 
 children. 
 
 
 
 71. 
 
 72. 
 
 73. 
 
 
 Minuend, 
 
 808 eggs. 
 
 967 birds. 
 
 842 pints. 
 
 
 Subtrahend, 
 
 527 " 
 
 209 " 
 
 763 " 
 
 
 Remainder, 
 
 eggs. 
 
 birds. 
 
 pints. 
 
 
 
 74o 
 
 75. 
 
 76. 
 
 
 Minuend, 
 
 400 horses. 
 
 502 cows. 
 
 $550 
 
 
 Subtrahend, 
 
 285 " 
 
 237 " 
 
 $118 
 
 
 Remainder, 
 
 horses. 
 
 cows. 
 
 $ 
 
 
 
 77. 
 
 78. 
 
 79. 
 
 
 Minuend, 
 
 800 bushels. 
 
 ? apples. 
 
 ? days» 
 
 
 Subtrahend, 
 
 269 " 
 
 372 " 
 
 18 " 
 
 
 Remainder, 
 
 bushels. 
 
 263 apples. 
 
 347 daya 
 
 
 
 80. 
 
 81. 
 
 82. 
 
 
 Minuend, 
 
 75 years. 
 
 $772 
 
 $324 
 
 
 Subtrahend, 
 
 ? " 
 
 $624 
 
 9 
 
 
 Remainder, 
 
 19 years. 
 
 $ 
 
 $136 
 
 
SUBTRACTION 99 
 
 REVIEW EXERCISES. 
 ADDITION AND SUBTRACTION. 
 80. 1. From a tank containing 935 gallons of water, 648 
 gallons were drawn off, after wMch. 247 gallons ran in. 
 How many gallons were there then in the tank ? 
 
 2. Two trains start from Chicago. One goes east 298 
 miles, and the other west 247 miles. How far apart are 
 they ? 
 
 3. A man purchased 8983 bricks, but used only 5362. 
 How many had he left ? 
 
 4. A man paid ^465 for a carriage. He spent $ 57 for 
 repairs on it, and sold it for $ 500. Did he gain or lose, 
 and how much ? 
 
 5. From the sum of 763 and 297 subtract their differ- 
 ence. 
 
 6o A horse and cow together cost $276. If the cow 
 cost f 81, what was the cost of the horse ? 
 
 7. Two trains start from cities 582 miles apart. If 
 they travel towards each other, one 123 miles and the other 
 238 miles, how far apart are they ? 
 
 8. Columbus discovered America in 1492, How many 
 years have passed since then ? 
 
 9. A merchant bought 30 pieces of cloth, containing in 
 all 979 yards. He afterwards sold all but 143 yards. How 
 many yards did he sell ? 
 
 10. The cost of building a house was $3972, and of fur= 
 nishing it $ 1597. What was the entire cost ? How much 
 more did it cost to build it than to furnish it ? 
 
 11. Cotton was first planted in the United States about 
 1759. How long ago was that ? 
 
 12. Ealph threw a ball 37 feet, and Jamie threw his 19 
 feet farther than Ralph. How many feet must Jamie go to 
 pick up his ball and return ? 
 
100 ELEMENTS OF ARITHMETIC. 
 
 13. A man OAvnecl a horse that cost $852, and a carriage 
 that cost $437. He was obliged to sell them both for 
 $ 1000. How much money did he lose by the sale ? 
 
 14. My neighbor has in his orchard 264 pear trees, which 
 is 187 less than the number of his apple trees. How many 
 apple trees has he ? How many trees has he in all ? 
 
 15. A farmer having 425 bushels of wheat, sold 127 
 bushels to one person, and 264 bushels to another. How 
 many bushels had he left ? 
 
 16. $24.03 + $17.75 + $19.63 + $10 + $18.09 + |7.99=? 
 
 17. $10 -($2.35 + $3.78 + $!) = ? 
 
 18. Mr. Jones earned during July $ 172. He paid a bill 
 for $ 28 to the grocer, $ 19 to the butcher, and $ 36 for rent. 
 How much of the money remained ? 
 
 19. $4700- ($462 + $2180)=? 
 
 $ 3900 - ($ 2614 + $ 854.50) = ? 
 
 20. A man gave to his son $ 575 ; to his daughter, $468; 
 and to his nephew, $249 less than to his daughter. How 
 much did he give to all ? 
 
 21. Two men, who are 425 miles apart, travel towards 
 each other, one 36 miles, the other 58 miles a day. How 
 far apart will they be at the end of two days ? 
 
 22. If the same men travel away from each other, how 
 far apart will they be at the end of two days ? 
 
 23. In a certain school there are 692 pupils. In the 
 primary grade there are 325; in the intermediate grade, 
 259: and the remainder are in the grammar grade. How 
 many are in the grammar grade ? 
 
 24. A man bought a lot of land for $ 1860, and built upon 
 it a house costing $7584. If he sold the property for 
 $ 10,000, how much did he make ? 
 
 25. If I buy at a grocery store sugar for 78 cents, cheese 
 for 39 cents, and nuts for 25 cents, giving in payment a $2 
 bill, how much change should I receive ? 
 
SUBTRACTION. 101 
 
 26. From 946 subtract 486 ; from the result subtract 284 ; 
 to this add 398; and from this answer subtract 199. 
 
 ^27. Lena's book has 482 pages. If she reads 168 pages 
 in the morning, and 89 pages in the afternoon, how many 
 more pages has she to read ? 
 
 28. Bert and Harry had each $1.40. Bert earned 28j^ 
 more, and Harry spent 12^ of his money. How much more 
 has Bert than Harry ? 
 
 29. In a school there were 278 girls and 215 boys. If 29 
 girls and 34 boys leave, how many pupils will there then be 
 attending the school ? 
 
 30. Into a bin that holds 825 bushels, I put 272 bushels 
 of corn, and afterwards 346 bushels more. How many more 
 bushels of corn will it hold ? 
 
 31. If I buy a hat for $ 6.75 and give in payment a $ 10 
 bill, what change should I receive ? 
 
 32. The entire expenses of a journey made by a family 
 were $391.86. The fare on the railroads was $192.15, 
 carriage hire was $ 37.25, fare on the steamers amounted to 
 $61.35, and the rest was paid for board. What did the 
 board cost ? 
 
 33. A merchant failed in business, owing debts to the 
 amount of $ 1837.85. His goods on hand were worth 
 $894.50, and there was due him from persons who had 
 bought goods from him, $ 591.75. Hoav much more did he 
 owe than he could pay ? 
 
 34. $ 68.95 + $ 81.36 - $ 29.15 - $ 42.36 + $ 25.84 = ? 
 
 35. $ 76.37 + $ 34.93 + $ 8.83 - $ 16.25 - $ 13.24 = ? 
 
 36. $ 59.23 + $ 18.76 - $ 32.17 + $ 7.08 - $ 41.25 = ? 
 
 37. $ 32.94 - $ 16.23 + $ 13.81 - $ 20.06 + $ 24.53 = ? 
 
 38. $ 68.92 + $ 23.75 - $ 15.27 - $ 6.35 + $ 15.18 = ? 
 
 39. $ 37.50 - $ 18.24 + $ 60.08 - $ 18.32 - $ 16.29 = ? 
 
 40. $51.23 + $67. + $59.38 -$62.81-$ 4.85 = ? 
 
 41. $ 24.69 + $ 41.80 - $ 23 + $ 27.42 _$8 + $15 = ? 
 
MULTIPLICATION. 
 
 81. 1. Henry worked 5 days per week for 5 weeks. 
 How many days did he work in that time ? 
 
 2o A boy delivered books to 12 houses^ leaving 6 books 
 at each house. How many books did he carry altogether ? 
 
 3. The wages of a laborer were $12 per week. How 
 much could he earn in 6 weeks ? 
 
 4. The price of a cord of wood is $5, How much 
 must be paid for 7 cords ? 
 
 5. The postman who delivers our mail travels 9 miles 
 per day. How far will he travel in 8 days ? 
 
 6» A lady bought 11 yards of cambric at 10 cents per 
 yard. How much did it cost her ? 
 
 7. A good walker can walk 4 miles per hour. How 
 far can he walk in 9 hours at that rate ? 
 
 8. How many are eight 3's ? Nine 5's? 
 
 9. How many are seven 7^s ? Eight 6's ? 
 
 10. How many are six 8's ? Nine 4's ? 
 
 11. How many are four 6's ? Five 4's ? 
 
 12. What have you been doing with the numbers in the 
 examples given above ? 
 
 82. A short process of finding the sum of several equal 
 numbers, or the process of taking one numbsr as many times 
 as there are units in another is called Multiplication. 
 
 83. The number taken or multiplied is called the Multi- 
 plicand. 
 
 102 
 
MULTIPLICATION. 
 
 103 
 
 84. The number which shows how many times the multi- 
 plicand is taken is called the Multiplier. 
 
 85. The result obtained by multiplying is called the 
 Product. 
 
 86. The multiplicand and multiplier are called the Factors 
 of the product. 
 
 87. The Sign of Multiplication is an oblique cross, x . It 
 
 is read multiplied by when the multiplicand precedes it and 
 
 times when the multiplier precedes it. 
 
 Thus 6 X 8 is read 6 multiplied by 8 wlien 6 is the multiplicand, 
 but it is read 6 times 8 wlien 6 is tlie multiplier. 
 
 
 
 
 MULTIPLICATIOM 
 
 TABLE. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 1 9 
 
 10 
 
 11 
 
 12 
 
 2 
 3 
 
 4 
 
 6 
 
 6 
 9 
 
 8 
 12 
 16 
 
 10 
 15 
 
 20 
 25 
 
 12 
 18 
 24 
 30 
 
 14 
 21 
 
 28 
 35 
 
 16 
 24 
 
 32 
 40 
 
 48 
 
 18 
 
 
 
 27 
 
 36 
 
 45 
 
 54 
 
 20 
 30 
 40 
 50 
 60 
 
 22 
 33 
 44 
 
 55 
 
 m 
 
 24 
 36 
 48 
 60 
 
 72 
 
 4 
 
 8 
 
 12 
 
 5 
 
 10 
 
 15 
 
 20 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 7 
 8 
 
 14 
 
 16 
 
 21 
 24 
 
 28 
 32 
 
 35 
 40 
 
 42 
 
 48 
 
 49 
 56 
 
 56 
 64 
 
 63 
 
 72 
 
 70 
 80 
 
 77 
 88 
 
 84 
 96 
 
 9 
 
 18 
 
 27 
 
 36 
 
 45 
 
 54 
 60 
 66 
 
 63 
 
 70 
 
 77 
 84 
 
 72 
 80 
 88 
 96 
 
 81 
 ^0 
 99 
 
 108 
 
 90 
 100 
 110 
 120 
 
 99 
 110 
 121 
 132 
 
 108 
 120 
 132 
 144 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 11 
 
 22 
 
 33 
 
 44 
 
 55 
 
 12 
 
 24 
 
 36 
 
 48 
 
 60 
 
 72 
 
 Explanation. — The numbers in the left-liand column may be 
 regarded as the multipliers, and the numbers across the top as the 
 multiplicands. The products will be found in the horizontal columns 
 opposite the multipliers. 
 
 Thus, 2 ones are 2 ; 2 twos are 4 ; 2 threes are 6 ; 2 fours are 8, etc. 
 
 The order may be changed so that the numbers in the upper hori- 
 zontal line may be regarded as the multipliers, and the numbers on the 
 left as the multiplicands. Each successive multiplier may be used 
 with each multiplicand. 
 
 Thus, 1 two is 2 ; 2 twos are 4 ; 3 twos are 6 ; 4 twos are 8, etc» 
 
104 ELEMENTS OF ARITHMETIC. 
 
 DRILL EXERCISES. 
 
 Tell instantly the following products : 
 
 8x4 5x11 12 x6 7x5 2x7 
 
 5x3 12 x7 11 x5 2x6 3x4 
 
 6x S 9x9 4x9 3x5 2x4 
 
 7x2 8x12 5x5 4x2 12 x3 
 
 9x8 3x3 7x6 5x2 9x4 
 
 8x7 4x10 9x7 6x9 10x12 
 
 3x9 5x7 10 x8 7x11 11 x4 
 
 4x6 6x5 11x10 7x3 2x3 
 
 5x6 11 x7 2x11 2x9 8x6 
 
 3x7 12 x9. 9x2 4x4 6x7 
 
 4x8 7x8 5x9 6x3 12 x8 
 
 5x12 9x6 7x4 10x11 10 x4 
 
 6 X 11 10 X 10 9 X 11 8 X 11 8 X 10 
 
 7x10 11x11 10 X 7 7x12 4x 7 
 
 8x9 12 X 12 8x8 9 X 10 10 x 3 
 
 9x5 7X'7 4x5 10 x6 12 x2 
 
 5x S 8x5 5 X 10 11 X 9 4 x 12 
 
 3x10 2x8 6x12 12x11 3x2 
 
 4x3 2x12 7x9 2x5 11 x3 
 
 3x11 9x3 6x10 3x8 11 x2 
 
 2 X 10 3 X 12 6 X 6 6x2 11 x 6 
 
 5x4 8x3 12 X 5 10 X 5 12 x 10 
 
 10 X 9 3x6 8x2 11 X 12 10 x 2 
 
 11 X 8 6x4 9 X 12 12 X 4 4 x 11 
 
MULTIPLICATION. 105 
 
 88. When the multiplier is not greater than 12. 
 
 ORAL EXERCISES. 
 
 1. When oranges cost 4 cents apiece, how much must 
 Mary pay for 6 oranges ? 
 
 2. If Lulu changes 6 half-dimes into cents, how many 
 cents will she have ? 
 
 3. Henry went to school every school-clay for 8 weekso 
 How many days was that ? 
 
 4. A girl bought 9 yards of ribbon for 8 cents a yard ? 
 How much did it cost her ? 
 
 5. An organ-grinder received 7 cents at each of 8 
 houses. How much did he get in all ? 
 
 6. A boy sold 9 pencils at 4 cents apiece. How much 
 did he get for them all ? 
 
 7. In a school-room there were 8 rows of seats, and 8 
 seats in a row. How many pupils could be seated there ? 
 
 8. A half-quire of paper contains 12 sheets. How many 
 sheets are there in 6 half-quires ? 
 
 9. A drawing book cost 12 cents. If each pupil in a 
 class of 9 pupils paid the same price for a drawing book, 
 how much did they all pay ? 
 
 10. Harry had to pay 9 cents per day for his fare in 
 going to and from school. How much was that per week ? 
 
 11. Alice planted 9 rows of asters in her flower bed, and 
 each row had 9 plants. How many asters did she set out ? 
 
 12. Each pupil in a class wrote 7 lines in his writing 
 book. If there were 7 pupils in the class, how many lines 
 did they all write ? 
 
 13. The cost of a railroad ticket from Ashburn to Gray- 
 • stone is 11 cents. How much will 8 such tickets cost ? 
 
 14. Some children are required to attend school 6 hours 
 per dayo How many hours will they be at school in 7 
 days? 
 
106 ELEMENTS OF ARITHMETIC. 
 
 15. If Mary goes to bed at 8 o'clock at night, and rises at 
 7 o'clock in the morning, how many hours does she spend 
 in bed in 9 days ? 
 
 16. Eeuben said that he saw 5 flocks of birds, each con- 
 taining 7 birds. How many birds did he see ? 
 
 17. A school week is 5 days. How many days are there 
 in 9 school weeks ? 
 
 18. Sarah made on her slate 9 rows of squares, each row 
 containing 7 squares. How many squares did she make ? 
 
 19. Henry brought home 5 dozen eggs. How many eggs 
 did he bring ? 
 
 20. Susie's father worked 9 hours per day for 7 days. 
 How many hours did he work in all ? 
 
 21. How many are two 2's ? Three 2's ? Four 2's ? 
 Five 2's ? etc. Give the table of 2's. 
 
 22. Give the table of 3's. Of 4's. Of 5's. Of 6's. Of 
 7's. OfS's. Of9's. OflO's. Ofll's. Of 12's. 
 
 23. Count by 5's from to 100 in this way : 0, 5, 10, 15, 
 20, 25, 30, etc. 
 
 24. Count by 3's from to 36. By Ts from to 84. 
 
 25. Count by 6's from to 72. By 9's from to 108. 
 
 26. Count by 4's from to 48. By 8's from to 96. 
 
 27. Count by lO's from to 110. By 12's from to 144. 
 
 WRITTEN EXERCISES. 
 89. 1 . How many are 6 times 397 ? 
 
 Multiplicand, 397 Explanation. — For convenience in maltiply- 
 
 - , ,,. ,. ^ ing, the multiplier is written under the multiplicand 
 
 Multiplier, O »' , . , . , , • i • .i ^ n 
 
 and we begm at the right to multiply m the follow- 
 Product, Z6'6Z ing manner: 
 
 6 times 7 units are 42 units. But 42 units are 4 tens and 2 units ; 
 therefore, the figure 2 is written in units' place in the product, and the 
 4 tens are reserved to add to the tens of the product. 
 
 6 times 9 tens are 54 tens, plus 4 tens reserved are 58 tens, or 5 
 hundreds and 8 tens ; therefore the figure 8 is written in tens' place in 
 
MULTIPLICATION. 
 
 107 
 
 the product, and the 5 hundreds are reserved to add to the hundreds 
 in the product. 
 
 6 times 3 hundreds are 18 hundreds, plus 5 hundreds reserved are 
 23 hundreds, or 2 thousands and 3 hundreds, which are written in 
 thousands' and hundreds' places in the product. Hence the product 
 is 2382. 
 
 Find the products of the following : 
 
 5x46. 
 6x37. 
 7x28. 
 6x93. 
 7 x86. 
 4x98. 
 3x69. 
 8x58. 
 9 X 43. 
 6x85. 
 
 12. 11 X 87. 
 
 13. 12 X 92. 
 
 2 
 3 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 
 14. 415 X 
 
 15. 372 X 
 
 16. 846 X 
 
 17. 364 X 
 
 18. 918 X 
 
 19. 325 X 
 
 20. 864 X 
 
 21. 325 X 
 
 22. 816 X 
 
 23. 932 X 
 
 24. 519 X 11 
 
 25. 837 X 12 
 
 26. 3245 X 
 
 27. 2861 X 
 
 28. 3942 X 
 
 29. 5687 X 
 
 30. 3834 X 
 
 31. 5279 X 
 
 32. 8168 X 
 
 33. 3934 X 
 
 34. 5468 X 
 35 o 3173 X 
 
 36. 4284 X 11 
 
 37. 1987 X 12 
 
 38. There are 5280 feet in a mile. How many feet does 
 a boy walk who walks 5 miles ? 
 
 39. A merchant sold Mrs. Allen 5 pieces of muslin, each 
 containing 39 yards. How many yards did he sell to her? 
 
 40. The average Avages of 5 clerks were $37.50 per 
 month. How much did they all earn in a month ? 
 
 41. When hay sells at $8.25 per ton, how much must I 
 pay for 8 tons ? 
 
 42. If a railway train runs 47 miles per hour, how far 
 will it run in 8 hours ? 
 
 43. Theodore Schuyler & Co. paid each of 5 bookkeepers 
 §85.25 per month. How much did they pay them all ? 
 
 44o A laborer found that his expenses were $ 23.85 per 
 month. How much will they be in 8 months ? 
 
108 ELEMENTS OF ARITHMETIC. 
 
 45. A carriage maker sold 8 carriages for $78.75 apiece. 
 How much did lie receive for them all ? 
 
 46. Mr. A loaned a man some money and received for 
 the use of it, or interest, $ 18.35 per year. How much 
 should he receive for the use of it for 5 years ? 
 
 47. A contractor built 8 houses for which he received 
 $ 7325 apiece. How much did he receive for all ? 
 
 48. The expenses of a large dry-goods store were $ 3215.25 
 per month. How much were they for 8 months ? 
 
 49. An organ manufacturer sold during 1891 nine church 
 organs, the average price of which Avas $ 7963. How much 
 did his sales amount to that year ? 
 
 50. The number of papers sold by the publisher of a 
 daily paper averaged 8432 each day. How many w^ere sold 
 in 6 days ? 
 
 51. A mechanic earned $87.35 per month for 8 months. 
 How much were his entire earnings in that time ? 
 
 52. A man sold 9 carriages for $92.55 apiece. How 
 much did he get for them ? 
 
 53. The steamship Koduna consumes 125 tons of coal 
 per day. How much will she consume in a voyage lasting 
 8 days ? 
 
 54. In one of the cities of the United States, the average 
 number of passengers carried upon the street-cars daily was 
 38,395. How many were carried in 6 days ? 
 
 55. The sales of tickets at a railway station for the 
 month of January amounted to $38,495. If they are the 
 same, on the average, for 7 months, how much will they be ? 
 
 56. The interest paid for a sum of money was $384.27 
 per year. How much interest money was paid in 5 years ? 
 
 57. The increase in population in a city for the year 
 1891 was 13,868. At that rate, what will be the increase 
 in 6 years ? 
 
 68. The average distance sailed by a steamship each voy- 
 
MULTIPLICATION. 109 
 
 age between New York and Liverpool, England, was 2831 
 miles o How many miles did she sail in 4 round trips ? 
 
 59. A gardener sold last year 9765 bunches of asparagus. 
 How many bunches would he sell in 7 years at that rate ? 
 
 Multiply each of the following numbers by 2, 3, 4, 5, 6, 
 7, 8, and 9 : 
 
 60. 5869. 63. 3954. 66. 86,973. 69. 18,572. 72. 48,362. 
 
 61. 7296 64. 3857. 67. 23,547. 70. 42,364. 73. 32,573. 
 
 62. 8164. 65. 5679. 68. 65,132. 71. 38,453. 74. 71,685. 
 
 90. When the multiplier is expressed by more than one figure. 
 
 ORAL EXERCISES. 
 
 1. How many are 10 times 6 ? 10 times 7 ? 10 times 8 ? 
 10 times 5 ? 10 times 9 ? 
 
 2. How many are 10 times 3 plus 3 times 3 or 13 times 
 3? 10 times 4 plus 6 times 4 or 16 times 4? 10 times 7 
 plus 3 times 7 or 13 times 7 ? 10 times 8 plus 5 times 8 
 or 15 times 8 ? 
 
 3. How many are 15 times 4 ? 18 times 3 ? 17 times 5 ? 
 16 times 6 ? 12 times 8 ? 14 times 3 ? 
 
 4. How many are 13 times 5 ? 13 times 6 ? 13 times 7 ? 
 
 5. What will 15 yards of ribbon cost at 5 cents per 
 yard ? At 8 cents per yard ? 
 
 6. How many days are there in 17 weeks ? 
 
 7. How many quarts are there in 16 gallons ? 
 
 8. How many feet are there in 13 yards ? 
 
 9. How much will 15 yards of calico cost at 6 cents per 
 yard ? , At 9 cents per yard ? 
 
 10. How much does a man save in 12 weeks if he saves 
 ^8 per week? 
 
 11. There are 9 square feet in a square yard. How many 
 square feet are there in 14 square yards ? 
 
110 ELEMENTS OF ARITHMETIC. 
 
 12. How many are 10 times 9 ? 10 times 5 ? 10 times 7 ? 
 
 13. What figure must be annexed to 9 to multiply it by 
 10? What to 5? What to 7? 
 
 14. How, then, is any number multiplied by 10? 
 
 15. How many are 10 sevens or 10 times 7 ? 100 sevens 
 or 100 times 7 ? ' 100 eights or 100 times 8 ? 100 times 6 ? 
 100 times 15? 100 times 35 ? 100 times 48? 100 times 56? 
 100 times 84 ? 100 times 97 ? 100 times 63 ? 
 
 16. How, then, is any number multiplied by 100 ? 
 
 17. How many are 100 times 9 ? 1000 times 9 ? 1000 
 times 7 ? 1000 times 15 ? 1000 times 35 ? 1000 times 43 ? 
 1000 times 54 ? 1000 times 75 ? 
 
 18. How, then, is any number multiplied by 1000 ? 
 
 19. Since a number is multiplied by 10 by annexing one 
 cipher to the number; by 100 by annexing two ciphers; by 
 1000 by annexing three ciphers, how may a number be mul- 
 tiplied by 10, 100, 1000, etc., or by 1 with any number of 
 ciphers annexed ? 
 
 91. A number is multiplied by 10, 100, 1000, etc., or by 1 
 with cmy number of ciphers annexed, by annexing to the multi- 
 plicand as many ciphers as there are in the multiplier. 
 
 WRITTEN EXERCISES. 
 
 92. What are the products of the following: 
 
 1. 45 X 10. 6. 324 X 100. 11. 5463 x 1000. 
 
 2. 67 X 10. 7. 416 X 100. 12. 3784 x 1000. 
 
 3. 38 X 10. 8. 723 x 100. 13. 5169 x 10000. 
 
 4. 46 X 10. 9. 342 X 100. 14. 4237 x 10000. 
 
 5. 76 X 10. 10. 518 X 100. 15. 8193 x 100000. 
 
 16. Multiply 832 x 3000. 
 
 ^^^ Explanation. —Since 3000 is 1000 times 3, 832 is first 
 
 3000 multiplied by 8, giving 2496 as a product, and this product 
 9J.Qrn00 ^^ multiplied by 1000 by annexing three ciphers. 
 
MULTIPLICATION. m 
 
 17. 
 
 45 X 20. 
 
 
 27. 
 
 346 X 500. 
 
 37. 
 
 3946 X 6000. 
 
 18. 
 
 37 X 30. 
 
 
 28. 
 
 279 X 300. 
 
 38. 
 
 5279 X 4000. 
 
 19. 
 
 68 X 40. 
 
 
 29. 
 
 518 X 700. 
 
 39. 
 
 8365 X 7000. 
 
 20. 
 
 73 X 80. 
 
 
 30. 
 
 264 X 900. 
 
 40. 
 
 7226 X 9000. 
 
 21. 
 
 56 X 60. 
 
 
 31. 
 
 385 X 400. 
 
 41. 
 
 5387 X 8000. 
 
 22. 
 
 90 X 40. 
 
 
 32. 
 
 498 X 600. 
 
 42. 
 
 7394 X 5000. 
 
 23. 
 
 30 X 50. 
 
 
 33. 
 
 341 X 800. 
 
 43. 
 
 2869 X 3000. 
 
 24. 
 
 23 X 90. 
 
 
 34. 
 
 957 X 200. 
 
 44. 
 
 9999 X 2000. 
 
 25. 
 
 84 X 70. 
 
 
 35. 
 
 691 X 500. 
 
 45. 
 
 8765 X 4000. 
 
 26. 
 
 65 X 60. 
 
 
 36. 
 
 444 X 400. 
 
 46. 
 
 9182 X 6000. 
 
 47. 
 
 Multiply 
 
 373 
 
 X 235. 
 
 
 
 
 First Process. 
 
 373 Second Process. 
 
 373 
 
 
 
 
 
 235 
 
 
 235 
 
 Ist partial product, 1865 1865 
 
 2d partial product, 11190 1119 
 
 3d partial product, 74600 746 
 
 Entire product, 87655 87655 
 
 Explanation of First Process, — For convenience, the multiplier 
 is written under the multiplicand with units under units, tens under 
 tens, etc. 
 
 Since we cannot multiply by 235 at one operation, we multiply by 
 the parts of 235, viz. : 5 units, 3 tens or 30, and 2 hundreds or 200, 
 and add the products. 
 
 5 times 373 is 1865, the first partial product ; 30 times 373 is 11190, 
 the second partial product ; 200 times 373 is 74600, the third partial 
 product. The sum of these partial products is 87655, the entire 
 product. 
 
 Explanation op Second Process. — The second process is the one 
 commonly used. It is the same as the first except that the ciphers are 
 omitted from the right of the partial products. The significant figures, 
 however, occupy their proper places. 
 
 Thus, in multiplying by 3 tens the product is 1119 tens, and conse- 
 quently, the right-hand figure of this product is placed under tens and 
 the others in their proper places in the product. 
 
 In multiplying by 2 hundreds or any number of hundreds the lowest 
 order of the product is hundreds, consequently the right-hand figure 
 of the product is written under hundreds. 
 
112 
 
 ELEMENTS OF ARITHMETIC. 
 
 93. From these processes the following general method of 
 multiplying i^ readily deduced : 
 
 1. The multiplier is imntten under the multiplicand, units 
 under units, tens under tens, etc. 
 
 2. Each figure of the multiplicand is multiplied by each 
 signijicant figure of the midtiplier and the right-hand figure of 
 each product is placed under the figure of the multiplier used 
 to obtain it. 
 
 3. The sum of the p)artial products luill be the entire product. 
 
 When there is a cipher in the multiplier, multiply by the significant 
 figures only, taking care to place the right-hand figure of each partial 
 product under the figure used to obtain it. 
 
 
 48. 
 
 
 49. 
 
 
 50. 
 
 
 5684 
 
 
 78645 
 
 
 59658 
 
 
 609 
 
 
 5008 
 
 
 3700 
 
 
 51156 
 
 
 629160 
 
 417606 
 
 
 34104 
 
 
 393225 
 
 1/ 
 
 '8974 
 
 
 3461556 
 
 
 393854160 
 
 220734600 
 
 ■ Multiply : 
 
 
 
 
 
 51. 
 
 267 by 24. 
 
 66. 
 
 3845 by 236. 
 
 81. 
 
 5698 by 792. 
 
 52. 
 
 836 by 27. 
 
 67. 
 
 7186 by 314. 
 
 82. 
 
 3179 by 865. 
 
 53. 
 
 918 by 61. 
 
 68. 
 
 5327 by 420. 
 
 83. 
 
 8169 by 734. 
 
 54. 
 
 309 by 38. 
 
 69. 
 
 6284 by 518. 
 
 84. 
 
 5234 by 826. 
 
 55. 
 
 465 by 27. 
 
 70. 
 
 4769 by 344. 
 
 85. 
 
 3186 by 839. 
 
 56. 
 
 732 by 55. 
 
 71. 
 
 6873 by 530. 
 
 86. 
 
 5279 by 918. 
 
 57. 
 
 817 by 63. 
 
 72. 
 
 2918 by 364. 
 
 87. 
 
 3864 by 609. 
 
 58. 
 
 398 by 84. 
 
 73. 
 
 4873 by 273. 
 
 88. 
 
 38,675 by 708. 
 
 59. 
 
 426 by 29. 
 
 74. 
 
 9275 by 731. 
 
 89. 
 
 42,308 by 692. 
 
 60. 
 
 918 by 76. 
 
 75. 
 
 8462 by 820. 
 
 90. 
 
 75,069 by 804. 
 
 61. 
 
 837 by 84. 
 
 76. 
 
 7319 by 394. 
 
 91. 
 
 83,427 by 3625- 
 
 62. 
 
 695 by 32. 
 
 77. 
 
 8425 by 972. 
 
 92. 
 
 80,069 by 5846. 
 
 63. 
 
 864 by 69. 
 
 78. 
 
 6685 by 873. 
 
 93. 
 
 58,325 by 3192. 
 
 64. 
 
 528 by 87. 
 
 79. 
 
 6517 by 762. 
 
 94. 
 
 38,164 by 8068. 
 
 65. 
 
 364 by 44. 
 
 80. 
 
 7395 by 469. 
 
 95. 
 
 49,234 by 9096. 
 
MULTIPLICATION. 113 
 
 96. f 48.27 by 584. 101. $543.79 by 3980. 
 
 97. $39.35 by 376. 102. $ 708.27 by 2900. 
 
 98. $ 73.86 by 705. 103. $ 496.83 by 5805. 
 
 99. $85.18 by 835. 104. $ 385.76 by 3760. 
 100. $49.66 by 597. 105. $483.74 by 5736. 
 
 106. A manufacturer sold 29 reapers for $ 96.85 ajoiece. 
 How much did he get for them ? 
 
 107. A man walked 37 miles. How many feet did he 
 walk, since there are 5280 feet in a mile ? 
 
 108. A tract of land containing 294 acres was sold for 
 $ 313 per acre. For ho^v much did it sell ? 
 
 109. It required 318 loads of bricks averaging 1387 bricks 
 to build a school building. How many bricks was that ? 
 
 110. A railroad employed 213 switchmen, whose wages 
 averaged $ 78.34 per month. How much did all the switch- 
 men receive per month ? 
 
 111. A merchant sold 7 pieces of cloth, each containing 
 34 yards, for $2.17 per yard. How much did he receive 
 for it ? 
 
 112. The Danton Leather Co. purchased 37 lots of hides, 
 each lot containing 384 hides. How many did they pur- 
 chase ? 
 
 113. A cloth dealer sold 54 pieces of cassimere, each con- 
 taining 48 yards, at $ 1.87 per yard. How much did he 
 receive for it ? 
 
 114. A fire destroyed 345 gross of pencils. Since a gross 
 is 144, how many pencils were destroyed ? 
 
 115. In a day there are 1440 minutes. How many min- 
 utes are there in a year ? 
 
 116. A farm of 95 acres was sold for $37.25 per acre. 
 How much was received for it ? 
 
 117. If your pulse beats 4320 times per hour, how many 
 times will it beat in 2 days of 24 hours each ? 
 
 118. The cost of building a railroad was $ 31,755 per mile. 
 
 EL. OF AU. — 8 
 
114 ELEMENTS OF ARITHMETIC. 
 
 If it was 118 miles long^ how much, did it cost to construct 
 it? 
 
 119. A pump can pump 365,864 gallons per day. How 
 many gallons can it pump in 31 days ? 
 
 120. The expense of an army in Europe is, on the average, 
 $ 195 per man. How much will it cost to keep an army of 
 32,596 men ? 
 
 121. If a locomotive costs $19,325, how much will 23 
 such locomotives cost ? 
 
 122. An army used 3285 barrels of flour each day for 49 
 days. How many barrels did it use altogether ? 
 
 123. A shoe manufacturer in Massachusetts sold 23,876 
 pairs of shoes at an average price of $3.15 a pair. How 
 much did he receive for them ? 
 
 REVIEW. 
 
 ORAL EXERCISES. 
 
 94. 1. Henry bought 3 pencils at 3 cents apiece, and 5 
 postage stamps at 2 cents apiece. How much did he pay 
 for all ? 
 
 2. The wages of a clerk were $ 15 per week. If he paid 
 $ 5 for board and $ 4 for other expenses, how much did he 
 save in 5 weeks ? 
 
 3. Nina spent 20 cents for a box of water-color paints, 5 
 cents for brushes, and had 10 cents left. How much money 
 had she at first ? 
 
 4. Two men travel in opposite directions from a starting- 
 point, one at the rate of 5 miles per hour, and the other at 
 the rate of 4 miles per hour. How far apart are they at 
 the end of 5 hours ? 
 
 5. Mary counted the petals on 6 flowers. All but 2 of 
 them had 6 petals each, and the others had 5 each. How 
 many petals had they all ? 
 
MULTIPLICATION. 115 
 
 6. Hannah was at school 6 hours per day for 4 days, and 
 3 hours on the other day. How many hours was she at 
 school that week ? 
 
 7. Warren had 5 hens and each hen had 7 chickens. 
 How many chickens were there ? 
 
 8. A grocer sold 5 pounds of sugar at 5 cents a pound, 
 and received in payment a fifty-cent piece. How much 
 change should he give back. 
 
 9. Henry counted a company of soldiers in this way: 
 he found that there were 8 rows of 9 men each and three 
 officers. How many soldiers did he see ? 
 
 10. Some children found 5 birds' nests containing 5 eggs 
 each, and 4 nests containing 4 eggs each. How many 
 eggs did they find ? 
 
 11. A farmer sold 5 calves at $ 5 each, and 5 sheep at 
 $ 4 each. How much did he receive for all ? 
 
 12. Mary paid 10 cents for a dozen peaches, and 2 cents 
 apiece for a dozen bananas. How much did they all cost 
 her? 
 
 13. A boy worked 6 days for 5 dimes per day, but he 
 spent 2 half-dimes for car fare each day. How much did 
 he save ? 
 
 14. Two brothers worked during the summer vacation of 
 10 weeks, one for f 6 per week and the other for $ 3 per 
 week. How much more did the one earn than the other in 
 that time ? 
 
 15. The pupils in a school were divided into 8 classes or 
 grades, each containing 9 pupils. If every class contained 
 5 girls and 4 boys, how many boys were there in the school ? 
 How many girls ? How many pupils ? 
 
 16. A merchant employed 5 clerks at $8 per week, and 
 2 at $ 10 per week. How much did he pay them all per 
 week? 
 
 17. The retail price of asparagus was 15 cents a bunch, 
 
116 ELEMENTS OF ARITHMETIC. 
 
 and the wholesale price was 12 cents. How much is a 
 retail dealer's profit on 20 bunches which he bought at 
 wholesale ? 
 
 18. A hatter bought 12 hats at $4 apiece, and sold them 
 at $ 6 apiece. How much did he gain ? 
 
 19. A clothier sold overcoats at $ 12.50 which cost him 
 $ 9.50. How much was his gain on 20 coats ? 
 
 20. After paying ^2 apiece for repairing clocks that 
 cost $ 10 apiece, a man sold 12 of them for $ 144. Did he 
 gain or lose by the sale, and how much ? 
 
 WRITTEN EXERCISES. 
 
 95. 1. A bookkeeper received $75 per month for his 
 services, but he spent in a year f 680. How much did he 
 save in a year ? 
 
 2. A train composed of 23 cars, containing 145 sheep 
 each, arrived at Boston and the sheep were sold at $ 5 a head. 
 How much was received for them ? 
 
 3. A boy had $350 when he was 15 years old, and he 
 saved $ 100 per year until he was 21. How much had he 
 then ? 
 
 4. A manufacturer employed 118 workmen whose aver- 
 age wages per month were $ 68. How much did they all 
 earn per month ? 
 
 5. A drover bought 83 cattle at $37.25 per head. He 
 sold 50 of them at $45 per head, and the rest at $35.25 per 
 head. How much did he gain ? 
 
 - 6. A tea merchant sold 348 pounds of tea at 57 cents 
 per pound and gained $ 13.55. How much did it cost him ? 
 7. A and B each began business with $4000. A gained 
 $600 the first year and B lost an equal sum. The second 
 year A gained $ 318 and B gained $ 895. How much was 
 each worth then ? 
 
MULTIPLICATION. 117 
 
 8. A farmer raised 895 bushels of oats. He kejjt 50 
 bushels for seed, and enough to keep 10 horses, allowing 15 
 bushels for each horse, and sold the rest at 35 cents per 
 bushel. How much did he receive for the amount sold ? 
 
 9. A coal dealer sold 315 tons of coal at $5.25 per ton. 
 He received in payment a note for $ 1000 and the rest in 
 cash. How much cash did he receive ? 
 
 10. The profits of a merchant were $ 5896.75. He paid 
 $ 1391 for his household expenses, and 3 times as much for 
 other expenses. What was his net gain? 
 
 11. What will be the cost of 485 chests of tea, each con- 
 taining 55 pounds, at 38 cents a pound ? 
 
 12. I had a farm of 313 acres. 185 acres yielded me an 
 income of $ 6.25 per acre, and the rest $ 6 per acre. How 
 much was my entire income from the farm ? 
 
 13. My agent sold 3 lots of goods. For the first he 
 received $ 945, for the second twice as much as for the first, 
 and for the third 3 times as much as for the other two. 
 How much did he receive for all ? 
 
 14o Three men formed a partnership. A contributed 
 $690, B twice as much, and C three times as much as A and 
 B. What was their entire capital ? 
 
 15. A drover bought 132 cattle at $45 per head, and 67 
 at $ 61 per head. He sold them all at $ 50 per head. Did 
 he gain or lose, and how much ? 
 
 16. A farmer sold his farm at a gain of $ 1685. The barns 
 cost him $ 1855, his house $ 8520, and the land twice as much 
 as the barns and house. How much did he get for the farm ? 
 
 17. A merchant bought 35 pieces of carpet, each con- 
 taining 49 yards, at $1.18 per yard. He paid cash for a 
 part of the cost and a note to settle the account for $ 1250. 
 How much cash did he pay ? 
 
 18. Three men began business at the same time. During 
 the first year A made a profit of $ S55, B of 3 times as 
 
118 ELEMENTS OF ARITHMETIC. 
 
 niuclij and C of 4 times as much as A. At the beginning 
 of the second year they formed a partnership in which each 
 man contributed his gain of the previous year,. What was 
 the entire capital ? 
 
 19. The daily sales cf a newspaper for 5 days were 
 16,845 copies. If the papers were sold for 3 cents each, 
 what was the amount received ? 
 
 20. I know a man whose income is $ 85.75 per day. How 
 much does he receive in a year, or 365 days ? 
 
 21. The annual expense of a school for 5 years was 
 $ 3825.25, exclusive of the wages of the teachers. In the 
 school there were 11 teachers whose average annual salary 
 was $ 815.S5. What was the annual expense of the school ? 
 
 22. It was discovered in taking a census that a certain 
 town contained 145 houses, that each house, on the average, 
 contained 2 families, and that each family averaged 6 per- 
 sons. How many people were there in the town ? 
 
 23. A man bought a farm of 189 acres. He sold 99 
 acres at $ 87 per acre, and thus reduced the cost of the rest 
 to $ 58 per acre. How much did the farm cost him ? 
 
