pi«M>H>>M O INES BROTHERS & COMPANY, Cincinnati, Philadelphia, Chicago. 2i^^^^^^r^%^ FROM A. L BANCROFT & CO., SAN FRANCISCO, CAL Publishers PACIFIC COAST READERS, and General Agents for the Pacific Coast for Harper's Geographies, Harper's Language Series, Milne's Arithmetics, Rid path's Histories, Forbriger's Pat Drawing Tablets, Requa's Combined Trial and Copy Page Writing Books. LIBRARV OF THE University of California. 4^1^ . 1^9 Received a^ ^-.-^l.^#-* . /og rj Accession No. ^y^'' /ff . Class No. ^^^m^mi --' Digitized by tine Internet Arciiive in 2007 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/firstlessonsinarOOmilnrich Inductiue Series, FIRST LESSONS AEITHMETIC INDUCTIVE PLAN, INCLUDING ORAL AND WRITTEN EXERCISES. BY WILLIAM J^MILNE, A.M. PRINCIPAL OF TiJJ(e(^^£~NORS^£^SieftpOI., GENESEO, N. Y. JONES BROTHERS & COMPANY: CINCINNATI, PHILADELPHIA, CHICAGO, MEMPHIS. 1878. COPYKIGHT, 1878, BY JOHN T. JONES. ELECTROTYPED AT FRANKLIN TYl'E KOUNDRY, CINCINNATI. This book is designed to teach the principles of Arithmetical Science as far as they are involved in the elementary processes, and to secure a reasonable degree of accuracy and rapidity in express- ing numbers and computing results. The pupils for whose use this book is intended can not be ex- pected, at the outset, to explain processes, give definitions, or assign reasons, though their notions of all of them may be quite correct; much less should they be expected to grasp the principles of a science by committing to memory statements which are the deduc- tions and generalizations of persons entirely familiar with the sub- jects treated. And, therefore, this work has been prepared upon the inductive and objective methods, so that the pupil may obtain his knowledge of arithmetic from actual work with numbers, rather than from the stereotyped statements of the book. The true method of awakening in the child a clear idea of num- bers is through a perception of objects, or by pictures or other repre- sentations of them. After he has become familiar with things and their names, the name will signify to him all that he needs to know about the objects so far as number is concerned; and it is no longer necessary that the objects should be present before him. The ne- cessity, too, of associating the same numbers with such a variety of things, causes him at length to cease to regard them in connec- tion with any particular thing, or as any thing more than abstrac- tions. (iii) IV P K E F A C E . In harmony with these laws of the development and growth of mind, this work presents at first the idea of number connected with visual representations of objects; afterward, when the pupils become familiar with the objects, concrete examples are given without illus- tration, and these are immediately followed by exercises upon ab- stract numbers, so that the pupil may become familiar with the various combinations that may be formed.* The first lessons are devoted to instruction in counting, reading, and writing numbers up to twenty. Immediately following these, are lessons upon Addition, in which abundant practice is given in combining numbers. The fundamental processes of Subtraction, Multiplication and Division, are treated upon a plan similar to that exemplified in Addition, including besides, numerous ex- amples combining two or more of the previous processes. The lessons on Fractions are so simple and yet so thorough, that the pupil is enabled to read, write, add, subtract, multiply, and divide fractions when the processes are not very complex. The work on Written Arithmetic, which follows, is believed to be simple, thorough, and practical; and the brief and practical treat- ment of the various classes of denominate numbers in common use, makes the work quite comprehensive within its scope. With the hope that the features of this book will commend it to all intelligent persons, and that instruction in the rudiments of arithmetic may be rendered more interesting and efiective through its agency, it is confidently offered to the public. W. J. M. State Normal School, Geneseo, N. Y., Januciry, 1878. i CONTENTSI ^1 7^\ I. Objects and numbers from 1 to 10 . II. Counting numbers from 1 to 20 III. Oral and slate exercises in Addition IV. Special review of Addition V. Oral and slate exercises in Subtraction VI. Special review of Subtraction and Addition VII. Heading and writing numbers to 100 . VIII. Oral and slate exercises in Multiplication IX. Special review of Multiplication, Subtraction Addition X. Oral and slate exercises in Division XI. Special review of Division, Multiplication tion, and Addition .... XII. Heading and writing Fractions XIII. Addition of Simple Fractions XIV. Subtraction of Simple Fractions . XV. Multiplication of Simple Fractions XVI. Division of Simple Fractions XVII. Notation and Numeration of Numbers . XVIII. Written exercises in Addition XIX. Written exercises in Subtraction . XX. Written exercises in Multiplication XXI. Written exercises in Division XXII. Exercises in Denominate Numbers XXIII. Table of Boman Notation Subtrac- and 16 30 32 48 50 53 70 72 91 93 96 99 103 104 107 113 118 122 128 134 144 (V) Jhe Inductive ^£rie^ OF MATHEMATICAL TEXT-BOOKS. I. FIRST LESSONS IN ARITHMETIC. II. PRACTICAL ARITHMETIC. III. KET TO PRACTICAL ARITHMETIC. IV. ELEMENTS OF ALGEBRA. {In preparation.-) V. ELEMENTS OF GEOMETRY. {In preparation.) (vi) «* > FIRST LESSOtI£BflVBB.»IT ilPOT^ t^i Jv -A" 7<- yi> ^ 1 NUMBERS FRmi^ TO 10 ■t- Dog One Two Trees 2 Three Horses 3 Four Sheep 4 Five Houses 5 I. II. III. IV. V. Sloe Ships 6 VI. Seven Birds 7 A^II. J5J/r//if Books 8 VIII, Nine Cherries 9 IX. Ten Stars 10 X. ,. -(vii) 8 FIRST LESSONS liESSOlS^ I. 1. Of what is this a picture? 2. How many tables are there in the picture? 3. How many balls are there? 4. Make as many marks on your slates as there are lamps in the picture. 5. The figure 1 represents one table, one ball, one lamp, or one object of any kind. 6. Make the figure one on your slates. /. 7. How many kittens are playing with the ball? 8. How many kittens are playing with the cat? 9. How many kittens are 1 kitten and 1 kitten? 10. How many chairs do you see in the picture? 11. Name two objects on the table. 12. Make two marks on your slates. 13. The figure 2 represents two chairs, two books, or two objects of any kind. 14. Make the figure two on your slates. ^. 15. Make the figures one and two. IN ARITHMETIC. LESSON II. 1. How many boats do you see in the picture? 2. How many fishes has one of the boys caught? 3. How many fishes has the other boy caught? 4. Two fishes and one fish are how many fishes? 5. How many fishing-rods do you see? 6. Make as many marks as there are fishing-rods. 7. The figure 3 represents three objects. 8. Make the figure three on your slates. */. 9. Make the figures one, two and three. 10. How many boys in the picture are fishing? 11. How many boys are digging for bait? 12. How many boys are three boys and one boy? 13. How many oars has each boat in the picture? 14. Make as many marks on your slates as there are boys in the picture. 15. The figure 4 represents four objects. 16. Make the figure four on your slates. ^. 17. Make the figures one, two, three and four. 10 FIRST LESSONS LESSOI^ III. 1. How many roses do you see on the bush? 2. How many birds are flying near the bush? 3. How many birds are there on the bush ? 4. How many birds are four birds and one bird? 5. How many eggs are there in the nest? 6. Make a mark for every egg in the nest. 7. The figure 5 represents five things. 8. Make the figure five on your slates. <^, 9. Make the figures one, two, three, four and five. 10. How many rose-buds are there near each other? 11. How many do you find growing alone? 12. How many rose-buds are five rose-buds and one rose-bud ? 13. Make six marks on your slates. 14. The figure 6 represents six things. 15. Make the figure six on your slates. ^ 16. Make the figures one, two, three, four, five, six. / J J J ^ ^ IN ARITHMETIC, 11 I^ESSON^ IV. 1. How many lambs are at the left in the picture? 2. How many trees are in a row in the picture? 3. How many trees are standing alone? 4. How many trees are six trees and one tree? 5. How many lambs are there in the picture? 6. Make seven marks on your slates. 7. The figure 7 represents seven things. 8. Make the figure seven on your slates. /^ 9. How many sheep are feeding in the field? 10. How many sheep are with the lambs? 11. Seven sheep and one sheep are how many sheep? 12. How many posts do you see in the picture? 13. Make as many marks as there are posts. 14. The figure 8 represents eight things. 15. Make the figure eight on your slates. J^c liESSOX XIII. 1. How many apples are 1 apple, 2 apples and 3 apples? 2 apples, 4 apples and 1 apple? 2. If you put 2 dol- lars, 3 dollars and 4 dol- lars into an empty purse, how many dollars will there be in the purse? 2+3+4=? 3. How many are 9 and 6 ? 4. How many are 8 and 5? 5. Walter caught 2 fishes, Harry caught 5, and Webster caught 3. How many fishes did they all catch? 6 and 9? 5 and 8? 22 FIRST LESSONS 6. Arthur had 7 chickens, and his father gave him 6 more. How many had he then? 7. 3 + 3 + 7=-? 4 + 1 + 6 = ? 3 + 2 + 8 = ? 8. How many are 3 books, 5 books and 6 books? 9. How many are 3 slates, 4 slates and 5 slates? 10. There are 5 birds in one nest and 4 in another. How many birds are there in both nests? 11. How many are 10 and 3? 3 + 10 = ? 12. How many are 10 and 4? 4 and 10? 13.2 + 3 + 5 = ? 2 + 2 + 2 = ? 1 + 5 + 6 = ? 14. What is addition ? What is the sign of ad- dition? 15. What is the sign of addition called? 16. Make the signs used in addition. 17. In one nest there were 5 little birds, in an-* other 4, and in another 4. How many birds were there in these nests? 18. In a school 5 pupils were tardy on Monday, 2 on Tuesday, 3 on Wednesday, 3 on Thursday, and on Friday. How many pupils were tardy during the week? Slate Exercises. Copy and add the following: 2 3 4 4 6 7 1 4 2 12 2 3 5 8 5 6 5 3 3 6 Copy and read the following: 20 18 15 17 13 12 11 19 IN ARITHMETIC. 23 LT]SSO]S^ XIV. 1. If a darning-needle costs 1 cent, and a skein of silk 7 cents, what will both cost? 2. How many skeins are 2 skeins and 7 skeins? 3. If there are 3 spools in one group — and 7 in another, how many spools are there in both groups? 4. ^ A woman who had 4 quarts of cherries, bought 7 quarts. How many did she then have? 4+7 = ? 5. There are 5 boys in one group and 7 in an- other. How many boys are there in both groups? 6. There are 6 cows in one field and 7 in an- other. How many cows are there in both fields? 7. How many are 7 and 7? How many are 8 and 7? 8. How many are 9 and 7? How many are 10 and 7? 9. If I have 1 dollar, and my brother gives me 8 dollars, how many dollars will I then have? 10. How many chairs are 2 chairs and 8 chairs? 11. How many are 3 and 8? How many are 4 and 8? 12. George walked a distance of 5 miles one day,, and 8 miles the next day. How far did he walk in both days? 5 + 8 = ? 24 FIRST LESSONS 13. 6 + 8 = ? 7 + 8 = ? 8 + 8 = ? 9 + 8=? 14. There are 10 girls in one class and 8 in an- other. How many girls are there in both classes? 15. Form the addition tables of sevens and eights. TABLES. 1 + 7= 8 6 + 7 = 13 1 + 8= 9 6 + 8 = 14 2 + 7= 9 7 + 7 = 14 2 + 8 = 10 7 + 8 = 15 3 + 7 = 10 8 + 7 = 15 3 + 8 = 11 8 + 8 = 16 4 + 7 = 11 9 + 7 = 16 4 + 8 = 12 9 + 8 = 17 5 + 7 = 12 10 + 7 = 17 5 + 8 = 13 10 + 8 = 18 :>U^c I^ESSO]^ XV. 1. How many peaches are 7 peaches, 2 peaches and 5 peaches? 7 + 2 + 5=? 2. A man gave 4 dollars for corn and 8 dollars for a ton of hay. How much did he pay for both ? 3. A man has 2 cows in one pasture, 4 in an- other, and 7 in another. How many cows has he in the three pastures? 4. A woman sold 5 dozen eggs at one time, 4 dozen at another time, and 8 dozen at another. How many dozen did she sell? 5. A woman sold 7 pounds of butter at one time, 3 at another time, and 5 at another. How many pounds did she sell? IN ARITHMETIC. 25 6. 6 + 3 + 2-=? 4 + 5 + 7 = ? 3 + 2 + 5 = ? 7. How many are 9 and 4? 4 and 9? 8. How many are 10 and 7? 7 and 10? 