^ A^^r^ y^r^ ^w^ Chicago, ^ 
 
 1870. 
 
 =^<2>*= 
 
Ivisofiy Blakenian^ Taylor dr» Co.^s Publications. 
 
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ROBINSON'S MATHEMATICAL SERIES. 
 
 -+* ■- : 
 
 FIRST LESSONS 
 
 m 
 
 MENTAL AND WRITTEN 
 
 ARITHMETIC 
 
 ON THE OBJECTIVE METHOD. 
 
 EDITED BY 
 
 SAMUEL D. BAEE, A.M. 
 
 NEW YOEK: 
 
 IVISON, BLAKEMAN, TAYLOR, & COMPANY, 
 
 138 & 140 Grand Street. 
 
 CHICAGO: 133 & 135 STATE STREET. 
 
 1871. . 
 
^5- 5-6" 
 
 Entered according to Act of Confess, in the year 1870, by 
 
 D. W. FISH, A. M., 
 In the office of the Librarian of Congress, at Washington. 
 
 EDUCATION DEPT. 
 
 Smith & McDougal, Electrotypers, 82 & 84 Beekman Street, New York. 
 
PEEF ACE. 
 
 The author calls attention to the following points as among the 
 claims made for this book. 
 
 1. It treats Number objectively y and by a method believed to be 
 new and simple. 
 
 2. The principles and processes are unfolded in natural order^ 
 as occasion demands them. 
 
 3. The greatest pains have been taken in the attempt to present 
 clearly the most elementary and fundamental principles, in the 
 firm belief that no sound scholarship can be erected on any other 
 basis, and that the elements are not only the most important but 
 are also the most difficult to teach successfully. If a pupil ever 
 needs guidance and help it is when he sets out in the path of 
 knowledge. Therefore, the aim has been so to instruct and en- 
 courage him in his first efforts that, gaining strength at each step, 
 he shall advance with increasing interest and delight. 
 
 4. Special attention has been devoted to Notation and Numera- 
 tion. Numbers consisting of three periods have been developed 
 objectively. It is believed that children can use numbers having 
 nine figures as readily and intelligently as those having three 
 figures, if Notation and Numeration be understood. A careful 
 examination of the method used is earnestly solicited. 
 
 5. Throughout the book the pupil is taught to use his reason 
 and common-sense, and, as a rule, is not required to memorize 
 until he perceives the truth of what is to be committed to 
 memory. 
 
 6. From the commencement mental and written work are com- 
 bined. At the outset the pupil is set to working on the black- 
 board. He first obtains his results mentally, and afterwards 
 reproduces his work with his hands, and looks upon it with his 
 eyes. It then becomes to him a reality. What is traced by the 
 muscles and pictured on the eye is the more indelibly imprinted 
 
 f7i24952;a 
 
IV PREFACE, 
 
 on the memory. It is for this reason that the book contains so 
 much work in the form of Equations, with the work so varied 
 and so full. It is believed that the pupil will thoroughly master 
 the Tables, learning readily and accurately to perform all the 
 elementary operations, by this process sooner than by any other ; 
 since he must use the Tables at every step. 
 
 7. Addition and Subtraction are treated in immediate connec- 
 tion ; also Multiplication and Division. Thus their correlations 
 are more clearly shown. Multiplication is at first worked up 
 under Graded Addition, and Division under Graded Subtraction, 
 till they are well understood, and Tables have been made and 
 used, before the terms Multiplication and Division are given. In 
 Division the three terms are written in precisely the same order 
 as the corresponding terms in Multiplication, that Long Multipli- 
 cation may illuminate and illustrate Long Division. 
 
 8. The Tables for Addition, Subtraction, Multiplication and 
 Division are developed objectively and progressively. 
 
 9. The principles of Factoring have been so applied as to sim- 
 plify Division and pave the way to Fractions. 
 
 10. The rule for Addition and Subtraction of Fractions having 
 unlike Denominators is developed objectively. 
 
 11. Names, Definitions and Rules are given with the full con- 
 viction that the best time to give them is when we give the 
 things named or defined, and the method of operation desoibed in 
 a Mule. If a child cannot comprehend a method of operation, he 
 should not be required to use it ; but if he can, then he can com- 
 prehend a clea.r statement of the method in a Rule ; and since he 
 understands the Rule^ and it is important, he should learn 
 THE Rule. 
 
 The aim has been not to load down the pupil with Arithmetic 
 as a burden from without, but to cause it to spring up within him 
 by a natural and healthful process ; that, growing and unfolding 
 with his intellect, it may be an organized, vital and indestructible 
 part of himself. 
 
 The author has written this book on his own plan, using his 
 own methods and illustrations, striving to adapt them to the 
 comprehension of children. He is not, however, so vain as to 
 presume that he has made no mistakes, and has produced a work 
 above criticism. 
 
 S. D. B. 
 
 June 1, 1870. 
 
SUGGESTIONS TO TEACHERS. 
 
 The author of this book requests most earnestly that teachers will seek 
 fully to comprehend his plan and methods, and strive to work by them, teach- 
 ing the book precisely as it is written. 
 
 It is not taken for granted that the pupil knows anything about Number. 
 The attempt is made to teach each thing in its proper place, accompanied by 
 such reasons and explanations that the pupil shall compreMnd tJie sui^ect^ and 
 not be required merely to memorize words and Tables to him almost without 
 meaning. The work is begun at the very basis, in the belief that the first and 
 most important work is to lay broad foundation-stones, so firmly that, whoever 
 shall build upon them, he shall build in confidence, knowing that he is on the 
 solid rock. 
 
 The aim is so to instruct the pupil at the commencement that he shall be 
 able to use the knowledge gained. The first few Lessons are quite simple, 
 since he is to be put to working on the blackboard at once. When each figure 
 and sign is first given he should be drilled in making it on'the blackboard till 
 he can write it neatly and with facility. 
 