 24. A single poppy has been known to contain 32,000 
 seeds. How many seeds will 335 such poppies contain ? 
 
 25 o In a codfish 9,334,000 eggs have been found. If 
 from each egg a fish was hatched, how many fishes were 
 hatched from the eggs of 27 such codfish ? 
 
 26. A man set out on foot to travel 600 miles. He 
 traveled during the first 5 days 45 miles per day ; during 
 the next 6 days 38 miles per day. How many miles had 
 he still to travel ? 
 
 27. A ship left port, sailing at the rate of 9 miles per 
 hour. After sailing for 23 hours she was driven directly 
 back by a storm at the rate of 12 miles per hour for 10 
 hours. How far v/as she from the port from which she sailed, 
 at the end of 30 hours ? 
 
DIVISION. 
 
 96. 1. How many groups of 2 squares can be formed 
 from 6 squares ? How many 2's are there in 6 ? 
 
 2. How many groups of 3 oranges can be formed from 9 
 oranges ? How many 3's are there in 9 ? 
 
 3. Henry earns 6 cents an hour. In how many hours 
 can he earn 42 cents ? How many 6's are there in 42 ? 
 
 4. Louise arranged 56 spools in rows, placing 7 in each 
 row. How many rows did she form ? 
 
 5 c Daisy's brother was ill for 49 days. How many weeks 
 was he ill ? How many 7's are there in 49 ? 
 
 6. Anna had 63 chickens which were hatched in broods 
 of 9 chickens each. How many broods were there ? How 
 many 9's are there in 63 ? 
 
 7. A lad rides each day 8 miles in going to and from 
 school. In how many days will he ride 96 miles ^ How 
 many times is 8 contained in 96 ? 
 
 8. Mollie arranged 12 books in 4 piles. How many 
 books were there in each pile ? 
 
 9. How many apples will each child receive, when 10 
 apples are divided equally among 5 children ? 
 
 10. What have you been doing with the above numbers ? 
 
 97. The process of finding how many times one number 
 is contained in another, or the process of separating a num- 
 ber into equal parts, is called Division. 
 
 98. The number to be divided is the Dividend. 
 
 119 
 
120 ELEMENTS OF ARITHMETIC. 
 
 99. The number by wbicli we divide is the Divisor. 
 
 100. The result obtained by division is the Quotient. 
 
 101. The part of the dividend remaining when the 
 division is not exact, is the Remainder. 
 
 102. The Sign of Division is -4-. It is read divided by. 
 When placed between two numbers, it shows that the one 
 at the left is to be divided by the one at the right. 
 
 Thus, 24 -T- 6 is read, 24 divided by 6. 
 
 103. Division is also indicated by writing the dividend 
 above the divisor, with a line between them. 
 
 Tims, '^f- is read, 24 divided by 6. 
 
 Another method of indicating division is by writing the 
 divisor at the left of the dividend, with a line between them. 
 Thus, 6)24 is read, 24 divided by 6. 
 
 DRILL EXERCISES. 
 
 104. Give the results instantly : 
 
 9x 
 
 2 = ? 
 
 
 9x ? 
 
 = 18 
 
 ? 
 
 X 
 
 2 
 
 = 18 
 
 
 18 
 
 — 2 = ? 
 
 6x7 = ? 
 
 
 6x? 
 
 = 42 
 
 ? 
 
 X 
 
 7 
 
 = 42 
 
 
 42 
 
 — 7 = ? 
 
 5x8 = ? 
 
 
 5x? 
 
 = 40 
 
 ? 
 
 X 
 
 8 
 
 = 40 
 
 
 40 
 
 -8 = ? 
 
 7x5=? 
 
 
 7x ? 
 
 = 35 
 
 ? 
 
 X5 
 
 = 35 
 
 
 35 
 
 -5 = ? 
 
 8x3 = ? 
 
 
 8x? 
 
 = 24 
 
 9 
 
 X 
 
 3 
 
 = 24 
 
 
 24 
 
 -3 = ? 
 
 4x9=? 
 
 
 4x ? 
 
 = 36 
 
 ? 
 
 X 
 
 9 
 
 = 36 
 
 
 36 
 
 -9=? 
 
 6x8 = ? 
 
 
 6 X ? 
 
 = 48 
 
 ? 
 
 X 
 
 8 
 
 = 48 
 
 
 48 
 
 -v-8=? 
 
 20- 
 
 -10 
 
 21 
 
 -^ 7 
 
 48- 
 
 12 
 
 
 
 50- 
 
 10 
 
 
 22—2 
 
 8- 
 
 - 4 
 
 48 
 
 ^ 4 
 
 24- 
 
 4 
 
 
 
 27- 
 
 9 
 
 
 32-8 
 
 20- 
 
 - 2 
 
 42 
 
 -f- 6 
 
 16- 
 
 2 
 
 
 
 66- 
 
 6 
 
 
 60- 5 
 
 96- 
 
 - 8 
 
 44 
 
 -11 
 
 24- 
 
 6 
 
 
 
 60- 
 
 12 
 
 
 88-11 
 
 6- 
 
 - 3 
 
 28 
 
 - 4 
 
 30- 
 
 5 
 
 
 
 63- 
 
 7 
 
 
 40-5 
 
 22- 
 
 -11 
 
 36 
 
 - 4 
 
 m^ 
 
 11 
 
 
 
 48- 
 
 8 
 
 
 44-4 
 
 12- 
 
 :- 2 
 
 70 
 
 -10 
 
 8- 
 
 2 
 
 
 
 49- 
 
 7 
 
 
 96 - 12 
 
 20- 
 
 - 4 
 
 63 
 
 - 9 
 
 81 -i- 
 
 9 
 
 
 
 35 -j- 
 
 7 
 
 
 24-3 
 

 
 DIVISIOX. 
 
 
 121 
 
 24-12 
 
 88- 8 
 
 35-5 
 
 . 90-10 
 
 45-9 
 
 18- 3 
 
 55-11 
 
 80-10 
 
 48-6 
 
 108- 9 
 
 60-6 
 
 84-7 
 
 10-2 
 
 110-11 
 
 121 - 11 
 
 15-3 
 
 16-4 
 
 77 - 11 
 
 80- 8 
 
 108 - 12 
 
 56-7 
 
 14-7 
 
 6- 2 
 
 99-9 
 
 21-3 
 
 90-9 
 
 4-- 2 
 
 36- 9 
 
 40-8 
 
 132 - 12 
 
 70-7 
 
 16-8 
 
 84-12 
 
 50-5 
 
 120 - 10 
 
 72- 8 
 
 72-12 
 
 18-2 
 
 27-3 
 
 110-10 
 
 20-5 
 
 30-3 
 
 54-9 
 
 100 - 10 
 
 120 - 12 
 
 12-6 
 
 18- 6 
 
 12-3 
 
 24-8 
 
 132 - 11 
 
 33-11 
 
 10-5 
 
 12-4 
 
 9- 3 
 
 144 - 12 
 
 40-4 
 
 15-5 
 
 33-3 
 
 36-6 
 
 42-7 
 
 25-5 
 
 30-10 
 
 32-4 
 
 77-7 
 
 60-10 
 
 30-6 
 
 24-2 
 
 40-10 
 
 99-11 
 
 56-8 
 
 18-9 
 
 72-6 
 
 64-8 
 
 28-7 
 
 36-3 
 
 54-6 
 
 36 - 12 
 
 55-5 
 
 72- 9 
 
 14- 2 
 
 105. When the divisor is not greater than i2„ 
 
 ORAL EXERCISES. 
 
 1. When coal is selling at $ 5 per ton, how many tons 
 can be bought for $ 45 ? 
 
 2. How many rows of trees are there in an orchard con- 
 taining 60 trees arranged in rows containing 6 trees each ? 
 
 3. If a railroad ticket to Chicago costs $ 7, how many 
 such tickets can be bought for $35? 
 
 4. A girl paid 96 cents for muslin at 8 cents per yard. 
 How many yards did she buy ? 
 
 5. How many dictionaries, at $S apiece, can be bought 
 for $U? 
 
 6. How long will it take a boy to save §48 if he saves 
 f 4 per month ? 
 
 7. Howard paid $9 for a suit of clothes. How many 
 such suits can be bought for $ 54 ? 
 
 8. How many gallons of molasses are there in 44 quarts ? 
 
122 ELEMENTS OF ARITHMETIC. 
 
 9. A peck measure holds 8 '][uarts of grain. How many 
 pecks are there in 72 quarts ? 
 
 10. If 8 nails are required to fasten a horseshoe, how 
 many horseshoes can be fastened with 80 nails ? 
 
 11. How many pairs of boots, at $ 6 a pair, can be bought 
 for $48? 
 
 12. A farmer had 63 bales of hay. If he drew it away 
 by drawing 7 bales at a load, how many loads did it make ? 
 
 13. A grocer put 96 quarts of vinegar into two-gallon 
 jugs. How many jugs did he use ? 
 
 14. A stick measured 72 inches in length. How many 
 feet long was it ? 
 
 15. Henry found that he had added 66 figures, and that 
 there were 6 figures in each column. How many columns 
 did he add? 
 
 16. Mary knew that there were 72 desks in her school- 
 room, and that there were 8 desks in a row, but she could 
 not tell how many rows there were. Can you tell? 
 
 17. A grocer had 76 eggs. How many dozen did he 
 have, and how many over ? 
 
 18. A lad saw 48 birds sitting upon 6 telegraph wires, 
 and he noticed that the same number sat upon each wire. 
 How many were there in each group ? 
 
 19. Lottie had saved 47 cents Avhich she wished to divide 
 equally, as far as possible, among four charitable societies. 
 How much did she give each, and how much had she left ? 
 
 20. How many times can a four-quart measure be filled 
 from 49 quarts ? How much will be left ? 
 
 21. A postman found that he walked 24 miles per day in 
 delivering mail. If he made 4 trips per day, how many 
 miles did he walk each trip ? 
 
 22. Susie spends -^ of the year with her aunt and the 
 rest of the year at home. How many months does she 
 spend with her aunt ? 
 
DIVISION. 123 
 
 WRITTEN EXERCISES. 
 
 106. 1. Divide 1702 by 3. 
 
 Divisor. Dividend, Explanation. — For convenience, the 
 
 3)1702 divisor is written at the left of the divi- 
 
 37^ ^ dend, and the quotient below it or above it. 
 
 bbi — i, Rem. ^^^ ^^^.^ ^^ ^^^^ ^^^^ ^^ divide. Thus, 3 
 
 ^^°^^^° ■ is not contained in 1 thousand any thousand 
 
 times, therefore the quotient cannot contain units of any order higher 
 than hundreds. Hence, we find how many times 3 is contained in all 
 the hundreds of the dividend. 
 
 1 thousand plus 7 hundreds are 17 hundreds ; 3 is contained in 17 
 hundreds 5 hundred times and 2 hundreds remainder. The 5 is writ- 
 ten in hundreds' place in the quotient, and the remainder, 2 hundreds, 
 is united with the tens. 
 
 2 hundreds plus tens are 20 tens ; 3 is contained in 20 tens 6 
 times, and a remainder of 2 tens. The 6 is written in tens' place in 
 the quotient, and the remainder, 2 tens, is united with the units. 
 
 2 tens plus 2 units are 22 units ; 3 is contained in 22 units 7 times, 
 and a remainder of 1. The 7 is written in units' place in the quotient, 
 and the remainder at a little distance to the right ; or it may be written 
 as a part of the quotient by writing the divisor under it. 
 
 Hence, the quotient is 567 and 1 remainder, or 567A. 
 
 Proof. — 567, the quotient, multiplied by 3, the divisor, plus 1, the 
 remainder, is equal to 1702, the dividend. Hence, the work is correct. 
 
 107. When examples in division are solved without 
 writing the products or remainders, the process is called 
 Short Division. 
 
 Short division is not generally employed when the divisor exceeds 12. 
 
 Divide by short division, and prove : 
 
 4. 5. 
 
 7 )59273 8 )614653 
 84674 76831f 
 
 3936-4. 16. $1125-9. 
 
 3234-3. 17. 31477-7. 
 
 5256^6. 18. $4986-9. 
 
 1362-3. 19. $6765-5. 
 
 1328-8. 20. $3836-7. 
 
 2. 
 
 3. 
 
 4)3624 
 
 6)65814 
 
 906 
 
 10969 
 
 6. 4872-4. 
 
 11. , 
 
 7 (3830 - 5. 
 
 12. ; 
 
 8. 2970-5. 
 
 13. , 
 
 9. 2976-6. 
 
 14. : 
 
 10. 4635-3. 
 
 15. : 
 
124 ELEMENTS OF ARITHMETIC. 
 
 21. 9872 -f- 6. 28. $9185-f- 4. 35. i-^/^. 
 
 22. 4567-5. 29. $ 84364 -^ 7. 36. i-V-- 
 
 23. $5 89.32 --6. 30. 1 38857 -J- 8, 37. ^7_37- 
 
 24. $ 81.74 -f- 9. 31. ^ 56878 -f- 6. 38. ^\^-^. 
 
 25. $83424 --8. 32. $91341-- 9. 39. Aie^s. 
 
 26. $73265 --5. 33. $72144-- 6. 40. ^f^i. 
 
 27. $58624 --7. 34. $31486 --11. 41. 5 2_6^^ 
 
 42. How many coats, at $5 each, can be bought for 
 $ 2250 ? 
 
 43. If a man spends $ 7 per week, how long will it take 
 him to spend $ 1477 ? 
 
 44. The salaries of 5 men amounted to $3765. What 
 was the average salary ? 
 
 45. If a ship sails 8 miles per hour, how long will it take 
 her to sail 1248 miles ? 
 
 46. A stage traveled 6 miles per hour. How many hours 
 would it require to travel 2352 miles ? 
 
 47. At a church collection, each person contributed, on 
 the average, 7 cents. If the entire sum collected was 4823 
 cents, how many persons contributed ? 
 
 48. How many yards of cloth, at 9 cents per yard, can be 
 bought for 29,439 cents ? 
 
 49. How many cloaks, at $5 each, can be bought for 
 $32,615? 
 
 50. In measuring the depth of the sea, 6 feet are called 
 a fathom. How many fathoms deep is the sea when it 
 measures 3516 feet ? 
 
 Divide each of the following numbers by 3, 4, 5, 6, 7, 8, 
 and 9. 
 
 51. 
 
 1950. 
 
 55. 
 
 5616. 
 
 59. 
 
 17286. 
 
 63. 
 
 277223. 
 
 52. 
 
 1476. 
 
 56. 
 
 6909. 
 
 60. 
 
 11615. 
 
 64. 
 
 548337. 
 
 53. 
 
 4360. 
 
 57. 
 
 2616. 
 
 61. 
 
 44854. 
 
 65. 
 
 389648. 
 
 54. 
 
 2943. 
 
 58. 
 
 1884. 
 
 62. 
 
 18809. 
 
 66. 
 
 261950. 
 
DIVISION. 125 
 
 108. When the divisor is expressed by more than one figure. 
 
 ORAL EXERCISES. 
 
 1. At 10 cents a yard, how many yards of cambric can 
 be bought for 40 cents ? 
 
 2. Tliere are 25 cents in a quarter of a dollar. To how 
 many quarter dollars are 50 cents equal ? 
 
 3. If I pay $ 60 for 20 lambs, how much do I pay for each? 
 
 4. A real estate dealer divided 30 acres into 10 equal 
 lots. How many acres were there in each ? 
 
 5. If a newsboy sells 40 papers per day, in how many 
 days will he sell 120 papers ? 
 
 6. If I sell books at 30 cents each, how many must I 
 sell to receive 90 cents ? 
 
 7. How many lO's are there in 60? In 90? In 120? 
 In 250? In 350? In 480 ? 
 
 8. Since in 60 there are 6 tens ; in 90, 9 tens ; in 120, 
 12 tens ; in 250, 25 tens, how may any number be divided 
 by 10 ? 
 
 9. How many lOO's are there in 600 ? In 900 ? In 
 1200 ? In 3600 ? In 7500 ? In 8400 ? 
 
 10. Since in 600 there are 6 hundreds ; in 900, 9 hun- 
 dreds ; in 1200, 12 hundreds ; in 3600, 36 hundreds, how 
 ma}^ any number be divided by 100 ? 
 
 11. How many lOOO's are there in 8000? In 25000 ? 
 
 12. Since in 8000 there are 8 thousands, in 25000 there 
 are 25 thousands, how may a number be conveniently 
 divided by 1000? 
 
 How may a number be divided by 10, 100, 1000, or by 1 
 with any number of ciphers annexed? 
 
 109. A number may be divided by 10, 100, 1000, or by 1 
 ivith any number of ciphers annexed, by cutting off from the 
 right of the dividend as many figures as there are ciphers on 
 the right of the divisor. 
 
126 
 
 ELEMENTS OF ARITHMETIC. 
 
 WRITTEN EXERCISES. 
 
 
 110. Divide: 
 
 
 
 
 
 
 1. 1|0)3968|0 
 
 
 2. 1|00)382|00 
 
 
 3. 1|00)468|57 
 
 
 3968 
 
 
 382 
 
 
 468^^ 
 
 4. 
 
 3720 by 10. 
 
 9. 
 
 57386 by 100. 
 
 14. 
 
 5283a by 1000. 
 
 6. 
 
 4860 by 10. 
 
 10. 
 
 42570 by 100. 
 
 15. 
 
 32685 by 1000. 
 
 6. 
 
 3879 by 10. 
 
 11. 
 
 29830 by 100. 
 
 16. 
 
 97237 by 1000. 
 
 7. 
 
 4265 by 10. 
 
 12. 
 
 34206 by 100. 
 
 17. 
 
 54862 by 1000. 
 
 8. 
 
 3842 by 10. 
 
 13. 
 
 51309 by 100. 
 
 18. 
 
 69154 by 1000. 
 
 19. Divide 38476 by 500. 
 
 Explanation, — The number is divided by 100 
 by cutting off the two right-hand figures of the 
 
 5[00 )384|76 
 
 '^^lo-f dividend. Dividing 384 by 5 is, therefore, dividing 
 
 38,400 by 500, The- quotient is, therefore, 76 and 
 
 4 hundreds remainder. This remainder, added to the remainder 
 
 obtained by dividing by 100, gives the entire remainder, 476 ; or the 
 
 quotient is 76|^f . 
 
 Divide : 
 
 20. 6835 by 20. 
 
 21. 7291 by 30. 
 
 22. 4863 by 60. 
 
 23. 3925 by 80. 
 
 24. 98746 by 300. 
 
 25. 38295 by 400. 
 
 26. 41283 by 700. 
 
 27. 61579 by 900. 
 32. Divide 15989 by 37. 
 
 Divisor. Dividend. Quotient. EXPLANATION. 
 
 37) 15989 (432-jV 
 148 
 
 28. 83426 by 500. 
 
 29. 31284 by 800. 
 
 30. 67395 by 700. 
 
 31. 81293 by 600. 
 
 118 
 111 
 
 When the divisor is 
 larger tlian 12 the steps in the sokition are 
 usually w^ritten. Thus, 37 is contained in 
 159 four times and a remainder of 11. The 
 4 is v^ritten in the quotient and the next 
 figure of the dividend annexed to the re- 
 mainder, making 118. 
 
 37 is contained in 118 three times and a 
 remainder of 7. The 3 is written in the 
 quotient and the next figure of the dividend 
 annexed to the remainder, making 70. 
 
 37 is contained in 79 two times and a remainder of 5. Therefore, 
 the quotient is 432 g^. 
 
 79 
 74 
 
 5 
 
DIVISION. 127 
 
 When tlie steps in the solution of an example are written, 
 the process is called Long Division. 
 
 111. From the solution of the example given, the follow 
 ing are seen to be the successive steps taken in solving an 
 example in Long Division : 
 
 Write the divisor at the left of the dividend, ivith a curved 
 line hetiueen them. 
 
 Find how many times the divisor is contained in the fewest 
 figures 07i the left hand of the dividend that ivill contain it, and 
 write the quotient at the right of the dividend or above it. 
 Multiply the divisor by this quotient, and place the product 
 under the figures divided. Subtract the result from the par- 
 tial dividend used, and to the remainder annex the next figure 
 of the dividend. 
 
 Divide as before until all the figures of the dividend have 
 been annexed to the remainder. 
 
 If any partial dividend will not contain the divisor, ivrite a 
 cipher in the quotient, then annex the next figure of the divi- 
 dend, and proceed as before. 
 
 If there is a remainder after the last division, write it after 
 the quotient, or with the divisor under it as part of the quotient. 
 
 Pkoof. — Multiply the divisor by the quotient, and to the 
 product add the remainder, if any. If the work is correct, the 
 result will equal the dividend. 
 
 1 . The quotient figure may generally be readily obtained by finding 
 how many times the first figure of the divisor is contained in the first 
 figures of the dividend tliat will contain it, making allowance for the 
 addition to the product from the second figure of the divisor. 
 
 2. If the product of the divisor by the quotient figure is greater 
 than the partial dividend from which it is to be subtracted, the quo- 
 tient figure is too large. 
 
 3. Each remainder must be less than the divisor ; otherwise the 
 quotient figure is too small. 
 
 4. When there is no remainder, the divisor is said to be exact. 
 
128 
 
 ELEMENTS OF ARITHMETIC. 
 
 33. 
 
 83)51875(625 
 498 
 
 207 
 166 
 
 34. 
 
 74)42683(576^1 
 370 
 
 56S 
 518 
 
 35. 
 
 52)45297(8715^ 
 416 
 
 369 
 364 
 
 415 
 415 
 
 503 
 444 
 
 59 
 Find the quotients of : 
 36. 4202-11. 40. 2310 --42. 
 
 37. 2556-12. 
 
 38. 1386-21. 
 
 39. 1488-31. 
 
 41. 3796-52. 
 
 42. 6840-15. 
 
 43. 1728-12. 
 
 57 
 
 "5 
 
 44. 2747-41 
 
 45. 5084-62. 
 
 46. 3367-91. 
 
 47. 3024-84. 
 
 Divide each of the following numbers by 21, 32, 43, 
 54, 65 : 
 
 48. 38425. 
 
 49. 65484. 
 
 50. 94653. 
 
 51. 45677. 
 
 52. 84732. 
 
 53. 64136. 
 
 54. 99874. 
 
 55. 20562. 
 
 56. 71117. 
 
 57. 40505. 
 
 58. 35794. 
 
 59. 13570. 
 
 60. 88992. 
 
 61. 46823. 
 
 62. 95895. 
 
 63. 703821. 
 
 64. 492532. 
 
 65. 669046. 
 
 66. 749638. 
 
 67. 838327. 
 
 Divide each of the following numbers by 37, 28, 19, 
 46, 35 : 
 
 68. 37905. 72. 29823. 76. 19198. 80. 469832. 
 
 69. 35911. 73. 50847. 77. 34440. 81. 269724. 
 
 70. 98634. 74. 72556. 78. 66984. 82. 123461. 
 
 71. 48208. 75. 96054. 79. 85745. 83. 777738. 
 
 Divide : 
 
 84. 50854 by 94. 
 
 85. 56394 by 78. 
 
 86. 559108 by 202. 
 
 87. 1547250 by 2134. 
 

 
 
 
 DIVISION. 
 
 
 129 
 
 
 
 
 
 Drill 
 
 Tahle. 
 
 
 
 
 A. 
 
 B. 
 
 a 
 
 D. 
 
 E. 
 
 F. 
 
 Cr. 
 
 1. 
 
 4 
 
 7 
 
 42 
 
 294 
 
 882 
 
 4410 
 
 9076352 
 
 2. 
 
 5 
 
 4 
 
 36 
 
 144 
 
 576 
 
 3456 
 
 4487063 
 
 3. 
 
 6 
 
 3 
 
 24 
 
 72 
 
 360 
 
 3240 
 
 6470342 
 
 4. 
 
 7 
 
 8 
 
 40 
 
 320 
 
 960 
 
 7680 
 
 1598788 
 
 5. 
 
 5 
 
 7 
 
 5(3 
 
 392 
 
 1568 
 
 15680 
 
 7421077 
 
 6. 
 
 2 
 
 8 
 
 72 
 
 576 
 
 1152 
 
 4608 
 
 6480713 
 
 7. 
 
 3 
 
 3 
 
 27 
 
 81 
 
 729 
 
 5103 
 
 2835108 
 
 8. 
 
 8 
 
 7 
 
 28 
 
 196 
 
 1960 
 
 9800 
 
 5306792 
 
 9. 
 
 4 
 
 8 
 
 04 
 
 512 
 
 1536 
 
 3072 
 
 4350787 
 
 10. 
 
 3 
 
 5 
 
 40 
 
 200 
 
 1800 
 
 5400 
 
 7863444 
 
 11. 
 
 
 
 G ^ 
 
 ^ 30 
 
 180 
 
 7200 
 
 28800 
 
 6095102 
 
 12. 
 
 7 
 
 9 
 
 27 
 
 243 
 
 1701 
 
 8505 
 
 7453745 
 
 13. 
 
 9 
 
 8 
 
 48 
 
 384 
 
 768 
 
 2304 
 
 6666006 
 
 14. 
 
 3 
 
 6 
 
 36 
 
 216 
 
 864 
 
 3456 
 
 8580437 
 
 15. 
 
 8 
 
 8 
 
 56 
 
 448 
 
 2240 
 
 4480 
 
 4654380 
 
 16. 
 
 3 
 
 7 
 
 63 
 
 441 
 
 3528 
 
 7056 
 
 7820194 
 
 17. 
 
 9 
 
 7 
 
 49 
 
 343 
 
 3773 
 
 15092 
 
 9003068 
 
 18. 
 
 4 
 
 6 
 
 42 
 
 252 
 
 1008 
 
 5040 
 
 3905746 
 
 19. 
 
 8 
 
 9 
 
 54 
 
 486 
 
 2430 
 
 7290 
 
 2849370 
 
 20. 
 
 4 
 
 5 
 
 40 
 
 200 
 
 1000 
 
 9000 
 
 6470181 
 
 21. 
 
 9 
 
 9 
 
 36 
 
 324 
 
 972 
 
 8748 
 
 5783900 
 
 22. 
 
 
 
 8 
 
 24 
 
 192 
 
 768 
 
 3840 
 
 4300125 
 
 23. 
 
 5 
 
 9 
 
 81 
 
 729 
 
 1458 
 
 3916 
 
 1758374 
 
 24. 
 
 3 
 
 8 
 
 72 
 
 576 
 
 2304 
 
 9216 
 
 9999999 
 
 25. 
 
 5 
 
 6 
 
 54 
 
 324 
 
 1296 
 
 6480 
 
 9876543 
 
 EXERCISES UPON THE TABLE. 
 
 ORAL EXERCISES 
 
 . 
 
 
 112. 1. ^xi?=? 2. C -. B = 
 
 ? 
 
 3. G-1 
 
 WRITTEN EXERCISES. 
 
 
 4. C+2) + ^=? 9. i)xC=? 
 
 
 14. E-^B=? 
 
 5. C+Z)+^+F = ? 10. Cxl0=? 
 
 
 15. E^ C=? 
 
 6. F - i> = ? 11. 2) X 400 zz: ? 
 
 
 16. F^ C=? 
 
 1. F- E=? 12. D X 203 = ? 
 
 
 17. F^ D=? 
 
 8. DxB=? 13. D^B=? 
 
 
 18. 100000 - F 
 
 19. G ^ D = ? 21. a - 
 
 {D + E+ F)=2 
 
 20. Fx D = ? 22. Ax 
 
 B 
 
 X C= ? . 
 
 EL. OF AR. — 9 
 
 
 
130 ELEMENTS OF ARITHMETIC. 
 
 REVIEW EXERCISES. 
 ORAL EXERCISES. 
 113. 1. A bookseller bought 15 books for $ 30 and sold them 
 at a gain of $ 15. How much apiece did he get for them ? 
 
 2. A miller sold 8 barrels of flour at $ 5 a barrel and 
 bought with the proceeds cloth at f 4 per yard. How many 
 yards did he purchase ? 
 
 3. The cost of 10 tons of coal was $ 50, but I sold it at 
 a gain of $ 20. How much did I receive per ton ? 
 
 4. A bookseller bought 12 dictionaries^ at $ 8 apiece 
 and sold them at $ 10 apiece. How much did he gain ? 
 
 5. A steam yacht ran up a river for 8 hours at 10 miles 
 per hour, and it came down in 5 hours. What was its rate 
 per hour coming down ? 
 
 6. The cost of 5 tons of coal at f 5 per ton was f 7 less 
 than the cost of 8 cords of wood. What did the wood 
 cost per cord ? 
 
 7* The expense of sending delegates to a convention 
 was $ 65. If the entire expense had been $ 1 less, the ex- 
 pense to each member would have been $ 2. How many 
 members were there ? 
 
 8. A man owed a debt amounting to $ 84. If he paid 
 $ 60 of the sum in 5 months, in how long a time will he 
 pay it all at that rate ? 
 
 9. A farmer sold a grocer 12 pounds of butter at 20^ 
 a pound and took in exchange brooms at 24 cents apiece. 
 How many brooms did he get ? 
 
 10. Two boys set out from the same place and traveled 
 in the same direction, Arthur traveling 5 miles per hour 
 and Bennie 3 miles per hour. How soon will Arthur be 18 
 miles ahead of Bennie ? 
 
 11. How far has each traveled when Arthur is 24 miles 
 ahead ? 
 
DIVISION. 
 
 
 12. Suppose they travel in opposite directions at those 
 rates, how far apart will they be in 8 hours ? 
 
 13. In how many days can 4 men do as much work as 8 
 men can do in 6 days ? 
 
 14. Howard's father gave him an acre of land to till so 
 that he might earn his own pocket money. He sold the 
 products of the land for $ 38, but he paid $ 5^ for seeds to 
 plant and $ 2i for help. How much money had he left ? 
 
 15. A boy bought 6 lead-pencils for 24j^ and exchanged 
 them for some erasers at 3p each. How many erasers did 
 he get ? 
 
 16. A man bought a carriage for $ 120. After spending 
 $ 40 in repairing and painting it, he sold it for $ 150. Did 
 he gain or lose, and how much ? 
 
 17. A newsboy bought 20 papers for 30 cents and sold 
 them at a gain of 30 cents. For how much apiece did he 
 sell them ? 
 
 18. Henry bought 6 car-tickets for 25 cents. How many 
 could he buy for 50 cents ? 
 
 19. A clerk saved $ 20 a month. In how many months 
 could he save enough to buy a library worth $ 240 ? 
 
 20. A jeweler bought 12 watches for $ 72 and sold them 
 at a gain of $ 3 each. At what price did he sell each of 
 them? 
 
 21. The wages of a machinist were $21 per week, and 
 his expenses were $ 10. In how many weeks can he pay a 
 debt of $ 75 and deposit $ 46 in a savings bank ? 
 
 WRITTEN EXERCISES. 
 
 114. 1. If a man paid ^ 7564 for 124 horses, how much 
 did each cost ? 
 
 2. At $ 9 a yard, how many yards of velvet can I buy 
 for $ 207 ? 
 
 3. How many weeks are there in 1736 days ? 
 
132 ELEMENTS OF ARITHMETIC. 
 
 4. If 12 horses cost $ 1560, what will one horse cost? 
 What will 5 horses cost at the same rate ? 
 
 5. How many dozen eggs are there in 1728 eggs ? 
 
 6. Divide $ 800.25 equally among 15 men. 
 
 7. In an orchard there are 25 rows of trees. If there 
 are 3200 trees in all, how many trees are there in a row ? 
 
 8. How many times 26 bushels are there in 22,568 
 bushels ? 
 
 9. How many acres of land will $22,575 buy, at $75 
 an acre ? 
 
 10. In 64 loads there were 143,360 pounds. How many 
 pounds were there in a load ? 
 
 11. How many days are there in 3600 hours ? In as 
 many minutes ? 
 
 12. How many hours are there in 4320 minutes ? How 
 many days ? 
 
 13. What number, multiplied by 23, will give 36,087? 
 
 14. Product, 62,397; multiplier, 27; find the multipli- 
 cand. 
 
 15. If the product of two factors is 21,015, and one fac- 
 tor is 45, what is the other factor ? 
 
 16. If you weigh 1476 ounces, how many pounds (of 16 
 ounces) do you weigh ? How many pounds does your sister 
 weigh, her weight being 133 ounces less than yours ? 
 
 17. If the receipts of a railroad for one year (52 weeks) 
 are $ 2,683,928, how much are they per week ? 
 
 18. The combined wealth of 405 men is $1,247,400. 
 How much would each have if the money were equally 
 divided among them ? 
 
 19. How many schooners, each carrying 8700 bushels of 
 wheat, will be required to carry 843,900 bushels ? 
 
 20. Suppose that two cans of equal size together hold 
 376 pints. How many gallons are there in each ? 
 
 21. How many poor families may be supplied from 37 
 
DIVISION". 133 
 
 barrels of flour, allowing 28 pounds to each, a barrel of 
 flour weighing 196 pounds ? 
 
 22. How many thousands make one million ? 
 
 23. If a railroad company pays $ 1750 for constructing a 
 car, how many cars can be made for $ 75,250 ? 
 
 24. The average daily receipts of a ferry-boat are $ 325. 
 In how many days will the receipts amount to $ 76,050 ? 
 
 25. The stock of a railroad company is $> 2,565,000. If 
 it is divided into shares of $ 500 each, how many shares 
 are there ? 
 
 26. Mount Everest, in Asia, is said to be 29,100 feet 
 high. Since there are 5280 feet in a mile, how many miles 
 high is it ? 
 
 27. In what number is 224 contained 28 times ? 
 
 28. Mr. Smith earns $28 while Mr. Allen earns $15. 
 How much will Mr. Smith earn while Mr. Allen earns 
 $120? 
 
 29. How many times will a cart-wheel, 15 feet in cir- 
 cumference, revolve in going a mile, or 5280 feet ? 
 
 30. How many posts and how many rails will be required 
 for a fence 312 feet long, if the posts are set 12 feet apart, 
 and the fence is 4 rails high ? 
 
 31. From a farm containing 1000 acres, there were sold 
 at one time 274 acres ; at another time, 238 acres. How 
 many acres remained unsold ? 
 
 32. Divide $432 between Mr. A. and Mr. B., so that ]Mr. 
 B. will have $ 16 more than Mr. A. 
 
 33. At an election, the sum of the votes received by two 
 opposing candidates was 6742. If the successful candidate 
 received 328 more votes than his opponent, how many votes 
 did each receive ? 
 
 34. Into a tank w^ere poured 76 barrels of kerosene, of 
 43 gallons each. If 2489 gallons are drawn from the tank, 
 how many gallons are left? 
 
184 ELEMENTS OF ARITHMETIC, 
 
 35. If I buy 8 boxes of starch, each box containing 24 
 pounds, at 8 cents a pound, and offer in payment a $ 20 bill, 
 what change shall I receive ? 
 
 36. Elsie and Lottie have together $5.70, and Lottie has 
 40 cents more than Elsie. How much money has each ? 
 
 37. If 14 men can do a piece of work in 56 days, how 
 long will it take 2S men to do half the work ? 
 
 38. I bought two horses. The first cost $312, and the 
 second f 40 less than the first. How much did the two 
 horses cost ? 
 
 39. If 19 tons of coal run an engine 703 miles, how far 
 will 17 tons run it ? 
 
 40. A farmer exchanged 4 cows, worth $79 each, for a 
 span of horses. What were the horses worth apiece ? 
 
 41. A horse cost $235, a carriage $219, and a hack 3 
 times as much as both. What did all three cost ? 
 
 42. A merchant bought 42 bales of cloth, each bale con- 
 taining 36 pieces, and each piece 38 yards, at 9 cents a yard. 
 He sold the whole at 11 cents a yard. How much did he 
 gain ? 
 
 43. If I take 764 bushels from 2000 bushels twice, how 
 much will remain ? 
 
 44. A drover with $ 2144 bought as many horses as pos- 
 sible for $ 165 each, and spent the remainder for sheep at 
 $ 4 a head. How many of each did he buy ? 
 
 45. HoAv many times 945 will make 23,625 ? 
 
 46. Twenty men built a school-house in 56 days. In how 
 many days could 70 men have built it ? 
 
 47. A man bought 163 barrels of flour at $9 a barrel; 
 15 barrels were spoiled, and the remainder he sold at $ 11 a 
 barrel. Did he gain or lose, and how much ? 
 
 48. A school uses 36 crayons a day. Since a gross is 12 
 dozen, how many gross will it use in 40 weeks, provided 
 there is no school on Saturday ? 
 
DIVISION. 135 
 
 49. If it takes 5 yards of cloth to make a pair of shirts, 
 what will 24 pairs cost at 15 cents per yard for the cloth, 
 45 cents apiece for bosoms, wristbands, and buttons, and 75 
 cents apiece for making ? 
 
 50. A clerk has a salary of f 42 a month, and his expenses 
 are f 27 a month. How many years will it take him to lay 
 up 1 900 ? 
 
 51. Find the cost of 267 yards of cotton at 13 cents a 
 yard, and twice as much at 12 cents a yard. 
 
 52. There are in a certain school 37 pupils 14 years old; 
 76 pupils 12 years old ; 114, 11 years old ; 149, 10 years 
 old ; and 168, 9 years old. What is their average age ? 
 
 53. A man paid $375 for a piano, $13 for freight and 
 cartage, and $2 for tuning it. He then rented it for 7 
 quarters at $ 15 a quarter, and afterwards sold it for $ 325. 
 Did he gain or lose, and how much ? 
 
 54. Two men leave the same place and travel in opposite 
 directions, one at the rate of 30 miles a day, and the other 
 at the rate of 37 miles per day. How far will they be apart 
 at the end of 14 days ? 
 
 55. If the men traveled in the same direction, how far 
 apart would they be in the same time ? 
 
 56. A, B, and C have together $198. A and C have 
 $ 145. A and B have $ 99. How much has each ? 
 
 57. Howard, Eoy, and Jamie went fishing, and together 
 caught 77 fish. Of these, Howard and Jamie together 
 caught 58. Howard's and Eoy's, when put together, num- 
 bered 54. How many did each catch ? 
 
 58. A farmer sold 18 calves at the rate of 3 for $33 
 How much did he get for them ? 
 
 59. If 117 tons of coal are worth $702, how much are 
 235 tons worth ? 
 
 60. If 12 barrels of flour are worth $132, what will 36 
 barrels cost ? 
 
136 ELEMENTS OF ARITHMETIC. 
 
 61. If a farmer receives f 2275 for 13 horses, for how 
 much would 27 horses sell at the same rate ? 
 
 62. A's farm contains 227 acres; B's lacks 12 acres of 
 being twice as much ; C's lacks 42 acres of being 3 times as 
 much as B's ; and D's contains as much as A's and C's. 
 How many acres are there in each farm ? 
 
 63. A farmer wished to obtain $ 260. He sold 72 bushels 
 of wheat at $1.50 per bushel, and enough apples at $2 per 
 barrel to obtain the sum required. How many barrels of 
 apples did he sell ? 
 
 64. If a bookkeeper receives $1400 per year for his ser- 
 vices, and his expenses are $ 840, in what time can he save 
 enough to buy 32 acres of land at $ 140 per acre ? 
 
 65. A man starting out on a journey took $200. He 
 paid for railroad fare $ 67 ; for berth in sleeping car, 4 
 days, $ 2 a day ; for hotel bills, 15 days, $ 3 a day ; and for 
 other expenses, $ 25. How much money had he left ? 
 
 66. The greater of two numbers is 4056, and their differ 
 ence is 3650. What is the less number ? 
 
 67. The divisor is 534, and the quotient 43. What u 
 the dividend ? 
 
 68. The subtrahend is 34,203, and the remainder is 8706 
 What is the minuend ? 
 
 115. The parenthesis, ( ), shows that the numbers in- 
 cluded within it are to be subjected to the same operation. 
 
 Thus, (15 — 3) X 3 means tliat 15 — 3, or 12, is to be multiplied by 
 3 ; but 15 — 3 X 3 means that 15 is to be diminished by 3 x 3, or 9. 
 
 (7 + 2) X 5 means that the sum of 7 and 2 is to be multii)liecl by 
 
 5 ; but 7 + 2x5 means that 7 is to be increased by 2 x 5. 
 
 7 + 5 
 (7 + 5)-^ (j, or — -";— , means that the sum of 7 aiul 5 is to be 
 G 
 divided by 0. 
 
 [(9 + 3) x 2 - 3] -^ 7 means that the sum of f) and 3 is to be multi- 
 plied by 2, the product diminished by 3, and the remainder divided 
 by 7. 
 
DIVISION. 137 
 
 (7 X 8) -(5 X 5) means that the product of 7 and 8 is to be 
 diminished by the product of 5 and 5. 
 
 116. The Vinculum, , or brackets [ ], may be used 
 
 instead of the parenthesis. 
 
 Thus, 5 — 3 X 3 or [5 — 3J x3 may be used instead of (5 — 3) x 3. 
 