9. 8 + 5 + 7 = ? 3 + 7 + 5 = ? 8 + 6 + 3 = ? 10. A farmer paid 5 dollars for sugar, 2 dollars for coffee, and 10 dollars for flour. What was the cost of the whole? 11. How many are 8 boys, 2 boys and 5 boys? 12. 8 + 3 + 2 = ? 3 + 2 + 6 = ? 5 + 2 + 4 = ? Slate Exercises. Copy and add the following : 673835648 256145221 223256343 3>a^c i.esso:n^ XVI. 1. There is 1 man on the steps and there are 9 men in the store. How many men are there in both places ? 2. Henry sold 2 books on Tuesday and 9 books on Wednesday. How many books did he sell in both days? 3. How many shells are 3 shells and 9 shells? 4. How many boats are 4 boats and 9 boats? 5. If Jennie is 5 years old, and her sister is 9 years older, how old is her sister? 3 26 FIRST LESSONS 6. How many are 6 and 9? 7 and 9? 7. There are 8 trees in one row and 9 in an- other. How many trees are there in both rows? 8. How many plants are 9 plants and 9 plants? 9. How many books are 10 books and 9 books? 10. A beggar had 1 cent, and a boy gave him 10 cents. How many cents did he have then? 11. If there are 2 straw- berries on one stem and 10 on another, how many are there on both stems? 12. How many are 3 or- anges and 10 oranges? 13. 4 + 10 = ? 5 + 10 = ? 6 + 10 = ? 14. A boy paid 7 shil- lings for oil and 10 shillings for a lamp. What was the cost of both ? 7 + 10 = ? 15. How many are 8 pigeons and 10 pigeons? 16. How many are 9 and 10? 10 and 10? 17. Form the addition tables of nines and tens. TABLES. 1+9=10 6+9=15 1+10=11 6+10=16 2+9=11 7+9=16 2+10=12 7+10=17 3+9=12 8+9=17 3+10=13 8+10=18 4+9=13 9+9=18 44-10=14 9+10=19 5+9=14 10+9=19 5+10=15 10+10=20 -^ IN ARITHMETIc/'g^JY^v27 liESSON XVII, 'k ■rm.tl 1. Ruth picked 4 pinks from one stem, 3 from another, and 2 from another. How many pinks did she pick? 2. If there are 6 boys in one class, 7 boys in another, and 5 boys in another, how many boys are there in the three classes ? 6 + 7 3. 2 + 2 + 2 = ? 3 + 3 + 3 = ? 5+5+5=? 6+6+6=? 4. Begin with 2 and count to 8 by 2's ; thus : Two, four, six, eight. 5. Begin with 3 and count to 15 by 3's. 6. Begin with 1 and count to 19 by 3's. 7. How many bells are 5 bells, 2 bells, 6 bells and 4 bells? 5 + 2 + 6 + 4 = ? 8.10+5 + 3 + 4 = ? 6 + 8 + 4 + 2 = ? 9. 2 + 3 + 6 + 1 = ? 3 + 2 + 4 + 7 = ? 10. Count to 20 by 5's. To 18 by 6's. 11. A merchant sold 5 yards of calico to one girl, 3 yards to another, and 6 to another. How many yards did he sell? 12. Begin with 2 and count to 17 by 3's. 13. How many are 7 barrels, 6 barrels and 4 barrels? 14. How many are 8 and 7 and 3? 15. Lucy had 8 pins, and found 10 more. How many had she then? 28 fikst lessons Slate Exekcises. Copy and add the following: 941268513 125510286 432423625 203331-544 3»<00- LESSON XTIII. 1. Jennie gave the teacher 6 hyacinths, Mary gave her 4, and Minnie gave her 3. How many hyacinths did they all give her? 6 + 4 + 3 = ? 2. In the park 3 boys are flying kites, 6 are play- ing ball, and 3 are rolling hoops. How many boys are there in the park? 3 + 3 + 6 = ? 3. There are 4 soldiers in one squad, 5 in an- other, and 6 in another. How many soldiers are there in the three squads? 4. How many are 4 guns, 5 guns and 6 guns? 5. There are 5 flags on one building, 2 on an- other, and 4 on another. How many flags are there on the three buildings? 5 + 2 + 4 == ? 6. There are 6 birds in one cage, 2 in another, and 7 in another. How many birds are there in the three cages? 6 + 2 + 7 = ? 7. 1 + 2 + 6 + 9==? 8 + 1 + 5 + 3 = ? 8. 7 + 1 + 8 + 2 = ? 3 + 3 + 3 + 3 = ? IN ARITHMETIC. 29 9. Jennie is 8 years old, and her brother is 5 years older. What is the age of her brother? 10. Delia has 1 doll, Mina has 2 dolls, Carrie has 3 dolls, and Ruth has 2 dolls. How many dolls have these girls? 11. How many are 6 and 3 and 5? 6 + 3 + 5 = ? 12. How many are 4 and 7 and 8? 4 + 7 + 8 = ? 13. There are 8 goblets on the shelf, 6 on the table, and 4 on the stand. How many goblets are there in all? 14. A wealthy gentleman had 3 draft-horses, 5 riding-horses, and 6 carriage-horses. How many horses did he have? 15. A gentleman spent 4 months of every year in Florida, 2 months in the White Mountains, 3 months in Europe, and the rest of the year at home. How many months was he away from home? 16. James wrote 6 lines in his writing-book on Monday, 3 lines on Tuesday, 7 lines on Wednesday, and 4 lines on Thursday. How many lines did he write in those four days? 17. 8 + 2 + 3 + 5 = ? 7 + 3 + 4 + 4 = ? Slate Exercises. Copy and add the following: 211648564 253152344 247321254 264235424 30 FIRSTLESSONS t REVIEW EXERCISES. Find the value of each of the following : 5 + 4 6^8 6 + 3 9 + 4 8 + 1 6 + 4 5 + 9 ^5 + 7 5 + 3 7 + 9 7 + 6 2 + 8 8 + 3 6 + 4 8 + 8 5 + 6 4 + 7 7 + 9 3 + 9 7 + 5 3 + 9 3 + 5 9 + 3 4 + 4 6 + 3 8 + 3 5 + 6 8 + 7 4 + 8 7 + 8 Q + 6 5 + 5 6 + 9 6 + 5 4 + 9 8 + 4 4 + 9 5 + 8 7 + 6 5 + 7 Add the following without using slate : 2 + 3 + 4 3 + 4 + 5 6 + 1 + 5 7 + 2 + 5 6 + 5+3 2 + 9 + 2 4 + 7 + 5 6 + 3 + 4 5 + 2 + 7 6 + 5 + 8 4+9 + 6 5 + 6 + 8 3 + 5 + 6 3 + 9 + 2 4 + 6 + 2 3 + 5 + 6 7 + 5 + 4 7 + 3 + 5 4 + 1 + 9 5 + 9 + 2 7 + 6 + 2 6 + 8 + 3 5 + 3 + 4 4 + 7 + 5 3 + 8 + 4 3+9 + 6 5 + 9 + 4 2+5 + 4 2 + 7 + 3 6+1 + 5 6 + 5 + 4 7 + 9 + 2 5 + 7 + 8 3 + 1 + 6 5 + 2 + 7 6 + 3 + 4 7 + 2 + 7 5 + 5 + 6 6 + 5 + 6 3 + 7 + 2 3+2 + 5 5 + 7 + 6 2 + 5 + 7 3 + 6 + 8 5 + 2+7 7 + 3 + 5 4 + 6 + 5 6+5 + 4 IN ARITHMETIC. - 31 REVIEW EXERCISES. Copy and add the following : 875356 5 67 4 6 9 8 4 4 12 2 352682246 42423783 5 3 6 5 5 4 6 3 2 4 4 2 3 9 7 4 2 1 2 5 3 8 4 3 1 5 6 7 4 5 1 1 8 4 4 6 2 2 3 2 5 1 2 5 1 Copy and add the following: 3 -f 4+2 + 6 2 + 4+2 + 3 2 + 6 + 8 + 4 3+2+1+6 4+8+2+4 2+6+1+7 4+2+3+9 8+1+7+4 8+4+2+5 4+7+4+3 6+4+2+7 5+2+7+6 7+2+9+1 5+6+4+3 9+2+6+3 4+2+6+5 4+9+2+4 5+2+7+6 5+2+4+3 7+4+6+1 9+1+4+2 8+2+6+3 9+1+4+5 4+2+6+7 5+2+7+4 5+9+2+3 8+7+2+2 4+3+5+8 4+7+6+2' 6+2+3+5 5+4+9+2 9+2+3+6 4+7+2+6 5+1+4+9 6+3+1+8 4+7+5+4 7+3+3+7 6+4+4+6 8+2+2+7 9+1+6+3 8+5+2+4 7+7+2+3 32 FIRST LESSONS T.ESSON XIX. 1. Make two marks on your slates. Erase one. How many marks are there left? 2. 1 mark from 2 marks leaves how many marks? 3. If there were 3 pencils on the table, and Helen took 1 away, how many were left? 4. How many are 3 pencils less 1 pencil ? 5. If 4 apples were put in a box, and afterward 1 was taken out, how many remained? 6. How many are 4 apples less 1 apple? 7. Hold up 5 fingers. Put one down. How many fingers are you holding up? 8. How many are 5 fingers less 1 finger? 9. Make 6 dots on your slates. Erase one. How many are there left? 10. How many are 6 dots less 1 dot? 11. How many steamships are represented in the picture ? 12. If 1 sails away, how many are left? 13. How many vessels are 7 vessels less 1 vessel? 14. How many boats do you see in the picture? IN ARITHMETIC 33 15. How many are being rowed away ? How many are left? 16. How many boats are 8 boats less 1 boat? 17. If there are 9 trees, and a man cuts 1 down, how many will be left? 18. How many trees are 9 trees less 1 tree? 3j*iC I^ESSON XX. 1. A rose-bush had 3 roses on it, but 1 dropped off. How many roses were there left on the bush? 2. How many are 3 less 2? 3. Make 4 marks. Erase 2. How many are left? 4. How many marks are 4 marks less 2 marks? 5. Helen had 5 pennies, and gave 2 to a beggar. How many had she left ? 6. How many are 5 less 2? The sign — takes the place of the word less. It is read minus. Thus, 4 less 2 equals 2, is written 4 — 2 = 2, and is read 4 minus 2 equals 2 ? 7. If you take 2 apples from a group of 6 apples, how many apples will be left? 8. How many apples are 6 apples less 2 apples? 9. Express 6 less 2 equals 4. 6 — 2 = 4. 34 FIRST LESSONS 10. William had 7 pennies, and gave 2 for a pen- cil. How many had he left? 7 — 2 = ? 11. How many slates are 8 slates less 2 slates? 12. How many are 9 less 2? How many are 10 less 2? How many are 11 less 2? 13. How many are 12 dollars less 2 dollars? Taking one number from another is called Sub- traction, 14. Form the subtraction tables of ones and twos. TABLES. 1 — 1=0 6-1 L = 5 2 — 2 = 7 — 2 = 5 2 — 1 = 1 7 — ] L = 6 3 — 2 = 1 8 — 2 = 6 3 — 1 = 2 8-1 L = 7 4 — 2 = 2 9 — 2 = 7 4 — 1=3 9-1 [ = 8 5 — 2 = 3 10 — 2 = 8 5 — 1 = 4 10—] L = 9 6 — 2 = 4 11 — 2 = 9 3>»»iC liESSO?^ XXTIII. 1. Begin with 1 and count to 11 by 2's. 2. Begin at 11 and count back to 1 by 2's. 3. Count 15 by 3^s. Count 16 by 4's. 4. Count back from 15 to by 3's. 5. Count back from 20 to by 5's. 6. Henry had fifteen dollars. He gave 2 dollars for a hat, 3 dollars for a vest, and 6 dollars for a coat. How much had he left? 7. What number added to 3 makes 10? 8. What number added to 9 makes 17? 9. A harness-maker having 19 harnesses, sold all but 9. How many did he sell? 19 — ? = 9. IN ARITHMETIC. 45 10. How many are 2 + 2? 2 + 12? 3 + 2? 3 + 12? 4 + 2? 4 + 12? 5 + 2? 5 + 12? 11. How many are 3 + 3? 3 + 13? 4 + 5? 4 + 15? 5 + 6? 5 + 16? 7 + 6? 7 + 16? 12. How many are 3 + 6? 3 + 16? 4 + 6? 4 + 16? 5 + 8? 5 + 18? 5 + 9? 5 + 19? 13. How many are 3 + 5? 3+15? 4 + 5? 4 + 15? 8 + 6? 8 + 16? 8 + 7? 8 + 17? 14. In the barn were 12 horses, but 5 of them were taken out. How many remained? 15. 10 chestnuts from 16 chestnuts leave how many chestnuts? 16 — 10 = ? 16. A grocer bought 19 brooms. He sold 3 at one time, 5 at another, and 4 at another. How many had he left? 19 — 3 + 5 + 4=-? 17. A lady purchased 4 dollars' worth of silk, 3 dollars' worth of velvet, 7 dollars' worth of lace, and 1 dollar's worth of linen. She gave in payment 20 dollars. How much should be returned to her? Slate Exercises. Copy and add the following: 4 5 3 2 6 1 4 5 3 2 6 1 4 5 3 2 6 1 Copy and subtract the following: 896789775 652523413 46 FIRST LESSONS LESSOINr XXIX. A man having 18 butter-tubs, sold 3 at one 5 at another, and 6 at another. How many tubs had he left? 2. In a room there are 14 flower- pots. There are 4 in one window, 5 in another, and the rest are in the third ^r window. How many are there in the <^li/,J^ third window? 3. On the first vine there are 12 bunches of grapes, on another there are 3 bunches, and on another 4 bunches. How many bunches more are there on the first vine than on the other two? 12 — (3-1-4)=? 4. How many are 7 and 5 less 3? 5. There are 13 boys playing ball, and 5 playing marbles. How many more are there playing ball than there are playing marbles? 13 — 5 = ? 6. What is meant by the term addi- )n? What is meant by the term subtraction? 7. Julia is 9 years old, and her sister is 5 years der. What is the age of her sister? 8. Charles is 6 years old, and James is 17 years d. How much older than Charles is James? 9. What number added to 7 will make 15? 10. What number added to 5 will make 12? IN ARITHMETIC. 47 11. Begin with 2 and count to 17 by 3's. 12. Count from 19 back to 1 by 3's. 13. What number subtracted from 18 will leave 10? 18 — ?=10. 18 — 10 = ? 14. What number subtracted from 16 will leave 9? 16 — ? = 9. 16 — 9=? 15. In a school of 18 pupils all but 3 study ge- ography. How many study geography? 16. James attended school 17 weeks, but Henry at- tended only 8 weeks? How many weeks did James attend more than Henry? 17 — 8 = ? 17. Julius had written all but 5 lines on a page of his writing-book. If the book had 19 lines on a page, how many had he written? 18. How many are 3 + 3 + 2 + 2 — 4+5 — 2 + 7? 19. How many are 5 + 7 + 1 — 3 — 2 + 5 — 9 + 8? 20. How many are 6 + 4 + 5 — 3 — 6 — 2 + 8 + 9? 21. How many are 8 — 3 + 2 — 3 + 6 + 2—6 + 2? 22. How many are 6 + 8 — 7 — 2 + 5 + 6 + 1—4? Slate Exercises. Copy and add the following: 56456532 4 2 2 12 9 4 3 35184657 Copy and subtract the following: 69864987 55712342 48 FIRST LESSONS REVIEW EXERCISES. Find the value of each of the following : 8-2 9-3 8- -4 9-6 6-2 7 — 5 8 — 3 7- -5 6-3 8 — 7 4 — 1 9 — 4 7- -2 9-2 8-3 6 — 6 6 — 1 8- -2 6-4 7 — 3 7 — 2 5 — 2 6- -4 8 — 3 9-7 8 — 5 9 — 4 8- -1 6 — 5 7-3 12-4 15 — 6 13- -6 14—9 16-6 15-6 17 - 9 12 - -8 11 — 4 15-7 13 — 4 10-5 14- -9 13-9 11 — 8 18 — 9 : 17—8 10- -8 11-8 16-9 15 — 6 13 _ 7 15- -6 12-7 14-5 ind the vaL Lie of the following, without using slat 3 + 2-4 4 + 6 — 5 6 + 3-2 5 + 6-8 4 — 3 + 6 9 — 4 + 2 6 + 5-8 5 + 6-9 5 + 8 — 7 6 + 9 — 5 9- -5 + 2 6 + 7 — 6 4+7 — 3 5 + 6-9 8 + 7 — 9 6 — 3 + 4 8 + 5-3 9-6-2 9- -8 + 6 8-3-4 9 — 6 + 5 9 + 5-6 8 + 2-6 9-3+2 7 — 2 + 5 8 — 3 + 4 6 + 3 — 8 7 + 5 — 9 8 — 4 + 3 6 — 4 + 2 7- -5 + 6 9 — 7 + 2 9 + 2-8 8 + 6 — 9 7 + 3-4 9 — 4+2 8 + 3 — 9 8 — 6 + 2 9- -7 + 3 8-6-2 8 — 2-4 9-3 — 6 8- -6 + 2 8 + 6 — 9 9 — 3 + 6 9 + 2-8 9- -6 + 4 8-3 + 7 6 — 4 + 2 6-4-2 8 + 3 — 9 8 — 3 — 4 IN ARITHMETIC. 