 The Signs +, — , =, and the Equation by Addition and that by Subtraction, 
 are carefully developed, with the full meaning and power of each item, so that 
 the pupil shall at once be able to use them understandingly. By their use he 
 will be able to write on the blackboard from day to day all he learns of Arith- 
 metic, and to reproduce it in every possible form. Pupils love blackboard- 
 work ; and in no other manner can they be taught so successfully. 
 
 Particular attention should be given to the relation between Addition and 
 Subtraction as shown in Lessons IX, X, XI, XII and XLIX. 
 
 The method of developing the Tables for Addition and Subtraction, and 
 those for Multiplication and Division, by the use of cubes, and the manner of 
 reading the Tables from the cubes, should be carefully explained. 
 
VI SUGGESTIONS TO TEACHERS, 
 
 The method of adding columns of numbers by 6's, shown on page 27, really 
 embraces the substance of Addition. The pupil should be well drilled also in 
 adding by 7's, 8's, 9's and lO's. This will lay a good foundation for subsequent 
 work in Notation. 
 
 The greatest possible pains should be taken in the development of Notation 
 and Numeration, as set forth on pages 46, 47, 62, 68, 74-76. 
 
 Addition by objects, as shown on pages 53 and 53, and Subtraction by 
 objects, as given on pages 56 and 57, should be explained till every pupil can 
 point out every step. 
 
 The Lessons on Graded Addition and Subtraction, commencing on page 81, 
 are specially hnportant, since they lay the foundation for Multiplication and 
 Division, and enable the pupil to see that what foUows is but a new applica- 
 tion of what he has already learned. 
 
 For each of the numbers 2 and 3 two Multiplication Tables are given ; but 
 for each of the numbers 4, 5, 6, 7, 8, 9, 10, only one Multiplication Table 
 is given. In each case a second Table should be written out by each piipil, in 
 the manner explained in Lesson LXIV, and be learned with the one given. 
 
 The Examples and Explanations showing the precise manner in which 
 Multiplication is derived from Addition, and Division from Subtraction, 
 should be dwelt upon till every pupil can solve the same Example both by 
 Addition and Multiplication, or by Subtraction and Division. 
 
 Great pains should be taken to show clearly the relations between Multipli- 
 cation and Division by the Solutions and Explanations given in the book. 
 The corresponding terms in Multiplication and Division are written in the 
 same order ^ that the pupil may the more readily understand the relations. 
 
 The difference between the two cases in Division given on pages 96 and 100 
 should be carefully pointed out. 
 
 The subject of Factoring as applied to Division, especially for the purpose 
 of deducing the General Principles of Division, should receive special atten- 
 tion, since these Principles are important in Fractions. 
 
 It is left to the Teacher to give all needed Explanations in Denominate 
 Numbers, and to supply any further Examples needed. 
 
 THE AUTHOR. 
 
rwTW' ' o'^s^'^^s^ 
 
 LESSON /. 
 
 1. In this picture, how many girls are in the swing ? 
 
 2. How many girls are pulling the swing ? 
 
 3. If you count both girls together, how many are 
 they? 
 
 One girl and one other girl are how many ? 
 Jf. How many kittens do you see on the stump ? 
 
 5, How many on the ground ? 
 
 6, How many kittens are in the picture ? 
 
 One kitten and one other kitten are how many ? 
 
 7, If you should ask me how many girls are in the 
 swing, or how many kittens are on the stump, I could 
 answer aloud, Oiie; or I could write One; orthus, 1. 
 
 8, If I write One, this is called the word One, 
 
 9, This, 1^ is named a figure One^ because it 
 means the same as the word Oiie, and stands for One. 
 
8 FIRST LESSONS IN 
 
 10, Write 1. What is this named ? Why ? 
 
 11, A figure one may stand for one girl, one kitten, or 
 one any thing. 
 
 12, When children first attend school, what do they 
 begin to learn ? Ans, Letters and words. 
 
 15, Could you read or write before you had learned 
 either letters or words ? 
 
 H-, If we have all the letters together, they are named 
 the Alphabet. 
 
 16, If we write or speak words, they are named Lan- 
 guage. 
 
 16, You are commencing to study Arithmetic ; and 
 you can read and write in Arithmetic only as you learn 
 the Alphabet and Language of Arithmetic. But little 
 time will be required for this purpose. 
 
 LESSON II. 
 
 1, If we speak or write words, what do we name them, 
 when taken together ? 
 
 2, What are you commencing to study ? Ans, Arith- 
 metic. 
 
 3, What Language must you now learn ? 
 If., What do we name this, 1 ? Why ? 
 
 5, This figure, 1, is part of the Language of Arith- 
 metic. 
 
 ^. If I should write something to stand for Two — 
 two girls, two kittens, or two things of any kind, what 
 do you think we would name it ? 
 
 7. A figure Two is written thus : 2. Make di figure 
 two, 
 
 8. Why do we name this 2i figure tivo 9 
 
 9. This figure two (2) is part of the Language of 
 Arithmetic. 
 
3IENTAL AND WRITTEN ARITHMETIC, 
 
 10. In this picture one boy is sitting, playing a flage- 
 olet. What is the other boy doing ? K the boy stand- 
 ing should sit down by the other, how many boys 
 would be sitting together? One boy and one other 
 boy are how many boys ? 
 
 11. You see a flageolet and a yiolin. They are mu- 
 sical instruments. One musical instrument and one 
 other musical instrument are how many ? 
 
 12. I will write thus: 112. We say that 1 boy 
 and 1 other boy, counted together, are 2 boys ; or are 
 equal to 2 boys. We will now write something to show 
 that the first 1 and the other 1 are to be counted 
 together. 
 
 13. We name a line drawn thus, — , a horizontal 
 line. Draw such a line. Name it. 
 
 H. A line drawn thus, | , we name a twrtical 
 line. Draw such a line. Name it. 
 
10 
 
 FIRST LESSONS IN 
 
 Plus means more; 
 
 15. Now I will put two such lines 
 together; thus, +. What kind of 
 a line do we name the first ( — ) ? 
 And what do we name the last ( | ) ? 
 Are these lines long or short ? 
 Where do they cross each other ? 
 