 117. In finding the value of expressions in which the 
 parenthesis, vinculum, or brackets are used, 
 
 First, perform the operations on the numbers that are ivrit- 
 ten icithin parenthesis and brackets, or under the vinculum, as 
 indicated by the signs. Next, multipily and divide, as indicated 
 by the signs x and -t-. Finally, add and subtract, as indi- 
 cated by the signs + and —. 
 
 rind the values of the following : 
 
 1. (4 + 7)x5. 8. l±^ + ^-^ 
 
 3 4 
 
 2. 4 + 7x5. 9. 2x12-2x3. 
 
 3. (8-3)x2. 10. 2x12 -(2x3). 
 
 4. 8-3x2. 11. i8-(3x4)-(2x3)^ 
 
 o 
 
 5. 4 + 16-4. 12. [(5 + 6)x4-5 x.S]x9. 
 
 6. (4 + 16) -4. 13. (3 X 4) X 5 -(9 + 7) -8. 
 7o (3 + 4) X (8-5). 14. 2 + 12-4+(8 + 8-4)-2„ 
 
 15. (312 + 36)— (381 -215) -65. 
 
 16. (214 -81) -(115 -18 + 6) + 10. 
 
 17. (413 _ 200) - (118 - 24 - 4 + 6) + 3. 
 
 18. (171 - S(\) - (3^r4 + 27) + 10. 
 
 19. (3x4)x0-(4 + 12)-f-4 + 5. 
 
 20. (5 + 2 + 6) X 4 -(7 + 8 -3) -6 + 10. 
 
 21. "(36-7)x 5 + (102 + 6) -9. 
 
 22. (99 - 3) - 8 - (86 + 10) - 12 + (3 + 6) - 3. 
 
FACTORING. 
 
 118. 1 . What is the product when 3 and 2 are multiplied 
 together ? What are 3 and 2 of their product ? (Art. 86.) 
 
 2. What factors will produce 9 ? 
 
 3. What numbers when multiplied together will produce 
 12 ? "What are 3 and 4, or 6 and 2 of their product ? 
 
 4. What are the factors of 20 ? Of 36 ? Of 15 ? 
 
 5. What are the factors of 27 ? Of 25 ? Of 32 ? 
 
 6. What are the factors of 21 ? Of 33 ? Of 50 ? 
 
 7. What are the factors of 63 ? What else may 7 be 
 called of 63 besides a factor ? 
 
 8. If 9 is one of two factors of 18, what is the other 
 factor ? If 3 is one of the factors ? If 6 is one of the 
 factors ? 
 
 9. What numbers will exactly divide 18 ? 25? 36 ? 
 
 10. Give the exact divisors of 42; 96; 35; 50; 27; 72. 
 
 11. Give the factors of 36 ; 40; 48; 70; 80. 
 
 12. Give the exact divisors of 44 ; 56; 64; 84; 96. 
 
 13. Name the exact divisors of 49; 88; 63; 24; 27. 
 
 14. What numbers between and 10 cannot be exactly 
 divided by any number except themselves and 1 ? What 
 numbers between 10 and 20 ? Between 20 and 30 ? 
 
 15. What numbers between and 10 can be exactly 
 divided by other numbers besides themselves and 1 ? Be- 
 tween 10 and 20 ? Between 20 and 30? 
 
 16. Select from the following the numbers that have no 
 exact divisors except themselves and 1: 35, 42, 63, 56, 61, 
 47, 49, 81, 37, 26, 18, 45. 
 
 138 
 
FACTORING. 139 
 
 17. Select from the following the numbers that have 
 exact divisors besides themselves and 1 : 24, 36, 41, 39, 27, 
 45, 33, 37, 50, 44, 60, 71, 72. 
 
 119. A number that expresses whole units is called an 
 Integer. 
 
 Thus, 5, 27, 35 are integers, or integral numbers. 
 
 120. The integers which, upon being multiplied together 
 will produce the number, are called Factors of the number. 
 
 Thus, 5 and 3 are the factors of 15. 
 
 121. An integer which will divide a number without a 
 remainder is called an Exact Divisor of the number. 
 
 Thus, 2, 3, 6, and 9 are exact divisors of 18. They are also factors 
 of 18. 
 
 122. A number that has no exact divisors except itself 
 and 1 is called a Prime Number. 
 
 Thus, 1, 3, 5, 7 are prime numbers. 
 
 123. A number that has exact divisors besides itself and 
 1 is called a Composite Number. 
 
 Thus, 24, 36, 40, 100 are composite numbers. 
 
 124. Factors that are prime numbers are Prime Factors. 
 
 Thus, 7 and 5 are the prime factors of 35. 
 
 125. A number that is exactly divisible by 2 is called an 
 Even Number. 
 
 Thus, 8, 12, 20, 24 are even numbers. 
 
 126. A number that is not exactly divisible by 2 is called 
 an Odd Number. 
 
 Thus, 15, 21, 35, 43 are odd numbers. 
 
 127. The process of separating a number into its factors 
 is called Factoring. 
 
140 ELEMENTS OF ARITHMETia 
 
 TESTS OP DIVISIBILITY^ 
 
 128. 1. Make a list of numbers from 1 to 126, which 
 have 2 for one or more of their factors or divisors, iSTotice 
 what the right-hand, or units' figure, of each is. 
 
 2. Make a list of numbers from 1 to 100 which have 5 
 foi one or more of their factors or divisors. Notice what 
 the units' figure of each is. 
 
 3. Make a list of numbers from 1 to 100 which have 3 
 for one or more of their factors or divisors. Divide the 
 sum of the digits of each of these numbers by 3, and notice 
 the remainder, if any. 
 
 4. Make a list of numbers from 1 to 200 which have 9 
 for one or more of their factors or divisors. Divide the sum 
 of the digits of each by 9, and notice the remainder, if any. 
 
 129. It is apparent, therefore, that: 
 A immher is divisible by 
 
 2 if the units' fgure is 2, 4, 6, 8, or 0. 
 5 if the units' figure is or 5. 
 
 3 if the sum of its digits is divisible by 3. 
 9 if the sum of its digits is divisible by 9. 
 
 130. Tell by inspection which of the numbers, on page 
 129, columns O and D, are divisible by 2; by 5 ; by 3 ; by 9. 
 
 Which in column E are divisible by 2 ; by 3 ; by 5 ; by 9. 
 
 WRITTEN EXERCISES. 
 
 131. 1. What are the prime factors of 336? 
 
 Explanation.— Since every fac- 
 tor of a number is a divisor of it, 
 we may find the prime factors of 
 336 by dividing by tlie exact divisors 
 that are prime numbers. Dividing 
 by 2, we find the factors of 330 to 
 be 2 and 168. But 168 has a factor 
 2, and, since a factor of a factor of 
 a number is a factor of the numbei 
 itself, we continue tlie process, and obtain tlie prime factors of 336. 
 
 2 
 
 336 
 
 
 2 
 
 168 
 
 
 2 
 
 84 
 
 
 2 
 
 42 
 
 
 3 
 
 21 
 
 
 7 
 
 
 336 = 2 X 2 > 
 
 <2xl 
 
 ? X 3 X 7 
 

 
 FACTORING. 
 
 
 ] 
 
 Vha 
 
 .t are the prime 
 
 ! factors 
 
 
 
 2. 
 
 Of 168? 
 
 12. 
 
 Of 625? 
 
 22. 
 
 Of 1280 ? 
 
 3. 
 
 Of 224 ? 
 
 13. 
 
 Of 912 ? 
 
 23. 
 
 Of 2625? 
 
 4. 
 
 Of 144 ? 
 
 14. 
 
 Of 832 ? 
 
 24. 
 
 Of 3204? 
 
 5. 
 
 Of 198 ? 
 
 15. 
 
 Of 990? 
 
 25. 
 
 Of 1155? 
 
 6. 
 
 Of 216 ? 
 
 16. 
 
 Of 1008? 
 
 26. 
 
 Of 2500 ? 
 
 7. 
 
 Of 484 ? 
 
 17. 
 
 Of 435 ? 
 
 27. 
 
 Of 1485? 
 
 8. 
 
 Of 576 ? 
 
 18. 
 
 Of 222? 
 
 28. 
 
 Of 1296? 
 
 9. 
 
 Of 432 ? 
 
 19. 
 
 Of 672? 
 
 29. 
 
 Of 2016 ? 
 
 10. 
 
 Of 672 ? 
 
 20. 
 
 Of 775 ? 
 
 30. 
 
 Of 2875? 
 
 lie 
 
 Of 396 ? 
 
 21. 
 
 Of 289 ? 
 
 31. 
 
 Of 2890? 
 
 141 
 
 CANCELLATION. 
 ORAL EXERCISES. 
 
 132. 1. How many times is 2 times 5 contained in 4 
 times 5 ? 2 times 3 in 4 times 3 ? 2 times any number in 
 4 times that number ? 
 
 2. How many times is 4 times 7 contained in 8 times 7 ? 
 
 4 times 35 in 8 times 35 ? 4 times a certain number in 8 
 times the same number ? 
 
 3. How many times is 6x12 contained in 18x12? 
 
 5 X 23 in 15 X 23 ? 7 x 47 in 21 x 47 ? 
 
 4. What is the quotient of (24 x 17)-^ (12 x 17)? Of 
 (63x24)-(9x24)? Of (48 x 61)- (24 x 61)? Of (36 
 times 19) divided by (18 times 19) ? 
 
 5. In determining the quotient, what numbers may be 
 omitted from both dividend and divisor ? 
 
 133. From the solution of the examples given, it is 
 evident that : 
 
 Rejecting equal factors from both dividend and divisor does 
 not alter the quotient. 
 
2 
 
 2 
 
 2 
 
 2 
 
 
 
 ^ 
 
 X0 
 
 xgx 
 
 w_ 
 
 16 
 '' 3 
 
 - .^1 
 
 % 
 1 
 
 x^ 
 1 
 
 X^ X 
 
 1 
 
 3 
 
 - ^3 
 
 142 ELEMENTS OF ARITHMETIC. 
 
 WRITTEN EXERCISES. 
 
 134. 1. Divide 4x6x8xl0by2x3x4xl5. 
 
 Explanation. — The dividend is 
 written above the divisor, with a line 
 between them as in division. 
 
 Since the factors 2, 3, and 4 are 
 factors of 4, 6, and 8 in the dividend, 
 we reject them from both dividend and divisor, leaving the factors 2, 
 2, and 2 in the dividend. Since the factor 5 is common to 10 in the 
 dividend and to 15 in the divisor, it is also rejected, leaving 2 in the 
 dividend, and 3 in the divisor. 
 
 The product of the uncanceled factors of the dividend is 16, and 
 of the divisor 3. Hence, the quotient is ^^, or 16 -;- 3, or 5^. 
 
 2. Divide 7 x 4 X 11 X 8 by 22 x 4 x 4 x 21. 
 
 -I Explanation, — In this example, the 
 
 . . ^ ^ common factor 7 is rejected from 7 and 21, 
 
 -*■ -^ ^ leaving 1 in the dividend and 3 in the 
 
 T X ^ X 11 X g ^ x^ divisor. 
 
 ^^ X ^ X ^ X ?J ^ Next, the common factor 4 is removed 
 
 <^ \ \ 2t ixom. 4 in the dividend and 4 in the divisor. 
 
 ? Then the common factor 4 is removed from 
 
 8 in the dividend and 4 in the divisor, leav- 
 ing 2 in the dividend and 1 in the divisor. Then the common factor 
 11 is rejected from 11 in the dividend and from 22 in the divisor, 
 leaving 1 in the dividend and 2 in the divisor. Then the common 
 factor 2 is removed from the dividend and the divisor. 
 The product of 1 x 1 x 1 x 1 in the dividend is 1. 
 The product of 1 x 1 x :l x 3 in the divisor is 3. 
 Hence, the quotient is 1 -^ 3, or J. 
 
 1. When the factor is contained once in any number, the 1 may 
 be omitted ; but it is better to write it until the process is clearly 
 understood. 
 
 2. When all the factors of both divisor and dividend are canceled, 
 the quotient is 1, for the dividend will then exactly contain the 
 divisor once. 
 
 Finxl the value of : 
 
 4x5x7x9 , 8x9x12x16 
 
 3. 
 
 2x2x6x7x3 * 4x3x5x6 X 20 
 
FACTORING. 143 
 
 2x3x8x12x24 ^ 15 X 20 X 25 X 27 
 
 6 X 4 X 36 X 4 * ' 10 X 15 X 18 X 25* 
 
 18 X 24 X 32 X 36 g 25 x 30 x 40 x 35 
 
 * 9 X 48 X 4 X 18 ' ' 20 X 15 X 18 X 7 * 
 
 Divide, using cancellation ; 
 
 9. 40 X 48 X 54 X 60 by 30 x 24 x 72 x 3. 
 
 10. 30 X 49 X 64 X 25 by 15 x 35 x 24 x 10. 
 
 11. 12 X 60 X 36 X 70 by 28 X 48 X 6 X 5. 
 
 12. 32 X 36 X 33 X 45 by 24 x 30 x 44 x 9. 
 
 13. 30 X 36 X 24 x 42 by 45 x 27 x 8 x 28. 
 
 14. 27 x 32 X 45 x 36 by 18 X 24 X 9 X 6. 
 
 15. 45 X 28 X 36 X 56 by 49 X 4 X 18 X 18. 
 
 16. 27 X 28 X 35 X 49 by 18 x 21 x 40 x 28. 
 
 17. 45 X 60 x 63 x 28 by 27 x 20 x 25 x 21. 
 
 18. 55 X 36 X 27 x 42 by 12 x 25 x 35 x 33. 
 
 19. 36 X 64 X 25 x 40 by 32 x 50 x 18 x 10. 
 
 20. 5() X 18 X 32 X 49 by 16 x 36 x 42 x 28. 
 
 21. 32 X 45 X 72 X 15 by 64 x 18 x 36 x 30. 
 
 22. 64 X 15 X 63 X 24 by 30 x 21 x 56 x 12. 
 
 23. 44 X 20 X 50 x 36 by 55 x 10 x 25 x 18. 
 
 24. 40 X 48 X 21 X 18 by 56 x 36 x 35 x 24. 
 
 25. 40 X 27 X 32 X 3 by 5 x 8 x 12 x 8 x 27. 
 
 26. A lady bougiit 9 yards of ribbon at 56 cents per 
 yard, but exchanged it for other ribbon at 32 cents per 
 yard. How many yards did she get ? 
 
 27. At the rate of $129 for 27 acres of land, what will 
 180 acres cost ? 
 
 28. At what price per yard will 5 bales of cloth, contain- 
 ing 12 pieces of 42 yards each, pay for 50 rolls of carpeting, 
 of 75 yards each, at $ 2.10 per yard ? 
 
 29. A merchant purchased 13 firkins of butter, each con- 
 taining 39 pounds, at 32 cents a pound, and paid for it with 
 
144 ELEMENTS OF ARITHMETIC. 
 
 4 patterns of silk of 13 yards each. How mucli was tlie 
 silk worth per yard ? 
 
 30. A grocer exchanged 27 chests of tea, each containing 
 48 pounds, worth 40 cents a pound, for 18 pieces of dress 
 goods, each containing 60 yards. What was the price per 
 yard of the dress goods ? 
 
 31 o A stationer had 28 packages of cards, each contain- 
 ing 24 cards, which he exchanged for 21 packages of note 
 paper, each containing 24 sheets, worth 2 cents per sheet. 
 AVhat were the cards worth apiece ? 
 
 32. Four farms, each containing 80 acres, worth $ 72 
 per acre, were exchanged for 5 farms, each containing 96 
 acres. What was the value per acre of the farms received 
 in exchange ? 
 
 33. A farmer exchanged 8 bushels of wheat, at f 2 per 
 bushel, for 3 pieces of muslin, worth 10 cents per yard. 
 How many yards were there in each piece ? 
 
 34 5x8x7x12x16x24x10x9 ^^ 
 3x16x18x4x20x6x5x8 
 
 _ 9 X 8 X 11 X 16 X 15 X 35 X 42 X 18 _ ^ 
 
 36. 
 
 37. 
 
 38. 
 
 39- 
 
 40. 
 
 6x21x7x5x25x32x4x9 
 
 12 X 9 X 30 X 25 X 16 X 36 X 48 
 18 X 6 X 15 X 35 X 72 X 24 X 9 
 
 _ ? 
 
 = 9 
 
 42 X 35 X 18 X 24 X 60 X 64 X 72 
 36 X 32 X 12 X 25 X 30 X 21 X 5 
 
 44 X 20 X 36 X 54 X 65 X 27 X 48 
 
 24 X 40 X 27 X 72 X 13 X 45 X 72 
 
 45 X 60 X 63 X 28 X 35 X 24 X 50 ^ ^ 
 
 25 X 21 X 56 X 21 X 40 X 48 X 36 
 
 36 X 48 X 21 X 35 X 42 X 96 X 56 ^ y 
 28 X 84 X 42 X 24 X 70 X 14 X 9 
 
FRACTIONS. 
 
 135. 1. Into Llow many parts are these squares divided? 
 
 2. How do the parts compare as to size ? 
 
 3. What is each part called ? 
 
 4. What is a half of anything ? 
 
 5. How many halves are equal to a whole one ? 
 
 6. How many halves are there in 2 apples ? In 3 pears ? 
 In 5 peaches ? In 1 orange and a half ? In 2 lemons and 
 a half ? In 4 and a half plums ? 
 
 7. Two half melons are equal to how many whole 
 melons ? 4 half melons ? 6 half melons ? 3 half melons 
 are equal to how many whole melons^ and how many half 
 melons besides ? 5 half melons ? 
 
 8. Draw a line. Divide it into halves. Then divide 
 each half into two equal parts. 
 
 9. Into how many parts is the whole line divided? 
 
 10. What is each part called ? 
 
 11. What do you call 2 of the parts ? 3 of the parts? 
 How many thirds are equal to a whole thing? 
 
 12. How many fourths are there in any whole thing? 
 
 13. How many fourths are there in 2 circles ? In 3 
 squares ? In 1 line and 1 fourth of a line ? In 2 lines and 
 3 fourths of a line ? 
 
 EL. OF AR. — 10 145 
 
146 ELEMENTS OF ARITHMETICo 
 
 14. How many fourths of an apple are there in half of an 
 apple ? 
 
 15. What part of a half is a fourth ? 
 
 16. Which is the greater^ 1 half of a yard or 2 fourths of 
 a yard ? 1 pineapple or 3 fourths of a pineapple ? 
 
 17. How many whole apples are equal to 8 fourths of 
 an apple ? To 12 fourths ? How many whole apples and 
 how many fourths besides are equal to 5 fourths ? To 
 7 fourths ? To 9 fourths ? To 10 fourths ? 
 
 18. Draw a circle. Divide it into fourths. Then divide 
 each fourth into 2 equal parts. 
 
 19. Into how many equal parts is the circle divided ? 
 
 20. What is each part called? 
 
 21. What do you call 2 of the parts? 3 of the parts f 
 5 of the parts ? 6 of the parts ? 7 of the parts ? 
 
 22. How many eighths are there in a pie ? In 2 pies ? 
 In 2 pies and 1 eighth of a pie ? In 3 pies and 3 eighths of 
 a pie ? In 1 fourth of a pie ? In 3 fourths of a pie ? In 
 1 pie and 3 fourths of a pie ? In half a pie ? In 2 and a 
 half pies ? 
 
 23. How many whole sticks of candy are equal to 8 
 eighths ? 16 eighths ? How many whole sticks of candy 
 and how many eighths besides are equal to 10 eighths ? 
 To 15 eighths ? To 17 eighths ? To 20 eighths ? 
 
 ^ 24. What is each part of this line 
 
 called ? 
 25 o What do you call 2 of these parts ? 
 
 26. How many thirds are there in the whole of anything ? 
 
 27. How many thirds are there in 3 lines ? In 5 lines ? 
 In 2 lines and 1 third of a line ? In 4 and 2 thirds lines ? 
 
 28. If I have 6 thirds of orange, to how many whole 
 oranges is that equal ? 9 thirds ? 12 thirds ? 
 
 29. How is this line divided ? 
 
 30. What is each part called ? What 
 
FRACTIONS. 147 
 
 do you call 2 parts ? What other name will express the 
 same part of the line that 2 sixths expresses ? 
 
 31. What do you call 3 parts of this line? What other 
 name will express the part of the line that 3 sixths 
 expresses ? , 
 
 32. What do you call 4 parts of this line ? What other 
 name will express the part of the line that 4 sixths 
 expresses ? 
 
 33. How many sixths are there in 2 sheets of paper? 
 3 sheets ? 4 sheets ? 5 sheets ? 
 
 34. Which is the greater, 1 third of a pie, or 1 sixth of 
 a pie? 
 
 35. What part of a third is a sixth ? 
 
 36. How many sixths are there in two thirds? In 1 
 peach and 1 third of a peach ? In 2 and a half peaches ? 
 In 3 and 5 sixths peaches ? 
 
 37. How many units and how many sixths besides are 
 there in 7 sixths ? 9 sixths ? 12 sixths ? 14 sixths ? 
 
 38. Draw a line. Divide it into sixths. Now divide 
 each sixth into 2 equal parts. 
 
 39. Into how many equal parts is the line divided ? 
 
 40. What is each part called ? 
 
 41. How many twelfths are there in the whole of any- 
 thing ? 
 
 42. How many twelfths of a line are equal to a sixth? 
 To a third ? To a half ? To 2 thirds ? To 5 sixths ? To 1 
 and 5 sixths ? To 2 and a half ? To 2 and 2 thirds ? 
 
 43. How many units and how many twelfths besides are 
 equal to 15 twelfths ? To 18 twelfths ? To 20 twelfths ? 
 
 44. What part of a sixth is a twelfth ? What other 
 name will express the part of the line that 2 twelfths 
 expresses? 4 twelfths? 6 twelfths? 8 twelfths? 10 
 twelfths ? 
 
 45. Draw three parallel lines of just the same length. 
 
148 ELEMENTS OF ARITHMETIC, 
 
 46o Divide the first one into halves, the second one into 
 fourths, and the third one into eighths. 
 
 47. Draw three other parallel lines of just the same 
 length as the first group. 
 
 48. Divide the first one into thirds, the second into 
 sixths, and the third into twelfths. 
 
 49. Which is the greater, 1 half or 1 third ? 
 
 50. Which is the greater, 1 half or 2 thirds ? 
 
 51. When you divide the halves into 2 equal parts, what 
 parts of the whole do you have ? 
 
 52. When you divide the thirds into 2 equal parts, what 
 parts of the whole do you have ? 
 
 53. One third is equal to how many sixths ? 
 
 54. Two thirds are equal to how many sixths ? 
 
 55. Three thirds are equal to how many sixths? 
 
 56. What is 1 half of 1 half ? 1 half of 1 third ? 
 
 57. Which is the greater, 1 third or 1 fourth ? 
 
 58. Which is the greater, 1 fourth or 1 sixth ? 
 
 59. Which is the greater, 1 third or 2 sixths ? 
 
 60. What parts of the whole do you get by dividing the 
 fourths into 2 equal parts ? 
 
 61. One fourth is equal to how many eighths ? 
 
 62. Two fourths are equal to how many eighths ? 
 
 63. Three fourths are equal to how many eighths ? 
 
 64. Which is the greater, 1 sixth or 1 eighth ? 
 
 65. How many sixths are equal to a half? 
 
 66. How many eighths are equal to a half ? 
 
 67. Which is the greater, 3 sixths or 3 eighths ? 
 
 68. What parts of the whole do you get by dividing the 
 sixths into 2 equal parts ? 
 
 69. One sixth is equal to how many twelfths ? 
 
 70. Two sixths are equal to how many twelfths ? 
 
 71. Three sixths aie equal to liow many twelfths? Four 
 sixths ? 5 sixths ? 6 sixths ? 
 
FRACTIONS. 149 
 
 72. Which is the greater, 1 twelfth or 1 eighth ? 
 
 73. How many twelfths are equal to a half ? 
 
 74. How many eighths are equal to a half ? 
 
 75. Which is the greater, 6 twelfths or 6 eighths ? 
 
 76. When anything is divided into o equal parts, what is 
 each part called '■ 
 
 77. What do you call 3 of the parts ? 
 
 78. How many fifths are there in the whole of anything ? 
 
 79. Draw a line and divide it into fifths. Divide each 
 fifth into 2 equal parts. Into how many equal jparts is the 
 line divided ? 
 
 80. What is each part called? 
 
 81. How many tenths are there in the whole of anything ? 
 
 82. How many tenths are equal to a fifth ? 
 
 83. Draw a line and divide it into thirds. Divide each 
 third into 3 equal parts. Into how many equal parts have 
 you divided the line ? What is each part called ? 
 
 84. How many ninths are equal to a third of the line ? 
 
 136. One or more of the equal parts of anything is a 
 Fraction. 
 
 137. Two numbers written one above the other with a 
 line between them are used to express a fraction. 
 
 138. The number which shows into how many equal 
 parts the unit has been divided is called the Denominator. 
 
 It is written below the line. 
 
 Thus, ill the fraction f, 5 is the denominator. It shows that the 
 unit has been divided into five equal parts. 
 
 139. The number which shows how many parts form the 
 fraction is called the Numerator. 
 
 It is written above the line. 
 
 Thus, in the fraction f , 3 is the numerator. It shows that the frac- 
 tion contains 3 of tlie 5 equal parts into which tlie unit was divided. 
 
150 ELEMENTS OF ARITHMETIC. 
 
 140. The numerator and denominator are called tke 
 Terms of a Fraction. 
 
 141. A fraction in which the numerator is less than the 
 
 denominator is called a Proper Fraction. 
 
 Thus, I, I, If are proper fractions. 
 
 The value of a proper fraction is, tlierefore, less than 1. 
 
 142. A fraction in which the numerator equals or exceeds 
 
 tlie denominator is called an Improper Fraction. 
 
 Thus, f, I, If are improper fractions. 
 
 The value of an improper fraction is, therefore, 1 or more than 1. 
 
 143. A number expressed by an integer and a fraction is 
 a Mixed Number. 
 
 Thus, 6|, 8^, 6\} are mixed numbers. 
 
 144. A fraction may be regarded as expressing unexecuted 
 division. 
 
 Thus, ^ is equal to 15 ^ 4 ; ^A is equal to 24 ■- 6. 
 
 145. 1. Eead and interpret the expression ■^^. 
 Explanation. — y\ represents 5 of the 13 equal parts into which 
 
 the unit has been divided. It also represents one thirteenth of 6, and 
 5 divided by 13. It is readj'??;e thirteenths. 
 
 Eead and interpret the following expressions : 
 
 2. f. 5. f 8. ^. 11. H- 14. If. 
 
 3. |. 6. f. 9. ^. 12. if 15. li. 
 
 4. 1. 7. f. 10. A. 13. H- 16. If 
 
 Express by figures: 
 
 17. Five sevenths. Ten fourteenths. Seven twelfths, 
 
 18. Eight ninths. Nine tenths. Five thirteenths. 
 
 19. Nineteen twenty-fifths. Thirty-eight fortieths, 
 
 20. Thirty thirty-tiinths. Forty-one sixty-eighths. 
 
 21. Forty-six eightieths. Twenty-seven forty-fifths. 
 
 22. Five two-hundred-sixths. Seven three-hundred-ninths, 
 
 23. One eighth of 17. One thirty-fifth of 59. 
 
 24. Fifteen divided by 1.7. Twenty-four divided by 83. 
 
FRACTIONS. 151 
 
 REDUCTION OF FRACTIONS. 
 
 146. To reduce integers or mixed numbers to fractions, 
 
 1. A man received $ 2 for a day's work. If he was paid 
 in half dollars, how many did he get ? 
 
 2. How many quarter dollars are equal to $ 2 ? 
 
 3. How many sixths are there in 2 pies ? Eiglitlis ? 
 
 4. In three things how many halves are there ? Fourths ? 
 Eighths? Thirds? Sixths? Twelfths? Fifths? Tenths? 
 
 5. How many fourths are there in 2 ? In 2J- ? In 2|? 
 
 6. How many eighths are there in 1 ? In 3f ? In 3|- ? 
 
 7. How many sixths are there in 3 ? In If ? In 2|? 
 
 8. How many fifths are there in 2 ? In 2f ? In 34 ? 
 
 9. How many sevenths are there in 1? In 1^? In 4f ?. 
 
 147. The process of changing the form of fractions with- 
 out changing their value is called Reduction of Fractions. 
 
 WRITTEN EXERCISES, 
 
 148. 1. Eeduce 12f to eighths. 
 
 -j9 _'96_ Explanation. — Since in 1 there are 8 eighths, in 
 
 9 6 ~" 3^ ° 9 9 ^^ there are 12 times' 8 eightlis, or ^^ ; and in 12 + f 
 "¥" " t ~ ~t'' there are ^/- + f , or -\^-. Hence, 12| is equal to -^/. 
 
 149. From the above solution it is seen that : 
 
 In changing a mixed number to a fraction, the integer is 
 multiplied by the given denominator, to this loroduct the 
 numerator of the fractioncd part is added, and the result is 
 ivritten over the given denominator. 
 
 Keduce the following to improper fractions : 
 
 2. 6/^. 7. 13f. 12. 36^. 17. 53^. 
 
 3. 9|. 8. 18|. 13. 44^^,. 18. 651^. • 
 
 4. 7f. 9. 21^^. 14 352V 19. 74|f. • 
 
 5. 83^. 10= 32-j7- 15. 37if. 20. 87f|. 
 
 6. 9^. 11. 18^-^ 16, 2% 21. 98i|. 
 
152 ELEMENTS OF ARITHMETIC. 
 
 150. To reduce improper fractions to integers or mixed 
 numbers. 
 
 1. To liow many oranges are 4 half oranges equal? 6 
 half oranges ? 8 half oranges ? 5 half oranges ? 
 
 2. To how many dollars are 4 quarter dollars equal ? 8 
 quarter dollars ? 16 quarter dollars ? 5 quarter dollars ? 
 
 3. In 10 half bushels how many bushels are there ? In 
 12 half bushels ? In 20 half bushels ? 
 
 4. How many units are there in 3 thirds ? In 6 thirds ? 
 In 12 thirds ? In 15 thirds ? In 7 thirds ? 
 
 5. How many units are there in f ? f? ^? i^? lA? 
 S. How many yards are there in f yards ? -i^ yards ? 
 
 ^82- yards? J^ yards? -L^- yards ? ^5_ y^rds ? 
 
 7. How many units are there in | ? | ? i£-? i-i 9 u, 9 
 
 1_I 9 13.9 lA 9 LS. ? 13-9 S_§. ? SJ. ? 4J. ? 4J. ? 
 6' 2* 3' 3* 4' 3* 7* 8* 9* 
 
 WRITTEN EXERCISES. 
 
 151o 1. Eeduce J^ to a mixed number. 
 
 i|-^ = 125 -T- T = 17-f-. Explanation. — Since 7 seventh* are 
 equal to 1 unit, 125 seventlis are equal to 
 as many units as 7 seventlis are contained times in 125 sevenths, or 
 17 f units. Therefore, ^^ = 17f. 
 
 152. From the above solution, it is seen that : 
 
 In reducing an imj)roper fraction to a mixed number the 
 numerator is divided by the denominator. 
 
 Eeduce to integers or mixed numbers. 
 
 2. 
 
 1A8. 
 
 7. 
 
 w- 
 
 12. 
 
 W- 
 
 17. 
 
 w- 
 
 3. 
 
 2^£ 
 
 8. 
 
 w- 
 
 13. 
 
 ¥^. 
 
 18. 
 
 m- 
 
 4. 
 
 ^K 
 
 9. 
 
 ¥^ 
 
 14. 
 
 4G * 
 
 19. 
 
 H^- 
 
 5. 
 
 L|i. 
 
 10. 
 
 W-- 
 
 15. 
 
 W- 
 
 20. 
 
 ^m- 
 
 6, 
 
 ¥/• 
 
 11. 
 
 899 
 44 • 
 
 J6. 
 
 w- 
 
 21. 
 
 m^ 
 
FRACTIONS. 153 
 
 153. To reduce fractions to smaller or lower terms. 
 
 1. Ill 1 of a yard how many fourths of a yard are there ? 
 How many sixths ? Eighths ? Tenths ? 
 
 2. In -1- of an orange how many sixths are there ? 
 How many ninths ? Twelfths ? 
 
 3. How are the terms of the fraction J obtained from 
 those of i ? f from i ? 
 
 4. Since the terms of the fraction | are obtained from 
 those of ^ by multiplying them by 2, how may the terms of 
 1 be obtained from those of J ? 
 
 5. Since the terms of the fraction | are obtained from 
 those of i by multiplying them by 3, how may the terms 
 of ^ be obtained from those of | ? 
 
 154. From these examples, it is seen that : 
 
 The value of a fraction is not changed by multiplying or 
 dividing both terms by the same number. 
 
 6. Change to twelfths : i; i; |; i; |; i; f 
 
 \ 7. Change to twentieths : i; J; f; i; |; tVJ iV * 
 * 8. Change f to halves ; f to fourths ; y\ to thirds. • 
 9. Change to thirds: f ; |; f ; f ; ^5 H^ HJ M; M^ 
 
 ' 10. Change to fifths: 3%; 3-^; A; A; A; Mj fi^ if 
 
 155. A number that is an exact divisor of two or more 
 numbers is a Common Divisor of those numbers. 
 
 156. A fraction is expressed in its Smallest or Lowest 
 Terms when its numerator and denominator have no common 
 
 divisor. 
 
 WRITTEN EXERCISES. 
 
 157. 1. Reduce J| to its smallest terms. 
 
 g\4S (*) ExTLANATiON. — Siiice the fractioii is to be Teduced to 
 
 qV7^\ ~ ^ its smallest terms, we divide the terms of the fraction by 
 
 ' '^ 8, and the terms of the resulting fraction by 2. The terms 
 
 '^)^^ _ 'J! of the fraction | have no common divisor ; therefore 
 
 2)^ "^ li» when reduced to its smallest terms, is equal to |. 
 
154 * ELEMENTS OF ARITHMETIC. 
 
 158. It is evident, therefore, that : 
 
 In reducing a fraction to its smallest terms, the numerator 
 and denominator are divided by any common divisor, and this 
 process is continued until the terms have no common divisor^ 
 
 Eeduce to their smallest terms : 
 
 2 18. 8 AA 14 _44_ 20 -5-4- 
 
 /5. 3g. O. gg. i'±. ;^QQ. /CU. j4^. 
 ^15. q lA if; 17 oi i6 
 
 A 14 10 32 ifi 84 oo 150 
 
 ^' 6^- ^1* "TS* •'-'''• TST- 2^* Tsf- 
 
 Cl2 iol5 1Q 60 94 350 
 
 b. 3-2-. 1/J. 3^. 10. -240- /i^- 49-0- 
 
 721 I'^-'i'^ 1Q 2 5 OK 375 
 
 '• 6&- ■L'5' T5- •^^- T2T° ^^- 10 5""0- 
 
 159. To reduce dissimilar to similar fractions. 
 
 1. If I bought 1 of a pound of one kind of tea, and i of 
 a pound of another kind, how many sixths of a pound of 
 each kind did I buy ? 
 
 2. Mr. Roe planted i of an acre with potatoes, and i of 
 an acre with corn. How many tenths of an acre did he 
 plant- with each ? 
 
 3. Mr. Lee has -| of a barrel of russet apples and J of a 
 barrel of Baldwin apples. How many twelfths of a barrel 
 has he of each ? 
 
 4. Change | and ^ to fractions having their denominators 
 alike, i and i i and i-. i and i i and i. i and i 
 h h and i i 1. and i. J-, i and ^V- h h an^i iV- i. h 
 and 1. i, 1, and 1. 
 
 160. T'ractions that have the same denominators are 
 called Similar Fractions. 
 
 161. Fractions that have not the same denomniators are 
 called Dissimilar Fractions^ 
 
 162. The denominator of simikir fractions is called a 
 Common Denominator^ 
 
FRACTIONS. 155 
 
 163. When similar fractions are expressed in their lowest 
 terms, they have their Least Common Denominator. 
 
 164. 1. Eeduce J, |, and |i to similar fractions. 
 
 Explanation. — Since the fractions are to 
 be changed to other fractions having a common 
 denominator, the terms of each fraction must 
 be multiplied by some number which will cause 
 them to have the same denominator. 
 
 By examining the denominators, 4, 8, and 12, 
 it is evident that the denominators of all the 
 fractions can be made 24, and the fractions will 
 then be similar. To make the denominators 24, the terms of the first 
 fraction must be multiplied by 6, the terms of the second by 3, the 
 terms of the third by 2. And thus the fractions are changed to the 
 similar fractions, if, if, f f . 
 
 3 3x6 
 
 4 4x6" 
 
 18 
 24* 
 
 5 5x3 
 
 8 8x3 
 
 15 
 "24' 
 
 1 11x2 
 
 22 
 
 2 12x2 
 
 24 
 
 Eeduce to similar fractions : 
 
 2. 
 
 h h A- 6. 
 
 3. 
 
 h h tV 7. 
 
 4. 
 
 h T% A- 8. 
 
 5. 
 
 i T% sV 9- 
 
 4. 
 
 Eeduce |, f, ^, 
 
 10. 
 
 h 
 
 t h A- 
 
 h 
 
 h ^, tV- 
 
 1 
 
 8"' 
 
 T2' -iv -iS- 
 
 f- 
 
 TT> 2T' Y- 
 
 3 2 3 
 4' 5' lo- 
 ss 4 11 
 6"? TO? 5- ■'•■'•• 
 
 5. _6_ 3. 12 
 
 8' 12' 4* -^'^' 
 
 4 _3_ 13 lo 
 
 75 21? 42- -^^• 
 
 and ^ to similar fractions having 
 their least common denominator. 
 
 Explanation. — The least common de- 
 nominator cannot always be easily found 
 by inspection. It may then be found as 
 in the margin. 
 
 1114 Since the least common denominator 
 
 3x2x2x4 = 48. must be the smallest number that will con- 
 tain each of the denominators, it must 
 contain each of the prime factors of the denominators and no other 
 factors. The prime factors are found as in the margin. 3 is a prime 
 factor of 3 and 12, and consequently a factor of the least common 
 denominator. Dividing by 3, and writing below, the quotients and num- 
 bers of which 3 is not a factor, we have 1,4, 4, 16. Dividing by 2, and 
 again by 2, the factors of the denominator are found to be the divisors 
 3, 2, 2, and the factor 4 in the last row. Their product is 48, the least 
 common denominator. The fractions thus become ||, |f , |f , f|. 
 
 3 
 
 3 4 12 16 
 
 2 
 
 1 4 4 16 
 
 2 
 
 12 2 8 
 
156 ELEMENTS OF ARITHMETIC. 
 
 165. Froiii these examples it is evident that : 
 
 In reducing dissimilar to similar fractioyis the least common 
 denomincdor is found by multiplying together all the priyne 
 factors of the given denominators. The terms of ectch frac- 
 tion are then midtiplied by such a member as icill cause each 
 fraction to have the least common denominator. 
 
 1. In finding the product of the different prime factors, each fac- 
 tor must be used the greatest number of times that it occurs in any 
 of the given numbers, 
 
 2. Reduce all mixed numbers to improper fractions. 
 
 Eeduce to similar fractions having the least common 
 denominator : 
 
 4 3. _7_ on 4.4 3 23 
 
 _2_j4j7 97^11413 
 
 fi. -4__ _7_ OQ K1 az 2 
 
 5> 125 3 0^- ^O. O3, O3-, gg-. 
 
 A. _J_ 0. OQ 07 43 3 1 
 
 y 3 6> 5- ^^- ^9> T2'J T'S' 
 
 ADDITION OP FRACTIONS. 
 
 166. 1. John walked i of a mile on Monday, | of a mile 
 on Tuesday, and f of a mile on Wednesday. How many 
 eighths of a mile did he walk in all ? 
 
 2. How many sixths are -f, f, and 1 ? 
 
 3. How many ninths are |-, i, and I ? 
 
 4. How many twelfths are ^, yVj ^.nd -f^ ? 
 
 5 4i2_9 3 I 1—9 JL-1_JL — 9 -5__1_-J_— ? 2_|_3_9 
 O- 7-1-7—- ^-Ts— • 10^10—- ll'll — • 4'4— • 
 
 5. J_ 3 _ 9 _5_ -1- J7_ = 9 i . JL — 9 
 
 6T^6 13^^13 — • 3U'30 — ' 
 
 6= A boy spent 1 of a dollar for a book, and i of •'^ dollar 
 for a cap. What part of a dollar did both cost ? 
 
 7 A farmer sold I of his sheep at one time and ^ of them 
 at another. What part of his sheep did he sell ? 
 
 8. How many sixths are there in 1 and i ? 
 
 9. How many ninths are there in J and 1 ? 
 
 15. 
 
 i 
 
 A. 
 
 9 
 To- 
 
 20. 
 
 16. 
 
 s 
 
 8> 
 
 f, 
 
 T^e- 
 
 21. 
 
 17» 
 
 1. 
 
 A. 
 
 3^. 
 
 22. 
 
 18. 
 
 i, 
 
 f. 
 
 2V 
 
 23. 
 
 19. 
 
 i, 
 
 *, 
 
 tV. 
 