49 REVIEW EXERCISES. Copy and find the value of the following: 6 + 3 — 4 + 2—6 7 — 3 + 8 — 6 + 2 5 — 3 + 7 — 2 — 6 8 — 3 + 6 — 5 — 4 5 + 9 — 8 — 3 + 2 7 — 5 + 4 + 2 — 8 9__3 + 2 — 6 + 4 9 — 3 — 4 + 6 + 2 5 + 6 + 3 — 3 — 9 9 — 3 + 6-3-8 8 + 3 — 7 — 4 + 2 5 + 2 — 3 + 6 — 4 9 — 3 + 6 — 4 + 8 8 — 3 + 7 + 2 — 6 9 — 4 + 3 + 2 — 6 9 — 4 + 3 + 2 — 8 5+2 — 3 + 4 — 8 2 + 6 — 3 + 7 — 5 9—3+4—3+2 6—3+4—6+2 4+2 — 3 + 6 — 4 7 — 3 + 2 — 6 + 4 When an example has several numbers inclosed in a parenthesis, find the value of those within the parenthesis first, and then of this result, and the rest of the numbers. Thus,4 + 6 — (3 + 2 — 4)=4 + 6 — 1; for (3 + 2 — 4) = 1. Copy and find the value of the following: 5 + 8— (3 + 6 — 2) 6 + 8 — (7 — 2 + 6) 6 + 3 + (4 — 3 + 2) 7 + 3 — (5 + 2 — 6) 5 + 4 — (6 + 2 — 5) 9 — 2 + (6 — 4 + 2) 5 — 4 — (8 + 2 — 9) 8 + 6--(2 + 3 — 1) 6 + 8 — (3 + 4 — 2) 8 — 4+ (8 — 6 + 4) 9 — 3 + (2 + 6 — 4) 9 + 8 — (8 — 3 + 2) 9 + 6 — (3 — 2 + 7) 9 — 7— (8 — 6— 2) 8 — 3 + (2 + 6 — 3) 9 + 2 — (7 — 4 — 2) 6 + 5— (6 — [8— 3]) 2 + 6 — (3 — [4 — 3]) 50 FIRST LESSONS LESSON- XXX. 1. How many quails are there in each group? 2. Express ten on your slates. 3. How many quails are there in two of the groups ? 4. How many are 1 ten and 1 ten? 5. Express 2 tens, or twenty : 20. What does the 2 represent? What does the show? 6. Express 2 tens and 1, or twenty-one : 21. 7. Express 2 tens and 2 ones, or twenty-two. 8. Express twenty-three, twenty-four, twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine. 9. Read the numbers you have written. 10. How many quails are there in the 3 groups? 11. Express 3 tens, or thirty: 30. What does the figure 3 represent? What does the show? 12. Express 3 tens and 1, or thirty-one. 3 tens and 2 ones, or thirty-two. 13. Express all the numbers from thirty-two to thirty-nine. 14. Express 4 tens, or forty. Express 4 tens and 1, or forty-one. IN ARITHMETIC. 51 15. Express the numbers from forty-one to forty- nine. 16. When two figures are written side by side, what does the figure at the right represent? 17. What does the figure at the left represent? 18. Write all the numbers from one to forty-nine. 19. Express in words: 24, 32, 38, 47, 18, 29, 10, 31, 49, 30. liESSON XXXI. 1. Express 5 tens, or fifty: 50. 2. How many figures are required to represent the number fifty? 3. What is the figure at the right? 4. What is the figure at the left? 5. How many tens are there in fifty? 6. Express 5 tens and 1, or fifty-one. 7. Write the numbers from fifty-one to fifty-nine. 8. Express 6 tens, or sixty : 60. 9. Write the numbers from sixty to sixty-ninco 10. Express the number seventy. 11. How many tens are there in seventy? 12. How many ones are there in seventy? How is this shown? 13. Write the numbers from seventy to seventy- nine. Read them. 14. Write the numbers from eighty to eighty- nine. Read them. 52 FIRST LESSONS 15. How many tens and ones are there in eighty- seven? In eighty-five? In eighty-eight? 16. Express in figures the numbers from eighty to eighty-eight. 17. Write 8 tens and 9 ones, or eighty-nine. 18. Express 9 tens, or ninety : 90. 19. Write the numbers from ninety to ninety-nine. 20. Ninety-nine and one more, make how many? One hundred. 21. How many tens are 9 tens and 1 ten? 22. How many tens are there in one hundred? 23. One hundred is expressed thus : 100, The figure at the right shows that there are no ones, the next to it shows that there are no tens, and the figure 1 shows that there is one hundred. When three figures are written side by side, the fig- ure in the first place at the right represents ones, the one next to it tens, and the one at the left hundreds. Slate Exercises. Express in figures the following : Thirty-two Eighty Forty-nine Ninety-six Eighteen Sixty-seven Eighty-four Seventy-one Thirty-three Forty-seven Twenty-nine Thirty-eight Copy and express in words the following: 36 52 28 77 78 84 81 39 29 55 IN ARITHMETIC. 53 liESSOlS^ XXXII. * 1. If you pay 1 cent for a pear, how many cents will you pay for 2 pears? 2. How many cents are 1 cent and 1 cent? 3. If you pay 1 cent for a slate-pencil, how many cents will you pay for 3 pencils? 4. How does the price of 3 pencils compare with the price of 1 pencil? 5. If a fig costs one cent, how many times 1 cent will 4 figs cost? 6. How many chairs are 5 times 1 chair? This sign X takes the place of the word times. Thus, 2 times 1 are 2, is written 2X1 = 2, and is read 2 times 1 equals 2. 7. If 1 marble costs 1 cent, what will 6 marbles cost? 6X1=^6. 8. Seven boys each own a sled. How many sleds do they all own ? 7 X 1 = ? 9. Each of 8 boys has a pair of skates. How many pairs do they all own ? 8 X 1 = ? 10. Nine men each own an acre of land. How many acres do they all own ? 9 X 1 = ? 11. How many are 10 times 1? 10 X 1=? 12. How many are 3 times 1 ? 3 X 1 =? 13. If you pay 1 cent for an apple, how many cents will you pay for 7 apples? 14. If each pupil, in a class of 9 pupils, answers 1 question, how many questions will they all answer? 54 FIRST LESSONS liESSON^ XXXIII. 1. A horse has 2 eyes. How many eyes have 2 horses? How many eyes are 2 times. 2 eyes? 2. A horse has 2 ears. How many ears have 3 horses? 3X2==? ~ 3. If 1 rocking-horse costs 2 dollars, what will 4 rocking-horses cost? 4 X2 = ? 4. If a boy goes to school only 2. days each week, how many days will he attend in 5 weeks? 5. A boy read 2 pages of his book each day for 6 days. How many pages did he read in that time? 6. If a boy earn 2 dollars a week, how many dollars will he earn in 7 weeks ? 7X2 = ? 7. What will 8 sheets of blotting-paper cost at 2 cents a sheet? 8X2=? 8. What will 9 pairs of shoes cost at 2 dollars a pair? 9X2 = ? 9. How many cents are there in 10 two-cent pieces ? 10. If it requires 2 yards of cloth to make a child's cloak, how many yards will it require to make ^ such cloaks? A short process of adding equal numbers is called Multiplication, IN ARITHMETIC 55 11. Form the multiplication tables of ones and twos. TABLES. 1X1-1 6X1- 6 1X2— 2 6X2 — 12 2X1 — 2 7X1— 7 2X2— 4 7X2 — 14 3X1—3 8X1— 8 3X2— 6 8X2 — 16 4X1=4 9X1— 9 4X2— 8 9X2 — 18 5X1 — 5 10X1 — 10 5X2 — 10 10X2 — 20 3>>iC IjESSOX XXXIV. 1. If a boy walks 3 miles in 1 hour, how many miles will he walk in 2 hours ? 2X3 — ? 2. What will 3 engravings cost at 3 dollars each ? - 3. Martha put 3 tulip bulbs in each of 4 pots. How many bulbs did she plant ? 4X3 — ? 4. Each of 5 boys threw 3 stones into the water. How many stones did they all throw ? 5X3 — ? 5. If the pupils learn 3 songs in a month, how many songs will they learn in 6 months? 6. If there are 3 cherries in 1 bunch, how many cherries are there in 7 such bunches? 7. If there are 8 boats, and 3 boys in each boat, how many boys are there in the boats? 8. If there are 3 globes on 1 chandelier, how many globes are there on 9 such chandeliers? 56 FIRST LESSONS 9. If a boy gives 4 cents for 1 orange, how many cents will he give for 2 oranges ? 2X4 = ? 10. A lumberman put 4 logs in the river each day for 3 days. How many logs did he put in the river? How many are 3 times 4? 3X4 = ? 11. If there are 4 crows sitting on each of 4 limbs of a tree, how many crows are there on the tree ? 4X4 = ? How many are 4 times 4? 12. What will 6 tons of hay cost at 4 dollars a ton ? 5X4 = ? 13.6X4 = ? 7X4 = ? 8X4 = ? 9X4 = ? 14. How many girls are there in 10 groups of 4 girls each ? 10X4 = ? 15. If a man can walk 4 miles in an hour, how far can he walk in 6 hours? 6X4 = ? 16. Form the multiplication tables of threes and fours. TABLES. 1X3= 3 6X3 = 18 1X4= 4 6X4 = 24 2X3= 6 7X3 = 21 2X4= 8 7X4 = 28 3X3= 9 8X3 = 24 3X4 = 12 8X4 = 32 4X3 = 12 9X3 = 27 4X4 = 16 9X4 = 36 5X3 = 15 10X3 = 30 5X4 = 20 10X4 = 43 IN ARITHMETIC. 57 liESSOlSr XXXY. 1. If 1 man builds 2 rods of fence in one day, how many rods can 5 men build in the same time? 2. If a shoe-maker makes 4 pairs of shoes in 1 day, how many pairs will he make in 6 days? 6X4-=? 3. If a turner can turn 3 rolling-pins in 1 min- ute, how many can he turn in 8 minutes? 8X3 = ? 4. If a woman sells 4 quarts of milk a day, how many quarts does she sell in 9 days ? 9X4 = ? 5. In 1 pint there are 4 gills. How many gills are there in 5 pints ? 5X4 = ? 6. A boy bought 4 pints of chestnuts at 3 cents a pint, and sold them for 20 cents. How much did he gain ? 4X3 = ? 7. (5X3)-f(3X4) = ? (4X2) + (6X3)=? 8. If a boy husks 3 bushels of corn in 1 hour, how many bushels will he husk in 7 hours ? 7X3 = ? 9. If 2 yards of lace will make 1 curtain, how many yards are required to make 4 curtains? 10. In 1 bushel there are 4 pecks. How many pecks are there in 7 bushels ? 7X4 = ? 11. How many are 9 times 2? 7X2? 5X2? 12. If there are 4 blades in 1 pocket-knife, how many blades are there in 10 such knives? 10 X 4 = ? 13. What will be the cost of 9 historical charts at 3 dollars each? 9X3 = ? 14. If in 1 field there are 4 acres, how many acres are there in 4 such fields ? 4X4 = ? 58 FIRST LESSONS Slate Exercises. Copy and multiply the following : 3 2 4 3 2 4 5 4 8 7 9 6 >>©4c IiESSO:N^ XXXTI. 1. John gave 5 cents apiece to each of 2 beggars. How much did he give to both ? 2X5 = ? 2. If a boat sails 5 miles in 1 hour, how far will she sail in 3 hours? 3X5 = ? 3. In each of 4 nests there are 5 eggs. How many eggs are there in the 4 nests? How many are 4 times 5 ? 4X5 = ? 4. What will 5 barrels of flour cost at 5 dollars a barrel ? 5X5 = ? 5. A cartman can draw 5 barrels of salt at 1 load, How many barrels can he draw in 6 loads? How many are 6 times 5? 6. What will 7 bunches of matches cost at 5 cents a bunch ? 7X5 = ? 7. How many are 8 times 5? How many are 9 times 5 ? IN ARITHMETIC. 59 8. How many cents are there in 10 five-cent pieces? 10X5 = ? 9.4X5 = ? 3X5 = ? 7X5 = ? 8X5 = ? 10. There are 6 lamps on each of 2 shelves. How many lamps are there on both shelves ? 2X6=? 11. If 6 bushels of wheat can be ground in 1 hour, how many bushels can be ground in 3 hours? How many are 3 times 6? 12. If a boy writes 6 lines in his writing-book ach day for 4 days, how many lines will he write? How many are 4 times 6? 13. If a man pays 6 dollars at 5 different times, how many dollars will he pay? 5X6 = ? 14. How many are 6 times 6? How many are 7 times 6? 15. In 1 package there are 6 drawing-pencils. How many pencils are there in 8 such packages? 8X6 = ? 16. How many are 9 times 6? 10 times 6 are how many ? 17. Form the multiplication tables of fives and sixes. TABLES. 1X5= 5 6X5 = 30 1X6= 6 6X6 = 36 2X5 = 10 7X5 = 35 2X6 = 12 7X6 = 42 3X5 = 15 8X5 = 40 3X6 = 18 8X6 = 48 4X5 = 20 9X5 = 45 4X6 = 24 9X6 = 54 5X5 = 25 10X5 = 50 5X6 = 30 10X6 = 60 60 FIRST LESSONS I.ESSON XXXTII. 1. If a croquet-player drives a ball through 2 arches at each stroke, through how many arches will he drive it by 3 strokes ? 3X2 = ? 2. How many are 9 sleds, 2 sleds, and 8 sleds? 3. A boy who shot 18 prairie-chickens, gave 2 to George, 3 to Henry, and 7 to Caspar. How many had he left? 18 — (2 + 3 + 7) = ? 4. A man bought 6 bushels of wheat at 2 dol- lars a bushel, and 8 bushels of barley at 1 dollar a bushel. What did he pay for both? 5. How many soldiers are there in 4 squads of 5 soldiers each? 6. (6X2) + (8X1)=? (3X5) + (4X1)=? 7. (4X2) + (3X3) = ? (2X5) + (2X3)=? 8. How many feet have 4 dogs? 7 dogs? 9. How many eyes have 6 horses? 9 horses? 10. How many legs have 6 rabbits? 8 rabbits? 11. How many wings have 10 birds? 7 birds? 12. From a bin containing 16 bushels of potatoes 4 bushels were taken at one time, and 6 at another. How many bushels remained in the bin? 13. A woman had a jar of butter, containing 20 pounds. She used 5 pounds one week, 6 pounds the next week, and the remainder the third week. How much did she use the third week? 14. There are 3 feet in 1 yard. How many feet are there in 7 yards? 