 IS. Each of you write thus: — , 
 
 11. This, +, is named JPlus. 
 and + also means more. 
 
 18. I will write, 
 
 One and One More Equal Two. 
 
 19. Now T will write part of this in the Language of 
 Arithmetic. I write the first One thus, 1; then the 
 other One thus, 1. Afterward I write, for the word 
 More, thus, +, placing the + between 1 and 1, so that 
 the whole stands thus : 1 + 1. As I write, I say, One 
 and One more. 
 
 W. Each of yon write 1 + 1. Eead what yon have 
 written. 
 
 21. This + , when written between the I's, shows that 
 they are to be put together, or counted together, so as 
 to make 2. 
 
 22. Because -f shows what is to be done, it is called 
 a Sign. If we take its name. Plus, and the word Sign, 
 and put both words together, we have Sign Plus, or 
 Phis Sign. In speaking of this we may call it Sign 
 Plus, or Plus Sign, or Plus. 
 
 28. 1, 2, + , are part of the Language of Arithmetia 
 
 Write the following in the Zanguage of Arithmetic : 
 
 2Jf. One and one more. 25. One and two more. 
 
 26. Two and one more. 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 11 
 
 LESSON IIL 
 OJV-B AJV^ OJVJ^ MO^Ii, BQZTAL 27fO. 
 
 1, I will write the above thus : 1 + 1 equal 2. 
 
 2, In length, are these lines, =, equal or unequal ? 
 
 3, We will use two lines thus drawn, =, to mean 
 equal, in place of the word equal. Writing them in 
 1 + 1 equal 2, we have 1 -{- 1 = 2. 
 
 Ji-, Because these two lines thus 
 written show something, they are 
 called a Sign. And because they 
 mean equal, we name them the 
 Sign of Equality. 
 
 5. Anything written that means 
 something, as, for instance, 1 + 1 
 = 2, is called an Expression. 
 
 6, An expression like the above, in which we use the 
 Sign of Equality, is named an Equation* 
 
12 
 
 FIE ST IjKJSSO.XS IiY 
 
 LESSON IV. 
 
 1, How many boys are in this boat ? 
 
 ^. If the boy in the end of the boat should fall out, 
 how many boys would be left in • the boat ? One boy 
 from two boys leaves how many boys ? 
 
 3, Instead of saying, One hoy from two hoys, we will 
 say, Tivo hoys less one hoy, which means the same. 
 Then, two boys less one boy will equal how many ? 
 
 ^. We will write, 2 less 1 = 1. Eead this. 
 
 5, We wish something that means less, to write be- 
 tween 2 and 1, for the word less. 
 
 6. Aline written thus, — ,is used 
 to mean less, instead of the word 
 
 7. The name of this line, — , 
 when thus used, is iif^mie^. Minus ^f^i^^^y^c^^ 
 means less. ^^^^^/^ 
 
 8, We may write 2 less 1 = 1, or 2 — 1 = 1. 
 
MENTAL AAJ} WRITTEN ARITHMETIC, 13 
 
 iQXJS^'flQ^-^- 
 
 9. Since this, — , shows that something is to be done, 
 what may we call it ? Ans. A Sign. 
 
 What is its name ? 
 
 10. If we put the two words Sign and Minus together, 
 they make Sign Minus, or Minus Sign, In speaking 
 of this Sign we may call it Sign Minus, or Minus Sign, 
 or Minus, 
 
 11. Each of you write, 2 — 1 = 1. Eead aloud what 
 you have written. 
 
 12. What is the name of this Expression : 2 — 1 = 1? 
 
 13. Write, and name in order, the following : 1, 2, +, 
 =, — . Of what do these form part ? 
 
 Write, in Arithmetical J^anguaae, the fottowinff : 
 
 H. One equals one. 
 
 15. One and one more equal two. 
 
 16, Two less one eaual one. 
 
14 
 
 FIRST LESSONS IN 
 
 LESSON V. 
 
 1, In this picture how many boys do you see fishing? 
 How many hunting ? How many in all ? 2 boys and 
 1 other boy are how many ? 
 
 2. We make a figure Three thus : 3. 
 
 S. 1 boy and 2 more are how many boys ? Are 1 boy 
 and 2 more just as many as 2 boys and 1 more ? 
 Jf, How many birds are on this tree ? 
 
 5. If 1 of them should fly away, how many would be 
 left ? 1 bird from 3 birds leaves how many birds ? 
 
 6. If, instead, 2 of the birds should fly away, how 
 many would be left ? 3 birds less 2 birds are how many ? 
 
 7. 1 from 3 leaves how many ? 2 from 3 ? 
 
 8. 3 boys are how many more than 2 boys ? Than 1 ? 
 
 9. Write, and read aloud, the following Equations: 
 1 + 1 = 2; 2-1 = 1; 2+1 = 3;. 
 
 1 + 2 = 3; 3-1 = 2; 3-2 = 1. 
 
MENTAL AND WBITTEN ARITHMETIC. 
 
 15 
 
 
 
 LESSON I/A 
 
 1, In this picture you see 1 driver? How many other 
 men in the wagon? 1 man and 3 other men are how 
 many? 3 men and 1 other man are how many ? 
 
 2, We write a figure Four thus: 4. 
 
 3, Are 3 men and 1 man more just as many as 1 
 man and 3 men more ? 
 
 Jf. If the driver should jump from the wagon, how 
 many men would be left in the wagon ? 1 man from 4 
 men leaves how many men ? 
 
 5. If, instead, the 3 other men should jump out, how 
 many would be left in the wagon ? 3 men from 4 men 
 leave how many men ? 
 
 6. 4 men are how many more than 3 ? Than 1 man ? 
 
 7. How many horses are in 1 span ? 
 
 8. How many spans of horses are drawing this wagon ? 
 
 9. How many horses are there in all ? 
 
16 FIRST LESSONS IN 
 
 10. 2 horses and 2 other horses are how many horses ? 
 