 24. 
 
FRACTIONS. 157 
 
 10. How many eighths are there in ^ and i ? 
 
 11. How many tenths are there in ^, i, and -^ ? 
 
 12. What kind of fractions can be added without chang» 
 ing their form ? 
 
 13. What must be done to dissimilar fractions before 
 they can be added ? 
 
 14. Add J and i; i and f ; J and |; ^ and |- ; |- and i; 
 ■!• and |- ; ^ and ^^2 ; i and -^ ; | and -^\. 
 
 15. Add i and i ; i and i ; ^ and l- ; i and -J- ; i and i. 
 
 16. Add 1 and | ; i and f ; | and | ; f and |- ; f and |-. 
 
 17. I paid f i for milk and $ Jq- for lettuce. How much 
 did I pay for both ? 
 
 18. James paid $4 for a fishing-rod and $yL- for a line. 
 What part of a dollar did both cost him ? 
 
 19. Lucy worked ^ of a day at her lessons and spent i 
 of a day at a picnic. What part of the day did she spend 
 at both ? 
 
 20. A farmer's son sold ^ dozen eggs to one nj.an, J dozen 
 to another, and f dozen to another. How many dozen did 
 he sell in all ? 
 
 21. A lad was presented with some money on his birth- 
 day. The next day he spent 4 of it, the day following, i of 
 it, and on the third day he spent -^ of it. W^hat part of 
 the money did he spend ? 
 
 WRITTEN EXERCISES. 
 167. 1. What is the sum of |, f, and |? 
 
 A = 2.5. ExpLAXATiox. — Siiice the fractions are not similar, 
 
 they must be changed to similar fractions before they 
 
 are added. 
 
 3 3. 0. 
 
 4 — 40 
 4^ g^ 2. 
 
 5 — 40^ The least common denominator of the given frac- 
 
 8 7 — o 7_ tions is 40, and 4 = M, 1 = M, '"uitl ^ = 32 Hence 
 the sum of the given fractions will be the sum of |f , 
 f§, and f-2, which is f^, or 2^^. 
 
158 ELEMENTS OF ARITHMETIC. 
 
 168. From the above solutions, it is evident that : 
 
 In adding fractions, dissimilar fractions are first changed to 
 
 similar ones, then their numerators are added, and the sum 
 
 is placed over the common denominator. 
 
 1. If the sum is an improper fraction, it should be reduced to an 
 integer or mixed number. 
 
 2. When there are mixed numbers or integers to be added, the 
 integers and fractions should be added separately and then those 
 results added. 
 
 Find the sum of the following : 
 
 10. 6|, 2|, and 4^^. 
 
 11. ^, 1^-^, and ^. 
 
 12. 4J, 3f, and 8^V 
 
 13. 5|, 101^, and ii. 
 
 14. 6f, 4f,'^andl2V 
 
 15. 121, H^ ^^^ h\' 
 
 16. 9^% i, and llij. 
 
 17. Ti^, 3U, and 6ii. 
 
 18. A farmer sold a load of hay for $8f, a load of oats 
 for $ 22|, and a load of wheat for $ 53|. How much did he 
 receive for all ? 
 
 19. James picked 5| bushels of apples from one tree, 
 7^ bushels from another, 6j\ bushels from another, and 8|- 
 bushels from another. How many bushels did he pick 
 in all ? 
 
 20. Three barrels of sugar contained respectively 225| 
 pounds, 232y''2- pounds, and 240^- pounds. What was the 
 weight of the whole ? 
 
 21 o A school-room is 32| feet long and 29 J feet wide. 
 What is the distance around the room ? 
 
 22. An express train starts from Boston at IJ p.m., 
 reaches Springfield 2J hours later, and Albany 4y^^ hours 
 later still. At what o'clock- does it arrive in Albany ? 
 
 2. 
 
 f. 
 
 A. 
 
 and 1^. 
 
 3. 
 
 A. 
 
 A, 
 
 and Y^2-- 
 
 4. 
 
 f. 
 
 f 
 
 and ^. 
 
 5. 
 
 h 
 
 TO"' 
 
 and ii 
 
 6. 
 
 A, 
 
 "2"0' 
 
 and |. 
 
 7. 
 
 f. 
 
 2% 
 
 and y3^. 
 
 8. 
 
 f, 
 
 A. 
 
 and |. 
 
 9. 
 
 A, 
 
 ^, 
 
 and I-. 
 
FRACTIONS. 159 
 
 SUBTRACTION OF FRACTIONS. 
 
 169. 1. Bertie earned $| and spent ^|. How much 
 had he left ? 
 
 2. Susie had | of a pound of candy, but she ate f of a 
 pound. How much had she left ? 
 
 3. There were f of a bushel of oats in a bin. If | of a 
 bushel was taken out, how much remained ? 
 
 ^•T T— • 12. 12—- 6 6— • 10 10 
 
 11 5 ? 11 8_ _ ? J_8 _ i.0 — '? 
 
 16 16~~- 13 13 ~ ' 25 25 ' 
 
 5. If I have $1 and give away $^, Avhat part of a dol- 
 lar shall I have left ? 
 
 6. A man who had J of a cord of wood sold i of a cord. 
 What part of a cord had he left ? 
 
 7. One pitcher held | of a quart of milk, and another 
 held f of a quart. How much more did the one hold than 
 the other ? 
 
 Q 1_1— ? 3_i_9 i_i— 9 5._1 — *? l — L—9 
 °-2" i ~ ' 4 2 — • 2 G"~* 6 2 ~ • 2 8~* 
 
 7._ i_ ? 
 
 8 2 — • 
 
 9. What kind of fractions can be subtracted without 
 changing their form ? 
 
 10. What must be done to dissimilar fractions before 
 they can be subtracted ? 
 
 11. l-t=? 1-| = ? l-f = ? 1-| = ? 1-^ = ? 
 
 19 1_1— ? 1_1— 9 1_2_'? 3._3._9 2 5__? 
 
 1^.^ 8"— -3 6 — -3 9— -4 8~-3 12—' 
 
 11 _ 5 _ 9 
 
 12 G — • 
 
 io 1 1 9 1 2 9 2 1—9 4_2_9 5_1_9 
 
 ^^' 2:^3 — • 2"~5"— • 3 4—- "5 3—- 6 4— • 
 
 7. 5_ _ 9 
 
 8 12 — • 
 
 14. A lad earned one day $f and spent $|. How much 
 had he left ? 
 
 15. A mechanic's wages were $3 per day, but he paid 
 $ I per day for his board. How much did he save daily ? 
 
 16. A boy agreed to work for a man 5 days, but lost If 
 days by sickness. How many days did he work ? 
 
160 ELEMENTS OF ARITHMETIC. 
 
 17. A boy raised vegetables which he sold for $3, The 
 cost of seed and other expenses were ^lyV How much 
 was his net profit ? 
 
 18. A man bought a hat for $2i a vest for $2|, and a 
 linen coat for $ li. How much change should he receive 
 from the merchant, if he paid for them with a ten-dollar 
 bill ? 
 
 WRITTEN EXERCISES. 
 
 170. 1. What is the difference between ^ and ^2" ^ 
 
 Explanation. — Since the fractions are not similar, they 
 JLA __. 4 5. must be made similar before they can be subtracted. The 
 
 ^ least common denominator of the given fractions is 48 ; 
 T 2^ ~ 15 i| = |f, and y7^ = ||. Hence, the difference between the 
 
 ^ given fractions is the difference between || and ||, which 
 
 io 1 7 
 IS 5g. 
 
 2. What is the difference between 15| and 8f ? 
 
 ^p.„ ir9 1121 Explanation. — Since the numbers are 
 
 ■f ~ ''^TT — T2" composed of integers and fractions, each may 
 
 8| = ^rf ~ ^1^2" ^^ subtracted separately. Reducing the frac- 
 
 (^ij. tions to similar fractions, it is evident that i| 
 
 ^ ^ cannot be subtracted from y\, hence 1, taken 
 
 from 15, is united with j%. 1, or i|,+ j% = fi. Subtracting 81f from 
 
 142 1, there is a remainder of Oil. 
 
 171. From the solution of the above examples it is seen 
 
 that : 
 
 In subtracting fractions, dissi7nila7' fractions are changed to 
 similar ones, then their 7iiimerators are subtracted, and the 
 difference is ivritteri over the common denominator. 
 
 When there are integers or mixed numbers to be subtracted, the 
 integers and fractions should be subtracted separately. 
 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 Out of 
 
 H 
 
 5 
 
 T 
 
 fV 
 
 T% 
 
 if 
 
 4t 
 
 H 
 
 Take 
 
 1 
 
 1 
 
 A 
 
 A 
 
 A 
 
 A 
 
 H 
 

 
 
 FRACTIONS. 
 
 
 
 161 
 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 14. 
 
 15. 
 
 16. 
 
 From 
 
 A 
 
 3\ 
 
 il 
 
 9 
 
 20 
 
 15 
 
 26 
 
 Take 
 
 t\ 
 
 A 
 
 H 
 
 2t\ 
 
 'A 
 
 1<^A 
 
 'H 
 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 21. 
 
 22. 
 
 23. 
 
 From 
 
 H 
 
 8A 
 
 m 
 
 IGf 
 
 20t\ 
 
 31tV 
 
 63t\ 
 
 Take 
 
 2f 
 
 2A 
 
 5| 
 
 lOf 
 
 16if 
 
 20|t 
 
 39ti 
 
 24. A box of soap weighed 75| pounds. Tlie weight of tlie 
 box alone was 3^^ pounds. How much did the soap weigh ? 
 
 25. From a farm of 120 acres there were sold 36Y^g acres. 
 How much was left ? 
 
 26. A flag-staff 50^^ feet high was broken off by a 
 storm, so that it measured 43|- feet. How much was 
 broken off ? 
 
 27. A grocer had 128|- pounds of sugar in one barrel, and 
 98| pounds in another. How much more was in the one 
 barrel than the other ? 
 
 28. A clerk earned $ 75 per month. His expenses during 
 that time were as follows: board, $22i-; washing, $5|; 
 and other expenses $16|. How much did he save per 
 month ? ' 
 
 29. A man spent -^^ of his money for a house, -^ for 
 furniture, and | for horses and carriages. What part of liis 
 money had he left ? 
 
 30. The sum of two numbers is llfV One of the num- 
 bers is 8J. What is the other number ? 
 
 31. From a web of cotton cloth containing 58j^g- yards, 
 8J yards were cut off at one time, and 15f yards at another 
 time. How many yards remained ? 
 
 32. Mr. Allen left f of his property to his wife, -^ to his 
 sons, i to his daughter, and the remainder to charitable 
 institutions. What part of his property was left to chari- 
 table institutions ? 
 
 EL. OF AR, — 11 
 
162 ELEMENTS OF ARITHMETIC. 
 
 Find the value of the following : 
 
 33. f + A + M + A- 40- 3i + 4| + 3f-lf 
 
 34- l+l + A + H- 41. 6f-4f + 3TV-lfV. 
 
 35- | + T'«-i-i- 42. 8t + 2J^-3i-3|. 
 
 36- A + A-A-sV 43. 2| + 3|-lf + 2|. 
 37. A-A + f-A- 44. 6i-2f-l| + 4A. 
 
 Qft _5 3_ I 6 _ 1 5 4.K 93 I A 5 Q 3 14 
 
 ^°- 2 1 42^ IT "84- ^^' ^ 5" + "TT '^TO -'-T5"- 
 
 39 i 5. _ 2. I 3. _7_ 4fi A5 I 23_15 I 94 
 
 ^^•16 3^4 12- ^'^- '^T ^ ^4 -•-¥ + ^9- 
 
 MULTIPLICATION OF FRACTIONS. 
 172. To multiply a fraction by an integer. 
 
 1 . At $ I" a yard, what will 2 yards of cloth cost ? 
 
 2. If a family uses f of a pound of butter each day, 
 how much will it use in 2 days ? 
 
 3. If a man earns ^^^ an hour, how much will he earn 
 in 3 hours ? 
 
 4. How much is 3 times |- ? 4 times |? 5 times -^? 
 3 times /q ? 2 times y4_ ? 7 times y2_ ? 6 times ^\ ? 
 
 6. How, then, may a fraction be multiplied by any 
 integer ? 
 
 6. If I gave i of an orange to each of 5 boys, how many 
 halves did I give away ? How many whole oranges Avas that ? 
 
 7. At $1 a yard, what will 2 yards of velvet ribbon 
 cost? 
 
 8. Mary's mother has 4 tumblers of jelly, each holdiug 
 I of a pint. How many pints of jelly has she ? 
 
 9. 5 times |=? 2 times |=? 3 times |=? 4 times -|=? 
 2 times f=? 3 times | = ? 4 times ^=? 6 times ^ = ? 
 
 10. How many are 4 times 1| ? 3 times 2| ? 5 times 3J ? 
 
 11. Multiply 4f by 5 ; 5A by 3; 94 by 8 ; 6f by 9. 
 
 12. How much is 5 times y^? Express the result in its 
 smallest terms. How may this result be obtained from y*^- ? 
 
FRACTIONS. 163 
 
 13. How much is 3 times yL? Express the result in 
 its smallest terms. How may this result be obtained from 
 
 14. In what other way, therefore, may we -sometimes 
 multiply a fraction by an integer ? Give both ways. 
 
 15. If a bookseller sells 15 books at ^^^ apiece, how 
 much will he get for them ? 
 
 16. When flour sells at ^4-J per barrel, how much will 9 
 barrels cost ? 
 
 17. A clerk's wages were $101 per week. How much 
 would he earn at that rate in 8 weeks ? 
 
 18. If the wages of a laborer are $ If per day, how much 
 does he earn in 10 days ? 
 
 19. A good walker can walk 4^ miles per hour. How 
 far can he walk at the same rate in 8 hours ? 
 
 173. It has been discovered, therefore, that : 
 A fraction can he multiplied by an integer by multiplying 
 the numerator or dividing the denominator by the integer. 
 
 WRITTEN EXERCISES. 
 
 174. 1. Mult: 
 
 Lpiy 
 
 «by8. 
 
 '■ 
 
 24^^- 24 • 
 
 _88 
 24 
 
 
 Explanation. — 8 times 
 
 = 3if,or3f. 
 
 iiisff, or 3i|, or3|. Or, 
 since dividing the denom- 
 
 «■•' i-' = 24 
 
 11 
 
 .-8 
 
 11 OO 
 
 inator multiplies the frac- 
 tion, 8 times ^\ is -y-, or 
 3|. 
 
 Multiply : 
 
 
 
 3 
 
 2. Aby5. 
 
 
 7. |byl6. 
 
 12. 3f by 6. 
 
 3. A by 4. 
 
 
 8. yVbyi5. 
 
 13. Gf by 12. 
 
 4. iXby6. 
 
 
 9. 11 by 14. 
 
 14. 10^^ by 15. 
 
 5. il by 5. 
 
 
 10. j§5by20. 
 
 15. 14f by 18. 
 
 6. H by a 
 
 
 11. A by 23. 
 
 16. le^by 8, 
 
164 ELEMENTS OF ARITHMETIC. 
 
 17. What will 5 pounds of cheese cost at 14f ^ per pound ? 
 
 18. If a train goes at the rate of 20| miles an hour, hov/ 
 far will it go in 12 hours ? 
 
 19. When potatoes are worth 66|^ a bushel, what must 
 I pay for 20 bushels ? 
 
 20 o How much will 15 tons of coal cost at $ 6^^^ a ton ? 
 
 21. How many feet are there in 10 rods, since there are 
 161 feet in 1 rod ? 
 
 22. A boy can chop if of a cord of wood a day. How 
 much can he choj) in 3 weeks, or 18 days ? 
 
 23. If a man works Sf hours per day, how many hours 
 will he work in 20 days ? 
 
 24. If the railroad fare between two places is $ 6|-^, what 
 will the fare of 10 persons cost ? 
 
 175. To multiply an integer by a fraction. 
 
 1. James had 12 chickens, but sold i of them. How 
 many did he sell ? What is f of 12 chickens ? 
 
 2. What will ^ of a pound of raisins cost at 16^ per 
 pound ? What is f of 16^ ? 
 
 3. Mr. Bird's horse can trot 5 miles in 20 minutes. 
 How many minutes will it take him to go 1 mile ? What 
 is I of 20 ? I of 20 ? I of 20 ? 
 
 4. What is f of 24? f of 14 ? | of 18 ? f of 28 ? 
 
 5. Two yards of ribbon were divided equally among 3 
 girls. How much did each girl get ? 
 
 6. If 1 yard of braid is worth 3^, what is J of a yard 
 worth ? What is f of 4^ ? 
 
 7. Four quarts of milk filled 5 pitchers of equal size. 
 How much did each pitcher hold ? What is | of 4 ? f of 
 
 4 ? I of 4 ? 
 
 8. What is i of 3 ? J- of 2 •? tV of 7 ? -J of 3 ? i of 5 ? 
 
 9. What is 1 of 4 ? I of 4 ? | of 7 ? i of 5 ? J of 
 
 5 ? I of 9 ? i of 8 ? f of 8 ? I of 15 ? i of 12 ? | of 12 ? 
 
FRACTIONS. 165 
 
 10. A coat cost $ 12^ and a x^air of trousers and a vest J 
 as much. How much did the suit cost ? 
 
 11. A schoolboy paid yL of his money for a pencil, f of 
 it for a grammar, and i of it for a penholder. If he had 
 40^ before he made his purchases, how much had he left ? 
 
 12. A lesson in arithmetic contained 30 examples. Henry 
 solved -f of them, and his sister solved f^ of them. How 
 many did each solve ? 
 
 13. The month of February has usually 28 days. If, in 
 1891, ^ of them were stormy, how many were pleasant ? 
 
 14. A paymaster paid his men only 4 of their monthly 
 wages. If each man earned $ 35, how much did he pay to 
 each ? 
 
 176. From these examples it is seen that : 
 Multiplying a number by a /inaction is finding a fractional 
 
 part of the number. 
 
 Thus, to multiply 12 by f we simply find f of 12. 
 
 WRITTEN EXERCISES. 
 
 177. 1. Multiply 13 by f, or find I- of 13. 
 
 ExPLANATiox. — To multiply 
 
 13 X - = ^^ ^ "^ = — or 101. ^^ ^y t '3 to lind 4 of 13. i of 13 
 
 5 5 5 ' ^ is y, and | of 13 is 4 times ^3^, 
 
 which is -\2_^ or lOf . 
 When the multiplier is a mixed number, multiply by the integer 
 and fraction separately, and add the results. 
 
 What is : 
 
 Multiply : 
 
 jVIultiply : 
 
 2. f of 15 ? 
 
 8. 54byx 
 
 14. ISbylJ. 
 
 3. fofl8? 
 
 9. 35by|. 
 
 15. 25 by 3f. 
 
 4. ^ of 34 ? 
 
 10. 48by|i. 
 
 16. 42by5f. 
 
 5. |of24? 
 
 11. 64 by If. 
 
 17. 34 by 71 
 
 6. ^oi2S? 
 
 12. 81byH. 
 
 18. 29by|. 
 
 1. ^of45? 
 
 13, 60 by 2V 
 
 19. m by 9f 
 
166 ELEMENTS OF ARITHMETIC. 
 
 20. Wlien corn is worth 75^ a bushel, what is the value 
 of f of a bushel ? 
 
 21. There are 125 acres in Mr. Swift's farm. How many 
 acres are there in |- of the farm ? 
 
 22. The expenses of a foundry for one month were $ 583, 
 What will f of the expenses be ? 
 
 23. What will 6f yards of silk cost at 88^ a yard ? 
 
 24. The average rate per hour of a very fast express 
 train is 52 miles. How far will it go in S^q hours ? 
 
 25. What will be the cost of 8^ pounds of tea at 30j^ a 
 pound, 6^ pounds of sugar at 5^ a pound, and 13^ pounds of 
 raisins at 16^^ a pound ? 
 
 26. What will be the cost of 6| dozen arithmetics at $8 
 per dozen, 5^ dozen grammars at $6 per dozen, and 6|- 
 dozen geographies at $ 9 per dozen ? 
 
 178. To multiply a fraction by a fraction. 
 
 1. If i of an apple is divided into 2 equal parts, what 
 part of the apple will each part be ? How much is i of -J 
 of an apple ? 
 
 2. If ^ of a yard is divided iiito 2 equal parts, what 
 part of a yard will each part be ? How much is i of J ? 
 
 3. If I of a pie is divided into 2 equal parts, what part 
 of the pie will each part be ? How much is i of ^ ? 
 
 4. If 1 of a pie is divided into 3 equal parts, what part 
 of the pie will each part be ? How much is -i- of |- ? 
 
 5. How much is 1 of 1? ^ of i ? i of i ? ^ofi? 
 i of 1 *? i of 1 '^ 
 
 Since -i- of i is -^^, how much 
 
 7. A man having i of an acre of land sold f of it. What 
 part of an acre did he sell ? 
 
 8. If a yard of crape costs $ J, what part of a dollar 
 will -| of a yard cost ? 
 
 i 
 
 on? iofi? 
 
 6. How much 
 
 iofi? 
 isiofi? 
 
 is 
 
 fofi? fofi 
 
 ? fofi? 
 
FRACTIONS. 167 
 
 9. I bought |- of a bushel of beans and planted f of them. 
 What part of a. bushel did I plant ? 
 
 10. How much is |^ of i ? Since i of -J- is yL^ how much 
 isiof-l? ioff? ioff? foff? |of|? 
 
 11. When cloth is worth $|- a yard, what will J of a yard 
 cost ? What will f of a yard cost ? 
 
 12. Mr. Stone, having a lot containing ^ of an acre, sold 
 i of it. What part of an acre did he sell ? If he had sold 
 -| of it, what part of an acre would he have sold ? 
 
 13. If a boy who had ^i spent i of it for candy, what 
 part of a dollar did he spend ? If he had spent | of it, 
 what part of a dollar would he have spent ? 
 
 14. Mr. Stearns, who owned | of a store, sold f of his 
 share. What part of the whole store did he sell ? 
 
 15. What is f of I? fof^? |of|? foff? 
 
 WRITTEN EXERCISES. 
 
 179. 1. Multiply | by f, or find f of |. 
 
 Explanation. — To multiply | by f, is to 
 ? X - = ^ ^^ = A find f of f. } of I is ^V (yJ-J; then, ^ of f 
 8 7 ^ X 7 14 will be (y|,-g) /^ ; and f of f will be 4 times 
 2 -5% . or It; or t\. 
 
 180. From the above it is seen that : 
 
 ' In multiplying a fraction by a fraction, the numerators are 
 multiplied together for the numerator of the p)roduct and the 
 denominators, for its denominator. 
 
 1. Reduce any mixed numbers to improper fractions. 
 
 2. Integers may be expressed as fractions by writing 1 as a denomi- 
 nator. Thus, 4 may be written f. 
 
 3. When possible use cancellation. 
 
 4. The word of between the fractions is equivalent to the sign of 
 multiplication. Such expressions are sometimes called Compound 
 Fractions. Thus, f of f is equal to f x f . 
 
168 ELEMENTS OF ARITHMETIC. 
 
 Find : 
 
 2. J off. 6. if off. 10. -.Voff 
 
 3. A off 7. If of||. 11. i-VofrV 
 
 4. 6 of 14. 8. ^oi^. 12. Aofi^. 
 
 5. fofff. 9. HoffI- 13. Ifofif. 
 
 Find : Multiply : 
 
 14. I of 41 18. fofi-byf 
 
 15. foflOf 19. fof^bySf. 
 
 16. -j-VofS^-V 20. 53 by 2 of 3^. 
 
 17. Hof7|. 21. 6fby|of4|. 
 
 Find the value of; 
 
 22. |x|X^Xf. 24. IxlX^Xif 
 
 23. fxfxHxif. 25. 2-^x|xHxf. 
 
 26. There are 16^ feet in a rod. How many feet are 
 there in J of a rod ? In 3|- rods ? 
 
 27. When hay is worth $20 J per ton, how much will j^ 
 of a ton cost ? 
 
 28. At $ 61- a ton, what must I pay for 8| tons of coal ? 
 
 29. A man purchased | of 420|- acres of land and then 
 sold I of what he had bought. How many acres did he 
 sell?' 
 
 30. At an auction sale a cow was sold for $ 39|, and a 
 horse for 3| times as much as the cow. For how much was ' 
 the horse sold ? 
 
 31. If a square foot of land is Avorth 16|j^, what is the 
 value of lOf square feet ? 
 
 32. The cloth of Kuth's dress cost ^^. The making 
 cost J as much, the trimming cost -J-^ as much as tlie making, 
 and the linings cost 4 as much as the trimmini^s. What 
 did the linings cost ? What was the whole cost of the 
 dress ? 
 
FRACTIONS. 169 
 
 DIVISION OF FRACTIONS. 
 181. To divide a fraction by an integer. 
 
 1. If I" of an orange is divided equally between 2 girls, 
 what part of the orange will each girl have ? How^ much is 
 
 4 _j_2 ? 
 
 2. A man divided | of an acre into 3 equal lots. How 
 large was each ? How much is | -^ 3 ? 
 
 3. If a horse ate -| of a bushel of oats in 4 days, how 
 much did he eat each day ? How much is f -;- 4 ? 
 
 4. In dividing a fraction by an integer, wdiat part of the 
 fraction is divided ? 
 
 5. If $ ^ is divided equally between 2 boys, what part of 
 a dollar will each boy get ? How much is i -r- 2 ? 
 
 6. When 3 pounds of sugar can be bought for $ i, what 
 is the price per pound ? How much is ^ -=- 3 ? 
 
 7. If J of a ship is owned equally by 5 men, what part 
 of the ship is owned by each ? How much is ^ -i- 5 ? 
 
 8. In what other way, besides dividing the numerator, 
 can a fraction be divided by an integer ? State both ways 
 in which a fraction may be divicled by an integer. 
 
 9. If 3 dozen oranges can be bought for $4, what do 
 they cost per dozen ? 
 
 10. If -| of a pound of tea is divided equally among 3 
 persons, what part of a pound will each receive ? 
 
 11. If f of a bushel of walnuts fills 4 bags of equal size, 
 what part of a bushel does each bag hold ? 
 
 12. 
 
 Mrs. Jay filled 5 tumblers with if of a gallon of 
 
 jelly. 
 
 How much did each tumbler hold ? 
 
 13. 
 
 Divide|by2; f by2; ^byS; tVl^V^; if by 5. 
 
 14. 
 
 Divide f by 2; i by 3; f by 4; A by 5; -J by 6. 
 
 15. 
 
 If 3 yards of cloth cost $ 14, what does 1 yard cost ? 
 
 16. 
 
 When 5 bushels of potatoes cost 34i, what does a 
 
 bushel cost ? 
 
170 ELEMENTS OF ARITHMETIC. 
 
 17. If 3 dozen oranges cost $ |, what does 1 dozen cost ? 
 
 18. If 4 brooms are worth $-j§g-, how much are they 
 worth apiece ? 
 
 19.. When I can buy 4 dozen eggs for ^^-^, what is the 
 price per dozen ? 
 
 20. If 5 yards of gingham sell for $^, what is the price 
 per yard ? 
 
 21. If 6 pounds of tea are worth $1|, what is a pound 
 worth ? 
 
 22. When 5 bushels of wheat sell for $4i, what is the 
 price per bushel ? 
 
 23. James earned $ 6| in 6 days. What were his daily 
 wages ? 
 
 24. A boy divided 6i- dozen marbles equally among his 5 
 sisters. How many did each receive ? 
 
 182. It is apparent from the solution of the above exam- 
 ples that : 
 
 A fraction may be divided by an integer by dividing the 
 numerator or multiplying the denominator of the fraction by 
 the integer. * 
 
 WRITTEN EXERCISES. 
 
 183. 1. Divide || by 5, or find ^ of f|. 
 
 Explanation. — Since dividing the nu- 
 
 20 20 -^ 5 4 nierator of a fraction divides the fraction, 
 
 ;tt -T- 5 = — ' = — -. the fraction f f may be divided by 5 by 
 
 ^^ ''^ dividing the numerator by 5. Hence the 
 
 result is /y. 
 
 2. Divide ^ by 6, or find J of j\. 
 
 Explanation. — Since multiplying the 
 
 5 rj !^ denominator of a fraction divides the frac- 
 
 :r7; -^ 6 = — — — - = ~. tion, the fraction j\ may be divided by 
 
 1^ Iw X i ^ 6 by multiplying the denominator by C. 
 
 Hence the result is y^^. 
 
Divide : 
 
 -B by 4. 
 
 f^by5. 
 «by8. 
 
 bv 
 
 FRACnONSc 
 
 
 
 7. If by 12. 
 
 11. 
 
 ijby23. 
 
 8. If by 16. 
 
 12. 
 
 If ^J 28. 
 
 9. ffbylS. 
 
 13. 
 
 tf^yss. 
 
 10. If by 15. 
 
 14. 
 
 if by 42. 
 
 171 
 
 113 = _4_7_ 
 
 15. Divide llf by 4. 
 
 Explanation. — The mixed number may 
 be reduced to an improper fraction, and tlie 
 division performed as before. 
 
 Or, tlie number may be divided wittiout be- 
 ing reduced to an improper fraction. Tlius, 4 
 is contained in 11| 2 times, and a remainder 
 of 3|, or ^^- ; and -Ls divided by 4 equals ^f. 
 Hence the result is 2 if. 
 
 Or, 
 4 )111 
 
 915 
 
 ^re- 
 
 Divide : 
 
 16. 17| by 7. 
 
 17. 15^ by 4. 
 
 18. 18f by 5. 
 
 22, 
 23. 
 24. 
 
 125f by 7. 
 
 153| by 8. 
 200f by 9. 
 
 19. 25fby8. 
 
 20. 31fby6. 
 
 21. 46f by 9. 
 
 25. Tlie cost of 5 tickets from Troy, N.Y., to Bethlehem, 
 K.H., was $ 404. How much did each cost ? 
 
 26. The rain interfered with a laborer's working 31- work- 
 ing days in 2 weeks. How many days did he work on an 
 average per week ? 
 
 27. A farmer forgot the price which he received for his 
 butter per pound, but he knew that he received $2| for 8 
 pounds. At that rate, what did he receive per pound ? 
 
 28. An estate of ^3956| was divided equally among 5 
 heirs. What sum did each receive ? 
 
 29. The earnings of 5 men for 10 days were $87|-. What 
 were the average earnings of each per day ? 
 
 30. If a man can reap a field in lOf days, in what time 
 can 12 men reap it ? 
 
172 ELEMENTS OF ARTTILAIETTC. 
 
 31. If a locomotive runs I'^^^q miles in 6 liours^ what is 
 the average rate of speed per hour ? 
 
 32. A field containing 125|- square rods was divided into 
 10 lots of equal size. How large were the lots ? 
 
 184. To divide an integer by a fraction. 
 
 1. How many tenths of a dollar are there in a dollar? 
 
 2. At $ yig- each, how many quires of paper can be bought 
 for $ 1 ? For $ 2 ? For f 3 ? For ^ 4 ? 
 
 3. When butter is worth $ i per pound, how many pounds 
 can be bought for $1? For $ 2 ? For $ 3 ? 
 
 4. How many penknives, at $i apiece, can be bought for 
 $ 1 ? For $ 2 ? How many at $ f apiece can be bought 
 for $2? For {^ 4 ? 
 
 5. If a boy spends -| of an hour in going to school, in 
 how many days will he spend 3 hours ? 6 hours ? 9 hours ? 
 
 6. A man gave his daughter ^f a month to spend as she 
 pleased. In how many months would he give her $ 9 ? 
 
 7. The retail price of one kind of molasses is $ f per 
 gallon. How many gallons can be bought for $S? 
 
 8. A young man spends | of each day in work and 
 recreation. In what time will he spend 12 days in this 
 way ? 
 
 9. How many times is i contained in 5 ? In 9? In 12? 
 
 WRITTEN EXERCISES. 
 
 185. 1. Divide 5 by f, or find how many times f is con- 
 tained in 5. 
 
 q ;- rr EXPLANATION. — Sliice \ IS contaliied 
 
 5 -f — =: — ^ — = 11|-. in 1 7 times, it is contained in 5, 6 times 
 
 '3 7 times, or 85 times, and 4 is contained 
 
 Or, in 6 I of 35 times, or llf times. Or, 
 
 K 3 5. 35_5_3_ii2 sincB in 5 there are Y, > is contained in 
 
 ^ - T- J -T- • 7 - ^^3- 3^5 111 times. 
 
 1. Cancellation can often be used advantageously. 
 
 2. Mixed numbers may be reduced to improper fractions. 
 

 
 FRACTIONS. 
 
 
 
 
 173 
 
 
 Find the quotients of : 
 
 
 
 
 
 
 
 2. 
 
 18 ^|. 
 
 7. 47 -- f. 
 
 
 12. 68- 
 
 -H' 
 
 
 17. 
 
 225 -ff. 
 
 3. 
 
 21 -f 
 
 8. 38 -3^^. 
 
 
 13. 71 H 
 
 . 1 2 
 " T3- 
 
 
 18. 
 
 387-^3. 
 
 4. 
 
 16 -f. 
 
 9. 21- f 
 
 
 14. 43- 
 
 ^tV 
 
 
 19. 
 
 477^^. 
 
 5. 
 
 24 ^|. 
 
 10. 36 --yV 
 
 
 15. 29- 
 
 ■^H- 
 
 
 20. 
 
 642-1^. 
 
 6. 
 
 36 -|. 
 
 11. 43 -A- 
 
 
 16. 35- 
 
 ^if 
 
 
 21. 
 
 936 -if 
 
 
 Divide the following : 
 
 
 
 
 
 
 
 
 22. 29 by 
 
 41 24. 
 
 36 
 
 by 8i. 
 
 
 26. 
 
 33 
 
 by 7|. 
 
 
 23. 32 by 
 
 31 25. 
 
 91 
 
 by 51 
 
 
 27. 
 
 54 
 
 by6f 
 
 28. Divide 24 by 5f 
 
 ^4\ 24 
 
 ^^ " Explanation. — By changing the dividend and 
 
 • • divisor to 7ths, the division is performed as in simple 
 
 39 )168(4,^ ^^^^^^^^^- 
 Divide the following : 
 
 29. 36 by 7 f. 32. 59 by 4 |. 35. 39 by 7|. 
 
 30. 46 by 3 |. 33. 36 by 3 |. 36. 35 by 16f 
 
 31. 52 by 6^. 34. 96by43-V 37. 46 by 8f. 
 
 38. When cloth is $.10| per yard, how many yards can 
 be bought for $3.36? 
 
 39. My lot has a frontage on the street of 385 feet. How 
 many rods is it in front, 161 feet being a rod ? 
 
 40. A cheap kind of flour sells for $3f per barrel. How 
 many barrels can be bought for % 76i ? . 
 
 41. A farmer bought a wagon for % 68, paying for it in 
 apples at $1 J per barrel. How many barrels did he give 
 for it ? 
 
 42. Boots are commonly sold in cases containing 12 
 pairs. If a dealer paid $ 99 for a quantity of boots at % 2J 
 a pair, how many cases did he purchase ? 
 
174 ELEMET^TS OF ARITHMETIC. 
 
 186. To divide a fractiou by a fraction. 
 
 1 . At $ -|- eachj how many books can be bought for $ | ? 
 At^l? 
 
 2. Among how many children can -| of a melon be 
 divided, if each child is given i ? | ? f ? 
 
 3. If cream is worth $^ per pint, how many pints can 
 be bought for $^? If $ j\ per pint ? If $ j\ ? 
 
 4. How many pitchers, each holding ^j of a gallon, can 
 be filled from if of a gallon ? 
 
 5. If a man uses -^^ of a pound of butter each day, how 
 many days will ^| of a pound last him ? 
 
 6. How many times is -f^ contained in i|-? y4_ jn -^-^? 
 _3_ in -9- ? -3_ in la (or l")*? -«- in 1 ^ -2- in 1 ^ -^ in 1 ^ -L 
 
 1 2 ^^^ 1 2 • 1 2 -"^V 1 2 \^^ ^) ■ 12^^^-^* 1 2 ^^^ -•- • 1 2 -"^^ -•- • 12 
 
 in -9- *? -2_ in -6- *? -3- in -«- ^ 
 
 ^^^12- 1 2 ^^M 2 • 1 2 -^^^ 1 2 • 
 
 7. If a train moves | of a mile each minute, how many 
 minutes will it require to go 3^ miles ? How many thirds 
 are there in 3|- miles ? 
 
 8. If it takes f of a yard of lace to trim a bonnet, how 
 many bonnets can be trimmed with 2|- yards of lace ? 
 
 9. How many aprons can be made from 4i yards of mus- 
 lin, if each apron requires 1|- yards ? 
 
 ? 
 
 How many half yards are there in li yards ? 
 How many half yards are there in 3^ yards 
 
 10. How many times are 3 half yards contained in 6 
 half yards ? 
 
 11. How many books can be bought with $6|, at $1| 
 apiece ? 
 
 12. How many pitchers, each holding 2 J pints, will be 
 required to hold 13| pints of milk ? 
 
 13. Among how many children can i of a pound of candy 
 be divided, if each child is given i- of a pound ? 
 
 14. How many eighths are there in ^ ? 
 
 15. If a coat requires | of a dozen buttons, how many 
 coats will require 4^ dozen buttons ? 
 
FRACTIONS. 17e5 
 
 16. With $f, how many yaids of cloth can be bought at 
 ^f per yard? 
 
 WRITTEN EXERCISESo 
 
 187. 1. Divide |- by |, or find how many times f is con- 
 tained in |-. 
 
 7:_j_2._7.N^3_ 2.1^ Explanation. — Since it is required to find 
 
 how many times f is contained in |, we first 
 
 find how many times | is contained in 1, and then find | of that result. 
 
 Since i is contained in 1 3 times, | is contained in 1 one half of 3 
 times, or | times. 
 
 Since |- is contained in 1 | times, in | it will be contained | of f 
 times, or f ^ times. 
 
 188. From the explanation given, it is apparent that : 
 
 A fraction may he divided by a fract'ion by 7nultii:)lying the 
 dividend by the divisor inverted. 
 
 1. Use cancellation whenever it is possible to do so. 
 
 2. Reduce mixed numbers and integers to improper fractions. 
 
 2. Divide | by ^. 
 
 Or, f = fi, and f = f f . There- 
 
 fore, 
 
 21 
 
 189. Hence it is seen that : 
 
 Another method of dividing a fraction by a fractioyi is to 
 reduce the fractions to similar fractions, and then to divide the 
 numerator of the dividend by the numerator of the divisor. 
 
 Divide : 
 
 3. 
 
 fby,-V 
 
 9. 
 
 H by f 
 
 15. 
 
 26i by 4f . 
 
 4. 
 
 Hby|. 
 
 10. 
 
 33-Vbyli 
 
 16. 
 
 19^ by 3f-. 
 
 5. 
 
 fbyf 
 
 11. 
 
 9A by 2|. 
 
 17. 
 
 361 by ^. 
 
 6. 
 
 Aby^. 
 
 12. 
 
 7A by 41 
 
 18. 
 
 l^ by f . 
 
 7. 
 
 A by f 
 
 13. 
 
 ^ by f 
 
 19. 
 
 16f by 3f . 
 
 8. 
 
 Hby^V 
 
 14. 
 
 12| by 7^. 
 
 20. 
 
 "\ by If. 
 
17G ELEMENTS OF ARITHMETIC. 
 
 21. How many times may 2}r gallons be drawn from a 
 barrel that holds 31|^ gallons ? 
 
 22. How many bushels of potatoes, at $ii a bushel, may 
 be bought with $ 5 ? 
 
 23. How many barrels of apples, each holding 2| bushels, 
 can be filled with 18 bushels ? 
 
 24. At $ 1|- a basket for peaches, how many baskets of 
 peaches shall I get for $ 8 ? 
 
 25. When flour is $ 7| per barrel, how many barrels of 
 ilour can be bought for $ 59 ? 
 
 26. How many pounds of beef, at 11|/ per pound, can be 
 bought for 871^? 
 
 27. How many books, at $31- per volume, can be pur- 
 chased for $31^? 
 
 28. When oranges are 121^ per dozen, how many dozen 
 can be bought for 98f ^ ? 
 
 29. How many plows, at ^ll^- each, can be bought for 
 $67^? 
 
 30. How many acres of land, at $43|- per acre, can be 
 bought for $237^? 
 
 31. A railroad 16^ miles long cost $66,937. What was 
 the cost per mile ? 
 
 32. If 4i hogsheads hold 25|- bushels, what will 1 hogs- 
 head hold ? 
 
 33. If a field containing 541 acres yielded 1165 bushels 
 of rye, what was the average crop per acre ? 
 
 34. Divide | of f of f of ^ by | of | of |. 
 
 3''4''7''9J • U 3''8j-^^^''y''9''^''2''^"9 
 
 Explanation. — In solving examples like this, aU the factors of the 
 divisor should be inverted. All the integers and mixed numbers should 
 be changed to improper fractions before the factors of the divisor are 
 inverted. Use cancellation to abridge the process. 
 