15. 10 + 6 — (4X2)=? 12 — 3 + (5X6) = ? IN ARITHMETIC. 61 16. A drover had 6 cows in one field, 8 cows in another, and 12 in another. How many cows had he in the three fields? 17. If a girl can pick 3 quarts of berries in 1 hour, how many quarts can she pick in 3 hours? Slate Exercises Copy and multiply the following : 6543273643 5798947987 >»-o^ ^{^ his father has 9 times as many. -. t2el„,, ''4'^^ How many sheep has his fa- ther? 4. How many sheep have Henry and his father? How many are 63 and 7? 5. An Indian sold 8 baskets at 10 cents apiece. How much did he receive for them? 8X 10 = ? 6. A boy having 6 rabbits, bought 8 more, and then sold 9. How many had he left? 7. Count to 48 by 6's. Count to 63 by 7's. 8. Count back from 45 to by 9's. 9. From a crock of butter weighing 25 pounds 7 pounds were taken at one time, 3 at another, and 6 at another. How many pounds remained? 10. A laundress bought at one time 3 flat-irons, each weighing 6 pounds; at another time 2 flat- irons, each weighing 8 pounds. What was the en- tire weight of the irons? 11. At 8 dollars a term each, how much will a music-teacher receive for 6 pupils ? 6X S = 2 12. There are 7 days in one week. How many days are there in 9 weeks? 7x9 = ? 9X7 = ? 68 FIRST LESSONS 13. If a horse trot 9 miles in one hour, how far will he travel in 10 hours? lOX 9 = ? 9 X 10 = ? 14. If one passenger-car has 8 wheels, how many such wheels will 5 cars have ? 5X8=^? 8X5 = ? Slate Exercises. Copy and multiply the following : 789689787 543595632 3^^C IjE8So:n^ xlii. 1. A cutler sold 6 sets of knives one day, 5 sets the next day, and 8 sets the next. How many sets did he sell? 2. What is addition ? What is the sign of addi- tion? 3. A painter having 13 pounds of paint, used 6 pounds one day and 3 the next. How many pounds had he left? 13 — (6 + 3)=? 13 — 6 — 3 = ? 4. What is subtraction ? What is the sign of sub- traction ? 5. A farmer sold 3 sheep at 4 dollars apiece, and 6 calves at 5 dollars a head. What did he receive for the whole? (3 X 4) + (6 X 5) =? 6. What is multiplication? What is the sign of multiplication ? IN ARITHMETIC. 69 7.8 + 5 + 6 = ? 24 — 3 — 6 = ? 8X2 = ? 8. In one yard there are 3 feet. How many feet are there in 4 yards? How many feet are there in 5 yards? 9. If a squirrel eats 8 nuts in one day, how many nuts will he eat in 9 days? 9X8 = ? 8X9 = ? 10. What will be the cost of 6 pears at 6 cents each ? 6X6 = ? 11. In one peck there are 8 quarts. How many quarts are there in 5 pecks ? 5 X 8 ^^ ? 12. How many are 9 times 5? How many are 7 times 6? 13. There are 4 classes in a school, each class con- taining 10 pupils. How many pupils are there in the school? 4X 10 = ? 14. In an orchard there are 7 rows of trees, and 9 trees in each row. How many trees are there in the orchard? 7X9 = ? 9X7 = ? 15. A man having 16 cords of wood, used all but 5 cords. How many cords did he use? 16 — 5 = ? 16. Helen gave 7 yards of cloth to a poor woman, and had 9 yards left. How many yards had she at first? 17. 17 + 3 — 8 = ? 21 — 6 = ? 21 — (6 + 5)=? Slate Exercises. Copy, add, subtract, and multiply the following : 5 8 9 8 ^5 6 9 - 8 37423244 70 FIRST LESSONS REVIEW EXERCISES. Find the value of the following : 7X9 3X4 3X8 8X6 6X6 6X5 8X5 5X4 7X3 3X9 5X8 6X7 6XS 4X9 7X5 4X7 3X9 2X9 3X5 8X9 6XQ 4X8 8X8 8X7 5X4 4X3 5X5 7X4 6X2 6X8 8X2 6X9 6X9 8X4 7X4 7X7 3X3 5X3 7x7 5X9 8X4 5X9 8X9 5X9 5X6 3X5 4X4 5X6 3X4 6X7 QXQ 3X6 3X8 8X7 4X9 5X8 6X2 7X2 9X4 8X9 Find the value of the following : (7 + 2)X9 8X(9~3) 4X(6 + 2) (3 + 4)X6 5X(8+1) 5X(7 + 3) (5 + 3)X8 4X(7-4) 3X(8-2) (9 + l)X7 9X(9-3) 7X(9 + 1) (8-3)X6 6X(8 + 2) 6X(6-4) - (8 + 2)X7 5X(9-6) 8X(7 + 2) (4 + 3)X5 4X(9-2) 4X(8 + 1) (5 + 5) X 5 8X(5 + 2) 5X(9-6) (4 + 3)X8 9X(3 + 7) 6X(9-2) (9-2)X3 7X(3 + 2) 7X(8 + 2) (8-4)X9 4X(9-4) 9X(7 + 2) (9-5)X8 6X(8-3) 4X(5H-4) J IN ARITHMETIC REVIEW EXERCISES. ■^ Off T. Copy and fiiK 1 the vahie of the J following : 2X3X4 7 + 3X(6 + 2) 4 + (3X9)-5 5X2X5 4 + 5X(9-3) 8+(7x2)-6 3X2X9 8 + 6X(4 — 2) 7 + (6X4) + 2 2X4X7 9 + 4X(9-3) 6 + (8X3)+5 4X2X8 5 + 5X(8-5) 6— (3X2) + 6 3X3X6 4+3X(6 + 4) 9 — (2X4) + 5 5X2X7 5 + 3X(8 + 4) 16 — (4X3) + 5 4X2X5 6 + 5X(9 + l) 20 + (2X6)-8 3X3X3 7 + 4X(8 — 3) 17 — (3X3) + 6 3X3X9 4 + 6 X ( 7— 2 ) 25-(7x3)-4 3X2X8 9 + 4X(5 + 2) 40 — (6X3)-8 2X5X9 2 + 3X (6-3) 29-(4X6)-5 2X4X 8 8 + 6 X (8-6.) 30 + (2X3) -9 3X2X6 6 + 4X(9-2) 50- (3X8) + 6 Copy and find tlie value of the i following : 6X8==40+? 3X9 = 20 + ? 3X?=20 + 1 7X5==30 + ? 5X8 = 33 + ? 5x? = 30 + 5 6X4 = 18 + ? 7X6 = 36 + ? 7X? = 37 + 5 9X3 = 22 + ? 5X9 = 40 + ? 6X? = 29 + 7 6X8 = 39 + ? 6X6 = 30 + ? 4X? = 22 + 6 5X7 = 30 + ? 7X7 = 50 — ? 9X? = 59 + 4 6X6 = 40- _? 8X4 = 36- ? 3X?=18 + 6 7X9=70- _? 7X5 = 40- ? 8X?=-59 + 5 8X8=60 + ? 9X6 = 60- ? 7X?-=41 + 8 5X9 = 51- -? 7X3 = 20 + ? 6X? = 49 + 6 ' \ )/t I 'i'/i/4" ' ' "-^y^^ ////'//. 72 FIRST LESSONS IjESSON XIjIII. 1. If Dick had 4 rabbits, how many times could he give away 2 rabbits to his companions? 2. Richard sold 6 apples, selling 2 at a time. How many times did he sell 2 apples ? 3, If William had 8 cents, and spent it 2 cents at a time for candy, how many times did he purchase candy? How many 2's are there in 8? 4. How many times is 2 contained in 8? 5. If you pay 2 cents for a slate-pencil, how many can you buy for 10 cents? How many 2^s are there in 10? 6. How many times is 2 contained in 10? 7. 10 is how many times 2? Finding how many times one number is contained in another is called Division. The sign -^ is used to indicate division. It is read divided by. Thus the expression 8^4=2, is read 8 divided by 4 is equal to 2. 8. Read the following: 6-^2 = S; 8-i-2-=4; 10-^-2 = 5; 12-^3 = 4. . 9. At 2 cents apiece, how many lemons can be bought for 12 cents? 12-^2=? IN ARITHMETIC 73 10. If a boy earns 2 cents an hour, how long will it take him to earn 14 cents? 11. How many 2's are there in 16? In 18? 12. If I have 20 cents in two-cent pieces, how many pieces have I? 13. Form the division tables of ones and twos. TABLES. l-f-l = l 6^1= 6 2-^2 = 1 12^2-= 6 2^1 = 2 7^1= 7 4-1-2 = 2 14-^2= 7 3--lr=3 8-^1== 8 6-^2 = S 16-^2= 8 4^1=^4 9-^1-= 9 8-f-2 = 4 18-^2= 9 5-1 = 5 10-^1 = 10 10^2 = 5 20-^-2 = 10 3j«»^C 1. If a boy rides one mile on the cars for 5 cents, how many miles can he ride for 10 cents? 10-i-5 = ? 2. If a boat sails 5 miles in one hour, in how many hours will it sail 15 miles? 15-^5 = ? 3. At 5 dimes apiece, how many tickets can be bought for 20 dimes? 20-^5 = ? 4. In how many hours will a horse travel 25 miles, if he travels 5 miles an hour? 25-^5 = ? 5. 30-^5=:? 35-^5==? 40-^5 = ? 6. Fannie attends school 5 days in a week. How many weeks will she attend if she is present 45 days? IN ARITHMETIC. 77 7. How many tons of coal at 5 dollars a ton, can be bought for 50 dollars? 50^5 = ? 8. A boy sold brackets at 2 dimes apiece, and got 12 dimes. How many did he sell? 12-v-2 = ? 9. When biscuits are 6 cents a pound, how many pounds can be bought for 18 cents? 18-^6 = ? 10. If it requires 6 yards of cloth for one cloak, how many cloaks can be made from 24 yards of cloth ? 24 -^ 6 =: ? 11. How long will it take a boy to read 30 pages, if he reads 6 pages every day ? 30 -^ 6 == ? 12. At 6 cents a quart, how many quarts of cher- ries can be bought for 36 cents? 36-^-6 = ? 13. How many fields of 6 acres each can be formed out of 42 acres ? 42 ^ 6 = ? 14. If there are 6 tea-spoons in one set, how many sets will 48 tea-spoons make ? 48 -^ 6 = ? 15. If there are 6 pairs of kid gloves in one pack- age, how many packages are there in 54 pairs? How many packages are there in 60 pairs? 16. Form the division tables of fives and sixes. TABLES. 5^5 = 1 30-^5= 6 6-^6 = 1 36-i-6= 6 10-^5 = 2 35^5= 7 12-^6=-2 42^6= 7 15-^-5 = 3 40-^5=- 8 18-^6 = 3 48^6= 8 20^5=-4 45^5= 9 24-^6 = 4 54--6=- 9 25^5 = 5 50-^5 = 10 30-^6 = 5 60-^6 = 10 78 FIBST LESSONS liESSON XI.VII 1. If a man earns 5 dollars in one week, in how many weeks will he earn 50 dollars? 50-4-5 = ? 2. How many times can you take 6 cents out of a purse that contains 48 cents? 48-^6 = ? 3. How many times 3 apples are 12 apples? 12 ^3 = ? 4. How many calves at 3 dollars each, can be bought for 18 dollars? For 21 dollars? For 24 dollars? 18h-3 = ? 21-^3 = ? 24-^3 = ? 5. There are 4 pecks in 1 bushel. How many pecks are there in 7 bushels? In 4 bushels? 6. In 32 pecks how many bushels are there? 7. A man gave to each of 4 boys 8 quails. How many quails did he give them ? 4X8 = ? 8. A clothier bought 10 coats at 5 dollars apiece, and sold them for 6 dollars apiece. How much did he gain? (10 X 6) - (10 X 5) =? 10X(6-5) = ? 9. How many bushels of potatoes at 1 dollar a bushel, must be given for 6 yards of cloth at 2 dol- lars a yard? 10. At 5 dollars a barrel, how many barrels of flour can be bought for 30 dollars? 30^5 = ? 11. How many balls at 6 dimes apiece, can be bought for 54 dimes ? For 30 dimes ? 54 -^ 6 = ? 12. If a cutler uses 18 blades, putting 3 blades in each knife, how many knives does he make? 13. A carman can draw 4 hogsheads of sugar at one load. How many loads will 16 hogsheads make? in arithmetic. 79 Slate Exercises. Copy and multiply the following : 5634875321 5387564689 Copy and divide the following : 6)42 5)35 3)21 2)18 3)27 4)36 >i:«>iC liESSON XI.TIII. • 1. In one week there are 7 days. How many weeks are there in 14 days? 14^-7 = ? • 2. In how many hours can Carrie solve 21 prob- lems, if she can solve 7 per hour ? 21 -^- 7 = ? 3. Helen sold her doll for 28 cents, and spent the money for drawing-pencils, paying 7 cents for each. How many pencils did she buy ? 28 -i- 7 = ? 4. At 7 dollars a picture, how many pictures can be bought for 35 dollars ? 35 ^ 7 = ? 5. If a fish weighs 7 pounds, how many such fish will be needed to weigh 42 pounds ? 42 -f- 7 = ? 6. If a man earns 7 dollars in one week, in how many weeks will he earn 49 dollars ? 7. If it requires 7 yards of cloth to make one cloak, how many cloaks can be made from 56 yards? - 8. If one box holds 7 quarts of nuts, how many boxes are required to hold 63 quarts ? 63 -^ 7 = ? 80 FIRST LESSONS 9. When flour is 7 dollars a barrel, how many barrels can be bought for 70 dollars ? 70 -r- 7 = ? 10. If a railroad coacjh has 8 wheels, how many such coaches will have 16 wheels? 16-^8 = ? 11. At 8 cents a yard, how many yards of calico can be bought for 24 cents ? 24 ^ 8 = ? 12. If a window requires 8 panes of glass, how many windows will 32 panes make ? 32 -^ 8 = ? 13. If there are 8 trees in a row, how many rows of trees are there in an orchard containing 40 trees? 40-^8-=? 14. How many gallons are there in 48 pints, since there are 8 pints in one gallon ? 48 -^ 8 = ? 15. 56 divided by 8 = ? 64 divided by 8 =- ? 16. How many bins will be required to hold 72 bushels, if one bin holds 8 bushels? 72-^8==? 17. In one pile of money there are 8 cents. How many such piles are there in 80 cents ? 80 h- 8 = ? 18. Form the division tables of sevens and eights. TABLES. 7h-7=-1 42^7== 6 8- -8=:1 48- -8= 6. 14-^7 = 2 49-f-7= 7 16- -8.-2 56- -8= 7 21-^7 = 3 56-^7= 8 24- -8-3 64- -8== 8 28-^7 = 4 63-^7= 9 32- -8 = 4 72- -8-: 9 35^7 = 5 70^7 = 10 40- -8=:5 80- -8-=10 IN ARITHMETIC. 81 liESSOK XlilX. 1. If 8 bushels of potatoes are needed to plant one acre of ground, how many acres will be planted if 40 bushels are used ? 40 -^ 8 ::= ? 2. If a blacksmith makes 6 horseshoes in one day, in how many days can he make 60 horseshoes? 3. If a man paid 5 dollars for taking one trip on the cars, how many such trips could he take for 30 dollars? 30^5 = ? 4. In a school of 56 pupils there are 7 pupils in each class. How many classes are there ? 56 -^ 7 =^ ? 5. In one pint there are 4 gills. How many pints are there in 36 gills ? 36 -^ 4 == ? 6. In one yard there are 3 feet. How many yards are there in 21 feet? In 30 feet? In 12 feet? 7. If one ball costs 2 dimes, how many balls can be bought for 6 dimes? For 12 dimes? For 18 dimes ? 8. How many oranges at 5 cents apiece must be given for 3 dozen of eggs at 10 cents a dozen ? 9. How many sleds at 3 dollars apiece can be bought for 27 dollars ? 