 11. If the 2 horses in front should be unhitched and 
 driven away, how many would be left ? 2 horses from 4 
 horses leave how many horses ? 
 
 12. 4 horses are how many more than 2 horses? 
 How many more than 3 horses ? Than 1 horse ? 
 
 IS. 3 horses and 1 horse are how many? 1 horse ' 
 and 3 horses ? 2 horses and 2 other horses ? 
 
 14^. 1 horse from 4 horses leaves how many ? 3 horses 
 from 4 leave how many ? 2 from 4 ? 
 
 15. 2 liorses are how many less than 4 horses ? 
 
 16. 1 horse is how many less than 4 horses ? 
 
 17. What Language have we commenced learning ? 
 
 18. Write these: 1, 3, 2, 4, +, —, =. Name each. 
 
 19. Of what Language are they part ? 
 
 Write, in ;Arith77tetical Za^iguage, the following : 
 
 20. One and one more equal two (1 + 1=2); 
 
 21. Two less one equal one ; . 
 
 22. Two and one more equal three ; 
 2S. One and two more equal three ; 
 2 If.. Three less one equal two ; 
 
 25. Three less two equal one ; 
 
 26, Three and one more equal four ; 
 21, One and three more equal four ; 
 
 28. Two and two more equal four ; 
 
 29, Four less one equal three ; 
 
 50, Four less three equal one ; 
 
 51, Four less two equal two ; 
 
 52, The Sign + shows that what is written at the 
 right of.it is to be coimted with what is written before it. 
 
 SS. The Sign — shows that what is written at the 
 right of it is to be tahen away from what is written be- 
 fore it. 
 
MENTAL AND WRITTEN ARITHMETIC, 17 
 
 LESSON VII. 
 
 i. In this picture, Minnie has 1 rose in her left hand ; 
 how many has she in her right hand ? 
 
 2. If she should put them all in her right hand, how 
 many would she have in her right hand ? 
 
 8, We make a figure Five thus : 5* Make one. 
 
 ^. 4 roses and 1 rose more are how many roses? 1 
 rose and 4 more roses are how many? 
 
 5. Are 4 roses and 1 more just as many as 1 and 4 
 more? 
 
 6. Willie has 3 roses in his right hand; how many 
 has he in his left hand ? If he should put them all in 
 his right hand, how many would he then have in his . 
 right hand? 3 roses and 2 roses are how many? 2 
 roses and 3 roses are how many? 
 
 7. Are 3 roses and 2 more just as many as 2 and 3 
 more? 
 
18 FIRST LESSONS IN 
 
 8, If Minnie should give her teacher the rose in her 
 left hand, how many would she have left ? 1 rose from 
 5 roses leaves how many roses ? 
 
 9, If, instead, she should give away the 4 roses in her 
 right hand, how many would she have left ? 4 roses 
 from 5 roses leave how many ? 
 
 10, If Willie should give his mother the 2 roses in 
 his left hand, how many would he have left ? 2 roses 
 from 5 roses leave how many ? 
 
 11, If, instead, he should give her the 3 roses in his 
 right hand, how many would he have left? 3 roses 
 from 5 roses leave how many ? 
 
 i^. How many more roses has Willie in his right 
 hand than in his left? 3 are how many more than 2 ? 
 
 13, How many more has Willie in his right hand 
 than Minnie in her left ? 3 are how many more 
 than 1 ? 
 
 14' How many more roses are in Minnie's right hand 
 than in Willie's ? 4 are how many more than 3 ? 
 
 15, How many more has she in her right hand than 
 Willie in his left? 4 are how many more than 2? 
 
 16, How many more has she in her right hand than 
 in her left ? 4 are how many more than 1 ? 
 
 17, How many roses are on the rose-hush ? 
 
 18, How many would be left if Minnie should pick 1 ? 
 If she should pick 4? If 2? If 3? 
 
 19, 5 are how many more than 4 ? Than 1 ? Than 
 3? Than 2? 
 
 Wtite, in A.rlthnietical language , the foUowing : 
 
 4 and 1 more equal 5 ; 5 less 1 equal 4 ; 
 
 3 and 2 more equal 5 ; 5 less 2 equal 3 ; 
 
 2 and 3 more equal 5 ; 5 less 3 equal 2 ; 
 
 1 and 4 more equal 5 ; 5 less 4 equal 1. 
 
MENTAL AND WRITTEN AKITHMETIC. 
 
 19 
 
 LESSON VIII. 
 
 1. How many chickens are on this hen's back? 
 
 2, How many other chickens are about her ? 
 S, 5 chickens and 1 chicken are how many? 
 
 4. We make a figure Six thus: 6. Make one. 
 
 5. How many ducklings are swimming in this stream? 
 
 6. How many are on shore ? 
 
 7. How many ducklings in all? 4 ducklings and 2 
 more are how many? 2 and 4 more are how many? 
 
 <^. How many swallows are on this gate ? 
 
 9. How many others are on the fence ? 
 
 10, 3 swallows and 3 other swallows are how many ? 
 
 11. If a hawk should fly away with the chicken on 
 the hen's back, how many chickens would be left? 
 1 chicken from 6 chickens leaves how many ? 
 
 12, If a fox should steal the 2 ducklings on shore, 
 how many ducklings would be left? 
 
20 FIRST LESSONS IN 
 
 13. If, instead, the 4 ducklings should float away, 
 how many would be left ? 2 ducklings from 6 duck- 
 lings leave how many ? 4 from 6 leave how many ? 
 
 14' If the 3 swallows on the gate should fly away, 
 how many would be left? 3 swallows from 6 leave 
 how many ? 
 
 JVrlte t?ie following hi £Jquations : 
 
 5 and 1 more equal 6 
 4 and 2 more equal 6 
 3 arid 3 more equal 6 
 2 and 4 more equal 6 
 1 and 5 more equal 6 
 
 6 less 1 equal 5 
 6 less 2 equal 4 
 6 less 3 equal 3 
 6 less 4 equal 2 
 6 less 5 equal 1. 
 