FRACTIONS. 177 
 
 Divide: 
 
 36. i of i Of I of tV by I of -I of ,\ of I of 6. 
 
 36. I of f of f of T^ by | of f of f of 8. 
 
 37. I of j\ of t\ of T^ by i* of f of Jj of 24. 
 
 38. Jjj of 4 of If of if by y\ of i of \l of 36. 
 
 39. I of f of iV of A by A of H of I of T*3 of 30. 
 
 40. tV of f of A of A by f of ^a of ^^ of ^ of 44. 
 
 41. I of f of JJ- of I- by i of J_ of if of ^ of 25. 
 
 42. I of t of i| of .*-, by I of T*5 of ff of I of 16. 
 
 43. f of JV of H of if- by 4 of J^ of if of A of 10. 
 
 44. T% of f of l-l of I by f of 4 of If of f | of 20. 
 46. f of f of if of /y by I of il of H of ^ of 45. 
 46. -rV of I of If of f by I of if of ^ of If of 18. 
 
 190. Expressions of unexecuted division of fractions are 
 frequently written in the form of fractions. Such expres- 
 sions are sometimes called Complex FractionSe 
 
 3 02 
 
 Thus, f -> f may be written | ; 6f -^f, -y-. 
 
 7 9 
 
 Ic Find the value of the fractional form, ^. 
 
 IT 
 
 4 
 
 Sot tttton ^ — 1 _:_ _3_ _ 44 ^ 2.1 _ 44 _ l l 7 
 
 TT 
 
 Find the value of the following : 
 
 2. 
 
 1 
 1' 
 
 5. 
 
 5i 
 4' 
 
 8. 
 
 2i 
 
 11. 
 
 n 
 
 H 
 
 3. 
 
 1 
 
 i 
 
 6. 
 
 
 9. 
 
 If 
 
 12. 
 
 -|of 8 
 
 4. 
 
 EL. 
 
 7 
 
 OF AR. 
 
 5 
 
 r 
 —12 
 
 10. 
 
 5| 
 3i' 
 
 13. 
 
 2t\ 
 iofi 
 
178 ELEMENTS OF ARITHMETIC, 
 
 191. To find the relation of one number to another. 
 
 1. There were 5 oranges on a plate. Willie took 1 
 orange, and Eddie took 2. What part of all the oranges 
 did Willie take ? Eddie ? 
 
 2. What part of 8^ is 1^ ? 3^ ? 5^ ? 
 
 7 is -J-Q of 10, because it is 7 of the 10 equal parts into 
 which 10 may be divided. What part of 10 is 9 ? 
 
 3. What part of 10 acres are 5 acres ? Of 9 quarts are 
 3 quarts ? Of 8 pounds are 2 pounds ? Of 10 inches are 2 
 inches ? 
 
 4. 1 is what part of 12 ? 2 of 12 ? 3 of 12 ? 8 of 12 ? 
 
 5. 5 melons are what part of 7 melons "^ 
 
 6. $ 5 are what part of $ 8 ? 
 
 7. 2 fifths are what part of 3 fifths ? 
 
 8. 3 eighths are what part of 7 eighths? 
 
 9. f is what part of f ? ^-^ of -f^ ? ^\ of \\ ? 
 
 10. What part of 2 ounces is f of an ounce ? 
 Suggestion. — 2 ounces = f ounces. What part of | is J? 
 
 11. What part of $ 5 are $ 21 ? 
 
 12. What part of \ of a gallon is i of a gallon ? 
 Suggestion. — i = f ; i = f . What part of f is | ? 
 
 13. What part of 10 quarts are 3|- quarts 
 
 14. What part of 7 acres are If acres ? 
 
 15. What part of |- of a rod is J of a rod ? 
 
 16. What part of 2| pounds are li pounds ? 
 
 17. What part of 10 is \ ? Of 10 is | ? Of 10 is 1\ ? 
 
 WRITTEN EXERCISES. 
 
 192. 1. What part of $ 100 are $ 72 ? 
 
 ^2 18 Explanation. — $1 is j^-^ of $ 100 ; consequently, 
 
 TOO — 25- $ 72 are f ^^^ of $ 100, or if of $ 100. 
 
 1. When one or both numbers are fractions not having a common 
 denominator, the numbers should be reduced to equivalent fractions 
 having a common denominator. 
 
 2. Mixed numbers should be reduced to improper fractions. 
 
FRACTIONS. 179 
 
 What part of : 
 
 2. 85 is 30? 7. 25is2i? 12. H is I ^ 
 
 3. 175 is 28? 8. 40is3|? 13. -^is^? 
 
 4. 12isf? 9.64is4i? 14. SiVisli? 
 
 5. 20 13 4? 10. 33isl|-? 15. 7iis4i? 
 
 6. 32is|-? 11. 72is2f? 16. 5J is 3| ? 
 
 17. Ill a farm of 120 acres there were 35J acres of pasture 
 land. What part of the farm was used for pasture ? 
 
 18. What part of 272^ square feet are 9 square feet ? 
 
 19. If a boy can do a piece of work in 25 days, what part 
 of the work can he do in 7f days ? 
 
 20. Miss Reed paid $20| for a gown, and $ 61- for a hat. 
 What part of the price of the dress did the hat cost ? 
 
 21. If the AYorking day is 10 hours long, how much will 
 a man who earns $ 3 a day get for 6 hours' work ? 
 
 ALIQUOT PARTS. 
 
 193. An aliquot part of a number is any integer or mixed 
 number that will exactly divide it. 
 
 ALIQUOT AND OTHER PARTS OF A DOLLAR. 
 
 50^ = 
 
 25^ = 
 20^ = 
 
 16|^ = fi. 
 
 io^ = f iV 
 
 66|^ = f|. 
 
 Hf = $^. 
 
 75)« = $|. 
 
 6if=9^. 
 
 40/ = «f 
 
 5^=$i^. 
 
 831/ = $|. 
 
 ill = ^j\. 
 
 37i/ = «|. 
 
 2^ = «!^V 
 
 871/ = fi. 
 
 194. 1. AYhat will 12 pounds of butter cost at 33^^ per 
 pound ? 
 
 Solution, — S3lf = $ }. Since 1 pound costs $ i, 12 pounds will cost 
 12 times fi,- which is $ i/-, or $4. 
 
180 ELEMENTS OF ARITHMETIC. 
 
 2. What will 20 readers cost at 25^ apiece ? At 50^ ? 
 
 3. What will 25 sponges cost at 20^ apiece ? 
 
 4. What will 36 yards of ribbon cost at 16|^ a yard ? 
 
 5. What will 48 yards of lace cost at 121)^ a yard ? 
 
 What is the cost of ; 
 
 6. 120 pounds of sugar at 5^ ? At SJ^ ? At W ? 
 
 7. 12 neckties at 75p ? At 66fj^ ? At 83^^ ? 
 
 8. 40 baskets at 37^^ ? At 40^ ? At 30^ ? 
 
 9. 80 gallons of molasses at 37J-^ ? At 62^^ ? At 87^^ ? 
 
 10. 28 dozen pencils at 12^^ ? At 16|^ ? At 20^ ? 
 
 11. 16 pairs of gloves at 66|^ ? At 83^^ ? At 75^ ? 
 
 12. 24 days' work at 1 1.25 ? At ^ 1.331 ? At $ 1.16| ? 
 
 13. At 16|^ per pound, how many pounds of raisins can 
 be bought with $ 2 ? 
 
 Solution. — 16|/ = $ |. Since 1 pound costs 1 1, as many pounds 
 can be bought with $ 2 as $ | is contained times in $ 2, or $ -^2, which 
 is 12 times. Therefore, 12 pounds of raisins can be bought with $2. 
 
 14. At 33^/ a dozen, how many dozen eggs can I buy 
 with $5? $3? $7? $4? 
 
 15. At 37^)^ apiece, how many books can be bought for 
 $21? $3|? $41? $51? 
 
 16. For $ 21, how many yards of muslin can be bought 
 at 10^ a yard ? At 12i^ ? At 16|^ ? 
 
 195. To find the whole when a part and the relation of the 
 part to the whole are given. 
 
 1 . If $ 5 is ^ of my money, how much money have I ? 
 10 cents is i of my brother's money. How much has he ? 
 
 2> A man sold 5 gallons of vinegar, which was i of all 
 he had. How many gallons had he ? 
 
 3. A farmer sold 10 quarts of milk per day, which was 
 ^ of what the cows produced. How much did they pro- 
 duce per day ? 
 
FRACTIONS, 181 
 
 4. A boy paid $ 6 for a coah, and that sum was J of his 
 wages for a montli. What weie his monthly wages ? 
 
 5. A lad purchased a box of cartridges, fired off 10 of 
 them, and had -f^ of them left. How many cartridges were 
 there in the box ? 
 
 6. A boy borrowed 50 feet of kite string, which was J-g 
 of what another boy had. How many feet of string did 
 the other boy have ? 
 
 7. A farmer estimated that a hail storm had caused him 
 a loss of 5 bushels of oats per acre, or yL of his crop. How 
 much did he think his land Tvould yield him per acre ? 
 
 8. A boy and his sister played backgammon for some 
 time, his sister beating him 15 games, which was \ of the 
 number of games played. How many did they play ? 
 
 9. If 2 thirds of a number is 6, hoAV much is i of it? 
 
 10. If 3 fourths of a number is 9, how much is i of it? 
 
 11. If 5 sevenths of a number is 10, how much is -^ of it? 
 Since ^ of the number is 2, what is the number ? 
 
 12. If I of a number is 8, how much is ^ of the number ? 
 Since ^ of the number is 4, what is the number ? 
 
 WRITTEN EXERCISES. 
 
 196. lo 320 is I of Avhat number ? 
 
 ^ = 320. Explanation. — Since 320 is f of a certain 
 
 1 n4 number, i of 320, or 64, is i of the number; and 
 
 ^ * since 64 is l of the number, the number must be 
 
 Number = 512. § times 64, or 512. 
 
 Find the number of which 
 
 2. 40 is |. 8. 125 is j%. 14. 
 
 3. 60 is f. 9. 340 is ^. 15. 
 
 4. 85 is y%. 10. 560 is f|. 16. 
 
 5. 75 is |. llo 300 is ^. 17. 
 
 6. 6o is f 12. 240 is i|. 18. 
 
 7. 81 is yV 13. 330 is if 19. 
 
 fis 
 
 f 
 
 ^iris 
 
 *• 
 
 II is 
 
 1- 
 
 If is 
 
 *• 
 
 f«is 
 
 f. 
 
 tfis 
 
 A- 
 
182 ELEMENTS OF ARITHMETIC. 
 
 20. A boy sold his bicycle at second hand for | of the 
 cost, receiving for it $ 80. How much did it cost him ? 
 
 21. Our tennis court is just f of what it was originally, 
 and it is now 120 feet. How long was it originally ? 
 
 22. The distance between Kochester and Syracuse, N.Y., 
 is 81 miles, and it lacks 4 miles of being -J- of the distance 
 from New York to Buffalo. What is the distance from 
 New York to Buffalo? 
 
 23. The width of a river near its source was 45 feet, and 
 that distance was -^ of what it was 20 miles farther down. 
 How wide was it at the latter point ? 
 
 24. A gardener set out some cabbage plants, but only 80 
 of them lived, which was just -^ of the number set out. 
 How many did he set out ? 
 
 25. A way train stopping at all stations averaged 25 
 miles per hour ; but the rate of the train was only ^ of that 
 of a fast express train. How many miles per hour did the 
 express train run ? 
 
 REVIEW EXERCISES. 
 ORAL EXERCISES. 
 197. 1. Change to improper fractions: 3|; 5|-; 6|; 
 
 2. Change to mixed numbers: -^; ^-^-', 4=-^-; ^; -^; 
 
 4 2 . 5J. 
 6 5 9 • 
 
 3. Find the sum of 3|, 4f, 51 6i 7f, and 3f . 
 
 4. Find the sum of 61 41, 2|, 3f, 41 and 2|. 
 
 5. A man had 3 lots containing respectively 3i acres, 
 2J acres, and 3|- acres. How many acres did they all 
 contain ? 
 
 6. The expense of a carpet was $12|, and it cost ^31 
 to put it down. What was the entire cost of the carpet 
 laid in the room ? 
 
FRACTIONS. 183 
 
 7. Terrence's uncle gave him $ 10 on Cliristmas, but he 
 spent $ of of it. How much had he left ? 
 
 8. A father allowed his son $40 per year for spending 
 money. At the end of the year the boy had $ 18|-. How 
 much did he spend ? 
 
 9. The expense of a Columbia bicycle was $ 140 ; but a 
 boy bought one that pleased him just as well for -| of that 
 price. How much did it cost him ? 
 
 10. The price of California peaches was $ | per dozen in 
 July. How much did 8 dozen cost? 
 
 11. A man had a field containing 4|^ acres, which he 
 divided into lots containing -J of an acre. How many lots 
 were there ? 
 
 12. A w^agon w^iich cost $ 50^ was sold for $ 46J. How 
 much was the loss ? 
 
 13. After purchasing a silver watch for f 19f , I had 
 $ 21| left. How much had I at first ? 
 
 14. A man owed for groceries $18|-; for meat, $ lOf , 
 and for clothes $S^. If he paid the debts out of $50, 
 which he has, how much did he have left ? 
 
 15. A boy went on a visit to his cousins with $ 35 to pay 
 his expenses. On returning he found that he had spent -f- 
 of his mone}^ How much had he left ? 
 
 16. The proprietor of a boarding house, after buying pro- 
 visions for the day, found that he had $ 351 left, which was 
 just -1- of the money he had before making the purchases. 
 How much did he spend? 
 
 17. An American flag was purchased by some school 
 children for $llf, and the flag-staff cost $61 What did 
 both cost ? 
 
 18. A man sold a horse for $ 160 and a cow for | as much. 
 How much did he get for both ? 
 
 19. A cow was sold for ^ 60, which was | of what a horse 
 was sold for. How much was received for both ? 
 
184 ELEMENTS OF ARITHMETIC. 
 
 20. After sj^ending J of my money, I had $35 left. 
 How much had I ? 
 
 21. A horse traveled 38^ miles in 7 hours. What was 
 his average rate of speed per hour ? 
 
 22. A man sold a cow for $40 and spent -J- of the sum. 
 Afterwards he bought a horse and paid in cash what he had 
 left from the sale of the cow. What did the horse cost 
 him, if the cash payment was ^ of the cost of the horse ? 
 
 23. A farm was sold in 2 parts. The first part contained 
 50 acres and that was -f- of the second part. How many 
 acres were there in the farm ? 
 
 24. I spent i of my money for books, i of it for furniture, 
 and had f 100 left. How much had I at first ? 
 
 25. James had a sum of money such that i of it was just 
 ^ of Henry's. What part of Henry's was the whole of 
 James's ? 
 
 26. One half of a mother's age was f of her daughter's. 
 What part of the mother's age was the daughter's ? 
 
 27. What part of the daughter's age was the mother's? 
 
 28. I multiplied by another fraction gives a product of f. 
 What is the multiplier ? 
 
 29. The difference between i and J of a number is 8. 
 What is the number ? 
 
 WRITTEN EXERCISES. 
 198. Find the value of : 
 
 1. 3i + 34 + G|-3|-A_5,^. 
 
 2. 5i + 6i + 3TV-6f-5A + 2.V 
 
 3. 7f + 5|-93L-|-8i-ii + 5^. 
 
 4. (7-| + 3|)x5i4-(3H-U)x3i 
 
 5. (3|-li)-l| + (2i-li)x6^-5. 
 
 6. (4f + 6} - 3f ) ~ A + (31 - 21) X 6. 
 
 7. (4i-3i+7TV + f + 4)-J^ + 8. 
 
 8. (0f + 4|)x(5i-3i)-(2i-lf). 
 
FRACTIONS. . 185 
 
 9. How much will 5 masons earn in 4 J- clays, at $ 3| per 
 day? 
 
 10. After selling i of liis load of potatoes to one person, 
 i to another, and 25 bushels to another, a farmer had sold 
 his entire load. How many bushels did he sell ? 
 
 11. One fifth of the expenses of an excursion were paid 
 by Mr, A., i of them by Mr. B., and the rest, which was 
 $ 17, by the rest of the party. What were the expenses of 
 the excursion ? 
 
 12. A man forgot how many bushels of rye he had sold, 
 but he remembered that he had received ^ 185 for the 
 rye at ^ J per bushel. How many bushels did he sell ? 
 
 13. A man loaned some money, receiving at the end of 
 the year $ 245 interest for it. If the rate of interest was 
 j-f of the sum loaned, how much did he loan ? 
 
 14. A man finds that he must save ^600 per year. He 
 takes boarders, and can accommodate 10. Suppose that he 
 pays $ 2520 for provisions, servants, and rent, and his rooms 
 are full all the year ; how much must he charge per week 
 to save $ 600 ? 
 
 15. A man of means spent -| of his life traveling in 
 America, \ of it in Europe, and the rest of it at his own 
 home. AVhat part of his life did he spend at his own 
 home ? 
 
 16. If he spent 30 years at his own home, how old was 
 he? 
 
 17. If A can do a piece of work in 6 da3^s, what part of 
 it can he do in 1 day ? If B can do the same piece of 
 work in 7 days, what part of it can he do in 1 day ? If 
 botli work together, what part of it can they do in 1 day ? 
 
 18. If both, working together, could only do -^ of it in 1 
 day, how many days would they need to complete it? 
 Since both together can do Jf in 1 day, what part of 42 
 days will they need to complete it ? 
 
186 ELEMENTS OF ARITHMETIC. 
 
 19. A man bequeathed J of his property to his brother, 
 ■1- of it to his sister, and the rest, which was $ 62500, to a 
 college. How much did he bequeath in all ? 
 
 20. A can do a piece of work in 3 days, and B can do it 
 in 5 days. In what time can both together do it ? 
 
 21. A can do a piece of work in 8 days, and B can do it 
 in 10 days. In what time can both do it together ? 
 
 22. It takes A and B 8 days to do a piece of work which 
 B can do alone in 12 days. In what time can A do it ? 
 
 23. Two men during a year earned together $1085; but 
 one of them only earned f as much as the other. How 
 much did each earn ? 
 
 24. Two brothers earn respectively $2 and $3 per day. 
 In what time can both together earn $ 13 ? 
 
 25. A man who owned f of a sash factory sold -| of his 
 share for $ 24000. How much was the factory worth at 
 that rate ? 
 
 26. Divide | of f of i| of 41 by f of | of ^j of if 
 
 27. Divide f of f of | of 6f by ^ of | of f of 2f. 
 
 28. Find the sum and the difference of 4iand 3f ; divide 
 the sum by the difference ; the difference by the sum ; and 
 then find the product of the quotients. 
 
 29. Multiply 3f by 2|; divide 3f by 2|; and find the 
 difference between the two results. 
 
 30. A man spent 1- of his money and $25 more, and then 
 had $ 225 left. How much money had he ? 
 
 31. A man after spending $2250 in repairing his house, 
 found that it cost him, including repairs, just J of the origi- 
 nal cost. What was the original cost ? 
 
 32. The estimated cost of a building was J of its actual 
 cost. If J of the estimated expense was $ 5000, Avhat was 
 the actual cost of the building ? 
 
 33. A school in one of our cities had the names of 1039 
 pupils on its rolls. If there had been one pupil more, there 
 
FRACTIONS. / A^^ 
 
 would have been 2\ times as many girls aJ^luo^lfin the 
 school. How many boys and how many girls were enrolled ? 
 
 34. A man engaged in manufacturing made a profit of 
 yl^Q on his capital, and this sum, added to his capital, made 
 i 27060. How much capital had he invested in the busi- 
 ness ? 
 
 35. The product of three numbers is 72|. Two of them 
 are 5f and 9|. What is the other number ? 
 
 36. If a man walks S^- miles in 2f hours, how far can he 
 walk in 5|- hours ? 
 
 37. A meter, the French unit of length, is very nearly 
 39f inches. What is the difference in length between 1 
 yard and 1 meter ? Wliat is the difference between 5^ 
 yards and 5|- meters ? 
 
 38. My annual income from money loaned at $ 6 on 
 f 100 is I 36.60. How much have I loaned ? 
 
 39. A company owned a cotton factory worth $ 350,000. 
 If A and B own -f- of it, what is the value of their share ? 
 
 40. Find the difference between 401 and 25|; divide the 
 difference by the sum and determine the quotient. 
 
 41. If 6 is added to both terms of the fraction |, is its 
 value increased or diminished, and how much ? 
 
 42. A merchant sold i i and ^ of his tea and had 130 
 pounds left. How much tea had he ? 
 
 43. The cost of a live turkey weighing 14 pounds was 
 $ 2. If the waste in dressing was f of the weight of the 
 turkey, for how much per pound must I sell it to gain $ 1 
 on the cost ? 
 
 44. One half of my goods was destroyed by fire ; i the 
 remainder was destroyed by water and the rest cost me 
 $ 8650. What was the entire cost of the goods ? 
 
 45. A carpenter alone can build a house in 25 days, but 
 with the assistance of his son he can build it in 15 days. 
 In how many days can the son build it ? 
 
DECIMAL FRACTIONS. 
 
 199. 1. If anything is divided into 10 equal parts, what 
 is each part called ? 
 
 2. If -f^ of a line is divided into 10 equal parts, what part 
 of the whole line is each 23art ? How much is -^ of -f^? 
 
 3. If y^-g- of a line is divided into 10 equal parts, what 
 part of the line is each part ? How much is yL of y^^- ? 
 
 4. How many hundredths are equal to J^-? 
 
 5. How many thousandths are equal to yi^- ? 
 
 6. The divisions of anything into tenths, hundredths, 
 thousandths, etc., are called decimal divisions. 
 
 200. One or more of the decimal divisions of a unit is 
 
 called a Decimal Fraction. 
 
 The word dea'mal is derived from the Latin word decern, wliich 
 means ten. 
 
 Decimal fractions are usually called decimals. 
 
 201. Since decimals have the same law of increase and 
 decrease as integers, the denonmiator of the fraction may 
 be indicated by the x>osition of the figures. 
 
 202. The figures in the first place at the right of units 
 represent tenths; in the second, hundredths ; in the third, 
 thousandths ; in the foiirth, ten-thousandths ; etc. 
 
 203. A period called the Decimal Point is i)laced before 
 the decimal. 
 
 Thus, .9 represents -^^ .48 represents -^^^ .059 represents j||o 
 .2 represents ^^ .08 represents yf^ .007 represents j-^^^ 
 
 188 
 
DECIMAL FRACTIONS. 
 
 189 
 
 .5 represents j% .04 represents jf ^ .006 represents yoW 
 
 .8 represents j\ 1.09 represents lyf ^ .029 represents yfli^ 
 
 1.1 represents ly^ l-^^ represents ly^^ 5.005 represents Sy^^^ff 
 
 Numeration Table. 
 
 
 
 
 <o 
 
 
 
 
 
 .c 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 to c 
 
 
 
 
 
 -= s 
 
 
 
 c 
 
 
 +J CO 
 
 
 
 
 
 ..: -o 3 
 
 
 dreds. 
 
 s. 
 
 :s. 
 
 "1 
 
 "5 
 .5 
 
 c/5 
 
 dredths. 
 usandths 
 •thousan 
 dred-tho 
 
 j= 
 
 c c .t: 
 
 "S 
 
 "c 
 
 c o c c 
 
 
 3 OJ c 
 
 s^ 
 
 <1> 
 
 Z3 J= <D = 
 
 
 3= (— =3 
 
 Q 
 
 H- 
 
 31 1- 1- = 
 
 s 
 
 7 9 6 
 
 
 5 
 
 8 4 6 9 
 
 4 
 
 INTEGERS. 
 
 
 
 DECIMALS. 
 
 
 The orders below millionths in their order are : 
 dred-millionths, billionths, ten-billionths, hundred 
 
 204. What order of decimals occupies 
 1st place ? 2d place ? 
 
 3d place ? 4tli place ? 
 
 5tli place ? 6th place ? 
 
 What decimal place is occupied by 
 Tenths ? Ten-thousandths ? 
 
 Thousandths ? Millionths ? 
 Hundredths ? Hundred-thousandths ? 
 
 ten-millionths, hun- 
 -billionths, etc. 
 
 7th place ? 
 2d place ? 
 4th place ? 
 
 Ten-millionths ? 
 Hundredths ? 
 Thousandths ? 
 
 EXERCISES IN NUMERATION. 
 
 205. 1. Eead 42.356. 
 
 Explanation. — The decimal part of the number expresses 3 tenths, 
 5 hundredths, 6 thousandths, or 356 thousandths. The whole expres- 
 sion is, therefore, read : 42 and 356 thousandths. 
 
 206. Hence it is evident that : 
 
 III reading a decimal, the decimal slioidd be read as an integer, 
 and the denomination of the rigJit-hand figure should he added. 
 
 In reading an integer and decimal, use the word and i»nly between 
 the integral and decimal parts of the number. 
 
190 ELEMENTS OF ARITHMETIC. 
 
 Eead the following : 
 
 2. 
 
 .31- 
 
 9. 
 
 2.63 
 
 16. 
 
 217.305 
 
 3. 
 
 .73 
 
 10. 
 
 3.10 
 
 17. 
 
 301.1031 
 
 4. 
 
 .281 
 
 11. 
 
 5.031 
 
 18. 
 
 130.2005 
 
 5. 
 
 .032 
 
 12. 
 
 7.003 
 
 19. 
 
 800.80061 
 
 6. 
 
 .0578 
 
 13. 
 
 72.070 ■ 
 
 20. 
 
 256.0032 
 
 7. 
 
 .5601 
 
 14. 
 
 26.303 
 
 21. 
 
 250.00071 
 
 8. 
 
 .5013 
 
 15. 
 
 31.120 
 
 22. 
 
 504.03062 
 
 EXERCISES IN NOTATION. 
 
 207. 1. Express decimally 54 thousandths. 
 
 Explanation. — Since 54 thousandths is equal to 5 hundredths and 
 4 thousandths, 4 is written in the thousandths' place, 5 in hundredths' 
 place, and since there are no tenths, in tenths' place. A decimal 
 point is then placed before the tenths. Hence 54 thousandths = .054. 
 
 208. Hence it is evident that : 
 
 In expressing a fraction decimally, the numerator of the frac- 
 tion is written and ciphers are prefixed, if necessary, to indicate 
 the denominator. The decimal point is then placed before tenths. 
 
 Express the following as decimals : 
 
 2. Six tenths. Four tenths. Nine tenths. 
 
 3. Thirteen hundredths. Thirty hundredths. 
 
 4. Fifteen thousandths. Two hundred one thousandths. 
 
 5. Seventeen thousandths. Six ten-thousandths. 
 
 6. Fourteen and three hundred thirty-two hundred- 
 thousandths. Five millionths. 
 
 7. Fifty-one and seven hundred forty-two millionths. 
 
 8. Fifty and seventy -five ten-thousandths. 
 
 9. Thirty-one and sixty-four hundred-thousandths. 
 
 10. Seventy-nine and nine ten-millionths. 
 
 11. Three hundred and three millionths. 
 
 12. Sixty and sixty-seven millionths. 
 
 13. 312 millionths. 201 ten-millionths. 
 
DECIMAL FRACTIONS. 191 
 
 14. 705 ten-millionths. 605 and 703 billiontlis. 
 
 15. 631 ten-billioiitlis. 25 and 25 ten-millionths. 
 
 16. ^Vo- 19- AVo- 22. 23710^^^. 
 17- jiih. 20. 210^11^^. 23. 6751^^^. 
 18. 6^^. 21. 601^1-1^. 24. 3002^VoW 
 
 REDUCTION OF DECIMALS. 
 
 209. To reduce dissimilar decimals to similar decimals. 
 
 1. How do the fractions y%, -f^-^, and yWo" compare in 
 value ? How do the decimals expressing them compare in 
 form ? What is the effect of annexing a cipher to a 
 decimal ? 
 
 2. How do the fractions -^-q, j^-q, and y-oVo compare in 
 value ? How do the decimals expressing them compare in 
 form ? What is the effect of prefixing a cipher to a decimal ? 
 
 3. How does the number of places in a decimal compare 
 with the number of ciphers in the denominator ? 
 
 210. Hence it is evident that: 
 
 1. Annexing ciphers to a decimal does not cdter its value. 
 
 2. Each decimal cipher prefixed to a decimcd diminishes 
 the value of the decimal tenfold. 
 
 3. The number of places in a decimcd is equcd to the number 
 of ciphers in the denominator of its equivcdent coinmon fraction. 
 
 WRITTEN EXERCISES. 
 
 211. 1. Eeduce .6, .46, and .095 to similar fractions. 
 
 Explanation. — Since the lowest order of the 
 
 PAA given decimals is thousandths, all the decimals must 
 
 = .DUU i^g changed to thousandths. Since annexing ciphers 
 
 46 = .460 to a decimal does not alter its value, we give to each 
 
 ^Q95 __ Q95 number three decimal places by annexing ciphers, 
 
 thus making them similar by reducing them all to 
 
 thousandths. 
 
192 ELEMENTS OF ARITHMEXrC. 
 
 2. Reduce to similar decimals : .0321, .00G23, .00025, .75. 
 
 3. Eeduce to ten-thousandths: .68, .0785, .063, .251, .27. 
 
 4. Express 8 tenths as hundredths ; as thousandths. 
 
 5. Change to similar decimals : .003, .02, .2100, .6530. 
 
 212. To reduce a decimal to a common fraction. 
 
 WRITTEN EXERCISES. 
 1. Eeduce .75 to its equivalent common fraction. 
 
 Explanation. — .75 expressed as a common frac- 
 .75 = yVo" ^^ I *^^^ ^® tW' which, being reduced to its smallest 
 terms, is |. 
 
 213. From the explanation it is apparent that : 
 
 A decimal may he reduced to a common fractioyi by omitting 
 the decimal point and siqyj^lying the proper denominator. 
 
 The result should be expressed in its lowest terms. 
 
 Reduce the following to common fractions in their smallest 
 terms : 
 
 2. 
 
 .75. 
 
 6. 
 
 .125. 
 
 10. 
 
 .0045. 
 
 14. 
 
 .00375. 
 
 3. 
 
 .50. 
 
 7. 
 
 .875. 
 
 11. 
 
 .075. 
 
 15. 
 
 .00625. 
 
 4. 
 
 .35. 
 
 8. 
 
 .625. 
 
 12. 
 
 .00875. 
 
 16. 
 
 .05375. 
 
 5. 
 
 .80. 
 
 9. 
 
 .375. 
 
 13. 
 
 .0250. 
 
 17. 
 
 .0875. 
 
 214. To reduce a common fraction to a decimal. 
 
 1. How many tenths are there in i? In i? 
 
 2. How many hundredths are there in \ ? In i ? In J? 
 
 3. How many hundredths are there in 1 ? In ^ ? In |- ? 
 
 4. How many thousandths are there in 1 ? In i? In |-? 
 
 WRITTEN EXERCISES. 
 
 215. 1. Reduce |- to an equivalent decimal. 
 
 ft\ ^ C\C\(\ Explanation. — | is \ of 5, or of 50 tenths ; and \ of 
 
 1 — ^ 50 tenths is 6 tenths, and 2 tenths remaining. 2 tenths are 
 
 .625 equal to 20 hundredths, and i of 20 hundredths is 2 hun- 
 dredths and 4 hundredths remaining. 4 hundredths are 
 equal to 40 thousandths, and i of 40 thousandths is 5 thousandths. 
 Hence | is equal to G tenths + 2 hundredths -1- 5 thousandths, or .625. 
 
DECIMAL FRACTIONS. 193 
 
 216. From the process it is evident that : 
 
 A common fraction may be reduced to a decimal by annex- 
 ing ciphers to the numerator and dividing by the denominator, 
 and pointing off as many decimal places in the quotient as 
 there are ciphers annexed. 
 
 In many cases the division is not exact. In such instances the 
 remainder may be expressed as a common fraction, or tlie sign + may 
 be employed after the decimal to show that the result is not complete ; 
 thus 1 = .166 1, or .166 + . 
 
 Reduce the following to decimals : 
 
 2. 
 
 3 
 
 To"' 
 
 6. 
 
 A- 
 
 10. 
 
 i*. 
 
 14. 
 
 f 
 
 18. 
 
 If 
 
 22. 
 
 f 
 
 3. 
 
 5 
 
 8- 
 
 7. 
 
 A- 
 
 11. 
 
 ¥k- 
 
 15. 
 
 1 5 
 4T- 
 
 19. 
 
 8|. 
 
 23. 
 
 5 
 T2- 
 
 4. 
 
 tV 
 
 8. 
 
 4V 
 
 12. 
 
 3^ 
 1 1- 
 
 16. 
 
 ^15- 
 
 20. 
 
 ^, 
 
 24. 
 
 A- 
 
 5. 
 
 A. 
 
 9. 
 
 A. 
 
 13. 
 
 A. 
 
 17. 
 
 3i. 
 
 21. 
 
 6A. 
 
 25. 
 
 "sf-o- 
 
 ADDITION OF DECIMALS. 
 
 217. lo What is the sum of ^^ and ^-^ ? Of .4 and .7 ? 
 
 2. What is the sum of j^ and -^\ ? Of .08 and .09 ? 
 
 3. What is the sum of ^L and j^^? Of .5 and .25 ? 
 
 WRITTEN EXERCISES. 
 218. 1. 'What is the sum of 2.85, 1.625, and 3.0235 ? 
 
 9 S^AA 9 Q^ Explanation. — The numbers are written so 
 
 Z.boOO or ^-oo^ ^^^^ ^j^i^g ^^^^-^^ under units, tenths under tenths, 
 1.6250 1.625 etc. The decimals may be made similar by annex- 
 3.0235 3.0235 i^g ciphers and then added, care being taken to 
 
 — place the decimal point in the sum directly under 
 
 7.4985 7.4985 the decimal point in the numbers added. Or, 
 since only units of one order are found in any 
 column, it is not necessary, in practice, to make the decimals similar. 
 
 Find the sum of the following : 
 
 2. 8.75 + .369 + 75.212. 4. 3.801 + 2.875 + 1-625. 
 
 3. 1.89 + 62.625 + 3.939. 5. 1.25 + 4.275 + 48.35. 
 
 EL. OF AR. — 13 
 
194 ELEMENTS OF ARITHMETIC. 
 
 6. .356 + 1.275 + 32.0325. 11. 3.495 + 1-87 + 3.25. 
 
 7. 15.815 + 17.21 + 31.41. 12. 38.835 + 8.9764 + 1.385„ 
 
 8. .2735 + 1.596 + 3.25. 13. 1.8035 + 3.00412 + 31.025, 
 
 9. 2.948 + 3.7642 + 85.035. 14. 62.589 + 3.875 + 1.3759. 
 10. 46.2 + 381.25 + 2.985. 15. 28.48 + 17.578 + 34.752. 
 
 16. What is the sura of 8 tenths, 12 hundredths, 75 thou- 
 sandths, 69 ten-thousandths, 21 hundredths ? 
 
 17. What is the sum of 312 ten-thousandths, 8 mil- 
 lionths, 65 thousandths, 413 ten-millionths, 9 tentlis ? 
 
 18. What is the sum of 82 hundredths, 510 thousandths, 
 6 tenths, 80 ten-thousandths ? 
 
 19. What is the sum of 303 thousandths, 4108 millionths, 
 635 ten-thousandths, 803 ten-millionths ? 
 
 20. A bicyclist traveled on his wheel 51.875 miles in 
 one day, 45.75 the next, and 35.375 the next day. How far 
 did he travel during these three days ? 
 
 21. A ticket agent sold in one month tickets to the value 
 of $ 18,756.75, in the next $ 17,983.25, in another $ 2001.875. 
 How much were his sales during those three months ? 
 
 SUBTRACTION OP DECIMALS. 
 
 219. 1. From j\ take -\. From .18 take .13. 
 
 2. From ji-^ take j^^. From .035 take .026. 
 
 3. From 4 take -f\. From 8 take .8. 
 
 WRITTEN EXERCISES. 
 
 220. 1. From 46.57 subtract 9.46325. 
 
 46 ^7000 — 4rC ^7 Explanation. — The numbers are written 
 
 ^ ~ * so that units stand under units, tenths under 
 
 y.46o2o = 9.463-.5 tenths, etc. The decimals may be made 
 
 oT^A/^r^-' o^^^TaT^^ similar and then subtracted, care beinff taken 
 oT.lObiO o<.10o<5 ^1 .1 J • 1 • ^ ' ^v- -J 
 
 to place the decimal ponit m the remainder 
 
 directly under the decimal point in the num- 
 ber subtracted. The ciphers may be supposed to be annexed when 
 we subtract, and consequently need not be written. 
 
DECIMAL FRACTIONS, 196 
 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 From 
 
 25.38 
 
 39.85 
 
 87.35 
 
 46.81 
 
 24.37 
 
 Take 
 
 14.05 
 
 29.755 
 
 59.355 
 
 21.35 
 
 12.358 
 
 7. 8. 9. 10. 11. 
 
 From 72.189 78.895 26.825 34.24 21.12 
 
 Take 35.976 53.5987 21.3452 30.16 12.31 
 
 12. What is the difference between 6.325 and 1.0345 ? 
 
 13. What is the difference between 6.45 and 2.3375 ? 
 
 14. From 65 hundredths subtract 65 thousandths. 
 
 15. From 162 thousandths subtract 261 millionths. 
 
 16. Find the value of 23.2 - .0325 + 6.95 - 3.4876. 
 
 17. Find the value of 6.594 - 2.0341 - .70325 + 8.963. 
 
 18. Find the value of 81.8 + 35.625 - 38.875 - 2.0034. , 
 
 19. Find the value of 4.6 + 2.635 - .8925 - .0359. 
 
 20. From a cistern that contained 30.5 barrels of water, 
 25.75 barrels were drawn off. How much water remains in 
 the cistern ? 
 
 21. A man bought a tub of butter for $35.75, paying 
 $ 10 in cash and the rest iii groceries. How much did he 
 pay in groceries ? 
 
 22. A man whose income was $ 10,000 spent in one year 
 $ 7375.87. How much did he save that year ? 
 
 MULTIPLICATION OP DECIMALS. 
 
 221. 1. AYhat is the product of ^\ x yV^ C)f .1 X .1 ? 
 Of .5 X .5 ? How does the number of ciphers in the de- 
 nominator of the product of these fractions compare with 
 the number of ciphers in the denominators of the factors ? 
 2. What is the product of j\ Xj^^? Of .1 x .01? 
 Of .5 x .05 ? How does the number of ciphers in the de- 
 nominator of the product of these fractions compare with 
 the number of ciphers in the denominators of the factors ? 
 
196 ELEMENTS OF ARITHMETIC. 
 
 3 How does the number of decimal places in a decimal 
 fraction compare with the number of ciphers in its denomi- 
 nator ? 
 
 4. How, then, does the number of decimal places in the 
 product of two or more decimals compare with the number 
 of places in the factors ? 
 
 222. It is apparent, therefore, that : 
 
 The number of decimal places in the product of tico or 
 more decimals is equcd to the number of decimal places in the 
 factors. 
 
 WRITTEN EXERCISES. 
 
 , 223. 1. What is the product of .275 multiplied by .17? 
 
 .275 Explanation. — Since 275 and 17 are the numerators of 
 
 ^rj tlie fractions, in mipltiplying tlie fractions, we must multiply 
 
 '' 275 by 17, which gives 4675 for a new numerator. 
 
 192o Since thousandths multiplied by hundredths produce 
 
 275 hundred-thousandths, the product of the fractions is 4675 
 
 ~C)AC7K hundred-thousandths, or .04675. 
 
 Or the number of decimal places in the product will be 
 five, the number of decimal places in both factors. Art. 222. 
 
 224. Hence, it is evident that : 
 
 To multiply decimcds tee midtipjly as if the numbers were 
 integers, and from the right of the product point off as many 
 figures for decimcds as there are decimal places in both 
 factors. 
 
 If the product does not contain a sufficient number of decimal places, 
 the deficiency must be supplied by prefixing ciphers. 
 
 Multiply : 
 
 2. .25 by .75. 5. .75 by 2.06. 8. .876 by .375. 
 
 3. .24 by 3.4. 6. .93 by .047. 9. .433 by 2.75. 
 
 4. .57 by 68o 7. .39 by .875. 10. 72.2 by .055. 
 
DECIMAL FRACTIONS. 197 
 
 11. 3.62 by .0037. 15. 2.37 by 7.563. 19. .1895 by .0436. 
 
 12. 1.98 by 2.034. 16. 2.53 by .00635. 20. 6.81 by 0.0875. 
 
 13. 25.7 by 3.887. 17. .3147 by .0052. 21. 578 by .0934. 
 
 14. 15.8 by .0855. 18. .1963 by .00638. 22. 0.765 by .067. 
 
 23. Multiply 5.372 by 100. 
 
 Explanation. — Since each removal of a figure one 
 
 5.372 place to the left increases its value tenfold, the removal of 
 
 100 the decimal point one place to the right multiplies by 10, 
 
 537.200 and its removal two places to the right multiplies by 100 ; 
 
 or, the' process may be performed in the ordinary manner. 
 