27-^3 = ? 10. How many yards of calico at 8 cents a yard must be given for 10 pounds of maple sugar at 8 cents a pound? (10 X 8)^8 = ? 11. If there are 4 pecks in one bushel, how many bushels are there in 40 pecks ? 12. How many 7's are there in 63? In 49? 13. How many 6's are there in 54? In 48? 82 first lessons Slate Exercises. Copy and multiply the following: 687846875 423525369 Copy and divide the following : 7)70 8)64 5)50 4)32 5)40 7)28 5^8^c 1. If one dress can be made from 9 yards of cloth, how many dresses can be made from 18 yards? 18-^9 = ? ^2. If a painter can paint 9 boards of fence in one hour, in how many hours can he paint 27 boards? 27-^-9 = ? 3. There are 9 square feet in 1 square yard. How many square yards are there in 36 square feet? In 45 square feet? 36-^9-=? 45-^9 = ? 4. If there are 9 candles in one pound, how many pounds are there in 54 candles? 54-^-9=? - 5. A man gave 63 bushels of corn to poor fam- ilies, giving 9 bushels to each family. How many families were relieved ? 63 -^ 9 = ? 6. If you travel 9 miles in one hour, in how many hours will you travel 72 miles? 81 miles? -7. 90 is how many times 9 ? 90 ^ 9 == ? IN ARITHMETIC. 83 - 8. If it requires 10 pickets to make one gate, how many such gates can be made using 20 pickets? 20^10=-? 9. How many ten-cent pieces are there in 30 cents? 30-^-10 = ? 10. There are 10 cents in one dime. How many dimes are there in 40 cents? 40-^10 = ? 11. If a man can drill 10 feet per day, how many days will it take him to drill 50 feet ? 50 ^ 10 == ? 12. James bought a Christmas gift for 60 cents, paying for it with ten-cent pieces. How many ten- cent pieces did he give for it ? 60 -^ 10 = ? 13. How many hours will it take a steamboat to make a trip of 70 miles, if it travels 10 miles in one hour? 70-^10 = ? 14. How many pupils, at 10 dollars a term, must a teacher instruct to receive 80 dollars ? 80 -f- 10 = ? 15. At 10 cents a pound, how many pounds of starch can be bought for 90 cents? 90-^10 = ? 16. How many tens are there in 100? 17. Form the division tables of nines and tens. TABLES. 9^9=1 54^9= 6 10-^10--1 60-^10= 6 18^9=2 63^9= 7 20-^-10=2 70-^10= 7 27^9=3 72^9= 8 30^10=3 80^10=- 8 36^9=4 81-^9= 9 40^10=4 90^10- 9 1 45^9=5 90-^9=10 50-^10=5 100-^10=10 1 84 FIRST LESSON L^g^^^-S LESSOlSr LI. 1. If there are 5 monkeys on one branch of a tree, 4 on another, and 3 on another, how many monkeys are there on the tree ? 5 + 4 + 3^? 2. How many more than 5 monkeys are 7 monk- eys ? 7 — 5=-? 3. How many more than 4 monkeys are 8 monk- eys? 8 — 4==? 4. How many more than 3 monkeys are 9 monk- eys? 9 — 3=-? 5. If you have 5 apples, how many more must you get to have 12 apples? 5 + ?= 12. 6. Mary is 6 years old, and her brother is' 15. In how many years will Mary be as old as her brother is now? 6-\-? = 16. 7. In one week there are 7 days. How many days are there in 4 weeks ? 4X7 = ? 8. Into how many groups of 5 oranges each can you divide 15 oranges ? 15-^5 = ? INARITHMETIC. 85 9. How many are 2 times 8? 4 times 8? 6 times 8? 10. How many times is 8 contained in 24? In 40? In 64? 11. If a cow gives 9 quarts of milk in the morn- ing, and 10 quarts at night, how many quarts does she give in one day? 12. George having 20 cents, paid 6 cents for candy and 8 cents for peanuts. How many cents had he left? 13. If one lily has three petals, how many petals have 9 lilies? 14.4-1-8 + 2 = ? 9 + 7-1-3 = ? 28 — 3 — 6 = ? 19_4=:? 7-1-8 — 6=? 15.7X7 = ? 9X8 = ? 45-^5 = ? 90-^10 = ? 16. In one dime there are 10 cents. How many cents are there in 4 dimes? 4 X 10=? 17. Mary bought 4 pencils at 5 cents apiece, and a drawing-book for 25 cents. What was the total cost of her purchases? 18. How many are 6 and 8 ? 16 and 8 ? 26 and 8? 36 and 8? 46 and 8 ? 56 and 8? 19. How many are 4 and 9? 14 and 9? 24 and 9 ? 34 and 9 ? 44 and 9? 54 and 9 ? 20. Count to 32 by 4's. Count to 54 by 6's. 21. Count to 90 by 9's. Count to 72 by 8's. 22. Count to 40 by 5's. Count to 63 by 7's. 23. How many 8's are there in 80 ? 80 -^ 8 = ? 24. How many 7's are there in 49 ? 49 ^ 7 = ? 25. How many 6^s are there in 54 ? 54 -r- 6 = ? 86 FIRST LESSONS IjESSON Hill. 1. Henry takes 9 steps in crossing the room once. How many times will he cross the room in taking 54 steps ? 54-^9 = ? 2. If a man works 10 hours in one day, how many hours will he work in 8 days? 8 X 10 = ? - 3. If a man works 10 hours in one day, in how many days will he work 100 hours? 100-^-10 = ? 4. Count to 27 by 3's. Count to 48 by 4's. 5. How many times is 5 contained in 30? In 45? 6. How many five-dollar bills will be required to pay for 5 tons of coal at 4 dollars a ton ? 7. A boy having 50 cents, paid 30 cents for a ball and 15 cents for a bat. How many cents had he left? 50-30—15=-? > 8. When clover seed is 8 dollars a bushel, how many bushels can be bought for 48 dollars? For 56 dollars? 48^8=? 56-^8=? 9. In one square yard there are 9 square feet. How many square feet are there in 6 square yards ? 10. 36 is how many times 4? How many times 6 ? How many times 9 ? 11. 24 is how many times 8? How many times 6 ? How many times 4 ? 1?. In an orchard there are 56 trees and 7 trees in a row. How many rows are there ? 56 + 7 — ? 13. In one yard there are 3 feet. How many feet are there in 8 yards? In 6 yards? In 9 yards? I N ARITHMETIC. 87 14. A house has 10 windows, and 8 panes of glass in each window. How many panes of glass are there in the house? 10X8 = ? 15. How many are 6 + 4 + 5? 8 + 3 + 7? 16. How many are 12 and 5? 22 and 5? 32 and 5? 42 and 5? 52 and 5? 62 and 5? 17. How many are 8 less 5? 18 less 5? 28 less 5? 38 less 5? 48 less 5? 58 less 5? 18. Two men start from the same place and travel in opposite directions, one at the rate of 6 miles an hour, and the other at the rate of 4 miles an hour. How far apart are they in 5 hours? 3^^C liESSON lilll. 1. If a blacksmith uses 8 nails in setting one horseshoe, how many nails will he uae in setting 8 horseshoes? 8X8=:=? 2. If one horse re- quires 4 horseshoes, how many horseshoes will be needed for 5 horses? For 6 horses? 5X4 = ? 3. How many times is 7 contained in 21 ? In 49? In 56? 4. How much can a wagoner earn in 9 days, if he earns 4 dollars per day? 9 times 4 = ? 88 FIRST TjESSONS 5. How many are 4 times 7 ? 5 times 7 ? 9 times 7? 6. How many crows are 5 crows, 3 crows, 6 crows, and 2 crows? 7. How many more than 10 crows are 18 crows? 8. How many more than 6 pens are 18 pens? 9. How many ten-dollar bills must be given for 8 boxes of handkerchiefs at 5 dollars a box? 10.6 + 7 — 4 = ? 8 + 5 — 7 — ? 9 + 3 — 6 = ? 11. (7X4)^2==? (8X6)H-4 = ? (6X6)-^9=-? 12. A farmer having 18 bushels of corn sold 5 bushels to one man and 3 to another. How many bushels had he left? 18 — (5 + 3) = ? 13. A woman put 20 quarts of strawberries into cans, putting two quarts into each can. How many cans did she use? 20h-2 = ? 14. In one pint there are 4 gills. How many gills are there in 7 pints ? 7X4 = ? - 15. In one gallon there are 4 quarts. How many gallons are there in 16 quarts? 16-i-4 = ? 16. In a school-room there are 48 desks arranged in rows containing 8 desks each. How many rows of desks are there in the room ? 48 -^ 8 = ? 17. A farmer sold some grain for 40 dollars, and took his pay in sheep at 5 dollars a head. How many sheep did he receive ? 40 ^ 5 = ? 18. How many dozens of eggs at 10 cents a dozen will pay for 6 yards of calico at 5 cents a yard? 19. If there are 6 forks in one set, how many forks are there in 9 sets? 9X6 = ? 6X9 = ? INARITHMETIC. 89 liESSOlS^ lilV. - 1. If a basket-maker can make 7 dinner-baskets in one day, in how many days can he make 42 such baskets ? 2. If one dining-table has 4 legs, how many legs have 9 such dining-tables ? 9X4 = ? 3. What will be the cost of 2 tops at 10 cents apiece, and 6 oranges at 5 cents each? 4. In one dime there are 10 cents. How many dimes are there in 90 cents? In 70 cents? In 40 cents ? 5. How many cents are there in 5 dimes? In 8 dimes? In 9 dimes? 6. What is the diiference in the cost of 6 peaches at 3 cents apiece, and 6 apples at 2 cents apiece? 7. 8 + 2 + 5 = ? 17— (8 + 4)=? 25— (5 + 2)=? 8. (3X2) + (7X3) = ? 27^(3X3) = ? 9. At 5 dollars apiece, how many hats can be bought for 25 dollars? 10. In one gallon there are 4 quarts. How many quarts are there in 4 gallons? -11. If one fish weighs 9 pounds, how many such fish will weigh 81 pounds? 81 ^9=? 12. If there are 3 joiners' planes on one bench, 8 on another, and 5 on another, how many planes are there on the 3 benches? -- 13. In a class, 36 questions were answered, each pupil answering 4 questions. How many pupils were there in the class? 36^4 = ? 90 FIRST LESSONS 14. There are 9 pins in the pin-cushion, 12 on the paper, and 5 in the box. How many pins are there in all? 9 + 12 + 5 = ? 15. A tailor earned 12 dollars one week, and 15 dollars the next, and after paying his expenses had 8 dollars left. How much money did he expend? (12 + 15) — 8-=? 16. How many more than 5 times 4, is 30? 17. How many more than 6 times 3, is 24? 18. How many are 7 + 3 + 5 divided by 3? Di- vided by 5? 19. How many are 7 + 9 + 2 divided by 6? Di- vided by 3 ? Slate Exercises. Copy and add the following : 32235537 5 435287626 7445645 17 679442442 Copy and multiply the following : 867987698 595388749 Copy and divide the following : 6 )36 8 )48 8 )72 9)81 7 )70 24-^6 36-^4 27-4-9 35^5 42^6 50-^5 56^8 60-^6 IN AKITHMETIC 91 REVIEW EXERCISES Find the value of the following : 81^9 45^5 36^6 72^8 40^5 27^3 18-^6 24^4 20-^2 32^8 63^7 60-4-6 28^4 18-^9 25-^5 15^3 18^2 36^4 28-^7 30 -f- 6 64^8 81-^9 20-^2 56^8 30-^3 16-^-2 45^5 24^8 36^9 32^4 35^5 48^6 35 -f- 7 40^8 32^4 12^2 35^5 16^4 27-^9 24^3 18^2 64^8 54^6 45^5 63^9 35^7 28-^4 27^3 Find the value of the following (25 — (36 + (41 + (51 + (67 + (18 + (28 + (29 + (13 + (60- (47- (33 + 5) 6) 7) 3) 5) 3) 4) 7) 3) 4) 2) 3) (37-2) (49-7) (33 + 3) ( 80 — 8 ) ( 60 — 6 ) (33 + 7) (18 + 3) (22 + 5) ( 43 + 6 ) (21+3) (31 + 5) (32 + 3) ^5 21^ ^7 36^ f-4 48^ ^-9 32^ ^6 35 4 i-8 28^ f-3 27-4 f-9 63^ f-7 45 4 ^6 56^ ^-4 49^ f-5 72-^ 28 H- 4 21^7 40 H- 5 54 H- 9 56^8 16^2 36^6 46^9 30^5 36^4 48-4-8 24 H- 6 (5 + 2) (7 + 2) (2 + 4) (3 + 5) (9-2) (8-4) (6-3) (7 + 2) (2 + 3) (3 + 4) (2 + 5) (6 + 3) 92 FIRST LESSONS REVIEW EXERCISES, Copy and find the value of the following (48^8)^2 (6+15)-^ -(6-3) 4+ (16^4)— 6 (81-^-9)-i-3 (84-24)^ -(9-1) 7+(35-^-5)— 4 (56^7)-^4 (7+18)-. -(7-2) 8-(36-f-9) + 2 (70-^7)^5 (8+28)- -(5+4) 9+(81-^9)-6 (36-h6)^3 (9+36)- -(3+2) 7+(32^4)-3 (20^5)-^2 (8+40)- -(7-1) 3+(49-i-7)— 5 (48-f-6)^4 (6+30)- -(8-4) 6+(64-^8)-4 (54-^6)-^3 (3+27)- ^(7+3) 5-(45-^5)+4 (36^9) -^2 (8+24)- K3+5) 9— (42^6)— 2 (70-^7)-^2 (7+43)- ^(2+3) 8+(16^4)— 6 (54-^9)-^3 (9+40)- -(5+3) 9+(21-^7)-6 (64-v-8)^4 (6+54)- K5+1) 3+(64^8)-7 (56-i-7)^2 (8+72)- ^(7+3) 3+(63-^-9)— 9 (72-^-8) -3 (9+47)^(2+5) 8+(60-^6)-5 Copy and find the value of the following 45-^5=-6+ ? 27-^ 3==6+ ? 45- -?=6+3 32-f-4==5+ ? 54-^ 6-^5+? 36- -?=4+2 81-^9-=3+? 80-f- 8=7+ ? 72- - ?=6+3 72h-9-:6+ ? 63-^ 9=3+? 70- -?=6+4 60-^6=8+ ? 21 : 3—4+? 42- - ?=3+4 49^7=4+ ? 14-f- 2=6-F? 40- -?=l+3 36-^4=3+? 64-^ 8=5+? 54-. -?=4+2 27 : 3—4+ ? 35^ 5=5+? 48- -?=5+3 164-2-=6+? 42^ 7=3+? 35- -?=6+l 24-^6-=3+ ? 48^ 6=2+? 70- - ?=8+2 IN ARITH METIC 93 P^^^^^^ 1. Into how many parts has one of the apples in the picture been divided? 2. How do these two parts compare in size? 3. What is one of these two equal parts called? One-half. 4. How many halves of the apple make the whole apple ? 5. Into what number of parts is one of the pears divided ? 6. How do these three parts compare in size? ~ ~ 7. What is one of these three equal parts called? One-third. 8. What are two of these three equal parts called ? 9. How many thirds are there in the whole pear? 10. Into how many parts has one of the oranges in the picture been divided? 11. How do these four parts compare in size? 94 FIEST LESSONS 12. What is one of these four equal parts called ? One-fourth, 13. What are two of the parts called? What are three of the parts called? 