 15. Eead aloud, in concert, what you have written. 
 
 16. James had 3 pennies, and his father gave him 2 
 more ; how many had he then ? He found 1 more ; how 
 many had he in all ? 
 
 11. Flora had 4 nice dresses for her doll, and her 
 mother made 2 new ones for it; how many dresses for 
 her doll had she then ? 
 
 18. Charlie caught 3 trout, and Willie 3 ; how many 
 did both catch ? 
 
 19. Mary had 2 canaries that sang, and 4 that were 
 not singers ; how many canaries had she in all ? 
 
 20. Albert had 6 pennies, and gave 3 of them for 6 
 r.pples ; how many pennies had he left ? He ate 2 of 
 the apples ; how many had he left ? His sister Helen 
 ate 2 more of them ; how many, in all, did Albert and 
 Helen eat ? How many apples had Albert then left ? 
 
 21. 4 and 1 are how many? 2 and 3? 4 and 2? 
 3 and 3 ? 1 and 5 ? 2 and 4? ' 3 and 2 ? 5 and 1 ? 
 
 22. 2 from 5 leave how many ? 3 from 5 ? 2 from 
 6 ? 4 from 6 ? 3 from 6 ? 1 from 6 ? 5 from 6 ? 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 21 
 
 LESSON IX. 
 
 L One, Two, Three, &c., are called Numbers ; and 
 
 because the figures 1, 2, 3, &c., stand for these numbers, 
 they are themselves commonly called numbers. 
 
 ^. When we put 2 and 3 together, or unite them, and 
 find that they equal 5, we are said to Add them. 
 
 3, Uniting two or more numbers, and finding what 
 number they equal, when taken together, is named 
 Addition, 
 
 Jf, When numbers are to be added, we usually write 
 them in a vertical column. Let us add 2, 3, and 1. 
 Writing the numbers as shown in the margin, and . 2 
 drawing a line below the column, we first find that 3 
 1 and 3 more equal 4 N^ext we add this 4 and the _ 
 remaining number, 2, and, finding that 4 and 2 more 6 
 equal 6, we write 6 below the line. 6 is named 
 the Amount^ or Sum^ of 2, 3, and 1. 
 
 Find and write the Sum in each of the following 
 
 Exercises for the Slate and Blackboard. 
 1321413222213 
 212.4 13 2321131 
 1111121122312 
 
23 
 
 FIKST LESSONS IN 
 
 LESSON X, 
 
 1. Wlien we take 2 from 5 and find that 3 remain, 
 or are left, we are said to Subtract 2 from 5. Subtract 
 means take away, 
 
 2. Taking one number from another, and finding 
 how many remain, is named Subtraction. 
 
 3. Since 5 is diminished, or made less, by subtracting 
 2 from it, we name 5 the Minuend. Minuend means 
 to be diminished. 
 
 Jf, Since 2 is subtracted from 5, 2 is named the Sub- 
 trahend. Subtrahend means to be subtracted. 
 
 5. Since 3 shows how many remain after subtracting 
 2 from 5, we name 3 the JEteinainder^ or the Dlf^ 
 ference between 5 and 2. 
 
 6. In performing Subtraction we 
 usually write the work in the form 
 shown at the right hand. 
 
 SUBTRACTION. 
 
 5 Minuend. 
 
 2 Subtrahend. 
 
 3 Remainder, 
 
 Exercises for the Slate akd Blackboard. 
 
 5 3 
 3 1 
 
 2 
 
 4 
 2 
 
 6 6 
 2 3 
 
MENTAL AND WRITTEN ARITHMETIC, 23 
 
 LESSON XI. 
 
 2 + 3 = 5. 
 
 i. The Sign of Equality divides every Equation into 
 two parts, named Memhei's. 
 
 2, The First Member and the Second Member of every 
 Equation are equal, and the Sign of Equality stands be- 
 tween them. 
 
 3, Since the Equation 2 + 3 = 5 is formed by Addi- 
 tion, we name it an JEquation by Addition. 
 
 4- In every Equation by Addition like the above, hav- 
 ing three numbers, with the greatest standing last, if 
 only one of the numbers be missing it is easy to find it. 
 
 Since the Sum of the first two numbers equals the 
 third, it is evident that if either of them be subtracted 
 from the third the Eemainder will equal the other 
 number. >fc 
 
 We may find any one of the three numbers thus : 
 
 Missing Numbers. Methods of Finding. 
 
 First. Subtract the Seco:n'd from the Third. 
 
 Second. Subtract the First from the Third. 
 Third. Add the First and Secon^d. 
 
 Write the proper numbers in place of (?) in these 
 Exercises for the Slate and Board. 
 
 2 + 3 = ? 
 
 3 + 1 =? 
 
 ? + 1 = 5 
 
 ? + 3 = 4 
 
 2 + 2 = ? 
 
 3 + 2 = ? 
 
 4 + ? = 5 
 
 2 + ? = 3 
 
 1+4 = ? 
 
 3 + ? = 5 
 
 2 + ? = 4 
 
 1 + ? =3 
 
 1 + 3 = ? 
 
 ? + 3 = 5 
 
 3 + ? = 4 
 
 ? + 2 = 3 
 
 2 + 1 = ? 
 
 1 + ? =5 
 
 ? + 2 = 4 
 
 2 + 4r=? 
 
 2 + ? =5 
 
 ? + 4 = 5 
 
 ? + 1 = 4 
 
 3 + ? = 6 
 
 3 + 3 = ? 
 
 4 + ? = 6 
 
 1 + ? = 6 
 
 ? + 4 = 6 
 
24 FIRST LESSONS IN 
 
 LESSON XII. 
 
 • 5-2 = 3.. 
 
 -/. Since the Equation 5 - 2 = 3 is formed by Sub- 
 traction, we will name it an Equation by Sub- 
 traction. 
 