 24. Multiply 27.65 by 10. 28. Multiply 6.8735 by 100. 
 
 25. Multiply 49.675 by 10. 29. Multiply .3768 by 1000. 
 
 26. Multiply 2.795 by 100. 30. Multiply 6.835 by 100. 
 
 27. Multiply 8.935 by 100„ 31. Multiply .7634 by 1000. 
 
 32. Multiply 89.92 by 10. By 100. By 1000. By 10,000. 
 
 33. Multiply 68.3025 by 100. By 1000. By 10,000. 
 
 34. Which is larger and how much, the product of .625 
 and .375, or their sum ? 
 
 35. What will 52.75 bushels of oats cost at ^.45 per 
 bushel ? 
 
 36. A merchant sold 24.25 yards of muslin at ^.125 a 
 yard. How much did he receive for it ? 
 
 37. A man bought 214 bags of barley. What did they 
 cost him, if each bag held 2.5 bushels, and the price was 
 $ .96 a bushel ? 
 
 38. A tailor bought 4.5 yards of cassimere at $2.25 a 
 yard, and 6.25 yards of broadcloth at $ 6.50 a yard. How 
 much more did the broadcloth cost than the cassimere ? 
 
 39. How many acres are left in a farm of 563.27 acres, 
 if 3 fields of 57.135 acres each are sold from it ? 
 
 40. A steamship in crossing the Atlantic sailed at an 
 average speed of 325.75 miles per day. If another steamer 
 sailed from the same port at the same time at the rate of 
 395.35 miles per day, how far were they apart in 5.75 days? 
 
198 ELEMENTS OF ARITHMETIC. 
 
 DIVISION OF DECIMALS. 
 
 1. When a number is divided by 3, what part of it 
 is found ? By 6? By 8 ? By 15 ? 
 
 2. How much is i of 9 ? 9--3? JoflS? 15^5? 
 
 3. How much is i of 9 tenths ? i of .8 ? i of .6 ? i of .6 ? 
 
 4. How much is i of 25 hundredths ? i of .24 ? i'^of .36 ? 
 
 WRITTEN EXERCISES. 
 226. 1. Find 1 sixth of 109.92, or divide' 109.92 by 6. 
 6)109.92 Explanation, — One sixth of 10 tens is 1 ten and 4 
 
 — ^ o 09 tens remaining ; 1 sixth of 4 tens remaining and 9 units, 
 or 49 units, is 8 units and 1 unit remaining ; 1 sixth of 1 
 unit remaining and 9 tentlis, or 19 tenths, is 3 tenths and 1 tenth 
 remaining ; 1 sixth of 1 tenth remaining and 2 hundredths, or 12 hun- 
 dredtlis, is 2 hundredtlis. Therefore, one sixtli of 109.92 is 18.32, or 
 the quotient of 109.92 .-^ 6 is 18.32. 
 
 2. What is 1 fifth of 86.4055 ? 5. $ 234.54 -- 9 = ? 
 
 3. What is 1 eighth of 94076.8 ? 6. $ 907.34 -- 7 = ? 
 
 4. What is 1 eleventh of 1086.415? 7. $324,784-8 = ? 
 8. Find 1 eighth of 9675 acres. 
 
 8)9675.000 Explanation, — We divide as in simple division 
 1209 375 ^^^^^^ ^® come to the remainder, 3 acres. This we 
 change to 30 tenths and continue the division thus: 
 1 eighth of 30 tentlis is 3 tenths and 6 tenths, or 60 hundredths, 
 remainder. 1 eighth of 60 hundredths is 7 hundredths and 4 hun- 
 dredths, or 40 thousandths, remainder ; 1 eighth of 40 thousandths is 5 
 thousandths. Therefore, 1 eighth of 9675 acres is 1209.375 acres. 
 
 How much is : 
 
 9. i of $23.50? 12. i of 392.25? 15. 8.643^6? 
 
 10. i of $36.75? 13. i of 468.36? 16. 85.86 -- 9 ? 
 
 11. i of $59.64? 14. -1- of 693.34? 17. 3.845 -- 5 ? 
 
 18. A man received $3278 for 5 acres of hand. What 
 was the price per acre ? 
 
 19. An estate vahied at $35,267 was divided equally 
 among 8 children. What was each child's share ? 
 
DECIMAL FRACTIONS. 199 
 
 227o The number of decimal places in the quotient. 
 
 1. What is the product of .5 x .7 ? How many decimal 
 places does it contain ? If .35 is divided by .7, what will the 
 quotient be ? How many decimal places does it contain ? 
 
 2. What is the product of .9 x .05 ? If .045, the product, 
 is divided by .9, one of the factors, what is the other factor ? 
 How many decimal places does it contain ? 
 
 3. Since the product contains as many decimal places as 
 there are in all the factors, if the product of two factors is 
 given and one of the factors, how may the number of decimal 
 places in the other factor be found ? 
 
 4. Since the dividend may be regarded as the product of 
 two factors, one of whicli is the divisor, and the other the 
 quotient, how may the number of decimal places in the 
 quotient be found from those in the dividend and divisor ? 
 
 228. It is therefore evident that : 
 
 The quotient tcill contain as many decimal places as the 
 number of decimal ]^laces in the dividend exceeds the number 
 in the divisor. 
 
 WRITTEN EXERCISES. 
 
 229. 1. Divide .15652 by .043. 
 
 .043). 15652 (3.64 Explanation. — Since the dividend is equal 
 
 -j^29 to the product of the divisor and quotient, if 
 
 - „ ■ 15652, the numerator of the dividend, is divided 
 
 by 43, the numerator of the divisor, the result 
 
 -^58 obtained will be the numerator of the quotient. 
 
 172 Dividing, it is found to be SG-l. Hence, the 
 
 -^^2 numerator of the quotient is found precisely as 
 
 in simple division. The denominator of the-' 
 
 quotient may be found by dividing the denominator of the dividend by 
 the denominator of the divisor, that is by dividing 100000 by 1000, 
 which gives 100. Therefore, the quotient is ff^, or 3.64. Or, 
 
 The quotient will contain as many decimal places as the number of 
 decimal places in the dividend exceeds the number in the divisor. 
 Art. 228. 
 
200 
 
 ELEMENTS OF ARITHMETIC. 
 
 230. Hence it is evident that : 
 
 In dividing decimals, ive divide as if the numhers ivere 
 integer's, and from the right of the quotient point off as many 
 places for decimcds as the number of decimcU places in the 
 dividend exceeds the number in the divisor. 
 
 1. If the quotient does not contain a sufficient number of decimal 
 places, the deficiency must be supplied hj prefixing ciphers. 
 
 2. Before commencing the division, the number of decimal places 
 in the dividend slioidd he made at least equal to the number of decimal 
 places in the divisor. 
 
 3. When there is a remainder after using all the figures of the 
 dividend, annex decimal ciphers and continue the division. 
 
 4. For business purposes, it is not necessary to carry the division 
 further than to four or five decimal figures in the quotient. 
 
 Divide : 
 
 2. 78.12 by 3.6. 
 
 3. 12.32 by 3.12. 
 
 4. 1675.8 by .49. 
 
 5. 11.904 by 3.1 
 
 6. 452.92 by .052. 
 
 7. 63.168 by .94. 
 
 8. .15275 by .325. 
 
 9. .12376 by .0364. 
 
 10. .0184428 by .5123. 
 
 11. .29184 by .0057. 
 12„ 4.5872 by .0122. 
 
 13. .056448 by 15.68. 
 
 14. .0816785 by 6.235. 
 
 15. .0983253 by .0673. 
 
 16. 5.97066 by 35.73. 
 
 32. Divide 325.47 by 100. 
 
 17. 14.91 by 70, 
 
 18. 8.25 by 1.5. 
 
 19. .00864 by .24. 
 
 20. 107.5 by .043. 
 
 21. .0009 by .125, 
 
 22. .07245 by .23. 
 
 23. 800 by .0004. 
 
 24. 12.126 by .235. 
 
 25. 167.544 by 71.6. 
 
 26. 56.112 by 0.56. 
 
 27. 2.1828 by .034. 
 
 28. 4066.2 by .648. 
 
 29. .08 by 1.611. 
 
 30. 114.87 by .0035. 
 
 31. 30000 by .00003. 
 
 100)325.47 Explanation. —Since each removal of a figure one 
 
 "3.2547 
 divides by 10, and the removal two places by 100 
 
 o 9r ir- place to the right decreases its vahie tenfold, the 
 ""^ removal of the decimal point one place to the left 
 
DECIMAL FRACTIONS. 
 
 201 
 
 231o Hence it is evident that ; 
 
 To divide by 1 ivith any numher of ciphers annexed, the 
 decimal point should he removed as many places to the left as 
 there are ciphers annexed to the 1. 
 
 33. Divide 392.5 by 100. 
 
 34. Divide 26.45 by 10. 
 
 35. Divide 369.5 by 1000. 
 
 36. Divide 4.825 by 100. 
 
 37. Divide 38.62 by 1000. 
 
 38. Divide 42.85 by 10000„ 
 
 39. How many yards of clotli, at $ 5.75 per yard, can be 
 bought for ^ 153.525 ? 
 
 40. How many barrels of apples, at $ 1.25 per barrel, can 
 be bought for $ i8.43| ? 
 
 41. How many cords of wood, at $ 5. 62 J per cord, can be 
 bought for $1495.121? 
 
 ACCOUNTS AND BILLS. 
 
 232. A record of the debts and credits between two 
 parties is called an Account. 
 
 233. A statement of the quantity and price of the arti- 
 cles sold, and the value of the whole, is called a Bill. 
 
 A bill is receipted when the words Received Payment, or 
 Paid, are written at the bottom, and the seller's name is 
 signed either by himself or by some authorized person. 
 
 234. The following abbreviations are in common use : 
 
 @, 
 
 At. 
 
 Do., 
 
 The same. 
 
 Mdse., 
 
 Merchandise 
 
 %, 
 
 Account. 
 
 Doz., 
 
 Dozen. 
 
 No., 
 
 Number. 
 
 Acc't, 
 
 Account. 
 
 Dr., 
 
 Debtor. 
 
 Pay't, 
 
 Payment. 
 
 Bal., 
 
 Balance. 
 
 Fr't, 
 
 Freight. 
 
 Pd., 
 
 Paid. 
 
 Bbl., 
 
 Barrel. 
 
 Hhd., 
 
 Hogshead. 
 
 Per, 
 
 By. 
 
 B(Vt, 
 
 Bought. 
 
 In.st. , 
 
 This month. 
 
 Rec'd, 
 
 Tfeceivedo 
 
 Co., 
 
 Company. 
 
 Int., 
 
 Interest. 
 
 Yd., 
 
 Yard. 
 
 Cr., 
 
 Creditor. 
 
 Lb., 
 
 Pound. 
 
 Yr., 
 
 Year. 
 
202 ELEMENTS OF ARITHMETIC. 
 
 1. BrooUyn, J^.Y., June 25, 1892. 
 
 Bought of JONES <Sc PURDY. 
 
 4- {>^ou(f^ <S.c^Pf&&, 
 
 800 /^. 
 
 @ f.Soh 
 
 f f06 
 
 60 
 
 d Jvtatoy S'&a^, 
 
 260 '' 
 
 .87\ 
 
 227 
 
 60 
 
 8 {^-aoc&Qy f^aui'i'nQ^, 
 
 200 ^' 
 
 - J2\ 
 
 i 26 
 
 00 
 
 2 6~6^t. (^afj&&-Q.iic^cLV, 
 
 ^32 '' 
 
 .06\ 
 
 27 
 
 00 
 
 2 6~6Z. ^la^k&w^, 
 
 /■50 '' " ,06h 
 
 8 
 
 26 
 
 R&(^&i 
 
 \/d9^ 
 
 26 
 
 Make ont in proper form, find the footings of the follow- 
 ing bills and accounts, and receipt them : 
 
 2. Miss Lucy Graham bought of Luckey, Piatt & Co. 15 
 yards of calico @ 10 cents; 36 yards of sheeting @ 18^ 
 cents; 2 pairs of gloves @ $1.50; 1 sun-umbrella @ % 3.75; 
 5 yards of Hamburg edging @ 25 cents ; 6 handkerchiefs @ 
 37i cents ; and 7 pairs of hose @ 50 cents. 
 
 3. Messrs. Henry Davis & Co. bought of James Harkness 
 150 barrels of flour @ % 5.90; 105 bushels of wheat @ f 1.20 ; 
 325 bushels of corn @ $.68; 675 bushels of oats @ | .40; 
 and 50 barrels of potatoes @ % 1.25. 
 
 4. Mrs. Charles Reid in account with Roger Caldwell: 
 August 7, 1892, 9 yards cashmere @ $ .75 ; \ yard velvet @ 
 $1.50; 12 yards lawn @ .121; li. yards silesia @ $ .30; | 
 yards silk at $ 1.50. 
 
 5. Darwin Howard bought of Stone & Bacon 15 pounds 
 nails @ 4ij^; 4 pairs of hinges @ 37|J^ ; 18 dozen screws @ 
 11^; 5 dozen milk-pans @ $2.75; 7 locks @ 75^; 2 saws 
 @ $ 1.35 ; 3 hammers @ '^^(f^ ; and 2 planes @ $ 1.50. 
 
REVIEW. 203 
 
 RBVIEWo 
 WRITTEN EXERCISES. 
 235. lo How much will 2 dozen pairs of kid gloves cost 
 at 1 1.25 a pair ? 
 
 2. A man paid out the following sums : For a pair of 
 horses, $375; for a carriage, $295; for harnesses, $115, 
 How much did he pay for all ? 
 
 3. A farmer sold 12.35 cords of wood at $ 1.75 per cord. 
 How much did he receive for it ? 
 
 4. A man purchased 8 tons of coal at $5.75 per ton, 
 and 7 cords of hard wood at $5.50 per cord. How much 
 did he pay for fuel ? 
 
 5. If illuminating gas is sold at $2.50 per thousand 
 cubic feet, for how much gas will $ 17.50 pay ? 
 
 6. Find the sum of 15 dollars 9 cents 3 mills, 12 dollars 
 5 cents 4 mills, 18 dollars 14 cents 7 mills, and 16 dollars 
 10 cents. 
 
 7. If 5.5 yards of ribbon cost $2.75, what will 9.75 
 yards cost ? 
 
 8. If a clerk earns $ 520 per year and spends $ 110 for 
 clothes, $215 for board, and $ 75.50 for other expenses, how 
 long will it take him to save $ 1000 ? 
 
 9. If 54,600 letters were mailed from a post-office in 
 30 days, what was the average number mailed per day ? 
 
 10. How many pairs of curtains can be bought for $414, 
 if 12 pairs are bought for $ 138 ? 
 
 11. A drover bought 280 head of cattle for $ 12,740, and 
 sold them at a gain of $8.75 per head. How much did he 
 receive for them per head ? 
 
 12. A man paid $87.50 for materials to paint his house, 
 and he paid 3 painters $ 2.50 per day for 4.5 days to do the 
 work. How much was the entire expense of painting his 
 house ? 
 
204 ELEMENTS OF ARITHMETIC. 
 
 13. The distance between two places is 168| miles. How 
 long will it take a bicyclist to ride the distance at 12.5 miles 
 per hour ? 
 
 14. How much will one dozen each of cups and saucers 
 cost, if the cups cost $ .74J and the saucers $ .67-i- apiece ? 
 
 15 „ What must I pay for 3 tables at $ 12.75 each, 2 chairs 
 at $9.50 each, 6 chairs at $3.75 each, and one sofa at 
 $25.50? 
 
 16. A merchant sold 35.5 yards of cambric at 25 cents 
 per yard and gained $ 1.775. How much did it cost him 
 per yard ? 
 
 17 o During the Civil War a dollar bill was, at one time, 
 worth only 65 cents in gold. How many dollars in gold 
 were 520 dollars in paper money worth at that time ? 
 
 18. A farmer received for his barley $151.37, for his 
 oats $ 196.84, for his corn as much as for barley and oats, 
 for his rye $ 75.93, and for his wheat as much as for all the 
 others. Eor how much did he sell his grain ? 
 
 19. A dealer bought 25 barrels of flour, each containing 
 196 pounds, for $112.50, How much did it cost him per 
 pound ? How much will be his entire gain if he sells it at 
 3J cents per pound ? 
 
 20. A merchant paid the following sums, on the average, 
 for expenses during each month : For rent, $ 275 ; for heat- 
 ing and lighting, $115; for clerk hire, $375; and for other 
 expenses, $ S5. What will be his net gain annually, if his 
 gross profits are $ 12,560 ? 
 
 21. An ice-dealer put into his ice-house 1315 tons of ice. 
 He sold at different times the following quantities : 216.75 
 tons, 396.5 tons, 542.7 tons. If the rest melted during the 
 summer, how many tons melted, and how many tons did he 
 sell? 
 
 22. Ha freight train runs 15.75 miles per hour, in how 
 many hours will it run 189 miles ? 
 
DENOxMINATE NUMBERS. 
 
 236. A number used in connection with some specified 
 thing is called a Concrete Number. 
 
 Thus, 5 days, 6 men, 4 feet, 8 pounds are concrete numbers. 
 
 237. A number used without reference to any particular 
 thing is called an Abstract Number. 
 
 Thus, 5, 6, 4, 8, are abstract numbers. 
 
 238. A concrete number in which the unit of measure is 
 established by law or custom is called a Denominate Number. 
 
 Thus, 5 gallons, 8 rods, 5 acres, are denominate numbers. 
 
 239. A denominate number which is composed of units of 
 one denomination only is called a Simple Denominate Number. 
 
 Thus, 3 feet, 8 ounces, 5 quarts, are simple denominate numbers. 
 
 240. A denominate number which is composed of units 
 of two or more denominations that are related to each other 
 is called a Compound Denominate Number. 
 
 Thus, 5 yards 2 feet 8 inches is a compound denominate number. 
 
 MEASURES OP LENGTH. 
 
 241. In measuring length or distance, Measures of Length 
 or Linear Measures are used. 
 
 Table. 
 12 inches (in.) = 1 foot (ft.). 
 3 feet =r 1 yard (yd.). 
 
 5J yards! ^^ rod (rd.). 
 
 16^ feet / ^ ^ 
 
 320 rods = 1 mile (mi.). 
 
 206 
 
206 
 
 ELEMENTS OF ARITHMETIC. 
 
 EXERCISES. 
 
 1. How many in. are there in 2 ft. ? In 3 ft.? In 4 ft.? 
 
 2. How many ft. are there in 4 yd. ? In 5 yd. ? 
 
 3. How many yd. are there in 4 rd. ? In 6 rd. ? In 8 rd.? 
 
 4. How many ft. are there in 2 rd. ? In 5 rd. ? In 10 rd.? 
 
 5. How many rd. are there in 2 mi. ? In 3 mi. ? 
 
 6. How many ft. are there in 24 in. ? In 36 in. ? 
 
 7. How many yd. are there in 9 ft. ? In 15 ft.? In 24 ft.? 
 
 8. How many rd. are there in 33 ft. ? In 66 ft. ? 
 
 9. How many mi. are there in 640 rd. ? In 960 rd. ? 
 
 10. How many in. are there in -J- ft. ? In | ft. ? In f ft. ? 
 
 11. How many ft. are there in i yd.? In li yd. ? 
 
 12. How many in. are there in 2 ft. ? In 2 ft. 6 in. ? 
 
 13. How many ft. are there in 1 yd. ? In 1 yd. 2 ft. ? In 
 2 yd. ? In 2 yd. 1 ft. ? In 3 yd. 2 ft. ? 
 
 14. How many in. are there in 1 yd.? In 1 yd. 1 ft.? 
 
 Angle. 
 
 MEASURES OF SURFACE. 
 
 242. Anything that has only length and breadth is called 
 a Surface. 
 
 Thus, this page, the side of a block, etc., are surfaces. 
 
 243. The difference in the direction 
 of two lines that meet is called an 
 Angle. 
 
 244. A figure that has four equal 
 sides and four equal angles is called a 
 Square. 
 
 1. A square inch is a square each of whose 
 sides is an ijich long ; a square, foot is a square 
 each of whose sides is a/ooi long. 
 
 2. The angles of a square are called right 
 angles. 
 
 Square. 
 
DENOMINATE NUMBERS. 
 
 207 
 
 Table. 
 
 144 square inches (sq. in. ) = 1 
 
 ft c^^no^.r^ f,^r^i■ 1 
 
 9 square feet 
 30|- square yards \ 
 272^ square feet / 
 160 square rods 
 640 acres 
 
 ) = 1 square foot (sq. ft.), 
 = 1 square yard (sq. yd.)o 
 
 = 1 square rod (sq. rd.), 
 
 = 1 acre (A.). 
 
 = 1 square mile (sq. mi.)o 
 
 EXERCISES. 
 
 1. How many sq. in. are there in 2 sq. ft. ? In 3 sq. ft. ? 
 
 2. How many sq. ft. are there in 3 sq. yd. ? In 4 sq. yd. ? 
 
 3. How many sq. rd. are there in 2 A. ? In 3 A. ? 
 
 4. How many sq. yd. are there in 18 sq. ft. ? In 36 sq. ft.? 
 6. How many sq. ft. are there in 288 sq. in.? In 432 sq. in.? 
 
 6. How many A. are there in J sq. mi. ? In i sq. mi. ? 
 
 7. How many sq. rd. are there in ^ A. ? In i A. ? 
 
 8. How many sq. ft. are there in i sq. yd. ? In | sq. yd. ? 
 
 9. How many sq. in. are there in ^ sq. ft. ? In i sq. ft. ? 
 10. How many sq. ft. are there in 2 sq. ft. 12 sq. in. ? 
 
 In 5 sq. ft. 80 sq. in. ? 
 
 MEASURES OP VOLUME. 
 
 245. Anything that has length, breadth, and thickness 
 is called a Solid. 
 
 246. The number of solid 
 units in a body is called its 
 volume or solid contents. 
 
 247. A solid having six equal 
 square sides, called faces, is a 
 Cube. 
 
 A sohd whose faces are each a 
 square inch is a cubic inch; a solid 
 whose faces are each a square foot is 
 a cubic foot. 
 
 Cube. 
 
208 ELEMENTS OF ARITHMETIC. 
 
 Table. 
 
 1728 cubic inclies (cu. in.) = 1 cu. ft (cu. ft.), 
 27 cubic feet = 1 cubic yard (cu, yd.), 
 
 128 cubic feet = 1 cord of wood or stone (C.)o 
 
 EXERCISES. 
 
 1. How many cu. in. are there in 2 cu. ft. ? In 5 cu. ft. ? 
 
 2. How many cu. ft. are there in 4 cu. yd. ? InScu. yd. ? 
 
 3. How many cu. ft. are there in 2 C. ? In 3 C. ? 
 
 4. How many cu. yd. are there in 54 cu. ft. ? In 81 cu. ft. ? 
 
 5. How many cu. in. are there in i cu. ft. ? In i cu. ft. ? 
 
 6. How many cu, ft. are there in \ C? In | C. ? 
 
 7. How many cu. ft. are there in 2 cu. yd. 3 cu. ft. ? 
 
 8. How many cu. in. are there in 1 cu. ft. 72 cu. in. ? 
 
 LIQUID MEASURES. 
 
 248. Measures used in measuring all fluids and liquids 
 are called Liquid Measures. 
 
 Table. 
 
 4 gills (gi.) = 1 pint (pt.). 
 2 pints = 1 quart (qt.). 
 
 4 quarts = 1 gallon (gal.). 
 
 1. In determining the capacity of cisterns, reservoirs, etc., 31 i gal- 
 lons are considered a barrel (bbl.), and 2 barrels, or 63 gallons, a 
 hogshead (hhd.). 
 
 2. A gallon contains 231 cubic inches. 
 
 EXERCISES. 
 
 1 . How many gi. are there in 2 pt. ? In 3 pt. ? In 5 pt.? 
 
 2. How many pt. are there in 3 qt. ? In 5 qt. ? In 8 qt. ? 
 
 3. How many qt. are there in 5 gal. ? In 6 gal. ? 
 
 4. How many gal. are there in 24 qt. ? In 40 qt. ? 
 
 5. How many qt. are there in 16 pt. ? In 20 pt. ? 
 
 6. How many gal. are there in 16 pt. ? In 20 pt. ? 
 
DENOMINATE NUMBERS. 209 
 
 7 How many pt. are there in ^ gal. ? In i gal. ? 
 
 8. How many pt. are there in 3 qt. ? In 3 qt; 1 pt. ? 
 
 9. How many gi. are there in 1 pt. 3 gi. ? In 1 pt. 2 gi. ? 
 10. How many pt. are there in 1 gal. ? In 2 gal. 3 qt, ? 
 
 DRY MEASURES. 
 
 249. Measures used in measuring fruit, roots, vegetables, 
 and grain are termed Dry Measures. 
 
 Table. 
 
 2 pints (pt.)=l (qt.). 
 
 8 quarts = 1 peck (pk.), 
 
 4 pecks = 1 bushel (bu. ) . 
 
 1. A bushel contains 2150.4 cubic inches. 
 
 2. A quart and a pint dry measure contain a little more than the 
 same liquid measures. 
 
 EXERCISES„ 
 
 1. How many pt. are there in 5 qt. ? In 7 qt. ? 
 
 2. How many qt. are there in 5 pk. ? In 4 pk. ? 
 
 3. How many pk. are there in 3 bu. ? In 6 bu. ? 
 
 4. HoAv many bu. are there in 24 pk. ? In 36 pk. ? 
 
 5. How many pk. are there in 40 qt. ? In 32 qt. ? 
 
 6. How many qt. are there in 16 pt. ? In 20 pt. ? 
 
 7. How many pk. are there in i bu. ? In i bu. ? 
 
 8. How many pk. are there in 2 bu. 3 pk. ? 
 
 AVOIRDUPOIS WEIGHT. 
 
 250. In weighing groceries and all heavy articles as hay, 
 grain, coal, including all metals except gold and silver, Avoir- 
 dupois Weight is used. 
 
 Table, 
 16 ounces (oz.) = 1 pound (lb,). 
 
 100 pounds = 1 hundred-weight (cwt.). 
 
 20 hundred-weight | _ ^ ^qj^ /rp ^ 
 2000 pounds i 
 
 EL. OF AR. — 14 
 
210 ELEMENTS OF ARITHMETICo 
 
 EXERCISES. 
 
 1. How many oz. are there in 2 lb. ? In 3 lb. ? In 4^ lb. ? 
 
 2. How many lb. are there in 4 cwt. ? In 6 cwt. ? 
 
 3. How many lb. are there in 3 T. ? In 5 T. ? In 61 T. ? 
 
 4. How many cwt. are there in 300 lb. ? In 400 lb. ? 
 
 5. How many T. are there in 2500 lb. ? In 3000 lb. ? 
 In 5000 lb. ? 
 
 6. How many lb. are there in i T. ? In i T. ? In f T. ? 
 
 7. How many oz. are there in yL T. ? In y-J-Q T. ? 
 
 TROY WEIGHT. 
 
 251. In weighing gold, silver, and jewels Troy Weight is 
 
 used. 
 
 Table. 
 24 grains (gr.) = 1 pennyweight (pwt.). 
 20 pennyweights = 1 ounce (oz.). 
 12 ounces = 1 pound (lb.). 
 
 The Troy lb. contains 5760 gr. ; the Avoirdupois lb. , 7000 gr. 
 The Troy oz. contains 480 gr, ; the Avoirdupois oz,, 437^ gr, 
 
 EXERCISES. 
 
 1. How many gr. are there in 2 pwt, ? In 4 pwt. ? 
 
 2. How many pwt. are there in 2 oz. ? In 4 oz. ?■ In 6 oz. ? 
 
 3. How many oz. are there in 4 lb.? In 4i- lb.? In 51 lb.? 
 
 4. How many gr. are there in 1 oz. ? In ^ oz. ? In i oz. ? 
 
 5. How many lb. are there in 24 oz. ? In 36 oz. ? 
 
 6. How many oz. are there in 40 pwt. ? In 50 pwt. ? 
 
 7. HoAv many pwt. are there in 48 gr. ? In 60 gr. ? 
 
 8. How many oz. are there in 1 lb. ? In 2 lb. 4 oz. ? 
 
DENOMINATE NUMBERS. 211 
 
 APOTHECARIES' WEIGHT. 
 
 252. In weighing medicines for prescriptions Apotheca- 
 ries' Weight is used by apothecaries and physicians. 
 
 Table. 
 
 20 grains (gr.)= 1 scruple (sc, or 3). 
 3 scruples = 1 dram (dr., or 3)- 
 8 drams = 1 ounce (oz., or §). 
 
 12 ounces = 1 pound (lb., cr lb). 
 
 EXERCISES. 
 
 1. How many gr. are there in 3 sc. ? In 4 sc. ? In 5^ sc. ? 
 
 2o How many sc. are there in 4 dr. ? In 5 dr. ? In 10 dr. ? 
 
 3. HoAV many dr. are there in 3 oz. ? In 8 oz. ? In 9^ oz. ? 
 
 4o How many oz. are there in 2 lb. ? In 4i- lb. ? In 5| lb. ? 
 
 5. How many lb. are there in 48 oz. ? In 60 oz. ? In 66 oz. ? 
 
 6. How many oz. are there in 24 dr. ? In 40 dr. ? In 52 dr. ? 
 
 7. How many dr. are there in 12 sc? In 18 sc. ? In 22 sc. ? 
 
 8. How many sc. are there in 40 gr.? In50gr. ? In90gr. ? 
 
 9. How many dr. are there in 2 oz. 3 dr. ? In 3 oz. 5 dr. ? 
 
 DIVISIONS OP TIME, 
 
 253. Table. 
 
 60 seconds (sec.) = 1 minute (min.). 
 
 60 minutes = 1 hour (hr.). 
 
 24 hours = 1 day (da.). 
 
 7 days = 1 week (wk,). 
 
 365 days — 1 year (yr.). 
 
 366 days = 1 leap year (1. yr.). 
 100 years = 1 century (C). 
 
 Centennial years whose number is exactly divisible by 400, and 
 other years whose number is exactly divisible by 4, are Leap Years. 
 
212 ELEMENTS OF ARITHMETIC. 
 
 There are 12 months in a year. The names and the num- 
 ber of days in each are as follows : 
 
 1. January (Jan.) 31 days. 7. July (July) 31 da. 
 
 2. February (Feb.) 28 or 29 da. 8. August (Aug.) 31 da. 
 
 3. March (Mar.) 31 da. 9. September (Sept.) 30 da. 
 
 4. April (Apr.) .30 da. 10. October (Oct.) 31 da. 
 
 5. May (May) 31 da. 11. November (Nov.) 30 da. 
 
 6. June (June) 30 da. 12. December (Dec.) 31 da. 
 
 The 29th day of February is the day added to make the leap year. 
 
 The following rhyme will serve to fix in memory the days 
 in each month : 
 
 Thirty days hath September, 
 
 April, June, and November. 
 
 All the rest have thirty-one, 
 
 Save February, w^hich alone 
 
 Hath twenty-eight, and one day more 
 
 "We add to it one year in four. 
 
 EXERCISES. 
 
 1. How many sec. are there in 5 min. ? In 6 min.? 
 In 10 min. ? 
 
 2. How many min. are there in 2 hr. ? In 3 hr. ? 
 
 3. How many hr. are there in 2 da. ? In 4 da. ? In i da. ? 
 
 4. How many da. are there in 3 wk. ? In 6 wk. ? 
 
 5. How many wk. are there in 14 da. ? In 35 da. ? 
 
 6. How many hr. are there in 240 min. ? In 180 min. ? 
 
 7. How many min. are there in 120 sec. ? In 300 sec? 
 
 8. How many da. are there in 4 wk. 3 da. ? In 5 wk. ? 
 
 9. Which of the following named years are leap years? 
 1880? 1892? 1900? 2000? 1864? 1870? 
 
 10. How many da. are there in June ? In Dec. ? In 
 Mar. ? In Aug. ? In Jan. ? In Sept. ? In Feb. ? 
 
DENOMINATE NUMBERS. 213 
 
 COUNTING. 
 
 254. The following denominations are used in countingo 
 
 Table. 
 12 things = 1 dozen (doz.). 
 12 dozen = 1 gross (gr. ) . 
 12 gross = 1 great gross (G. gr.). 
 20 things = 1 score. 
 
 STATIONERS' TABLE. 
 
 255. The following table is used in the paper trade '. 
 
 Table. 
 24 sheets = 1 quire (qr.). 
 20 quires = 1 ream (R.). 
 
 2 reams = 1 bundle. 
 
 5 bundles = 1 bale. 
 
 EXERCISES. 
 
 1. How many things are there in 3 doz. ? In 6 doz. ? 
 
 2. How many are 3 score ? 3 score and 10 ? 
 
 3. How many doz. are there in 3 gr. ? In 1 G. gr. ? 
 
 4. How many sheets are there in 2 qr. ? In 5^ qr. ? 
 
 5. How many qr. are there in 3 K. ? In 6 R. ? Ine^R.? 
 
 6. How many qr. are there in 48 sheets ? In 96 sheets? 
 
 256. 
 
 
 MISCELLANEOUS EXERCISES. 
 
 
 1. 4 bu. 
 
 
 = ? pk. 
 
 11. 
 
 IJcu. 
 
 ft. 
 
 = ? cu. in. 
 
 2. 3 ft. 
 
 
 = ? in. 
 
 12. 
 
 32 qt. 
 
 
 = ? pk. 
 
 3. 5 1b. 
 
 
 = ? oz. (Ap.) 
 
 13. 
 
 2iA. 
 
 
 = ? sq. rd. 
 
 4. 10 qt. 
 
 
 = ? pt. 
 
 14. 
 
 3 yd. 
 
 
 = ? in. 
 
 5. 18 sq. 
 
 ft. 
 
 = ? sq. yd. 
 
 15. 
 
 48 qt. 
 
 
 = ? gal. 
 
 6. 6 oz. 
 
 
 = ? pwt. 
 
 16. 
 
 31 hr. 
 
 
 = ? min. 
 
 7. 70 da. 
 
 
 = ? wk. 
 
 17. 
 
 10 R. 
 
 
 = ?qr. 
 
 8. GO units 
 
 = ? doz. 
 
 18. 
 
 72 gr. 
 
 
 = ? pwt. 
 
 9. 3 qr. 
 
 
 = ? sheets 
 
 19. 
 
 32 oz. 
 
 (Av 
 
 .) = ?ib. 
 
 10. 54 cu. 
 
 ft. 
 
 = ? cu. yd. 
 
 20. 
 
 4 score 
 
 = ? units. 
 
214 ELEMENTS OF ARITHMETIC. 
 
 REDUCTION OP DENOMINATE NUMBERS, 
 
 257. The process of changing a denominate number from 
 one denomination to another without altering its value is 
 called Reduction. 
 
 258. The process of changing a denominate number to 
 an equivalent number of a lower denomination is called 
 Reduction to Lower Denominations, or Reduction Descending, 
 
 259. The process of changing a denominate number to 
 an equivalent number of a higher denomination is called 
 Reduction to Higher Denominations, or Reduction Ascending. 
 
 REDUCTION TO LOWER DENOMINATIONS. 
 
 260. 1. How many inches are there in 7 yd. 1 ft. 4 in. ? 
 
 7 
 
 3 Explanation. — Since there are 3 ft. in 1 yd., in 7 yd. 
 
 21 there are 7 times 3 ft., or 21 ft., and 21 ft. + 1 f t. = 22 ft. 
 
 1 Since there are 12 in. in 1 ft., in 22 ft. there are 22 times 12 in., 
 
 or 264 in., and 264 in. + 4 in. = 268 in. Therefore 7 yd. 1 ft. 4 
 
 in. = 268 in. 
 
 The business method of reducing denominate numbers to 
 264 lower denominations is more convenient than that given in the 
 
 4 explanation. It is illustrated by the process in the margin. 
 
 268 
 
 261. Hence the business method is as follows : 
 
 To reduce denominate numbers to loiver denominations, 
 multiply the number of the highest denomination given by the 
 7iumber indicating hoio many units of the 7iext lower denomi- 
 nation are equal to one of the higher, and to the product add 
 the number given of this lower denomination. 
 
 Proceed in like manner with this cmd each successive result 
 thus obtained, until the number is reduced to the required 
 denomination. 
 
 22 
 12 
 
DENOMINATE NUiMBERS, 215 
 
 Reduce to lower denominations : 
 
 2. 4 yd. 2 ft. 8 in. 23. 4 rd. 3 yd. 2 ft. 6 in. 
 
 3. 12 yd. 1 ft. 7 in. 24. 5 rd. 2 yd. 1 ft. 8 in. 
 
 4. 15 yd. 2 ft. 10 in. 25. 8 rd. 1 yd. 2 ft. 10 in. 
 
 5. 27 yd. 2 ft. 11 in. 26. 3 sq. yd. 4 sq.ft. 18 sq. in. 
 
 6. 6 gal. 3 qt. 1 pt. 27. 5 sq. yd. 8 sq.ft. 100 sq. in. 
 
 7. 13 gal. 2 qt. 1 pt. 28. 2 A. 110 sq. rd. 5 sq. yd 
 
 8. 15 gal. 1 qt. 1 pt. 2 gi. 29. 8 A. 25 sq. rd. 4 sq. yd. 
 
 9. 18 gal. 3 qt. 1 pt. 3 gi. 30. 2cu.yd.l0cu.ft.85cu.in. 
 
 10. 4 hr. 15 min. 15 sec. 31. 4cu.yd.6cu. ft. 714cu. in. 
 
 11. 10 lir. 12 min. 18 sec. 32. 3 lb. 2 oz. 7 dr. 1 sc. 
 
 12. 18 hr. 20 min. 35 sec. 33. 4 lb. 8 oz. 5 dr. 2 sc. 
 
 13. 4 cwt. 20 lb. 8 oz. 34. 3 R. 10 qr. 18 sheets. 
 
 14. 8 cwt. 75 lb. 12 oz. 35. 4 R. 8 qr. 10 sheets. 
 
 15. 2 T. 5 cwt. 80 lb. 12 oz. 36. 3 lb. 6 oz. 10 pwt. 
 
 16. 4 T. 7 cwt. 35 lb. 9 oz. 37. 5 lb. 10 oz. 3 sc. 
 
 17. 4 lb. 6 oz. 18 pwt. 10 gr. 38. 6 da. 10 hr. 25 min. 
 
 18. 8 lb. 4 oz. 10 pwt. 8 gr. 39. 8 bu. 3 pk. 7 qt. 1 pt. 
 
 19. 5 lb. 5 oz. 15 pwt. 9 gr. 40. 8 rd. 4 yd. 2 ft. 6 in. 
 
 20. 3 bu. 3 pk. 5 qt. 1 pt. 41. 30 bu. 3 pk. 6 qt. 
 
 21. 5 bu. 2 pk. 7 qt. 1 pt. 42. 5 A. 140 sq. rd. 3 sq. yd. 
 
 22. 6 bu. 1 pk. 4 qt. 1 pt. 43. 7 mi. 100 rd. 13 ft. 6 in. 
 
 44. Reduce .75 yd. to feet and inches. 
 
 „75 
 3 Explanation. — Since in 1 yd. there are 
 
 2^25 3 ft., in .75 yd. there are .75 of 3 ft. or 2.25 ft. 
 
 ^2 Since in 1 ft. there are 12 in., in .25 ft. there 
 
 are .25 of 12 in., or 3 in. 
 
 Therefore, .75 yd. = 2 ft. 3 in. 
 
 3.00 
 
 J5 yd. = 2 ft. 3 in 
 
 45. I cwt. 48. f yd. 51, .4 cu. yd. 54. .73 wk. 
 
 46. I rd. 49. .6 bu. 52. f mi. 55. .3125 gSi 
 
 47. -I wk. 50. .45 T. 53. f A. 56. .475 yd. 
 
216 ELEMENTS OF ARITHMETIC. 
 
 REDUCTION TO HIGHER DENOMINATIONS. 
 
 262. 1. Keduce 235 pints to bushels, pecks, etc. 
 
 235 pt. 
 
 Explanation. — Since there 
 -\-\r7 f -j ^f are 2 pt. in 1 qt., in 235 pt. there 
 Ll -t * are as many quarts as 2 pt. are 
 
 ^^ P^- ^ ^t- contained times in 235 qt., or 117 
 
 3 bu. 2 pk. qt. 1 pt. 
 
 235 pt. = 3 bu. 2 pk. 5 qt. 1 pt. ^, ^^^^^^ ^^^'^ ^^^ ^ f ^^ ,\^\^ 
 ^ jr X X there are as many pecks m 1 1 / qt, 
 
 as 8 qt. are contained times in 117 qt., or 14 pk. 5 qt. 
 
 Since there are 4 pk. in 1 bu., in 14 pk. there are as many bushels 
 as 4 pk. are contained times in 14 pk., or 3 bu. 2 pk. 
 
 Therefore, 235 pt. are equal to 3 bu. 2 pk. 5 qt. 1 pt. 
 
 263. Hence it is evident that : 
 
 To reduce a deyiominate number to units of higher denomina- 
 tions, the given number is divided by the number {ndicating 
 how many units of the given denomination malie one of the 
 next higher denomination. 
 