14. How many fourths are there in the whole orange ? One or more of the equal parts of any thing is called a Fraction, Two numbers written one above the other, with a line between them, are used to express a fraction. The number below the line shows the number of equal parts into which the thing is divided. It is called the Denominator, The number above the line shows how many parts form the fraction. It is called the Numerator, One-half is therefore expressed thus : \. One-third is therefore expressed thus: \. One-fourth is therefore expressed thus: \. Two-fourths is therefore expressed thus : J. Three-fourths is therefore expressed thus: J. Slate Exeecises. Copy and read the following : i * i I I Express in figures the following : One-half Two-thirds One-fourth One-third Three-fourths Two-fourths IN ARITHMETIC. 95 U?f.. I i- I i IT I i I feli- 111 i-l il i I 1. Draw a line and divide it into 5 equal parts. 2. What is one of these parts called ? One-fifth. 3. What are two of the parts called? What are four of the parts called ? 4. How many fifths make the whole line? 5. Draw a line and divide it into 6 equal parts. 6. What is one of these parts called ? One-sixth. 7. What are three of the parts called ? What are five of the parts called ? 8. How many sixths make the whole line? 9. If a pie is divided into seven equal pieces, what part of the pie is one piece ? Three pieces ? Four pieces ? How many sevenths make the whole pie ? 10. If a pound of candy is equally divided among eight boys, what part of the pound will one boy receive ? 11. How many eighths of it will four boys receive ? Three boys ? Six boys ? 12. A miller divided a barrel of flour equally among nine poor families. What part of the barrel of flour did he give to each family? 13. How many parts did he give to five families? To eight families? 14. If a dollar is divided into ten equal parts, what is one of the parts called ? What are foi+r oF=-^ the parts called? W^hat are ten of the pac/^ called? (us 96 FIRST LESSONS Slate Exercises, Copy and read the following : 1 f lift i i i ^ i i All Express in figures the following: 3 fifths 4 fifths 3 eighths 3 sixths 2 thirds 5 eia^hths 5 tenths 8 tenths 6 eighths 2 ninths 7 ninths 5 sevenths 3j#4C 1. Mary had 1 half of a dollar, and her mother gave her 1 half of a dollar more. How much money had she then? 2. How many thirds are 1 third and 2 thirds? 3. If John gave ^ of an orange to his sister, and I to his cousin, how many fourths did he give both ? 4. Henry earned ^ of a dollar in one day, and f of a dollar the next day. How much money did he earn in both days? How many fifths are \ and f ? 5. Jane bought |^ of a yard of ribbon, and Caro- line bought I" of a yard? How many sixths did both buy ? How many sixths are |-, |- and ^ ? 6. A farmer sold |- of a bin of wheat to one man, and f to another. How many sevenths did he sell? 7. How many eighths are f, |- and J? I IN ARITHMETIC 97 8. William bought 2J yards of cloth for a pair of pantaloons, f of a yard for a vest, and 4^ yards for a coat. How many yards of cloth did he buy? 9. A man plowed 1 j^ acres one day, and lA acres the next day. How many acres did he plow ? 10. A laborer bought f of a ton of coal at one time, f of a ton at another, and ^ of a ton at another. How much coal did he buy? Slate Exercises. Copy and add the following: i + i + 4 i + f + l l + f + l 3j^C LESSON LTIII, 1. Lucy had ^ of a dollar, and her brother gave her :^ of a dollar more. How much money did she have then ? 2. How many halves of a dollar are equal to one dollar ? 3. How many fourths of a dollar are equal to one dollar? 4. How many fourths of a dollar are equal to one-half of a dollar? How many fourths are there ini? 9 1 - - ■ 98 FIRST LESSONS 5. How many fourths of a dollar are there in ^ and J of a dollar? How many fourths in ^ and ^? 6. Draw two lines of the same length. 7. Divide the first line into 2 equal parts. 8. Divide the second line into 6 equal parts. 9. What is one part of the first line called ? 10. What is one part of the second line called? 11. What are three parts of the second line called? 12. How do three parts of the second line compare in length with one part of the first line ? 13. How many sixths of a line are equal to one- half of a line ? 14. How many sixths are there in ^ and ^? How many sixths are there in J and |- ? In ^ and f ? 15. How many eighths are equal to one-fourth? Hcfw many eighths are there in ^ and f ? ■ 16. Mary bought 2^ yards of lace one day, and 2^ yards the next day. How many yards did she buy ? 17. John bought 2^ pounds of raisins, and Joseph bought 3f pounds. How many pounds did they buy ? 18. How many ninths are ^ and ^? f and |^? 19. How many eighths are ^ and f ? Slate Exercises. Copy and add the following : i+i+i i+l+l i+i+i *+i+f i+i+i i+l+A IN ARITHMETIC. 99 liESSOlS^ I.IX. 1. Henry had 3 fourths of an orange, but gave 1 fourth of it to his cousin. How much had he left? 2. If a boy had f of a pound of candy, but gave ^ of it away, how much had he left? -| — i = ? 3. If I have f of a dollar, and spend ^ of a dol- lar, what part of a dollar will I have left? 4. A girl having -j^ of a dollar, paid ^^ of a dollar for thread. How much had she left? 5. From ^ take ^. From |^ take ^. 6. If Julia divides a loaf of cake into 8 equal parts, and gives away -| of it, how many eighths will she have left? 7. A man chopped f of a cord of wood in the forenoon, and f of a cord in the afternoon. How much more did he chop in the afternoon than in the forenoon ? f — f ^^ ? 8. Gilbert owns f of a boat, and Carl the re- mainder. What part does Carl own? 9. Oscar bought a pair of skates for -^^ of a dol- lar, and sold them for -^-q of a dollar. What part of a dollar did he lose? -^ — to"^^^ 10. How much more than |^ of a pound is f of a pound? f — f = ? 11. In a school f of the pupils are girls, and the rest are boys. What part of the school is boys ? 12. How many more than 3^ are 8^ ? 13. A grocer having lOf pounds of butter, sold 4 pounds of it. How much had he left? 100 FIRST LESSONS 14. A tailor had 12|- yards of cloth, and sold 3f yards. How many yards had he left? Slate Exercises. Copy and subtract the following : 4 1 7 2. 6. 3 4 1 •6 "6 ¥ 8 9 9 Z 5" 6A-¥o H-H 8f-2J 9f-3f 3j^C 1. Into how many parts is the first melon in the picture cut? How do the parts compare in size? 2. Into how many equal parts is each of the halves of the second melon cut? 3. How many such parts are there in the two halves^ or the whole melon? 4. What are these- parts called? 5. How many sixths are there in one-half? 6. From -l- subtract ^. From ^ subtract f. 7. Draw a line and divide it into 3 equal parts. 8. Draw another line of the same length, and divide it into 3 equal parts. 9. Divide each part of the second line into 3 equal parts. 10. How many of these smaller parts are there in the whole line? What are these parts called? I IN ARITHMETIC. 101 11. How many ninths are there in one third? 12. How many ninths are there in two thirds? 13. From ^ of an orange take f of an orange. 14. Belle having f of a yard of velvet, gave away I of a yard. How much had she left? 15. A grocer who had ^ of a barrel of beans, sold f of a barrel. What part of a barrel did he have left? 16. How many tenths of a dollar are there in one- fifth of a dollar? In two-fifths? 17. From f of an hour take -^ of an hour. 18. A milliner having 4f yards of silk, used 2-| yards. How many had she left? Slate Exercises. Copy and subtract the following: l-i f-f l-i f-A ^-^ H-H 5i-2i 71-lA ^-^ 5i-2i ^-^ 6i-6A 5^#^c LESSON liXIII 1. At |- of a dollar each, how many books can be bought for 1 dollar? How many times is ^ contained in 1? 2. At 2 fifths of a dollar per yard, how much cloth can be bought for 4 fifths of a dollar? How many times are f contained in |^? 3. If a boy can earn 2 eighths of a dollar per day, how long will it take him to earn 6 eighths of a dol- lar? How many times are f contained in f? 4. If a man can mow f of an acre in one hour, how long Will it take him to mow j- of an acre? How many times are f contained in -f^? 5. A boy having f of a pound of candy, divided it equally among his playmates, giving to each f of a pound. How many playmates had he ? 6. A grocer having ^ of a barrel of vinegar, sold 3^ of a barrel each day until it was all sold. In how many days did he sell it? IN AKITHMETIC. 105 7. How many times are -^ contained in 3^, or what is the vahie of 9 tenths -^ 3 tenths? 8. How many times are -^ contained in 3^, or what is the value of it "^ IT ^ 9. How long will it take a boy to earn ^ of a dollar, by earning ^ of a dollar per day? How many times are -fy contained in 3^ ? -fs "^ A ^^ ^ 10. How many times are f contained in ^? 3^ in M? A in if? A in if? ^ in H? 11. How long a time will be required to fill a cis- tern, if ^ of it is filled every hour ? lor^-^-^ = ? Slate Exercises. Find the value of the following : ¥ 2 3>®^C liESSOK I.XIV. 1. If William divides 4 oranges equally between his 2 sisters, how many will each receive ? 2. What is one-half of 4 cents ? What is one-half of 4? 3. By what number do you divide to find one-half of any number ? 4. How many pears is 1 third of 3 pears ? 5. How many pears are 2 thirds of 3 pears? ^1- ¥^*- ¥^l- ¥-^l ^i- ¥^i- ¥^f- ¥-^1 H- i^h 8^1. 7-^1 106 FIRST LESSONS 6. By what number do you divide to find one- third of any number? How do you find 2 thirds of a number? 7. How many roses are one-half of 8 roses? 8. If a woman divides 5 loaves of bread equally between 5 persons, what part will each person receive ? 9. How many loaves are 3^ of 5 loaves? f of 5 loaves? f of 5 loaves? 10. How is ^ of a number found ? f of a number ? 11. If 14 marbles are equally divided between 2 boys, how many marbles does each boy receive? 12. How many are ^ of 14 ? How many are ^ of 14? 13. One-fourth of 20 cents is how many cents? 14. By what number do you divide to find one- fourth of any number ? 15. A man shot 12 pigeons and sold ^ of them. How many had he left? 16. A boy saAv 18 fowls, |- of which were white. How many were white ? 17. A boy having 40 cents, paid 4 of his money for pencils and -^ of it for paper. How much did he have left? 18. A farmer sold 7 bushels of wheat for 28 dollars. What did he receive for one bushel? 19. How many bushels are ^ of 28 bushels? 20. How many bushels are ^ of 28 bushels? 21. What is J^ of 20? What is | of 56 ? 22. If 25 dollars is divided equally among 5 men, what part of the money will each receive? How much will each receive? IN ARITHMETIC. 107 ^ NGTATiONA NUMERATION tif LESSOIS^ I.XV. A single thing is called a Unit, How many units make one ten? Two tens? Copy and read 10 12 16 64 27 73 35 42 90 59 99 4. Express in figures the following : Ten. Forty-eight. Eighty -five. Thirteen. Sixty-fi3ur. Ninety. Twenty-five. Seventy-nine. Ninety-three. Thirty-two. Eighty-one. Ninety-nine. 5. Express in figures the One ten, no units. Five tens, six units. Six tens, four units. Two tens, seven units. Seven tens, one unit. Three tens, five units. Eight tens, six units. Foui; tens, five units. Nine tens, nine units. following : One ten, one unit. Five tens, two units. Six tens, six units. Two tens, one unit. Seven tens, three units. Three tens, four units. Eight tens, one unit. Four tens, six units. Nine tens, one unit. 108 FIRST LESSONS 6. When two figures are written side by side what does the one at the right represent? What does the one at the left represent? 7. How many tens and units in ninety-nine ? 8. If one is added to the 9 units, what is the result? 9. How many are 9 tens and 1 ten ? 10. How many tens make one hundred ? 11. Copy and reac 1: 100 104 108 116 138 149 157 168 173 123 188 199 205 316 423 216 304 229 12. Express the following in figures ; One hundred, two tens, and five units. Four hundreds, five tens, and no units. Seven hundreds, five tens, and six units. Eight hundreds, nine tens, and one unit. Three hundreds, nine tens, and eight units. Five hundreds, three tens, and seven units. Two hundred. Nine hundred. Two hundred, forty. Nine hundred, forty. Eight hundred, five. Six hundred, seventy. Three hundred, fifty. Five hundred, eighty. Three hundred, forty. Five hundred, twenty. Five hundred, ninety. Seven hundred, thirty. 