 2, In this, and in every Equation by Subtraction 
 having only three numbers, and the largest of the num- 
 bers standing first, the first number is equal to the Sum 
 of the two others. 
 
 S, If the second number be subtracted from the first, 
 the Difference will equal the third ; and if the third be 
 subtracted from the first, the Difference will equal the 
 second. 
 
 From this it is evident that in any such Equation by 
 Subtraction, we may find any one of the numbers thus : 
 
 Missing Numbers. Methods of Finding. 
 
 EiRST. Add the Second and Third. 
 
 Second. Subtract the Third from the First. 
 Third. Subtract the Second from the First. 
 
 Find and write the missing numbers in the following 
 
 Exercises for the Slate and Board. 
 
 5-2 = ? 5-? = 4 ? - 1 == 3 S-l=? 
 
 5_4=,? ?_3 = 2 4-1 = ? 3-?=l 
 
 5-?=:3 ?-2 = 2 4-3 = ? 3-2 = ? 
 
 5-?=l ?-2=3 4-2=? 3-?=2 
 
 5-3=? ?-4=l 4-?=2 6-2=? 
 
 5-1=? ?-l=4 4-?=l 6-?=3 
 
 5_?^2 ?-3=l 4-?=3 ?-2=4 
 
 6-3 = ? 6-4 = ? ?-3=:3 6-?=4 
 
 ?-4=2 6-1=? 6-5=? 6-?=2 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 25 
 
 UPPER COUNTERS. 
 
 
 1 
 
 L 
 
 3 
 
 4 
 
 5 
 
 1 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 5 
 
 
 _. 
 
 
 ^m 
 
 ^ 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 ^ 
 
 
 k^^ 
 
 ^^ 
 
 
 1^^ 
 
 ^=n-H 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 
 
 4 
 
 5 
 
 6 
 
 
 
 
 5 
 
 ^ 
 
 ^ 
 
 
 ^ 
 
 
 LESSON XIII. 
 
 We now proceed to make a Table, on the plan shown 
 above. The Table will help ns in adding and sub- 
 tracting, in performing the work in all Examples such 
 as we have had. 
 
 First, we arrange, side by side, 5 rows of little cubes, 
 with 5 cubes in each row. Then we place another 
 row of 5 cubes a little above these, and a like row a 
 little to the left of the 5 rows. We number and write 
 on the cubes of the last two rows, " 1, 2, 3, 4, 5,^' as 
 shown. These two rows, thus numbered, are to be used 
 as Counters. 
 
 Suppose we wish to add 2 and 3 more, and write 
 the Sum in the Table. We take from the Lower Count- 
 ers the cubes numbered "1, 2," and from the Upper 
 Counters the cubes numbered "1, 2, 3," and, adding 
 them, find their Sum is 5. We then write this Sum on 
 the cube which stands at the right of the cube num- 
 bered "2" in the Lower Counters, and below the cube 
 numbered " 3 '' in the Upper Counters. 
 
 In the same manner we find and write the Sum of 
 any other two numbers written on the Counters. 
 
26 FIRST LESSOJVS IN 
 
 The Table is filled as far as 6. We read it thus : 
 1 and 1 are 2, 2 and 1 are 3, 3 and 2 are 5, 
 
 1 and 2 are 3, 2 and 2 are 4, 3 and 3 are 6 ; 
 
 1 and 3 are 4, 2 and 3 are 5, 4 and 1 are 5, 
 
 1 and 4 are 5, 2 and 4 are 6 ; 4 and 2 are 6 ; 
 
 1 and 5 are 6 ; 3 and 1 are 4, 5 and 1 are 6. 
 
 We will name this an Addition Table. We may 
 also use it as a Subtraction Table. 
 
 If we add 2 and 3, their Stc7n is 5 ; as appears in the 
 Table. If we Subtract 2 from 5, the Difference is 3. 
 We obtain this result from the Table, thus : 1st, we find 
 in the Lower Counters 2, which is to be subtracted ; 2d, 
 we pass from this 2 along to the right, and find 5, from 
 which 2 is to be subtracted ; 3d, directly above this 5, 
 in the Upper Counters, we find the 3, which is the Dif- 
 ference between 2 and 5. 
 
 We read this as a Subtraction Table thus : 
 
 1 from 2 leaves 1, 2 from 3 leave 1, 3 from 5 leave 2, 
 
 1 from 3 leaves 2, 2 from 4 leave 2, 3 from 6 leave 3 ; 
 
 1 from 4 leaves 3, 2 from 5 leave 3, 4 from 5 leave 1, 
 
 1 from 5 leaves 4, 2 from 6 leave 4 ; 4 from 6 leave 2 ; 
 
 1 from 6 leaves 5 ; 3 from 4 leave 1, 5 from 6 leave 1. 
 
 Exercises eor the Slate and Boaed. 
 
 2+3=? l+?=6 4-1=? 3+?=6 
 
 24-4 = ? 4 + ? = 6 3-2 = ? 1 + 4=? 
 
 3_1=^? 2+?=6 1 + 3 = ? 5 + ? = 6 
 
 4 + ?::^5 14.5:=? 5__9^2 4-2=? 
 
 Writtei^ Exercises. i 
 
 1, Frank had 4 peaches, and Henry 2; how many 
 had both boys ? 
 
 2, Emma had 6 pinks, and gave 3 of them to Walter; 
 how many had she left ? 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 27 
 
 LESSON XIV. 
 
 By Objects. | 
 
 3l 13 
 
 r-. 
 
 >A- e 
 
 sR 02 
 
 - 
 
 *E ^^'! 
 
 If 
 
 20 02 
 
 ^a^i 
 
 i 
 
 \=\- 6 
 
 4- -4 
 
 
 J Q^ 
 
 
 By Mgtires. 
 