 We proceed in like manner imth this, and each successive 
 quotient, till the whole is reduced to the required denomination. 
 
 The last quotient, with the remainders {when there are any) 
 annexed, will be the required answer. 
 
 2. Eeduce 3265 inches to rods, yards, etc. 
 
 Suggestion. — In dividing by 5^ it is 
 usually more convenient to reduce the 
 divisor to halves and the dividend also 
 to halves before dividing. 
 
 It should be remembered then, that 
 the remainder, if there is any, is half- 
 yards, which must be changed to yards. 
 „. ^ —.J 
 
 Reduce : 
 
 3. 4213 gi. to gal., etc. • 6. 3539 in. to rd., etc. 
 
 4. 3185 pt. to gal., etc. 7. 2456 in. to rd., etc. 
 
 5. 5724 gi. to gal., etc. 8. 1605 in. to rd., etc 
 
 12 
 
 3265 in. 
 
 3 
 
 272 ft. 1 in. 
 90 yd. 2 ft. 
 2 
 
 11 
 
 180 [or 2 yd. 
 16 rd. 4 half-yd.. 
 
DENOMINATE NUMBERS. 217 
 
 9. 3564 pt. to bu., etc. 18. 9894 min. to da., etc. 
 
 10. 4683 pt. to bu., etc. 19. 3943 sq. in. to sq. yd., etc. 
 
 11. 3947 pt. to bu., etc. 20. 5937 sq. in. to sq. yd., etc. 
 
 12. 5164 oz. to cwt., etc. 21. 1894 sheets to K., etc. 
 
 13. 8127 oz. to cwt., etc. 22. 2315 sheets to K., etc. 
 
 14. 5983 lb. to tons, etc. 23. 7864 cu. in. to cu. ft., etc. 
 
 15. 3795 lb. to tons, etc. 24. 6754 gr. to lb. Troy, etc. 
 
 16. 4285 hr. to wk., etc. 25. 8386 gr. to lb. Troy, etc. 
 
 17. 8168 sec. to hr., etc. 26. 9138 gr. to lb. Apoth., etc. 
 
 27. Keduce 2 qt. 1 pt. to the fraction of a gallon. 
 
 Explanation. —2 qt. 1 pt. = 5 pt. ; 1 gal. = 8 pt. Therefore 2 qt. 
 1 pt., or 5 pt., are f of 1 gal., or 8 pt. 
 
 28. Keduce 1 ft. 3 in. to the fraction of a yard. 
 29- Reduce 3 qt. 1 pt. to the fraction of a peck. 
 
 30. Reduce 4 pk. 3 qt. to the fraction of a bushel. 
 
 31. Reduce 2 qt. 1 pt. 3 gi. to the fraction of a gallon. 
 
 32. Reduce 2 pk. 4 qt. 1 pt. to the fraction of a bushel. 
 
 33. Reduce 4 yd. 2 ft. 6 in. to the fraction of a rod. 
 
 ADDITION OP DENOMINATE NUMBERS. 
 
 264. 1. Find the sum of 3 gal. 3 qt. 1 pt. ; 5 gal. 2 qt. 
 
 1 pt. ; 6 gal. 2 qt. 1 pt. ; 8 gal. 3 qt. 
 
 Explanation. — The numbers are written so that 
 units of the same denomination stand in the same col- 
 umn, and we begin at the lowest denomination to add. 
 
 The sum of the pints is 3 pt. , which are equal to 1 qt. 
 1 pt. ; the 1 pt. is written under pints and the 1 qt. re- 
 served to add to the quarts. 
 nA o ^ The sum of the quarts is 11 qt., which are equal to 2 
 
 gal. 3 qt. ; the 3 qt. are written under the quarts and the 
 
 2 gal. reserved to add to the gallons. 
 
 The sum of the gallons is 2-1 gal., which are written under the 
 gallons. 
 
 gal. 
 
 qt. 
 
 pt. 
 
 3 
 
 3 
 
 1 
 
 5 
 
 2 
 
 1 
 
 6 
 
 2 
 
 1 
 
 8 
 
 3 
 
 
218 ELEMENTS OF ARITHMETIC. 
 
 Find the sum of : 
 
 2. 3 gal. 3 qt. 1 pt. ; 18 gal. 2 qt. 1 pt. ; 12 gal. 3 qt. 
 
 3. 8 bu. 3 pk. 5 qt. ; 16 bu. 2 pk. 7 qt. ; 12 bu. 3 pk. 4 qt. 
 
 4. 7 da. 5 hr. 20 miii. ; 8 da. 14 hr. 10 min. ; 6 da. 18 hr. 
 
 5. 24 lb. 6 oz. (Av.) ; 35 lb. 14 oz. ; 26 lb. 10 oz. ; 83 lb. 
 
 6. 13 lb. 4 oz. 10 pwt. 8 gr. ; 11 lb. 5 oz. 11 pwt. 16 gr. ; 
 
 10 lb. 10 oz. 15 pwt. 21 gr. 
 
 7. 10 yd. 2 ft. 10 in. ; 15 yd. 1 ft. 9 in. ; 8 yd. 2 ft. 7 in. ; 
 18 yd. 1 ft. 11 in. ; 16 yd. 2 ft. 8 in. 
 
 8. 15 gal. 3 qt. 1 pt. 3 gi. ; 18 gal. 2 qt. 1 pt. 2 gi. ; 28 
 gal. 2 qt. ; 21 gal. 1 pt. 3 gi. 
 
 9. 12 A. 35 sq. rd. ; 14 A. 110 sq. rd. ; 15 A. 132 sq. rd. 
 
 11 A. 96 sq. rd. ; 25 A. 100 sq. rd. 
 
 10. 5 T. 6 cwt. 14 lb. 10 oz.; 7 T. 15 cwt. 36 lb. 15 oz. 
 17 T. 8 cwt. 84 lb. 12 oz. ; 10 T. 9 cwt. 94 lb. 11 oz. 
 
 11. 5 yd. 2 ft. 8 in. ; 4 yd. 1 ft. 11 in. ; 4 yd. 2 ft. 9 in. 
 3 yd. 1 f t. ; 4 yd, 8 in. ; | of a yard and | of a foot. 
 
 SUBTRACTION OP DENOMINATE NUMBERS. 
 
 265. 1. From 12 lb. 6 oz. 7 pwt. take 3 lb. 8 oz. 10 pwt. 
 
 Explanation. — The numbers are written so that 
 
 units of the same denomination stand in the same 
 
 cokmm, and we begin at the lowest denomination to 
 
 8 9 17 subtract. Since 10 pwt. cannot be subtracted from 
 
 7 pwt., a unit of the next higher order is united with 
 
 the 7 pwt., making 1 oz. 7 pwt., or 27 pwt. Then 27 pwt. — 10 pwt. = 
 
 17 pwt., which we write under pwt. in the remainder. 
 
 Since 1 oz. was united with the 7 pwt., only 5 oz. are left in the 
 minuend. Inasmuch as 8 oz. cannot be subtracted from 5 oz., a unit 
 of the next higher order is united with it, making 1 lb. 5 oz., or 17 oz. 
 Then 17 oz. — 8 oz. == 9oz., which we write under oz, in the remainder. 
 
 Since 1 lb, was changed to ounces, there are but 11 lb. left in the 
 minuend. Then 11 lb. - 3 lb. = 8 lb. 
 
 Therefore, the remainder is 8 lb. 9 oz. 17 pwl. 
 
 2. From 8 bu. 4 pk. 3 qt. take 3 bu. 2 pk. 7 qt. 
 
 lb. 
 
 oz. 
 
 pwt. 
 
 12 
 
 6 
 
 7 
 
 3 
 
 8 
 
 10 
 
DENOMINATE NUMBERS. 219 
 
 3. From 12 gal. 2 qt. 1 pt. 2 gi. take 5 gal. 3 qt. 1 pt. 3 gi. 
 
 4. From 10 da. 5 hr. 20 min. take 7 cla. 7 hr. 10 min. 
 
 5. From 31 lb. 4 oz. 15 pwt. take 8 lb. 6 oz. 12 pwt. 
 
 6. From 15 yd. 2 ft. 7 in. take 4 yd. 2 ft. 10 in. 
 
 7. From 12 rd. 4 yd. 1 ft. take 8 rd. 4 yd. 2 ft. 
 
 8. From 13 lb. 8 oz. 5 dr. take 7 lb. 10 oz. 7 dr. 
 
 9. From 18 A. 18 sq. rd. take 7 A. 110 sq. rd. 
 
 10. From 40 sq. yd. 8 sq. ft. 51 sq. in. take 15 sq. rd. 
 6 sq. ft. 100 sq. in. 
 
 11. From 25 T. 8 cwt. 75 lb. 10 oz. take 10 T. 11 cwt. 
 35 lb. 15 oz. 
 
 12. How many years, months, and days intervened be- 
 tween Dec. 24, 1892, and May 26, 1842 ? 
 
 Explanation. — Since the later date expresses 
 the greater period of time, it is written as the 
 minuend, and the earlier date is, therefore, the 
 „Q subtrahend. In writing the dates the number of 
 the month is given instead of its name. The sub- 
 traction is then performed as in denominate numbers, considering a 
 month as 30 days, and a year as 12 months. 
 
 yr. 
 
 mo. 
 
 da. 
 
 1892 
 
 12 
 
 24 
 
 1842 
 
 5 
 
 26 
 
 ? 
 
 13. How long was it from Dec. 10, 1835, to May 15, 188 
 
 14. How long was it from June 13, 1864, to Jan. 1, 1890 ? 
 
 15. How long was it from Apr. 8, 1839, to July 7, 1891? 
 
 16. How long was it from Sept. 23, 1881, to Aug. 10, 1892 ? 
 
 MULTIPLICATION OP DENOMINATE NUMBERS. 
 
 266. 1. Multiply 3 bu. 3 pk. 7 qt. by 7. 
 
 Explanation, — 7 times 7 qt, are 40 qt., or 6 pk. 
 1 qt. The 1 qt. is written under qt. in the product, 
 and the pk. reserved to add to the pecks, 7 times 
 ~[ 8 pk. are 21 pk. ; 21 pk. + 6 pk. reserved are 27 pk,, 
 or 6 bu, 3 pk. The 3 pk, are written under pk. in 
 the product, and the 6 bu. reserved to add to tlie bushels. 
 
 Seven times 3 bu, are 21 bu, ; 21 bu. + 6 bu, reserved are 27 bu. 
 Therefore, the product is 27 bu. 3 pk. 1 qt. 
 
 bu. 
 
 pk. 
 
 qt. 
 
 3 
 
 3 
 
 7 
 7 
 
220 ELEMENTS OF ARITHMETICo 
 
 2. Multiply 4 bu. 3 pk. 5 qt. by 7 
 
 3. Multiply 5 gal. 1 qt. 3 pt. 2 gi. by 6, 
 
 4. Multiply 5 lb. 6 oz. 10 pwt. 8 gr. by 8. 
 
 6. Multiply 7 lb. 8 oz. 5 dr. 2 sc. 10 gr. by 7, 
 
 6. Multiply 3 hr. 20 niiu. 35 sec, by 5, 
 
 7. Multiply 2 T. 5 cwt. 48 lb. 15 oz. by 8. 
 
 8. Multiply 2 rd. 3 yd. 2 ft. 10 in. by 9. 
 
 9. Multiply 12 cu. yd. 15 cu. ft. 1115 cu. in. by 
 
 10. Multiply 8 sq. yd. 2 sq. ft. 45 sq. in. by 5. 
 
 11. Multiply 5 da. 8 hr. 15 min. 25 sec. by 4. 
 
 12. Multiply 12 rd. 4 yd. 2 ft. 8 in, by 8. 
 
 13. Multiply 5 R. 8 qr. 16 sheets by 9. 
 
 14. Multiply 6 bar. 10 gal. 3 qt. 1 pt. by 7, 
 
 DIVISION OP DENOMINATE NUMBERS. 
 
 267. 1. Divide 14 gal. 3 qt. 1 pt. by 4. 
 
 gal. qt. pt. Explanation. — One fourth of 14 gal. is 3 gal. 
 
 4)14 3 1 and 2 gal. remainder. The 3 gal. are written ia 
 3 2 1^ the quotient, and the 2 gal. remainder are united 
 ^ with the 3 qt., making 11 qt. 
 
 One fourth of 11 qt. is 2 qt. and 3 qt. remainder. The 2 qt. are 
 written in the quotient, and the 3 qt. remainder united with the 1 pt. , 
 making 7 pt. 
 
 One fourth of 7 pt. is 1| pt. 
 
 Therefore, the quotient is 3 gal. 2 qt. If pt. 
 
 2. Divide 21 gal. 3 qt. 1 pt. 3 gi. hy 6. 
 
 3. Divide 15 bu. 3 pk. 5 qt. 1 pt. by 4. 
 
 4. Divide 13 yd. 2 ft. 6 in. by 7. 
 
 5. Divide 23 cwt. 68 lb. 10 oz. by 5. 
 
 6. Divide 42 lb. 8 oz. 15 pwt. by 9. 
 
 7. Divide 22 lb. 7 oz. 5 dr. 2 sc. by 8. 
 
 8. Divide 19 hr. 10 min. 36 sec. by 6. 
 
 9. Divide 21 sq.^ yd. 7 sq. ft. 45 sq. in. by 4, 
 10. Divide 15 rd.'4 yd. 2 ft. 8 in. by 5. 
 
DENOMINATE NUMBERS. 
 
 221 
 
 PRACTICAL MEASUREMENTS. 
 
 268. To compute the area of rectangular surfaces. 
 
 1. What is a square inch? Draw a figure 6 in. long and 
 1 in. wide with all its angles equal. How many sq. in. 
 does it contain ? 
 
 2. If the figure were 2 in. wide, how many sq. in. would 
 it contain? How many if it were 3 in. wide ? 4 in. wide ? 
 5 in. wide ? 6 in. wide ? ^ 
 
 3. What is a square foot? How many sq. ft. are there 
 in a strip of ground 12 ft. long and 1 ft. wide, with all its 
 angles equal ? How many, if it is 2 ft. wide ? 3 ft. wide ? 
 4 ft. wide ? 5 ft. wide ? 6 ft. wide ? 
 
 4. How many sq. ydo are there in a walk 15 yd. long 
 and 5 yd. wide, having its angles all equal ? 
 
 269. A figure that has four straight sides 
 and four equal angles is called a Rectangle. 
 
 The angles of a rectangle are all right angles. Uectaugie. 
 
 270. The number of square units that a surface contains 
 is called its Area. 
 
 Thus the area of a rectangle 4 in. long 
 and 2 in. wide is 8 sq. in., for it may be 
 divided into 2 rows, each containing 4 sq. in. 
 
 For additional instruction in measure- 
 ments, consult the author's Standard Arithmetic. 
 
 WRITTEN EXERCISES. 
 271. 1. What is the area of a rectangular walk 22 ft. 
 long and 5 ft. 6 in. broad ? 
 
 Explanation. — Since the walk 
 is 22 ft. long and 5i ft. wide, its 
 area in feet is 121 sq. ft. 
 
 The same result can be obtained 
 by expressing-the length and breadth 
 in inches, and reducing the result 
 obtained to square feet. 
 
 22x51 = 121 
 or, 
 264 x(j6 = 17,424 
 17,424 sq. in. = 121 sq. ft. 
 
222 ELEMENTS OF ARITHMETIC. 
 
 272. Hence it is evident that : 
 
 The area of any rectangular surface is obtained by finding 
 
 the product of the numbers expressing its length a^id breadth. 
 
 The length and breadth must be expressed in units of the same 
 denomination. 
 
 2. What is the area of a rectangular room 17 ft. long 
 and 23 ft. wide ? 
 
 3. How many sq. ft. of surface are there in the side wall 
 of a rof)m 16 ft. long and 12 ft. high ? How many sq. yd. ? 
 
 4. The floor of a rectangular room 18 ft. wide and 24 
 ft. long contains how many sq. yd. ? 
 
 5. A farm in the form of a rectangle has a length of 
 160 rd. and a breadth of 120 rd. How many sq. rd. of land 
 does it contain ? How many acres ? 
 
 6. How many acres are there in a rectangular field 60 
 rd. wide and 80 rd. long ? 
 
 7. A farmer paid $2.50 per acre for plowing a rec- 
 tangular field that was 2^ rd. wide and 45 rd. long. What 
 was the entire expense of plowing the field ? 
 
 8. A plasterer received 23<^ per sq. yd. for plastering 
 the walls and ceiling of a room 18 ft. long, 15 ft. wide, and 
 12 ft. high, no deduction being made for the openings. 
 How much did lie receive ? 
 
 9. How much wall it cost to carpet the floor of the 
 room with carpeting 1 yd. wide at 85^ per yd. ? 
 
 10. The farm of Mr. Hudson is 160 rd. long and 150 rd. 
 wide, and he purchases an adjoining farm 160 rd. long and 
 90 rd. wide. How many acres has he after the purchase ? 
 
 11. A room 18 ft. wide and 24 ft. long was carpeted with 
 carpeting 1 yd. wide at $ 1.25 per yd. How much did it cost ? 
 
 12. The length of a field was 80 rd. and its breadth was 
 60 rd. If half of it was planted with corn and half the 
 remainder with potatoes, how many acres were planted to 
 each ? 
 
DENOMINATE NUMBERS. / ' 223 
 
 273. To compute the volume of rectangular sc 
 
 1. Wliat is a cubic inch ? A cubic foot ? A cubic yard ? 
 
 2. How many cubic feet of stone are there in a rec- 
 tangular block of stone which is 6 ft. long, 1 ft. wide, and 1 
 ft thick? 
 
 3. How many cubic feet would there be if the block 
 were 2 ft. Avide ? 2 ft. thick ? 2 ft. wide and 2 ft. thick ? 
 How many if it were 7 ft. long, 2 ft. wide, and 2 ft. thick ? 
 
 4. How many cubic feet are there in a similar block 
 which is 6 ft. long, 3 ft. wide, and 1 ft. thick ? In a block 
 6 ft. long, 3 ft. wide, and 2 ft. thick ? In a block 6 ft. 
 long, 3 ft. wide, and 3 f t„ thick ? In a block 6 ft. long, 4 ft. 
 wide, and 3 ft. thick ? In a block 6 ft. long, 3 ft. wide, and 
 4 ft. thick? In a block 3 ft. Jong, 4 ft. wide, and 6 ft. 
 thick ? 
 
 274. The number of solid units 
 that a body contains is its Volume 
 or Solid contents 
 
 The solid units are cubic inches, 
 cubic feet, cubic yards, etc. 
 
 If a sohd is 4 ft. long, 3 ft. wide, and 
 3 ft. thick, its volume is 36 cu. ft. , for 
 it may be divided into 3 blocks, each containing 12 cu. ft., as shown 
 by the illustration. 
 
 ^^e 
 
 ^fP^^ 
 
 1 ^ 
 
 Cubic 
 
 ft. 
 
 > 
 
 I 1^ 
 
 Cubic 
 
 ft. 
 
 ilill^ 
 
 Cubic 
 
 ft. 
 
 ^ 
 
 (1 
 
 ^. 
 
 275. A pile of wood 
 or stone 8 ft. long, 4 ft. 
 wide, and 4 ft. high is 
 called a Cord. 
 
 A pile that is 1 ft. 
 long, 4 ft. wide, and 4 
 feet high is called a 
 Cord Foot. 
 
 Stone walls are sometimes measured by the Perch, which 
 contains 24.75 cu. ft. 
 
 
224 ELEMENTS OF ARITHMETIC, 
 
 WRITTEN EXERCISES. 
 
 276. 1. How many cubic feet of stone are there in a 
 rectangular block 8 ft. long, 5 ft. 3 in. wide, and 3 ft. 6 in. 
 thick? 
 
 o p-i oi -I A J Explanation. — ^ Since a block 8 ft. long, 
 
 'f 2 I ft. wide, and 1 ft. thick contains 8 cu. ft., a 
 
 block 5} ft. wide and 1 ft. thick contains 5^ times 8 cu. ft., or 42 
 cu. ft. 
 
 Since a block 8 ft. long, 5} ft. wide, and 1 ft. thick, contains 42 cu. 
 ft., a similar block, that is 3^ ft. thick, contains 3 J times 42 cu. ft., or 
 147 cu. ft. 
 
 277. Hence it is evident that : 
 
 The volume or solid contents of a rectangular solid are 
 obtained hy finding the product of the numbers expressing its 
 length, breadth, and thickness. 
 
 The length, breadth, and thickness must be expressed in units of 
 the same denomination. 
 
 Find the volume of solids with dimensions as follows i 
 
 2. Length 10 ft., breadth 7 ft., thickness 6 ft. 
 
 3. Length 9 in., breadth 11 in., thickness 8 in. 
 
 4. Length 5 yd., breadth 8 yd., thickness 10 yd. 
 
 5. Length 14 ft., breadth 10 ft., thickness 15 ft. 
 
 6. Length 8 ft. 6 in., breadth 8 ft., thickness 10 ft. 
 
 7. Length 12 ft. 4 in., breadth 9 ft., thickness 8 ft. 
 
 8. Length 10 ft. 6 in., breadth 10 ft., thickness, 12 ft 
 
 9. Length 12 ft. 4 in., breadth 10 ft., thickness 8 ft. 
 
 10. Length 8 ft. 4 in., breadth 10 ft 6 in., thickness 8 ft. 
 
 How many cords of wood are there in the following piles ? 
 
 11. Length 24 ft., breadth 4 ft., height 4 ft 
 
 12. Length 32 ft., breadth 4 ft., height 8 ft. 
 
 13. Length 48 ft., breadth 16 ft, height 12 ft. 
 
 14. How much will the wood in the last pile cost at 
 ^ 5.50 per cord ? 
 
DENOMINATE NUMBERS. 225 
 
 15. How much will it cost to excavate a cellar which is 
 40 ft. long, 22 ft. wide, and 8 ft deep, at 30j^ per cubic yard ? 
 
 16. How much will be the expense of laying a stone wall 
 40 ft. long, 1 ft. 6 in. thick, and 6 ft. high, at $ 2.50 per 
 perch ? 
 
 278. To measure lumber. 
 
 In measuring lumber, boards 1 inch thick or less are 
 
 estimated by the number of square feet of surface. 
 
 Thus a board 1 foot wide and 10 feet long contains 10 feet board 
 measure, if it is 1 inch or less in thickness. 
 
 Tlie length and width must both be expressed in feet. 
 
 279. When lumber is more than 1 inch in thickness, the 
 number of feet board measure is obtained as follows : 
 
 The length in feet is multiplied by the breadth in feet, and 
 
 this X)roduct by the number of inches in thickness. 
 
 Thus, the number of feet board measure in a timber 12 ft. long, 15 
 in. wide, and 2^ in. thick is obtained as follows: 
 
 12 X 1^ X 2^ =: 37], tlie number of feet board measure. 
 
 How many feet are there in the following boards : 
 
 1. 16 ft. long, 18 im wide. 4. 20 ft. long, 14 in. wide. 
 
 2. 12 ft. long, 15 ill. wide. 5. 15 ft. long, 10 in. wide. 
 
 3. 18 ft. long, 9 in. wide. 6. 20 ft. long, 16 in. wide. 
 
 7. What will be the cost of 10 boards, each 12 ft. long 
 and 15 in. wide, at $ 20 per thousand feet ? 
 
 8. How many feet board measure are there in 20 joists 
 18 ft. long, 16 in. wide, and 3 in. thick ? 
 
 9. What will those joists cost at $ 18 per thousand feet ? 
 
 10. What will be the cost of 20 planks 16 ft. long, 14 in, 
 wide, and 2 in. thick, at f 22 per thousand feet ? 
 
 11. What will be the cost of 20 planks 12 ft. long, 15 in. 
 wide, and 1^ in. thick, and 40 scantlings 6 in. wide, 12 ft. 
 long and 4 in. thick, at $ 18 per thousand feet ? 
 
 EL. DF AR. — 15 
 
INTEREST. 
 
 280c When a person borrows money, he usually has to 
 pay a certain number of hundredths, or a certain per cent^ of 
 the sum for the use of it for each year. 
 
 281. The character % is often used instead of the words 
 hundredths or per cent. 
 
 Thus, 5 hundredths, or 5 per cent, is expressed 5%. 
 
 ORAL EXERCISES. 
 
 282. 1. When a sum equal to 6% of the money loaned 
 is paid for the use of it for each year, how much is paid for 
 the use of f 100 for 1 yr., or what is the interest for 1 yr. ? 
 
 2. When the rate of interest is 5% per year, how much 
 wall be the interest of $ 100 for 1 yr. ? For 2 yr. ? For 3 yr. ? 
 
 3. When the rate of interest is 8% per year, what will 
 be the interest of $ 100 for 1 yr. ? ^ 200 ? $ 500 ? 
 
 4. When the rate of interest is 6% per year, what will 
 be the interest of $ 400 for 1 yr. ? For 2 yr. ? For 2^ yr. ? 
 For i yr. ? 
 
 6. When the rate of interest is 4% per year, how much 
 will $ 200 amount to in 1 yr. ? In 2 yr. ? In 2-1- yr. ? 
 
 6. When $400 is loaned for 3 yr. at 5% per year, how 
 much is the interest ? What is the amount due ? 
 
 What is the interest of : 
 
 7. $ 600 for 1 yr. at 5% ? d. $ 800 for 2 yr. at 4% ? 
 
 8. $500for 1 yr. at C% ? ]0. $400 for 3 yr. at 5% ? 
 
 226 
 
INTEREST. 227 
 
 11. $600forliyr.at3%? 15. $800 for 11 mo. at 6% ? 
 
 12. $200for2iyr. at5%? 16. $600 for 1 mo. 10 da. at 6% ? 
 
 13. $800foriyr.at6%? 17. $600 for 2 mo. 10 cla. at 6% ? 
 
 14. $ 600 fori mo. at 6%? 18. $800 for Syr. at 5% ? 
 
 283. The sum paid for the use of money is called Interest, 
 
 284. The sum for the use of which interest is paid is 
 called the Principal. 
 
 285. The sum of the principal and interest is called the 
 Amount. 
 
 286. In computing interest it. is usual to regard a year 
 as 12 months, and a month as 30 days. 
 
 WRITTEN EXERCISES. 
 
 287. 1. What is the interest of $ 420.50 for 5 yr. at 5% ? 
 
 Solution. 
 $420.50 Principal. 
 .05 Rate. 
 
 $21.0250 Interest for 1 yr. 
 5 
 
 $105.1250 Interest for 5 yr. 
 Find the interest of : 
 
 2. $212.34 for 3 yr. at 4%. 5. $219.20 for 4 yr. at 7%. 
 
 3. $426.12 for 4 yr. at 5%. 6. $324.16 for 5 yr. at 4%. 
 
 4. $ 324.25 for 5 yr. at 6%. 7. $ 582.30 for 6 yr. at 5%. 
 8. What is the interest of $ 366.20 for 3 yr. 8 mo. at 6%? 
 
 Solution. 
 
 $366.20 
 .06 
 
 $21.9720, int. for 1 yr. 
 
 3 
 
 $65.9160, int. for 3 yr. 
 I of the int. for'l yr. = 10.9860, int. for 6 mo. 
 
 I of the int. for 6 mo. = 3.6620, int. for 2 mo. 
 
 $80.5640, int. for 3 yr. 8 mo. 
 
228 ELEMENTS OF ARITH^METIC. 
 
 Find the interest of : 
 
 9. $412.20 for 2 yr. 8 mo. at 4%. 
 
 10. $ 318.36 for 3 yr. 6 mo. at 6%. 
 
 11. $425.24 for 4 yr. 5 mo. at 5%. 
 
 12. $ 685.60 for 3 yr. 7 mo. at 7%. 
 
 13. $437.28 for 5 yr. 2 mo. at 8%. 
 
 14. $ 314.26 for 4 yr. 4 mo. at 3%. 
 
 15. $ 736.16 for 2 yr. 10 mo. at 6%. 
 
 16. Find the amount of $ 524.36 for 2 yr. 5 mo. 18 da. 
 
 at 7%. 
 
 Solution. 
 
 $524.36 
 .07 
 
 $36.7052, int. for 1 yr. 
 2 
 
 $ 73.4104, int. for 2 yr. 
 1 of the int. for 1 yr. = 12.2350, int. for 4 mo. 
 
 I of tlie int. for 4 mo. = 3.0587, int. for 1 mo. 
 
 I of the int. for 1 mo. = 1.5293, int. for 15 da. 
 
 A of the int. for 15 da. = .3058, int. for 3 da. 
 
 $ 90.5392, int. for 2 yr. 5 mo. 18 da. 
 524.36 , principal. 
 
 $614.8992, amount. 
 Find the interest and amount of : 
 
 17. $360.24 for 2 yr. 7 mo. 15 da. at 5%. 
 
 18. $415.48 for 3 yr. 5 mo. 10 da. at 6%. 
 
 19. $217.36 for 2 yr. 6 mo. 15 da. at 4%. 
 
 20. $423.35 for 3 yr. 8 mo. 15 da. at 6%. 
 
 21. $720.16 for 3 yr. 7 mo. 17 da. at 5%. 
 
 22. $468.36 for 3 yr. 9 mo. 10 da. at 4%. 
 
 23. $219.47 for 2 yr. 5 mo. 12 da. at 6%. 
 
 24. $325.34 for 1 yr. 1 mo. 13 da. at 7%. 
 
 25. $436.33 for 4 yr. 2 mo. 10 da. at 5%. 
 
 26. $297.45 for 5 yr. 3 mo. 15 da. at 6%. 
 
GENERAL REVIEW, 
 
 288. lo What will 50 sheep cost if 75 are worth |?375? 
 
 2. How many years is it since the Declaration of Inde- 
 pendence was signed ? 
 
 3o A rope 13 yd. 2 ft. 6 in. long was cut into 3 equal 
 parts. How long were they ? 
 
 4. A farmer sold at one time 3 bu. 3 pk. 7 qt. of clover 
 seed, and at another 5 bu. 2 pk. 6 qt. How much did he 
 sell? 
 
 5o A wagon wheel is 15 ft. 8 in. in circumference. How 
 many revolutions \vill it make in going 5 miles ? 
 
 6. James spent 57 cents, which was -| of all the money 
 he had. How much money had he ? 
 
 7. A father gave his children $8 apiece. If he had 
 given them $12 apiece it would have taken $40 more. 
 How many children were there ? 
 
 8. The larger of two fractions is \^, and the difference 
 between them is f . What is the smaller fraction ? 
 
 9. If I of an acre of land produces 125 bushels of 
 potatoes, how many bushels will 3^ acres produce ? 
 
 10. A can do a piece of work in 10 days, and B can do it 
 in 8 days. In what time can both do it ? 
 
 11. A man who owned -f of a factory sold | of his share 
 for $ 3500. What w^as the factory worth at that rate ? 
 
 12. Two boys did some work for 65 cents. If one of 
 them earned 15 cents more than the other, how much did 
 each earn ? 
 
 220 
 
230 ELEMENTS OF ARITHMETIC. 
 
 13. A man who spent ^ of his money and ^12 more 
 found that he had $ 15 left. How much had he at first ? 
 
 14. A man died leaving to his 3 children a farm of 317 A. 
 115 sq. rd. to be divided equally among them. What was 
 each one's share of the land ? 
 
 15. How many acres of land, at $45 per acre^ can I get 
 ^or 90 oxen at $37.75 per head ? 
 
 16. Mr. David Bounds purchased from D. Henderson & 
 Co. 13 yd. muslin @ 8^, 8 yd. sheeting @ $ .111, 25 yd. 
 ribbon @ 7^ per yd., 10 lb. cotton batting at 8^^, 2 pr. 
 gloves @ $ 1.25, 3 doz. pearl buttons @ $ .27. Make out a 
 receipted bill. 
 
 17. How long will it take a person to count $1,000,000 
 if he counts $ 5 per second for 10 hours per day ? 
 
 18. If a railroad train runs at the rate of 521 nii. per 
 hour, how far will it run in 8 hr. 40 min. ? 
 
 19. How many feet board measure are there in 45 joists, 
 each 18 ft. long, 16 in. wide, and 2 in. thick? 
 
 20. A farmer sold 3 T. 18 cwt. 85 lb. of hay at $ 9.50 per 
 ton. Hew much did he get for it ? 
 
 21. A cask of wine contained 5S gab 3 qt. 1 pt. How 
 much was it worth at $ 2.75 per gallon ? 
 
 22. A silversmith sold some silver plate at $ 1.50 per 
 ounce, receiving for it $ 93. How much did he sell ? 
 
 23. A merchant had 320 yd. of cloth, which he made 
 into suits which required 6 yd. 2 ft. each. How many suits 
 did he make ? 
 
 24. What will be the expense of painting a gable roof 
 48 ft. long and 25 ft. wide at $ .30 per sq. yd. ? 
 
 25. How many yards of matting, 1|- yd. wide, will be re- 
 quired to cover a floor 7 yd. 2 ft. 7i in. wide, and 18 ft. long ? 
 What will be the cost of it at $ .70 per lineal yd. ? 
 
 26. A man has a lot containing 10 acres that is 25 rods 
 wide. How hm": is it ? 
 
GENERAL REVIEWo 231 
 
 27. A money-lender loaned $520 for two years at 6%. 
 How much money should he receive as principal and interest 
 at the end of the time ? 
 
 28. One number is ^ of another, and their sum is 490. 
 What are the numbers ? 
 
 29. A man who owned f of a vessel sold ^ of his share 
 for $ 12,500. At that rate, what was the vessel worth ? 
 
 30. Two men engaged in business, one furnishing 4 of 
 their capital, and the other the rest. If they gained $ 1477, 
 what was each one's share of the gain ? 
 
 31. Mr. Davis loaned his nephew $ .350 on Jan. 1, 1890, 
 at 6 % interest. If the debt was paid Jan. 1, 1893, how 
 much was the amount ? 
 
 32. How much will it cost to carpet a room 18 ft, wide 
 and 24 ft. long at $ 1.15 per sq. yd. ? 
 
 33. A druggist sold 1 lb. of quinine, which was made into 
 two-grain pills. How many pills were there ? 
 
 34. A stationer bought 10 reams of paper at $ 1.90 per 
 ream, and sold it at 1 cent per sheets How much was his 
 gain ? 
 
 35. Henry bought at a grocer's 15 lb. 12 oz. of butter at 
 32 cents per pound. How much did it cost him ? 
 
 36. Two brothers earned during their vacation $ 110, but 
 one of them earned only |- as much as the other. How much 
 did each earn ? 
 
 37. If I of a bbl. of flour costs $4.20, what will 7 bbl. 
 cost ? 
 
 38= Divide I of I of I of II of 3^ by I of if of | of f 
 of 9. 
 
 39. Sound travels about 1140 feet per second. If a flash 
 of lightning is seen, but the thunder accompanying it is not 
 heard until 35 seconds later, how far away is the lightning ? 
 
 40. The product of two numbers is 3.35, and one of them 
 is .25. What is the other number ? 
 
ANSWERS, 
 
 Page90.— 2. 283. 3.254. 4.231. 5.212. 6.136. 7.185c 
 8. 300. 9. 198. 10. 198. 11. 280. 13. 315. 13. 274. 14. 213. 
 15. 152. 16. 107. 17. 260. 18. 226. 19. 289. 20. 233. 21. 313. 
 22. 293. 23. 197. 24. 179. 25. 271. 
 
 Page 91. — 26. 40 mi. 27.^57. 28. 88 yr. 29. 100 articles. 
 30. 38 boys; 41 girls ; 79 children. 31. 134 fishes. 32. 180 pounds. 
 33. 262 pages. 34. 168 acres. 35. 31 mi. 36. 44 examples. 37. 79^. 
 38. 87 bu. 
 
 Page 92. — 39. 50 chickens. 40. $57. 41. 9Sf. 42. $1.32„ 
 43. 105 cows. 44. $ 1-.83. 45. 112 things. 46. 126 marbles. 
 
 47. 97 leaves. 48. 121 loaves. 49. 262 qt. 50. 148 lb. 
 
 Page 93. — 51. 88 animals. 52. $25. 53. $40. 54.252. 55.266 
 yd. 56. .$88, horse; $176, together. 57. $ 1.30. 58. 123 lines. 
 59. $ 1821. 60. 1335 acres. 61. 3198 pupils in all ; 1401, 4th school. 
 
 Page94.— 62. 593. 63.470. 64.541. 65.467. 66.461. 67.365. 
 68. 542. 69. 5116. 70. 5048. 71. 3681. 72. 4129. 73. 5415. 
 74. 5311. 75. 71,425. 76. 67,053. 77. 64,794. 78. 48,551. 
 
 79. 23,437. 
 
 Page 96.— 2. 532. 3. 723. 4. 351. 5. 413. 6. 214. 7. 441. 
 8. 633. 9. 201. 10. 515. 11. 543. 12. 410. 13. 662. 14. 126. 
 
 15. 456. 16. 634. 17. 613. 18. 245. 19. 724. 20. 654. 
 
 Page 97. — 24. 17. 25. 17. 26. 37. 27. 23. 28. 41. 29. 38. 
 30. 16. 31. 23. 32. 54. 33. 24. 34. 34. 35. 48. 36. 3. 37. 68. 
 38. 55. 39. 37. 40. 614. 41. 414. 42. 222. 43. 27. 44. 513. 
 45. 223. 46. 110. 47. 208. 48. 129. 49. 126. 50. 317. 51. 428. 
 52. 781. 53. 292. 54. 221. 
 
 Page 98. — 55. 166. 56.323. 57.468. 58.227. 59.414. 60.601. 
 61. 209. 62. 153. 63. 625. 64. 434. 65. 657. 66. 609. 67. 641. 
 68. $49. 69. $721. 70. 169 children. 71. 281 egg.s. 72. 758 birds. 
 73. 79 pints. 74. 115 horses. 75. 265 cows. 76. $^432. 77. 531 bu. 
 78. 635 apples. 79. 365 da. 80. 56 yr. 81. $ 148. 82. $ 188. 
 
 Page 99. — 1. 534 gal. 2. 545 mi. 3. 3621 bricks. 4. Lost $22. 
 5. 594. 6. $ 195. 7. 221 mi. 8. Answers differ. 9. 836 yd. 
 10. $ 5569, entire cost ; $2375 more. 11. Answers differ. 12. 112 ft. 
 
 Page 100. — 13. $289. 14. 451 apple trees ; 715 trees. 15. 34 bu„ 
 
 16. $97.49. 17. $2.87. 18. $89. 19. $2058; $431.50. 20. $1262. 
 21. 237 mi. 22. 613 mi. 23. 108 pupils. 24. $556. 25. 580^. 
 
 Page 101. —26. 375. 27. 225 pages. 28. $ 1.00. 29. 430 pupils. 
 30 207 b;i. 31. $3.25. 32. $101.11. 33. $351.(i0. 34. $ 104.64. 
 
 232 
 
ANSWERS. 233 
 
 35. $90.64. 36. .$11.65. 37. $34.99. 38. 1 86.23. 39. $44.73. 
 40. $109.95. 41. $77.91. 
 
 Page 107. — 2. 230. 3. 222. 4. 196. 5. 558. 6. 602. 7. 392. 
 8. 207. S. 464. 10. 387. 11. 510. 12. 957. 13. 1104. 14. 2905. 
 15. 2976. 16. 4230. 17. 1456. 18. 2754. 19. 1625. 20. 7776. 
 
 21. 1300. 22. 4896. 23. 6524. 24. 5709. 25. 10,044. 26. 12,980. 
 
 27. 14,305. 28. 11,826. 29. 34,122. 30. 30,672. 31. 21,116. 
 
 32. 73,512. 33. 27,538. 34. 27,340. 35. 0,346. 36. 47,124. 
 
 37. 23,844. 38. 26,400 ft. 39. 195 yd. 40. $187.50. 41. $66.00. 
 42. 376 mi. 43. $ 426.25. 44. $ 190.80. 
 
 Page 108.— 45. $630.00. 46. $91.75. 47. $ 58,600. 48. $25,722.00. 
 49. $71,667. 50. 50,592 papers. 51. $698.80. 52. $832.95. 53.1000 
 tons. 54. 230,370 passengers. 55. $269,465. 56. $1921.35. 57.83,208. 
 58. 22,648 mi. 
 