13. When three figures are written side by side, what does the one at the right represent? ^ The one next to it? The one at the left? IN . A.IIITHMETIC . Copy and read : 314 268 572 681 372 465 200 2G0 350 580 673 856 371 426 842 730 307 406 184 755 862 985 422 999 109 15. Express in figures the following: Two hundred, ten. Five hundred, fifty. Two hundred, forty. Seven hundred, five. Eight hundred, four. Nine hundred, fifteen. Five hundred, sixteen. Eight hundred, twelve. Five hundred, fifty-five. Four hundred, sixty-six. Nine hundred, forty -five. Six hundred, forty-seven. Eight hundred, fifty-four. Seven hundred, forty-one. Nine hundred, ninety-nine. Eight hundred, sixty-three. Three hundred, ninety-nine. Seven hundred, twenty-nine. The method of expressing numbers by figures is called the Arabic notation. The method of reading numbers is called NumeV" ation. 110 FIRST LESSONS LESSOIS^ I.XTI. 1. If 1 be added to nine hundred ninety-nine, how many hundreds will there be? 2. To what are 10 hundreds equal ? One thousand, 3. One thousand is expressed thus: 1^000, 4. Copy and read the following : 1,000 1,004 1,016 1,323 1,548 2,000 2,159 3,872 5,625 7,822 8,465 6,372 4,189 9,786 9,999 5. Express in figures the following: Seven thousand, two hundred, fifty-six. Nine thousand, nine hundred, ninety-five. Five thousand, three hundred, twenty-eight. 6. How many are 9 thousand and 1 thousand? 7. Ten thousand is expressed thus: 10^000, 8. Copy and read the following: 10,000 18,000 ' 22,005 35,012 87,482 96,856 13,729 88,543 9. Express in figures the following: Ninety thousand, nine hundred, fifty-six. Eighty -four thousand, five hundred, forty. Seventy-five thousand, eight hundred, fifty. Fifty-three thousand, two hundred, twenty. Eighty thousand, seven hundred, forty-four. Ninety thousand, nine hundred, ninety-nine. IN ARITHMETIC. Ill 10. How many are 90 thousand and 10 thousand? 11. One hundred thousand is expressed thus: 100,000. liESSON I.XTII. 1. Copy and read the following: " 100,000 120,000 562,000 159,725 540,805 723,016 986,784 684,379 2. What do the three figures at the right repre- sent? Units, tens, and hundreds, 3. What do the three figures at the left repre- sent? Units, tens, and hundreds of thousands. 4. A group of figures containing the units, tens, and hundreds of any denomination is called a Period. 5. How many periods are there in the number 132,654? 6. What is the denomination of the first period? 7. What is the denomination of the second period? 8. Copy and read the following: 354,728 639,955 845,762 876,953 423,872 635,476 82,675 400,800 8,005 264,361 315,008 216,040 9. How many figures must there be in a period? 10. How are the periods separated from each other ? 112 FIEST LESSONS Express in figures the following: 11. Two hundred eight thousand, five hundred sev- enty-four. 12. Seven hundred fifty-four thousand, two hundred twenty-five. 13. Eight hundred sixty-three thousand, four hun- dred eighty-nine. 14. Eight hundred sixty-five thousand, three hun- dred thirty-seven. 15. Eighty-four thousand, five hundred eighty-four. 16. Nine hundred thirty-three thousand, six hun- dred twelve. 17. Three hundred thousand, six hundred. 18. Eight hundred ninety-four thousand, six hun- dred twenty-four. 19. Five hundred eighty thousand, seven hundred ninety-seven. 20. Three hundred eighty-nine thousand, seven hundred eighty-one. 21. Seven hundred eighteen thousand, nine hundred eighty-seven. 22. Six hundred thousand, one. 23. Six hundred twenty-four thousand, three hun- dred twenty-nine. 24. Eight hundred thirty-nine thousand, five hun- dred sixteen. 25. Nine hundred, thirty-five thousand, eight hun- dred twenty-six. 26. Nine hundred sixty-eight thousand, nine hun- dred thirty-seven. IN ARITHMETIC. 113 -^=7^^ LESSON I.XVIII. 1. Helen solved 6 examples, Martha solved 5, and Jennie solved 9. How many examples did they all solve ? 2. How many goblets are 3 goblets, 8 goblets, and 9 goblets? 3. James learned 5 lessons on Monday, 6 lessons on Tuesday, and 3 lessons on Wednesday. How many lessons did he learn in the three days? 4. How many marbles are 5 marbles, 4 marbles, 6 marbles, and 2 marbles? 5. After spending 2 cents for a slate-pencil and 5 cents for a lead-pencil, James had 5 cents left. How much had he at first? 6. How many dollars are 10 dollars and 8 dollars? DEFINITIONS. The process of finding one number that is equal to two or more numbers is called Addition, The result in addition is called the Sum, The Sign of Addition is an upright cross : -f- , called plus. The Sign of Equality is two short horizontal lines: =. 10 114 FIRST LESSONS 1. What is the sum of 5, 4, 7, and 6 ? PROCESS. Explanation. — For convenience the numbers n to be added are arranged so that tliey stand in a column. "* We begin at the bottom to add, thus: 6, 13, 7 17, 22, and write the sum beneath. /» Hence, the sum is 22. In adding say 6, 13, 17, 22, instead of 6 and 7 22 are 13 and 4 are 17 and 5 are 22. Slate Exercises. Copy and add the following : (2.) (3.) (4.) (5.) (6.) 4 5 6 8 3 2 6 5 2 2 3 3 3 3 5 5 4 2 3 1 7. William shot 6 squirrels, Henry shot 8, John shot 5, and James shot 7. How many squirrels did they shoot? 8. Oscar owned 4 cows, Herbert owned 5, George owned 3, and Clarence owned 6. How many cows did the four boys own? 9. A merchant received 5 boxes of oranges one day, 6 the next, and 4 the next. How many boxes of oranges did he receive? 10. A farmer sold 6 cords of wood to one man, 3 cords to another, 8 to another, and 2 to another. How many cords of wood did he sell ? IN ARITHMETIC. 115 Slate Exercises. Copy and add the following : (11.) (12.) (13.) (14.) (15.) 4 6 3 4 6 7 5 6 7 2 8 7 9 2 3 2 8 2 3 9 3 5 6 5 3 4 1 2 4 2 3j^C liESSON I.XIX. 1. What is the sum of 217, 853, and 365? PROCESS. Explanation. — For convenience the num- o 1 Y bers to be added are arranged so that units Q ^ ey stand under units, and tens under tens, etc. Beginning at tiie right hand, each column *^ o " ' is added separately. 2^ z|. 3 5 Thus, 5 + 3-1-7 = 15, the sum of the units. 15 units are equal to 1 ten and 5 units. The 5 is written under the column of units, and the 1 is reserved to add with the tens. 1 reserved +6 + 5 + 1=13, the sum of the tens. 13 tens are equal to 1 hundred and 3 tens. The 3 is written under the column of tens, and the 1 is reserved to add to the column of hundreds. 1 reserved +3 + 8 + 2 = 14, the sum of the hundreds. 14 hundreds are equal to 1 thousand and 4 hundreds, which are written in thousands' and hundreds' places in the sum. In adding, name results only. Thus, instead of saying, 5 and 3 are 8 and 7 are 15, say 5, 8, 15. 116 first lessons Slate Exercises. Copy and add the following : (2.) (3.) (4.) (5.) 678 852 338 852 153 365 466 738 261 472 ^53 645 (6.) (7.) (8.) (9.) 235 512 832 893 415 416 142 416 268 304 624 842 317 497 615 249 10. A man traveled 250 miles by railroad, 325 by steamboat, and 190 by stage. How many miles did he travel ? 11. What is the sum of three thousand five hundred twenty-six, eight thousand one hundred twelve, six thousand nine hundred eighteen? 12. A butter-buyer bought 326 pounds of butter from one dairy, 518 pounds from another, and 168 pounds from another. How many pounds of butter did he buy? 13. Find the sum of six hundred seventy-nine, three hundred twenty-eight, and fourteen. 14. Three persons deposited money in a bank. The first put in 192 dollars, the second 466 dollars, and the third 825 dollars. How much money did they deposit ? I IN ARITHMETIC. 117 15. A man paid 400 dollars for a span of horses, 150 dollars for a carriage, and 75 dollars for a harness. How much money did he pay for all ? Copy and add: (16.) (17.) (18.) 6,231 27,638 25,789 1,354 13,925 32,644 4,872 53,164 58,793 3,125 41,538 62,461 3,258 31,604 58,261 4,123 58,643 21,469 (19.) (20.) (21.) 6,843 13,462 46,845 2,754 26,541 27,891 6,281 32,716 84,263 4,156 42,683 38,419 7,123 94,275 32,981 8,142 24,239 19,497 (22.) (23.) (24.) 4,265 68,465 48,634 3,789 31,892 31,299 4,573 47,468 87,109 4,684 39,210 46,296 4,591 40,009 37,415 8,743 32,567 45,984 4,268 26,849 51,846 5,745 45,265 74,695 118 FIRST LESSONS SUBTRACTION liESSOI^ I.XX 1. Herbert earned 19 dollars and gave 7 dollars for a coat. How much had he left? 2. Mary painted 18 pictures and sold 6 of them. How many had she left? 3. If a man earns 15 dollars a week and uses 8 dollars, how many dollars will he have left? 4. A grocer having 22 pounds of lard, sold 11 pounds of it. How many pounds had he left? 5. William had 12 marbles, but gave away 5 of them. How many had he left? 6. 25 pencils are how many more than 10 pencils? 7. From a piece of cloth containing 22 yards, 16 yards were cut. How many yards remained? 8. From a cistern containing 40 gallons of water, 10 gallons were drawn out. How many gallons re- mained ? 9. There are 24 pupils in a school, of whom 13 are boys. How many are girls? 10. One piece of cloth contained 26 yards, and another contained 13 yards. How many yards more were there in the larger piece ? 11. A farmer who had 23 horses, sold 11 of them. How many horses had he left? IN ARITHMETIC. 119 DEFINITIONS. The process of finding the difference between two numbers is called Subtraction. The greater of the two numbers is called the Minuend. The lesser of the two numbers is called the Sub- trahend. The result obtained by subtracting is called the Meniainder or Difference, The Sign of Subtraction is a short horizontal line : — , called minus. 1. From 768 subtract 425. PKOCESS. Explanation, — For convenience the less - ^ „ number is written under tlie greater, units under units, tens under tens, etc. 4 2 5 Beginning at the right hand, each figure 3 4 3 ^^ ^^^^ subtrahend is subtracted from the cor- responding figure of the minuend. 8 units minus 5 units equal 3 units, which are written under the units. 6 tens minus 2 tens equal 4 tens, which are written under the tens. 7 hundreds minus 4 hundreds equal 3 hundreds, which are written under the hundreds. Hence the remainder is 343. Slate Exercises. Copy and subtract the following : (2.) (3.) (4.) (5.) (6.) 635 876 957 543 665 421 453 835 320 543 120 FIUST LESSONS I.ESSOX LXXI. 1. From 7425 subtract 2536? PROCESS. Explanation. — For convenience the .,. , ^ . f^ r less number is written under the greater. Minuend 7425 .^ , .^, ,, ,,/ units under units, tens under tens, hundreds Subtrahend 2536 under hundreds, etc. Remainder 4 8 8 9 Beginning at the right hand, each fig- ure of the subtrahend is subtracted from the corresponding figure of the minuend. Since 6 units can not be subtracted from 5 units, 1 ten is united with the 5 units. 1 ten is equal to 10 units. 10 units and 5 units are 15 units. 6 units from 15 units leave 9 units. The 9 is writ- ten in units' place. Since 1 of the tens was united with the units, there is but 1 ten left. Because 3 tens can not be subtracted from .1 ten, 1 hundred is united with the 1 ten. 1 hundred is equal to 10 tens. 10 tens and 1 ten are 11 tens. 3 tens from 11 tens leave 8 tens. The 8 is writ- ten in tens' place. Because 5 hundreds can not be subtracted from 3 hundreds, 1 thousand is united with the 3 hundreds. 1 thousand equals 10 hundreds. 10 hundreds and 3 hundreds equal 13 hundreds. 5 hun- dreds from 13 hundreds leave 8 hundreds. The 8 is expressed in hundreds' place. Since 1 thousand was added with the hundreds, there are but 6 thousands left. 2 thousands from 6 thousands leave 4 thousands. The 4 is written in thousand's place in the remainder. Hence the remainder is Slate Exercises. Copy and subtract the following : (2.) (3.) • (4.) (5.) (6.) 6825 8541 3412 1694 3728 4376 7285 1589 469 1963 I IN ARITHMETIC. 121 Copy and subtract the following : (7.) (8.) (9.) (10.) (11.) 4328 6351 4211 6253 4101 2579 5472 3587 4897 2853 12. A paid 8,728 dollars for his farm, and B paid 5,935 dollars for his. How much more did A^s farm cost than B\s? 13. There were manufactured in a cotton factory 1,857 yards on Monday, and 2,139 yards on Tuesday. How many yards more were manufactured on Tues- day than on Monday ? 