 3-3 
 
 - 4H 
 
 -6 
 
 2=2 
 
 >=6 
 
 -6 
 
 We will now 
 add 3, 2, 4, 2, 3 
 and 4, as shown at 
 the right hand, 
 and lind how 
 many times we 
 can make the 
 number 6 from 
 them. 
 
 Beginning at 
 the bottom, the 
 first number is 4. 
 From the 3 next 
 above this we take 
 
 2, which, added to 
 4, make 6. Hav- 
 ing taken 2 from 
 
 3, we have 1 left. 
 We now add this 
 1 to the 2 stand- 
 ing above the 3, and have 3 for the Sum. Taking 3 of 
 the 4 ones next above, and adding them to this Sum, 3, 
 we have 6. [N'ext we add the 1, left from 4, to the 2 
 standing above the 4, and find that their Sum is 3. 
 Adding together this Sum, 3, and the last number, 3, 
 we find their Sum to be 6. Thus we make the number 
 6 three times from the whole column. We write each 6 
 in the Sum, and write + between the 6's. 
 
 While performing the work we say thus : 4 and 2 are 
 6 ; 2 from 3 leave 1, 1 and 2 are 3, 3 and 3 are 6 ; 3 from 
 4 leave 1, 1 and 2 are 3, 3 and 3 are 6. The entire col- 
 umn is equal to 3 times 6. 
 3 
 
 6+6+6. Sum. 
 3 times 6 
 
28 FIRST LESSONS IN 
 
 In the same manner perform the work in each of 
 these 
 
 Examples tor the Slate akd Board. 
 
 1 
 
 4 
 
 2 
 
 2 
 
 3 
 
 3 
 
 5 
 
 4 
 
 4 
 
 1 
 
 5 
 
 4 
 
 5 
 
 4 
 
 1 
 
 3 
 
 2 
 
 5 
 
 5 
 
 5 
 
 4 
 
 3 
 
 3 
 
 5 
 
 5 
 
 4 
 
 2 
 
 4 
 
 2 
 
 2 
 
 4 
 
 2 
 
 5 
 
 4 
 
 3 
 
 5 
 
 2 
 
 5 
 
 5 
 
 3 
 
 3 
 
 4 
 
 4 
 
 3 
 
 1 
 
 5 
 
 4 
 
 3 
 
 4 
 
 3 
 
 3 
 
 3 
 
 5 
 
 5 
 
 3 
 
 4 
 
 2 
 
 4 
 
 5 
 
 4 
 
 2 
 
 3 
 
 4 
 
 3 
 
 4 
 
 3 
 
 2 
 
 4 
 
 • 
 
 4 
 
 1 
 
 3 
 
 5 
 
 4 
 
 3 
 
 2 
 
 5 
 
 4 
 
 3 
 
 LESSON XV. 
 
 1. In this scnool, one boy has 6 books on his desk, 
 and another has 1 ; how many books have both on their 
 desks ? 6 books and 1 book are how many ? 
 
 2, We make a figure Seven^ thns : 7. Make one. 
 S, How many caps are hanging in the upper row? 
 
 How many in the lower row ? How many caps are 
 there in all ? 5 caps and 2 caps are how many ? 
 
 4. How many bojs are standing? How many are 
 sitting ? How many boys are there in all ? 4 boys and 
 3 boys are how many ? 
 
 5. How many girls in this school? If 1 of them 
 should go home, how many would be left ? 1 girl from 
 7 girls leaves how many ? 
 
 6. If 2 of the 7 boys should go and take their caps 
 and go home, how many boys would be left ? 2 boys 
 from 7 boys leave how many ? 2 caps from 7 caps leave 
 how many ? 
 
 7. How many girls are standing? If 3 of them 
 should sit down, how many would be left standing ? 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 29 
 
 If the girls so left standing should go home, how 
 many girls would be left ? 4 girls from 7 girls leave 
 how many ? 3 girls from 7 leave how many ? 
 
 8, How many books are on the teacher's desk ? If 5 
 of them should be taken away, how many would be left ? 
 5 books from 7 leave how many ? 
 
 Wkitten" Exercises. 
 
 1. James had 3 marbles, and Harry gave him 4 more ; 
 how many had he then ? He lost 2 marbles ; how many 
 had he left ? 
 
 2. Charlie had 2 peaches, and his mother gave him 5 
 more ; how many had he in all ? He ate 4 ; how many 
 had he remaining ? 
 
 3. Walter had 4 cents, and his father, gave him 3 
 more ; how many had he then ? He spent 3 cents for 7 
 plums; how many were left? He ate 5 plums; how 
 many had he left ? 
 
30 
 
 FIRST LESSONS IN 
 
 LESSON XVL 
 
 1. Eecite this Table as 
 both an Addition and Sub- 
 traction Table. 
 
 2, Write the following in 
 Equations, and read them : 
 
 5 and 2 are 7 (5 + 2 = 7) 
 4 and 3 are 7 
 3 and 4 are 7 
 2 and 5 are 7 
 7, less 4 equal 3; 
 7 less 5 equal 2 ; 
 
 6 less 3 equal 3 ; 
 
 rPPER COUNTERS. 
 
 
 J_ 
 
 J_ 
 
 3. 
 
 £ 
 
 J^ 
 
 ^ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 
 
 
 5 
 
 6 
 
 7 
 
 
 
 
 
 6 
 
 ^ 
 
 _™ 
 
 
 
 
 
 ^ 
 
 
 7 less 3 equal 4 ; 
 5 less 2 equal 3 ; 
 
 7 less 2 equal 5 ; 
 5 less 3 equal 2. 
 
 EXEECISES FOE THE SlATE AKD BoARD. 
 Jidditlon, 
 
 222311251123 
 132153214442 
 322312311211 
 
 SiibtracHon. 
 
 6 6 7 
 14 3 
 
 LESSON XVIL 
 
 ci'D'DITIOJV A.T SIGHT, 
 
 1. If I write letters, thus, ox, dog, horse, you can 
 name the toords, which they form, at first sight, without 
 stopping to Sjjell them. 
 