 Page 109.— 59. 68,355 bmiches. 60. 11,738; 17,607; 23,476; 
 29,345; 35,214; 41,083; 46,952; 52,821. 61. 14,592; 21,888; 29,184; 
 36,480; 43,776; 51,072; 58,368; 65,664. 62. 16,328; 24,492; 32,656; 
 40,820 ; 48,984 ; 57,148 ; 65,312 ; 73,476. 63. 7908 ; 11,862 ; 15,816 ; 
 19,770; 23,724; 27,678; 31,632; 35,586. 64. 7714; 11,571; 15,428; 
 19,285; 23,142; 26,999; 30,856; 34,713. 65. 11,358; 17,037; 22,716; 
 28,395; 34,074; 39,753; 45,432; 51,111. 66. 173,946; 260,919; 
 
 347,892; 434,865; 521,838; 608,811; 695,784; 782,757. 67. 47,094; 
 70,641; 94,188; 117,735; 141,282; 164,829; 188,376; 211,923. 
 68. 130,264; 195,396; 260,528; 325,660; 390,792; 455,924; 521,056; 
 586,188. 69.37,144; 55,716; 74,288; 92,860; 111,432; 130,004; 
 148,576; 167,148. 70. 84,728; 127,092; 169,456; 211,820; 254,184; 
 296,548; 338,912; 381,276. 71. 76,906; 115,359; 153,812; 192,265; 
 230,718; 269,171; 307,624; 346,077. 72. 96,724; 145,086; 193,448; 
 241,810; 290,172; 338,534; 386,896; 435,258. 73. 65,146; 97,719; 
 130,292; 162,865; 195,438; 228,011; 260,584; 293,157. 74. 143,370; 
 215,055; 286,740; 358,425; 430,110; 501,795; 573,480; 645,165. 
 
 Page 110. —1. 450. 2.670. 3.380. 4.460. 5.760. 6.32,400. 
 7. 41,600. 8. 72,300. 9. 34,200. 10. 51,800. 11. 5,463,000. 
 12. 3,784,000. 13. 51,690,000. 14. 42,370,000. 15. 819,300,000. 
 
 Pagelll. — 17. 900. 18. 1110. 19. 2720. 20. 5840. 21. 3360. 
 
 22. 3600. 23. 1500. 24. 2070. 25. 5880. 26. 3900. 27. 173,000. 
 
 28. 83,700. 29. 302,600. 30. 237,600. 31. 154,000. 32. 298,800. 
 
 33. 272,800. 34. 191,400. 35. 345,500. 36. 177,600. 37. 23.676,000. 
 
 38. 21,116,000. 39. 58,555,000. 40. 65,034,000. 41. 43,096,000. 
 42. 36,970,000. 43. 8,607,000. 44. 19,998,000. 45. 35,060,000. 
 46. 55,092,000. 
 
 Page 112. - 
 
 -51. 6408. 52. 
 
 22,572. 
 
 53. 55,998. 
 
 54. 11,742. 
 
 55. 12,555. 
 
 56. 40,260. 57. 
 
 51,471, 
 
 58. 33,432. 
 
 59. 12,354. 
 
 60. 69,768. 
 
 61. 70,308. 62. 
 
 22,240. 
 
 63. 59,616. 
 
 64. 45,936. 
 
 65. 16,016. 66 
 
 . 907,420. 67. 2,256,404. 
 
 68. 2,237,340. 69. 3,255,112. 
 
 70. 1,640,536. 
 
 71. 3,642,690. 
 
 72. 
 
 1,062,152. 73. 
 
 1,330,329. 
 
 74. 6,780,025. 
 
 75. 6,938,840. 
 
 76. 
 
 2,883,686. 77. 
 
 8,189,100. 
 
 78. 5,836,005. 
 
 79. 4,965,954. 
 
 80 
 
 3,468,255. 81. 
 
 4,512,816. 
 
 82. 2,749,835. 
 
 83. 5,996,046. 
 
 84. 
 
 4,323,284. 85. 
 
 2,673,054. 
 
 86. 4,84-6,122. 
 
 87. 2,353,176. 
 
 88. 
 
 27,381,900. 89. 
 
 29,277,136. 
 
234 ELEMENTS OF ARITHMETIC. 
 
 90. 60,355,476. 91. 302,422,875. 92. 468,083,374. 93. 186,173,400, 
 94. 307,907,152. 95. 447,832,404. 
 
 Page 113. — 96. $28,189.68. 97. #14,795.60. 98. $52,071.30. 
 99. $71,125.30. 100. $29,647.02. 101. $2,164,284.20. 102. $2,053,- 
 983.00. 103. $2,884,098.15. 104. $ 1,450,457.60. 105. $2,774,732.64. 
 106. $2808.65, 107.195,360 ft. 108. $92,022. 109. 441,066 bricks. 
 110. $16,686.42. 111. $516.46. 112. 14,208 hides. 113. $4847.04. 
 114. 49,680 pencils. 115. 525,600 min. 116. $ 3538.75. 117. 207,360 
 times. 118. 3,747,090. 
 
 Page 114. — 119. 11,341,784 gal. 120. $ 6,356,220. 121. $ 444,475. 
 122. 160,965 bbl. 123. $75,209.40. 
 
 Page 116. — 1. $220. 2. $16,675. 3. $950. 4. $8024. 5. $321.50. 
 6. $184.81. 7. A, $4918; B, $4295. 
 
 Page 117. — 8. $243.25. 9. $653.75. 10. $3-32.75. 11. $ 10,136.50. 
 12. $1924.25. 13. $11,340. 14. $8280. 15. $77 loss. 16. $32,810. 
 17. $773.70. 18. $6840. 
 
 PagellS. — 19. $2526.75. 20. $31,298.75. 21. $12,799.60. 
 
 22. 1740 people. 23. $ 13,833. 24. 10,720,000 seeds. 25. 252,018,000 
 fishes. 26. 147 mi. 27. 123 mi. 
 
 Page 123. —6. 1218. 7. 1366. 8. 594. 9. 496. 10. 1545. 11. 984. 
 12. 1078. 13. 876. 14. 454. 15. 166. 16. 125. 17. 211. 18. 554. 
 
 19. 1353. 20. 518. 
 
 Page 124. — 21. 1645^. 22. 913 J. 23. $14.88|. 24. $9.08|. 
 
 25. $10,428. 26. $14,653. 27. $8374f. 28. $22961-. 29. $ 12,052. 
 30. $4857|. 31. $9479i. 32. $10,149. 33. $12,024. 34. $2862/^ 
 35. 454. 36. 237. 37. 391. 38. 459. 39. 13,170|. 40. 17,897. 
 41. 4390j3... 42. 450 coats. 43. 211 weeks. 44. $753. 45. 156 hr. 
 46. 392 hr. 47. 689 people. 48. 3271 yd. 49. 6523 cloaks. 50. 586 
 fathoms. 51.650; 487| ; 390; 325; 278f ; 243 1 ; 216%. 52.492; 369; 
 2951; 246; 210f ; 184f ; 164. 53. 1453i ; 1090; 872; 726i ; 622f- ; 
 545 ; 484|. 54. 981 ; 735| ; 588| ; 490| ; 420f ; 367| ; 327. 55. 1872 ; 
 1404 ; 1123J ; 936 ; 8021 ; 702 ; 624. 56. 230.3 ; 1727^ ; 13814 ; llolf ; 
 987; 863|; 767f. 57. 872; 654; 523 i ; 4-36; 373f ; 327; 290|. 
 58. 628; 471; 376f ; 314; 2691; 2.35^; 209|. 59. 5762; 4321 1 ; 
 3457i ; 2881 ; 24692 ; 2160| ; 1920f,. 60. 3871| ; 2903| ; 2323 ; 1935^ ; 
 16592; 1451|; 1296|. 61. 14,951'; 11,213|; 8970| ; 7475| ; 6407f ; 
 
 If. 62. 6269f ; 4702^ ; 37614 ; 3134| ; 2687 ; 2351 1 ; 2089|. 
 55,444f; 46,203f ; 39,603f; 34,652| ; 30,802|. 
 
 64. 182,779 ; 137,084i ; 109,667f ; 91,389| ; 78,333f ; 68,542,^ ; 60,926|. 
 
 65. 129,882|; 97,412; 77,929| ; 64,941| ; 55,664; 48,706; 43,294|. 
 
 66. 87,3162; 65,487f ; 52,.390 ; 43,658| ; 37,421f; 32,743| ; 29,105|. 
 Pagel26. — 4. 372. 5.486. 6.387^^. 7. 426i&^. 8.384^2^-. 
 
 9. 573/^0- 10. 425Ji>,. 11. 298^^^',. 12. 342x§o. 13. ^IS^^- 
 14. 52,8^,. 15. 32/gVo- 16- Q^^Wo- 17- 54j-V\fj. 18. 69rWo- 
 
 20. 3411;]. 21. 2433V 22. 81^^. 23. 49/^. 24. 329/^0. 25. 95|2i. 
 
 26. 58^^. 27.68IH- 28- 1<>«M&- 29.39/^. 30. 96|ff . 31.135m. 
 Page 128.-36. 382. 37. 213. 38. m. 39, 48, 40. 55, 41. 73. 
 
A^^SWERS. 235 
 
 42. 456. 43. 144. 44. 07. 45. 82. 46. 37. 47. 36. 48. 1829if ; 
 1200P; 893||; 711^; 691i§. 49. SllS^^; 2046i-|; 1522|| ; 1212f f ; 
 1007||. 50. 4507/'r; 2957||; 2201^0; 1752i| ; 1456if. 51. 2175.y2j-; 
 1427i| ; 1062^1 ; 8"454t ; 702i|. 52. 4034,^ ; 2647f I ; 1970|| ; 15693''^ ; 
 1303||. 53. 3054^2.; 2004/.,; 1491||; li87||; 986ff. 54. 4755.^^^; 
 312I322 ; 232211 ; i849|| ; 1536|i. 55. 979^\ ; 642p ; 478/3 ; 3801| ; 
 316||. 56. 338611; 22221 1; 1653|| ; 1316f|; lOO^V 57. 1928^; 
 126511; 941i|; 750^^; 623ia, 58. 1704^f- ; 1118i? ; 832i| ; 662f | ; 
 550|f. 59. 646^^; 424 32^ ; 315|| ; 251|| ;"'208|S. 60. 4237i|; 2781; 
 206911; 1648; 1369^^3. ^61. 2229.ii; 14633-^; 1088|| ; 867 i- ; 720|f. 
 62. 4566^^ ; 2996|| ; 2230^^3 ; 1775i| ; 1475||."' 63. 33,515,*^ ; 21,994,i| ; 
 16,367|f ; 13,033|^|; 10,828^5- 64. 23,453 .^ f ; 15,391|^; ll,454|f ; 
 9120if; 7577||. 65. 31,859./r; 20,907f|; 15,559^^/; 12,.389|^ ; 
 10,2936V 66. 35,697 Jf; 23,426/^; 17,433|-| ; 13,882l|; ll,532|f, 
 67. 39,9202V; 26,19711;" 19,495i|; i5,524fi-; 12,897|f. 68. 1024i-f; 
 13532-i; 1995; S2A^\; 1083„ 69. 970fi.; 12821 f; 1890 ^V ; 780|i; 
 1026iv 70. 266511; 3522i|; 5191t\; 2144Jf; 28I83V 71. 1302|4 ; 
 172111; 2537^5^; 1048; 1377if. 72. 8O63V; 1065.^38 ; 1569i| ; 648i| ; 
 8523V 73. 1374/^; 1815|| ; 26763?-; 1105^; 1452f|. 74. 
 
 3818i|; 1577^1; 20733^. 75. 2596/^; 34301 f ; 5055j^ 
 
 -5 * 
 
 76. 518f f ; 685J I ; lOlO^^g ; 417^1 ; 548if • 77. 930|^ ; 
 1230; 1812i|; 748|| ; 984. 78. 1810L|; 2-392 ^V ^ 3525 j-\ ; 1456 /^ ; 
 1913f f. 79. 2317if ; 30622^8 ; 4512i| ; 1864 J^ ; 2449ff. 80. 12,698/^ ; 
 16,77911; 24,728; 10,213|| ; 13,4232|. 81. 7289f f ; 9633; 14,196; 
 
 83. 21,019|l; 27,776if ; 40,933ii; 16,907if; 22,2213^ 84. 541. 
 85. 723. 86. 2767^-^f. 87. 726^\o^o^. 
 
 Page 131. — 1. $61. 2. 23 yd. 3. 248 weeks. 
 
 Page 132. —4. $130; $650. 5. 144 doz. 6. $53.35. 7.128 
 trees. 8. 868 times. 9. 301 acres. 10. 2240 lb. 11. 150 da.; 2if da. 
 12. 72 hr.; 3 da. 13. 1569. 14. 2311, multiplicand. I5" 467. 
 16. 92j-\lb.; sister, 83if lb. 17. $51,614. 18. $3080. 19. 97 
 schooners. 20. 23f gal. 21. 259 families. 
 
 Page 133. — 22. 1000. 23. 43 cars. 24. 234 da. 25. 5130 shares. 
 26. 5|.Tfgmi. 27. 6272. 28. $224. 29. 352 times. 30. 27 posts ; 
 104 rails. 31. 488 acres. 32. A, $208; B, $224. 33. 3535; 3207. 
 34. 779 gal. 
 
 Page 134. — 35. $4.64. 36. Elsie. $2.65; Lottie, $3.05. 37. 14 
 da. 38. $584. 39. 629 mi. 40. $ 158. 41. $ 1816. 42. $1149.12. 
 43. 472 bii. 44. 12 horses ; 41 sheep. ' 45. 25 times. 46. 16 da. 
 47. Gained $ 161. 48. 50 gross. 
 
 Page 135. — 49. 8 75.60. 60. 5 yr. 61. $98.79. 62. lO^jf vr. 
 53. Gained .S40. 54. 938 mi. 55. 98 mi. 56. A. $46 ; B,' $ 53 ; 
 C. $99. 57. Howard, 35 tishes ; Jamie, 23 fishes; Roy, 19 fishes. 
 58. $198. 59. $1410. 60. $396. 
 
236 ELEMENTS OF ARlTHMETICo 
 
 Page 136. —61. -$4725. 62. B, 442 acres ; C, 1284 acres ; D, 1511 
 acres. 63.76 1b. 64. 8 yr. 65. .$55. 66.406, 67.22,962. 
 68. 42,90a 
 
 Page 137. — 1. 55. 2. 30. 3. 10. 4. 2. 5. 8. 6. 5. 7. 2L 
 
 8. 5. 9. 36. 10. 4. 11. 16. 12. 36. 13. 58. 14. 10. 15. 117. 
 16. 40. 17. 112. 18. 56. 19. 109. 20. 60. 21. 157. 22. 7. 
 
 Page 141. —2. 2, 2, 2, 3, 7, 3. 2, 2, 2, 2, 2, 7. 4. 2, 2, 2, 2, 3, 3, 
 5. 2, 3, 3, 11. 6. 2, 2, 2, 3, 3, 3. 7. 2, 2, 11, 11. 8. 2, 2, 2, 2, 2, 2, 
 3, 3. 9. 2, 2, 2, 2, 3, 3, 3. 10. 2, 2, 2, 2, 2, 3, 7. 11. 2, 2, 3, 3, 11. 
 12. 5, 5, 5, 5. 13. 2, 2, 2, 2, 3, 19. 14. 2, 2, 2, 2, 2, 2, 13 15. 2, 3, 
 3, 5, 11. 16. 2, 2, 2, 2, 3, 3, 7. 17. 3, 5, 20. 18. 2, 3, 37. 19. 2, 2, 
 
 2, 2, 2, 3, 7. 20. 5, 5, 31. 21. 17, 17. 22. 2, 2, 2, 2, 2, 2, 2, 2, 5. 
 33. 3, 5, 5, 5, 7. 24. 2, 2, 3, 3, 89. 25. 3, 5, 7, 11. 26. 2, 2, 5, 5, 
 5, 5. 27. 3, 3, 3, 5, 11. 28. 2, 2, 2, 2, 3, 3, 3, 3. 29. 2, 2, 2, 2, 2, 3, 
 
 3, 7. 30. 5, 5, 5, 23. 31. 2, 5, 17, 17. 
 Page 142. —3. 2^. 4. 1||. 
 
 Page 143. — 5. 4. 6.16. 7.3. 8. 27|. 9.40. 10. 18|. 11.45. 
 12. 6. 13. 4. 14. 60. 15. 40. 16. 3J5. 17. 16 i. 18. 6if. 19. 8. 
 20. 2[. 21. 1\. 22. U. 23. 6f. 24. 2. 25. 1. 26. 15| yd. " 27. §860. 
 
 28. .$3. 12 V. 29. .$3.12. 
 
 Pagel44. — 30. 48^. 31. IJf. 32. $48. 33. 53i yd. 34.7. 
 35. 39|. 36. 2||. 37. 161/^. 38. If. 39. 4]-V. 40. 13|. 
 Page 151.-2. tf 3. -V-. 4. %«. 5. -W- G- H- ?• ¥• »• H^. 
 
 9. -w. 10. W. 11- ¥/- 12- ¥/• 13. -y#. 14. W-- 15- W. 
 
 16. \V. 17. iop. 18. i||i. 19. ^tyK 20. ^1^2-. 21. ^^K 
 Page 152. — 2. 17|. 3" 42f. 4. 49|. 5. 46|. 6. 52j8g. 7. 22/3. 
 
 8. 16.l;'. 9. 12||. 10. 22i«. 11. 20t|. 12. Uj?,. 13. 19j?y. 
 
 14. 21||- 15. 3f|. 16. 10||. 17. 15||. 18. 24ff. 19. 49/^. 
 
 20. 56^8. 21. 93i|. 
 
 Pagel54.— 2. '1. 3. f. 4. i. 5. }. 6. f. 7. i 8. f. 9. |. 
 
 10. /.. 11. 2. 12. 3. 13. ii_^ 14. 11. 15. _.!_. 16. ^_. 17, |. 
 
 18. J. 19. I.' 20. f. 21. 2V 22. |. 23. ji. 24. f. 25. ^V 
 
 Paip-p 15 «5 Q H 9 .5 ,^ 1 •'■' 4 4 4 30 9 8 K 16 6 
 
 X-age J.OO. ,i. j.T, 12, J2« «*• 18' 18 » T8- *• ¥2? T2» T2- *'• 3 65 3^» 
 
 _5_ R15 8 6 71. 5 9 24 615 1218 024 6 13 lA 8 
 
 36' "• 20? 20' 20' •• ■3O' 30' 30' **• 2l' 24' 2-f ^- 42' i2' 12' •*•"• ^?' 
 
 6 3 2 11 1.5 12 6 4 10 6 2014 5 103320 6 66 
 
 2¥' 2¥' 2T- ■•■•*•• 60' 60' 60' 60- ■*''^- ¥8' ?8' 48' ?F ' ■'■^- ¥¥' ¥¥' ¥T' TT* 
 
 Pop'p 156 IR 4.5 3.5 27 Ifi 70 96 63 I7 36 2.5 14 
 
 i-age X3D. 10. go, go, go". lb. jj^, j^^y tT2- ^*- eo' eo' eff- 
 
 1Q _4 8_^ __8 1 16 IQ 7 5 7 2 .5 6 Ofk 14 12 35 01 160 J7 5 
 
 ■*■**• 10,S' 10.'<' 108° •*-^' T2O' T20' T20* *"• ¥2' ?2' ¥2' *•'•• 2l)0' 20(J» 
 
 5 6 90 100 176 3 5 OO 7 2 2 14 OA 8 3 5 2 16 OK 192 
 
 20-0' '*'*• 220' 220"' 220- *"• 60' 60' "50* **• T80' T8^' ISO' ***• T205 
 
 8 8 5 2 5 OR 2 5 6 2 1 9 2 Of 4 6 5 4 2 6 5 OQ 10 2 118 8 10 
 
 r20' T20'' **"• ~5G"' 56' 56- «•• 'Y'5'1 7o' 75' ^^' T80 ' Tf ' TSO* 
 
 29. -Mt^-S ^l^- -'^4-*. 
 
 Page 158.-2! If. 3. lit. 4.11^. 5. 1^- 6. If J. 7. ^f. 
 
 8. m 9. li^. 10. l3fo\. 11. 10^9. 12. 162,1. 13. 16|^ 14. 12|. 
 
 15.23U. 16. 21fii. 17. 172Vo- 18. |85,V. 19. 28' J bu. 20. 698rVo 
 Ib. 21. 125xVft. 22. 
 
ANSWERS. 237 
 
 Page 160.-3. i. 4. 3%. 5. j\. 6. J,. 7. f^. 8, H 9. f|. 
 
 Page 161. - 10. 3\V. 11. tI 0. 12- tVo- 13- 6/^. 14. 12|i. 
 15. 4,\. 16. 18^3,. 17. 43^. 18. 5|f. 19. 5^. 20. 5|t. 21. 3i|§. 
 22. 10 /f.. 23. 23-,\. 24. 71||lb. 25. 83 j^V acres. 26. 6^Jft. 
 27. 30 5L lb. 28. $30 .14. 2Q. ^}^. 30. 2^ |. 31. 34|f yd. 32. ^V 
 
 Page 162. -33, If 34. 2if. 35. ff 36. y^^fo- 37. ilJ. 38. |i 
 39. t\. 40. 9it|. 41. 43^,. 42. 4,V,. 43. TfJ. 44. ^U- 
 45. 41^. 46. lOfif. 
 
 Page 163. — 2. If. 3. fa. 4. 2*. 5. 3J3. 6. 8.1. 7. 10. 
 8. lOi. 9. 12|. 10. lOf. 11. 12i|. 12. 2Q\. "'l3. 782." 14. 1583, 
 15. 259f. 16. 130|. 
 
 Page 164. — 17. 71f^. 18. 247^ mi. 19. $13.33|. 20. $ lOOi. 
 21.165 ft. 22. 16| cords. 23.'' 175 hr. 24. #67|. 
 
 Page 165. — 2, 6a. 3. 15. 4. 24/^. 5. 21. 6. 12^. 7. 20|^. 
 
 . 13. 19i. 14. 22^. 
 
 15. 85„ 16. 2252 . 17. 241|. 18. 21|. 19. 528. 
 
 Page 166. — 20. 45^. 21. 100 acres. 22. $249f. 23. $5.61. 
 24. 452f mi. 25. $5.03.K 26. $148^. 
 
 Page 168.- 2. f. 3. f. 4. |. 5. |f. 6. [S. 7. |. 8. ^ 
 9. ^^0- 10. If. 11. j\%. 12. iff. 13. iff. 14. 3. 15. 6ff. 
 
 16. 51ft. 17. 6|f. 18. if. 19. ItV^- 20. 4|f. 21. 26,%. 22. J. 
 
 23. 2^3° 24. If. 25. if. 26. 14/^ ft. ; 57f ft. 27. $ 12j\. 
 
 28. |58J. 29. I684 acres. 30. $145i. 31. $1.81}. 32. Lining 
 $2j-V; Dress $22f. 
 
 Page 171.-3. tV 4. /^o 5./^. G. ^\. 7. .V g. .VV 
 9. ,\. 10. 3V5. 11. i^V 12. life. 13. 3I2. 14. tMt- 16= ^h 
 
 17. 3|f. 18. 3|f. 19. 3if. 20. 5/7. 21. 5^. 22. 17||. 23. 19if. 
 
 24. 22ii. 25. $8^\. 26. 4i da. 27. $x%perlb. 28. $791ii. 
 
 29. $1|. 30. II da. 
 
 Page 172. — 31. 23 |f mi. per hr. 32. 12|^ sq. rd. 
 
 Pagel73. — 2. 30„ 3. 36|. 4. 423. 5, 43^, e. 54. 7. 62|. 
 8. 42f. 9. 29|. 10. 791. 11. 621. 12. 72j^V 13. 76ii. 14. 47J. 
 15. 36ii. 16. 45if. 17. 129.21. 18. 13931. 19. 25754. 20. 898f. 
 21. 1486if. 22. 614. 23. "91. 24. 4^^. 25. 17|!-. 26. 43^. 
 27. 7||. 29. 4|f 30. I220. 31. 7^. 32. 12/,^. 33. 9||. 
 34. 20|f. 35. 522^. 36. 2J-V-*' 37. 53%. ' 38. 31^1 yd. 39. 23^ rd. 
 40. 21-2«v bbl. 41. S6j\ bbl. 42. 3 cases. 
 
 Page 175.-3. Iff. 4. l^V 5. 2^. 6. 3i. 7. I3V 
 8. lif. 9. lOf. 10. 2^\. 11. 3r\9^. 12. 1||. 13. 6i|. 
 14. Uff. 15. 5i|. 16. 6hl 17. 40^^. 18. 30^-°-. 19. 4r\%. 
 
 6.3 
 
 Page 176.— 21. 12f times. 22. 7t3j- bu. 23. 711 bbl. 
 
 24. 4j\ baskets. 25. 8 bbl. 26. 7|i lb. 27. 10 books. 28. 7y^o doz. 
 29. 6 plows. 30. 5if| acres. 31, $4056ff. 32. 5| bu. 33. 21 ^V? bu. 
 
238 ELEMENTS OF ARITHMETIC, 
 
 143 
 
 Page 177.- 35. .2^V 36. J V 37.^1^. 38. .VVs- 39. 3/^^, 
 
 40. 3^. 41. n. 42. 3f^. 43. if- 44. ^0- 45. jh- 46. A\- 
 2. 22. 3. IJ3. 4. Hi 5. H. 6. 6|. 7o llf. 8. 2||. 
 
 9. 11. 10. Iff. 11. l^^V 12. If. 13. 1\. 
 
 Page 179.-2. tV 3. .V 4.3V 5.3V 6. .f «. 7« ^V 
 8- 2V0. 9. 3¥t. 10. /.. 11. 2V8. 12. t\. 13" I- 14. U- 
 15. 33_. 16.30. 17. tV^-. 18. t#t. 19. -r\V 20.^1- 21- $1.80. 
 
 Page 181. — 2. 64. 3. 100. 4. 170, 5. 100. 6. 91. 7. 90. 
 a. 325. 9. 400. 10. 600. 11. 850. 12. 380. 13. 510, 14. 5, 
 
 15. .^''o. 16. ItV- 17. Hf. 18. h\' 19- ^h 
 
 Page 182. — 20. $120. 21. 100 ft. 22. 425 mi. 23. 150 ft. 
 24. 192 plants. 25. 55 mi. 
 Page 184.-1. 431,, 2. 63VV 3, 16|H. 4. 78f|i. 
 
 6. 42|. 7. 765. 8. 34^1' 
 
 Page 185. —9. $84?. 10. 40 bu. 11. $25jV. 12. 211f bu. 
 13. $40831. 14. .|6. 15. i'.. 16. 72 yr. 17, A, i of work ; 
 B, } of work ; both, ts of work. " 18. 42 da. ; J- of 42 da. = 3fV da. 
 
 Page 186. — 19. .$150,000. 20. 1,? da. 21. 4f da. 22. 24 da. 
 23. $620; $465. 24. 2| da. 25. $90,000. 26.' 14.i^. 27. 2||. 
 28. 1. 29. 8fV2. 30. $312.50. 31. $9000. 32. $26,666f. 
 33. 719 girls; 320 boys. 
 
 Pagel87. — 34. $25,771v.V 35. lyW/?. 36. 20//^ mi. 
 
 37. 3|in. ; 19|| in. 38. $610. 39. $250,000. 40. ^Vfg. 
 
 41. Increased yV 42. 600 1b. 43. $ | per lb. 44. $34,600. 
 45. 37ida. 
 
 Pagel92. — 2. f. 3.^. 4. ^V. 5. |. 6.1. 7o |. 8. f. 
 
 9. f. 10. ,-^^. 11. j\. 12. «|o. 13. fo- 14. sh' 15' rb- 
 
 16. ,%%. 17, sV 
 
 Page 193.— 2. .3. 3. .625. 4. .1875. 5. .25. 6. .106 + . 
 
 7. .15. 8. .025. 9. .0606 + . 10. .316+. 11. ,7105263+. 
 
 12. .45 + . 13. .108+. 14. .428751 + . 15. .36585 + , 16. .0064. 
 
 17. 3.5. 18. 1.857142 + . 19. 8.66 + . 20. 3.166 + . 21. 6.-35,. 
 22. .571. 23. .15625. 24. .0375. 25. .006. 
 
 2. 84.331. 3. 68.454. 4. 8.301. 5. 53.875. 
 
 Pagel94. — 6. 33.6635. 7. 64.435. 8. 5.1195. 9. 91.7472. 
 
 10. 430.435. 11. 8.615. 12. 49.1964. 13. 35.83262. 14. 67.8399. 
 15. 80.81. 16. 1.2119. 17. .9962493. 18. 1.938. 19. .3706883. 
 20. 133 miles. 21. $38,741,875. 
 
 Page 195. — 2. 11.33. 3.10.095. 4.27.995. 5.25.46. 6.12.012. 
 7. 3<).213. 8. 25.2963. 9. 5.4798. 10. 4.08. 11. 8.81. 12. 5.2905. 
 
 13. 4.1125. 14. .585. 15. .161739. 16. 26.629!). 17. 12.81965. 
 
 18. 76.5466. 19. 6.3066. 20. 4.75 bbl. 21. $25.75. 22. $2624.13. 
 Page 196. — 2. .1875. 3. .816. 4 38.76. 5. 1.545. 6. .04371. 
 
 7. .34125. 8. 3285. 9. 1.19075. 10. 3.971. 
 
ANSWERS. 239 
 
 Page 197. — 11. .013394. 12.4.02732. 13.99.8050. 14.1.3509, 
 15. 17.92431. 16. .0160655. 17. .00103644. 18. .001252394. 
 
 19. .0082622. 20. .595875. 21. 53.9852. 22. .051255. 24. 276.5. 
 
 25. 496.75. 26. 279.5. 27. 893.5. 28. 687.-35. 29. 376.8. 30. 683.5. 
 31. 763.4. 32. 899.2; 8992; 89920; 899200. 33. 6830.25; 68302.5; 
 6&3025. 34. .765625 Sum greater. 35.23.7375. 36.3.03125. 
 37.513.60. 38. $30.50. 39. 391.865 acres. 40. 400.2 mi. 
 
 Pagel98. — 2. 17.2811. 3. 11759.6. 4. 98.765. 5. S26.06. 
 
 6. $129.62. 7. $40,598. 9. $5,875. 10. $7.35. 11. $7,455. 
 12. 98.0625. 13. 58.545. 14. 138.668. 15. L4405. 16. 9.54, 
 17. .769. 18. $655.60. 19. $4408.375. 
 
 Page200. — 2. 21.7. 3. 3.9487 + . 4. 3420. 5. 3.84. 6. 8710, 
 
 7. 67.2, 8. .47. 9. 3.4. 10. .036. 11. 51.2. 12. 376. 13. .0036. 
 
 14. .0131. 15. 1.461. 16. .1671044-, 17. .213. 18. 5.5. 19. .036. 
 
 20. 2500. 21. .0072. 22. .315. 23. 2,000.000. 24. 51.6. 25. 2.34. 
 
 26. 100.2. 27. 64.2. 28. 6275. 29. .0490 + . 30. 32,820. 
 31. 1,000,000,900. 
 
 Page201. — 33. 3.925. 34. 2.645. 35. .3695. 36. .04825, 
 
 37. .03802. 38. .004285. 39. 26.7 yd. 40. 14.75 bbl. 41. 265.8 cords. 
 
 Page 202. —2. $21.82. 3. $1564.50. 4. $10.20. 5. $31,405. 
 
 Page 203. —1. $30. 2. $785. 3. $58.6625. 4. $84.50, 
 
 5. 7000 ft. 6. $61.-394. 7. $4,875. 8. 8.368+ yr, 9. 1820 letters. 
 
 10. .36 pairs. 11. $54.25. 12. $121.25. 
 
 Page204. — 13. 13.5 hr. 14. $17.07. 15. $105.25. 16. 20^. 
 17. $338. 18. $1544.70. 19. $.023f gain, $ 59.00. 20. $2360. 
 
 21. 1155.95, tons sold ; 159.05, tons melted. 22. 12 hr. 
 Page215. — 2. 176 in. 3. 451 in. 4. 574 in. 5. 1007 in. 
 
 6. 220 d. 7. 436 gi. 8. 494 gi. 9. 607 gi. 10. 15,315 sec. 
 
 11. .36,738 sec. 12. 66,035 sec. 13. 6728 oz. 14. 14,012 oz. 
 
 15. 73,292 oz. 16. 139,769 oz. 17. 26,3(>2 gr. 18. 48.248 gr. 
 19. 31,-569 gr, 20. 251 pt. 21. -367 pt. 22. 409 pt. 23. 930 in. 
 24. 1082 in. 25. 1654 in. 26. 4482 sq. in. 27. 7732 sq. in. 
 28. 13,012^ sq. yd. 29. -39,480] sq. yd. 30. 110,677 cu. in. 
 31. 197,706 cu. in. 32. 18,680 gr. 33. 27,220 gr. 34. 1698 sheets. 
 35. 2122 sheets. 36. 20,400 gr. 37. 33,660 gr. 38. 555,900 ct. 
 39. 575 pt. 40. 1758 in. 41. 1980 pt. 42. 28,438 sq. yd. 
 43. 463,482 in. 45. 37 lb. 8 oz. 46. 3 yd. 1 ft. 3| in. 47. 5 da. 
 10 hr. 40 min. 48. 1 ft. 3i in. 49. 2 pk. 3.2 qt. 50. 9 cwt. 
 61. 10 cu. ft. 1382.4 cu. in. ' 52. 266 rd. 3 yd. 2 ft. 53. 68 sq. rd. 
 17 sq. yd. 2 sq. ft. 82^ sq. in. 54. 5 da. 2 hr. 38 min. 24 sec. 
 55. 1 qt. 2 gi. 56. 1 ft. 5.1 in. 
 
 Page 216. —3. 131 gal. 2 qt. 1 pt. 1 gi. 4. 398 s^al. 1 pt. 5. 178 
 gal. 3 qt. 1 pt. 6. 17 rd. 4.^ yd. 11 in. 7. ^2 rd. 2 yd. 8 in. 
 
 8. 8 rd. 1 ft. 9 in. 
 
 Page 217. —9. 55 bu. 2 pk. 6 qt. 10. 73 bu. 5 qt. 1 pt. 11. 61 bu. 
 2 pk. 5 qt. 1 pt. 12. 3 cwt. 22 lb. 12 oz. 13. 5 cwt. 7 lb. 15 oz. 
 14. 2 T. 19 cwt. 83 lb. 15. 1 T. 17 cwt. 95 lb. 16. 25 wk. 3 da. 
 13 hr. 17. 2 hr. 16 min. 8 sec. 18. 6 da. 20 hr. 54 min. 19. 3 sq. yd. 
 55 sq. in. 20. 4 sq. yd. 5 sq. ft. 33 sq. in. 21. 3 R. 18 qr. 22 sheets. 
 
 22. 4 R. 16 qr. 11 sheets. 23. 4 cu. ft. 952 cu. in. 24. 1 lb. 2 oz. 
 
240 
 
 ELEMENTS OF ARITHMETIC. 
 
 1 pwt. 10 gr. 25. 1 lb. 5 oz. 9 pwt. 10 gr. 26. 1 lb. 7 oz. 18 gr. 
 28. i\yd. 29. Jspk. 30. l/.ybu. 31. p gal. 32. |-i bu. 33. ff rd. 
 
 Page 218. — 2. 36 gal. 1 ql 3. 38 bu. 2 pk. 4. 22 da. 13 hi. 
 30 mill. 5. 169 lb. 14 oz. 6. 35 lb. 8 oz. 17 pwt. 21 gr. 7. 70 yd. 
 
 2 ft. 9 in. 8. 84 gal. 1 qt. 1 pt. 9. 79 A. 153 sq. rd. 10. 41 T. 31 lb. 
 11. 23 yd. 1 ft. lOi in. 2. 5 bu. 1 pk. 4 qt. 
 
 Page 219. — 3. 6 gal. 2 qt. 1 pt. 3 gi. 4. 2 da. 22 hr. 10 min. 
 5. 22 lb. 10 oz. 3 pwt. 6. 10 yd. 2 It. 9 in. 7. 3 rd. 4i- yd. 2 ft. 
 8. 5 lb. 9 oz. 6 dr. 9. 10 A. 68 sq. rd. 10. 25 sq. yd. 1 sq. ft. 95 sq. in. 
 11. 14 T. 17 cwt. 39 lb. 11 oz. 13. 51 yr. 5 mo. 5 da. 14. 25 yr. 
 6 mo. 18 da. 15. 52 yr. 2 mo. 29 da. 16. 10 yr. 10 mo. 17 da. 
 
 Page 220. — 2. 34 bu. 1 pk. 3 qt. 3. 34 gal. 1 pt. 4. 44 lb. 4 oz. 
 2 pwt. 16 gr. 5. 54 lb. 1 oz. 2 sc. 10 gr. 6. 16 hr. 42 min. 55 sec. 
 7. 18 T. 3 cwt. 91 lb. 8 oz. 8. 24 rd. 2 yd. 1 ft. 6 in. 9. 75 cu. yd. 
 12 cu. ft. 1506 cu. in. 10. 41 sq. yd. 2 sq. ft. 81 sq. in. 11. 21 da. 
 9 hr. 1 min. 40 sec. 12. 103 rd. 1 ft. 10 in. 13. 48 R. 18 qr. 
 14. 44 bbl. 13 gal. 1 pt. 
 
 2. 3 gal. 2 qt. 1 pt. li gi. 3. 3 bu. 3 pk. 7 qt. f pt. 4. 1 yd. 2 ft. 
 11} in. 5. 4 cwt. 73 lb. llf oz. 6. 4 lb. 8 oz. 19 pwt. 10| gr. 
 7. 2 lb. 9 oz. 7 dr. 2 sc. 2^ gr. 8. 3 hr. 11 min. 46 sec. 9. 5 sq. yd. 
 4 sq. ft. Hi sq. in. lO;" 3 rd. 2 ft. llj in. 
 
 Page 222. — 2. 391 sq. ft. 3. 192 sq. ft. ; 211 gq. yd. 4. 48 sq. yd. 
 
 6. 30 A. 7. $17,578+. 8 
 
 12. 15 A. corn 
 3. 792 cu. in. 
 
 7. 888 cu, ft. 
 11. 3 cd. 12 
 
 7i 
 4. 
 8. 
 
 8 cd. 
 
 5, 19,200 sq. rd. ; 120 A. 
 9. $25.50. 10. 240 A. 11. $60. 
 Page 224.-2. 420 cu. ft. 
 
 5. 2100 cu. ft. 6. 680 cu. ft. 
 9. 986f cu. ft 10. 700 cu. ft. 
 14. $396. 
 
 Page 225. — 15. $78,222. 16. $36.3636 + . 
 1. 24 ft. 2. 15 ft. 3. 13L ft. 4. 23i ft. 5. 12* ft. 
 7. $3.00. 8. 1440ft. 9. $25.92. 10. $16,426+": 
 Page 227.-2. $25.4808. 3. $85,224. 4. $97,275. 
 
 6. $64,832. 7. $174.69. 
 Page 228.-9. $43,968. 10. $66.8556. 11. $93,907. 
 
 $27.14. 
 
 A. potatoes. 
 
 400 cu. yd. 
 
 1260 cu. ft. 
 
 13. 72 cd. 
 
 5. 26 1 ft. 
 
 $ 25.38. 
 
 $61,376. 
 
 12 
 
 13. 
 
 Amt. 
 Amt. 
 Amt. 
 Amt. 
 Amt. 
 
 180.742 
 
 $407,521. 
 
 $239,458. 
 
 $850,888. 
 
 $251,732. 
 
 $ 527.838. 
 
 Page 229. —1. 
 
 4. 9 bu. 2 pk. 5 qt. 
 
 8. !«-. 9. 5831 bu. 
 
 Page 230.— 13 
 
 14. $40 
 
 18. $85,865 
 20. $94,195 
 22. $70,774 
 24. $25,493 
 26. $94.44; 
 ! 250. 2. 
 5. 1685:fV 
 10. 4i da. 
 $36, 
 
 15. $125,147. 
 
 Amt. $501,345. 
 Amt. $ 517.545. 
 Amt. $539,134. 
 Amt. $350.8339. 
 \mt. $391.89. 
 Answers differ. 3. 
 revolutions. 6. 95^\ 
 
 11. $26250. 12. 40/' 
 14. 105 A. 145 sq. rd. 
 
 17. 
 19. 
 21. 
 23. 
 25. 
 
 $171.97. 
 $47.2815; 
 $22,098; 
 $130,728; 
 $32,262; 
 $91,508; 
 
 4 yd. 1 ft. 10 in. 
 7. 10 children. 
 ; 25</. 
 
 15. 75.* A 
 
 16. $ 7.87 amount of bill. 17. 5 da. 5 h. 33 min. 20 sec. 18. 453^ mi. 
 19.2160 ft. 20. $37,453 + . 21. $ 161.906 + . 22. 5 lb. 2 oz. 23.48 
 suits. 24. $40. 25. 42 yd.; $29.40. 26. 64 rd. long. 
 
 Page 231. — 27. $582.40. 28. 210 ; 280. 29. $62500. 30. $633; 
 $844. 31. $413. 32. $55.20. 33. 2880 pills. 34. $29. 35. $ 5.04. 
 36. $50; $60. 37. $39.20. 38._f|. 39. 7|f mi. 40. 13.4. 
 ^\ S B A rt y 
 
 UNIVERSITY 
 
 Q/:: 
 
 ,^\y:. 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 
 THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 I APR t.0 19(6 
 
 FEB 6 19'7 
 - JUL 16 1917 
 
 0CT21M8 
 SEP 30 19t9 
 
 f" 25 1926 
 
 jin-> O-^ ?«■?« 
 
VB I74C3 
 
 Ms- 
 
 % 3 r