14. A man who owned 8,754 acres of land, sold 3,215 acres to one company, and 2,154 acres to an- other. How many acres were unsold? 15. From eight thousand, five hundred sixteen, subtract three thousand, six hundred twenty-four. 16. A mill was purchased for 13,850 dollars, and afterward sold at a loss of 2,175 dollars. How much was received for it? Copy and subtract the following: (17.) (18.) (19.) (20.) (21.) 15,862 14,684 28,465 35,469 99,846 9,839 13,299 13,984 24,983 31,985 (22.) (23.) (24.) (25.) (26.) 85,724 27,869 35,321 73,814 125,269 46,689 13,780 15,094 15,209 48,736 li 122 FIRST LESSONS MULTIPLICATION --^- LESSON I.XXII. 1. What will 9 sheep cost at 6 dollars a head ? 2. What will 10 quarts of milk cost at 6 cents a quart ? 3. What will be the cost of 4 wagons at 30 dol- lars apiece? 4. Since 12 inches make one foot, how many inches are there in 3 feet? 5. At 11 cents a quart, what will 9 quarts of ber- ries cost? 6. If a laborer can work but 13 days per month, how many days can he work in 3 months? 7. In one week there are 7 days. How many days are there in 12 weeks? 8. If one hogshead of sugar is worth 40 dollars, what are 2 hogsheads of sugar worth ? 9. If one apple-tree yields 14 bushels of apples, how many bushels will 2 such trees yield? 10. If a man plants 20 apple-trees in one row, how many trees will he plant in 5 such rows? 11. When loaf-sugar is worth 15 cents a pound, how much will 4 pounds cost? 12. What will 9 dozen of peaches cost at 10 cents per dozen? I IN ARITHMETIC. 12»3 13. What will be the cost of 3 copy-books at 20 cents each? 14. What will be the cost of 4 yards of muslin at 30 cents a yard ? 15. If you give 25 cents to each of 3 beggars, how much will you give them? DEFINITIONS. The process of taking one number as many times as there are units in another, is called Jlultiplication, The number to be taken or multiplied is called the Mu ltix> Ilea iid. The number which shows how many times the multiplicand is taken is called the 3£ultiplie7\ The result obtained by multiplying is called the JProduct. The Sign of Multiplication is an oblique cross : X . It is read times or multiplied by, 1. Multiply 421 by 3. ■ PROCESS. Explanation. — For convenience the i ct 1 multiplier is written under the multipli- Multiplicand 421 jr.-- ... -ixi-ji cand. Beginning at the right hand each Multiplier 3^ figure of the multiplicand is multiplied Product 12 6 3 ^y t^^e multiplier. Thus 3 times 1 unit are 3 units. The 3 is written in units' place in the product. 3 times 2 tens are 6 tens. The 6 is written in tens' place in the product. 3 times 4 hundreds are 12 hundreds, which equal 1 thousand and 2 hundreds. The 1 thousand is written in thousands' place and the the 2 hundreds in hundreds' place in the product. Hence the product is 1263. 124 first lessons Slate Exercises. Copy and multiply the following : (2.) (3.) (4.) (5.) 6102 5233 5212 7342 4 3 3 2 (7.) (8.) (9.) 5234 3123 4123 2 3 3 liESSOX I.XXIII 1. How many are 8 times 196? PROCESS. Explanation. — For convenience the mul- 1 Q ^ tiplier is written under the multiplicand. Be- ginning at the right each figure of the multi- ^ plicand is multiplied by the multiplier. ;[ 5 g g Thus, 8 times 6 units are 48 units. 48 units are equal to 4 tens and 8 units. The 8 is written in units' place in the product, and the 4 is reserved to add with the tens. 8 times 9 tens are 72 tens, plus the 4 tens reserved, are 76 tens. 76 tens are equal to 7 hundreds and 6 tens. The 6 is written in tens' place in the product, and the 7 is reserved to add with the hun- dreds. 8 times 1 hundred are 8 hundreds, plus 7 hundreds reserved, are 15 hundreds. 15 hundreds are equal to 1 thousand and 5 hundreds, which are written in hundreds' and thousands' places in the product. Hence the product is 1568. The accuracy of the work may be tested by reviewing it. IN ARITHMETIC. 125 Copy and multiply : (2.) (3.) (4.) (5.) 1345 3824 6325 7863 6 7 (7.) 4 (8.) 5 (6.) (9.) 7581 6274 6235 7341 3 9 8 2 10. If a ship sails 368 miles in one week, how many miles will she sail in 7 weeks? 11. There are 5280 feet in a mile. How many feet are there in 9 miles? 12. What will 6 yoke of cattle cost at 184 dollars a yoke? 13. If 1 horse is valued at 275 dollars, what is the value of 6 such horses? 14. If a ship sails 895 miles in 1 week, how far will she sail in 9 weeks? 15. What is the product of 457 multiplied by 7? 16. What is the product of 784 multiplied by 4? 17. If a clerk receives a salary of 972 dollars a year, how much will he receive in 5 years? 18. Since there are 365 days in 1 year, how many days are there in 9 years? 19. Multiply 3452 by 3. Multiply 1472 by 5. 20. Multiply 6841 by 6. Multiply 5261 by 7. 21. Multiply 7186 by 3. Multiply 9167 by 8. 22. Multiply 2041 by 8. Multiply 9380 by 9. 126 FIRST LESSONS, 23. Multiply 4926 by 6. Multiply 4829 by 7. 24. Multiply 3624 by 4. Multiply 1684 by 2. 25. If it takes 5428 shingles for the roof of a house, how many shingles will it take for 5 such houses? Copy and multiply : (26.) (27.) 4753 6529 5 3 (28.) 8881 6 (29.) 9573 4 (30.) 2469 8 (31.) 6382 6 (32.) 4629 7 (33.) 3265 9 (34.) 8475 8 (35.) 8463 4 3^#^C liESSON^ I.XXIV. 1. Multiply 435 by 37. FIRST PROCESS. EXPLANATION. — The numbers are written as before. Since in multiply- "^^^ ing we multiply by the parts of a 3 7 number, to multiply by 87 we multi' 7v425= 2975 ^^^ ^^ ^^ ^"^ ^^ ^' ^"^ ^^^ ^^^ ^^' 30X425 --^ 12750 7 ^j^^^g 425 ^re 2975, the first par- 37X425=:15725 tial product ; 30 or 3 tens times 425 are 1275 tens or 12750, the second partial product. The sum of these partial products will be the entire product. Hence the entire product is 15725. IN ARITHMETIC. 127 SECOND PROCESS. EXPLANATION. — In this process the cipher at the right of the partial prod- ^ uct is omitted and the significant figures 3 7 placed in their proper places. Thus, in 7y42.5= 2975 multiplying by 3 lois, the product is _ ^' 1275 tens or 12 thousands 7 hundreds o A 4 -^ L Z i ^^^ g j.gj^g^ which are written in their 15 7 25 proper places in the partial product. In multiplying by tens the lowest denomination of the product is tens, in multiplying by hundreds it is hundreds, by thousands it is thousands. Hence, The first figure of each product should he placed under the figui^e by which we multiply. Slate Exercises. (2.) Multiply 3468 By 12 (3.) 4126 31 (4.) 7824 23 (5.) 6846 43 (6.) 7125 37 (7.) Multiply 4685 By 76 (8.) 7235 83 (9.) 4986 49 (10.) 5843 121 (11.) 1987 316 (12.) Multiply 5964 By 73 (13.) 6842 39 (14.) 2723 64 (15.) 8962 135 (16.) 8462 216 (17.) Multiply 8964 By 62 (18.) 3562 36 (19.) 7825 47 (20.) 3265 234 (21.) 9268 327 128 FIRST LESSONS uMMm D I V I S I O N tzz=i 1. How many barrels of flour at 8 dollars a barrel, can be bought for 72 dollars? 2. If there are 3 feet in a yard, how many yards are there in 27 feet? 3. At 4 dollars a cord, how many cords of wood can be bought for 40 dollars? 4. If a man can walk 4 miles per hour, how long will it take him to walk 36 miles? 5. If 5 paces are a rod of distance, what is the length in rods of a piece of land which is 30 paces long ? 6. If paper is put in packages containing 6 sheets each, how many packages can be made out of 48 sheets ? 7. If 5 bushels of wheat make a barrel of flour, how many barrels of flour can be made from 40 bushels of wheat? 8. At 6 dollars a ton, how many tons of hay can be bought for 54 dollars? 9. If a man can build 7 rods of fence in one day, in how many days can he build 35 rods? 10. If a teacher earns 9 dollars a week, in how many weeks can she earn 90 dollars? . IN ARITHMETIC. 129 DEFINITIONS. The process of finding how many times one num- ber is contained in another is called JDivision, The number to be divided is called the Divi- dend. The number by which the dividend is to be di- vided is called the Divisor. The result in division is called the Quotient, The part of the dividend remaining when the di- vision is not exact is called the Remainder, The Sign of Division is -f-, read divided by. Til us, 24 -f- 6 is read, 24 divided by 6. 1. How many times is 3 contained in 369? FIRST PROCESS. EXPLANATION. — For Convenience Divisor. Dividend. Quotient. the divisor is written at the left of the 3)36 9(100 dividend, and the quotient at the right. Beginning at the left we divide. 3 is contained in 369 I hundred times and a remainder. The 100 is placed in the quotient, and the divisor is multiplied by it. The product 300 is written under the dividend. Sub- tracting there is a remainder of 69. 3 is contained in 69, 20 times and a remainder. The 20 is placed in the quotient and the divisor multiplied by it. The product 60 is written under the partial dividend. Subtracting the remainder is 9. 3 is contained in 9, 3 times and no remainder. The 3 is placed in the quotient, and the divisor multiplied by it. The product 9 is writ- ten under the partial dividend. Subtracting, there is no remainder. The true quotient is therefore the sum of the partial quotients, which is 123, and 3 is contained in 369, 123 times. 300 20 69 3 60 123 9 9 130 FIRST LESSONS SECOND PROCESS. EXPLANATION. — The second process is •§ g i the same as the first, except that the ciphers QNQAQ/'IOQ ^^^ omitted from the right of the quotient o)oov[iAo figures and partial products, the value of ^ figures being indicated by their position. g Thus, the figures of the quotient are 1 hun- Q dred, 2 tens, and 3 units, which are written — — ' in succession so that each figure represents " its proper value. ^ The products 3 hundreds, 6 tens and 9 units are placed under hundreds, tens, and units in the partial dividends. Slate Exercises. Copy and divide the following : (2.) (3.) (4.) (5.) 4)488( 5)550( 3)699( 2)264( liESSOI^ I.XXVI. 1. How many times is 6 contained in 1398 ? PROCESS. Explanation. — For convenience the di- fi^l ^98^2*^'^ visor is written at the left of the dividend -I 9 and the quotient at the right. Beginning at the left of the dividend we divide. 1 «^ 6 is not contained in 1 thousand any 1 8 thousand times, therefore the quotient can 2 g not be higher than hundreds. Hence we -I Q must find how many times 6 is contained in all the hundreds of the dividend. 1 thousand is equal to 10 hundreds. 10 hundreds plus 3 hundreds equal 13 hundreds. 6 is contained in IN ARITHMETIC, 131 13 hundreds 2 hundred times and a remainder. The 2 is written in hundreds' place in the quotient and the divisor multiplied by it. Subtracting this product from the partial dividend, 13 hundreds, there is a remainder of 1 hundred. 1 hundred remainder is equal to 10 tens. 10 tens plus 9 tens equal 19 tens. 6 is contained in 19 tens 3 tens times and a re- mainder. The 3 is written in tens' place in the quotient, and the divisor multiplied by it. Subtracting this product from the partial dividend, 19 tens, there is a remainder of 1 ten in the quotient. The 1 ten remainder equals 10 units. 10 units plus 8 units equal 18 units. 6 is contained in 18 units 3 times and no re- mainder. The 3 is written in units' place in the quotient, and the divisor multiplied by it. Hence the quotient is 233. Slate Exeecises. Copy and divide the following: (2.) (3.) (4.) 5)32860( 8)72880( 6)63546( (6.) 4) 87564 ( (10.) 6) 38412 ( (7.) 7) 67326 ( (11.) 5) 42865 ( (8.) 9) 27819 ( (12.) 8) 32848 ( (5.) 3) 56841 ( (9.) 7) 58562 ( (13.) 4) 48264 ( (14.) (15.) (16.) (17.) 7) 54285 ( 9) 32616 ( 8) 42856 ( 9) 49842 ( The solution of examples having a small divisor, may be shortened by performing the multiplications and subtractions without waiting the results. This process is called Short Division. 132 FIRST LESSONS PROCESS. 6)1398 233 The solution of Example 1, by short division, is: Explanation. — 6 is contained in 13 hun- dreds 2 hundred times and 1 hundred remain- der. The 2 is written in the quotient under units of the same order in the dividend. 1 hundred remainder and 9 tens make 19 tens. 6 is contained in 19 tens 3 tens times and 1 ten remainder. The 3 is written in the quotient under tens in the dividend. 1 ten remainder and 8 units make 1