 2 3 5 
 
 2, If I write numters, thus, 3, 4, 2, you may become 
 
 able to name their Sums, without stopping to add, as 
 readily as you name words. 
 
MENTAL AND WRITTEN ARITHMETIC. 31 
 
 S, Copy the Exercises on your slate, and name the 
 Sums, going from the left to the right ; tlien from right 
 to left; and linally name them by skipping, in every 
 possible manner. Do not write the Sums. 
 
 At recitation the Exercises will be written on the 
 blackboard, and you will name , the Sums instantly, as 
 your teacher points to the Exercises, one by one. 
 
 Addition at Sight, 
 
 415233546223 
 261534231322 
 
 The Sums are the same m all cases wjiere the figures 
 are the same, though the order of the figures be changed. 
 Hence it is necessary only that we l:now what figures 
 we have, without regarding the order in which they 
 stand. 
 
 Write the missing numbers in the following 
 
 Exercises por the Slate k^d Board. 
 
 5+?=:6 44-? = 6 3+? = 7 6+?=r7 
 
 ?-2=:5 4-3=? 5-4=? ?_2 = 4 
 
 2+? = 7 6-5=:? 6-3=? 2 + 2=? 
 
 5+?=7 4+?=7 ?+2=7 7-4=? 
 
 5-2=? 5-3=? 6-2=? 7-?=2 
 
 3+4=? 2+5=? 6-4=? ?-4=3 
 
 L How many more are 7 than 4 ? Than 2 ? 6 ? 1 ? 
 3? 5? 4? 
 
 2, How many less are 2 than 7 ? Than 5 ? 3 ? 6 ? 
 4? 7? 
 
 S, How many less are 3 than 5?4?7?6? 
 
 Jf. How many are 3 and 3 ? 2 and 5 ? 4 and 2 ? 
 3 and 4 ? 
 
32 FIRST LESSONS IN 
 
 LESSON XVIIL 
 
 BY ADDITION. 
 
 2 + 3=3^ a,id 3 + 2=^3. 
 
 BY SUBTRACTION. 
 e> — 2 ^^3^ and O — 3 = 2* 
 
 1. In the first of the aboye Equations, 2 + 3 is the 
 FiKST Membee, and 5 is the Second Member. The 
 two Members of an Equation are always equal, and the 
 Sign of Equality stands between them. 
 
 2, Since the Sum of 2 and 3 is 5, we may write two 
 Equations by Addition: 2 + 3 = 3^ and 3 + 2 = 3. 
 
 S, Again, since the Sum of 2 and 3 is 5, it is evident 
 that if 2 be subtracted from 5, the Eemainder will be 
 3 ; and that if 3 be subtracted from 5, the Eemainder will 
 be 2. Hence, we write two Equations by Subtraction : 
 3 - 2 = 3, SLTid 3 - 3 = 2. 
 
 U* Thus, from the three numbers, 2^ 3^ and 3, we 
 have formed four Equations : 2 -\- 3 = 3, 3 + 2 = 3; 
 3- 2=3, and 3 -3 --=2. 
 
 5, From any three unequal numbers, such that the 
 Sum of the two smaller ones equals the largest, we may 
 form tzvo Equations by Addition and tivo Equations by 
 Subtraction. 
 
 Eule I. 
 To Form Equations by Addition: 
 
 I. — For the First Member of an Equation, ivrite tlie 
 ttuo smaller numbers with the Sign Plus between them. 
 
 II. — For the Second Member, write the largest of the 
 three numbers, placing the Sign of Equality bettveen the 
 Members, 
 
 III. — Form the second Equation by Addition from the 
 first, by changing the places of the two smaller numbers. 
 
MENTAL AND WRITTEN ARITHMETIC, 33 
 
 EULE 11. 
 
 To Form Equations by Subtraction: 
 
 I. — For the First Member of mi Equation^ tvrite the 
 largest of the three numbers, a?id after it one of the ttvo 
 smaller numbers, ijlacing the Sig7i Minus between them, 
 
 II. — For the Second Member, ivrite the other of the two 
 smaller numbers, placing the Sign of Equality between 
 the Members, 
 
 III. — Form the second Equation by Subtraction from 
 the first, by changijig the places of tjie two smaller 
 numbers. 
 
 Note. — When the two smaller numbers are eqnal, the two 
 Equations by Addition will be precisely alike, and also those by 
 Subtraction. 
 
 In the manner directed by the preceding Kules, form 
 and write four Equations from each of the following 
 
 Groups of Three Numhers, 
 
 1, 3 and 4 ; 1, 5 and 6 ; 1, 6 and 7 ; 2, 5 and 7 ; 
 
 2, 4 and 6 ; 1, 4 and 5 ; 3, 4 and 7 ; 2, 3 and 5. 
 
 If the three numbers are given in an Equation, it is 
 plain that we can form three more Equations from this. 
 
 Form three other Equations from each of the fol- 
 lowing 
 
 Equations, 
 
 44-2 = 6; 5 + 1 = 6; 7-3 = 4; 7-1 = 6; 
 7-2 = 5; 6-1 = 5; 3+2 = 5; 4 + 1 = 5. 
 
 
 
 
 
 Addition at Sight. 
 
 
 
 
 
 3 
 
 2 
 
 3 
 
 2 
 
 1 
 3 
 
 3 
 3 
 
 4 3 3 1 
 
 3 3 4 6 
 
 5 
 
 1 
 
 3 
 5 
 
 3 
 4 
 
 4 
 3 
 
34 
 
 FIRST LESSONS IN 
 
 LESSON XIX. 
 
 1, In this picture, how many peaches are on Willie's 
 table ? How many on Mary's table ? How many on 
 both ? 7 and 1 are how many ? 
 
 2, How many oranges are on Willie's table ? How 
 many on Mary's ? How many on both ? 6 oranges and 
 2 oranges are how many ? 
 
 3, How many pears are on Mary's table ? How many 
 on Willie's ? How many on both ? 5 pears and 3 pears 
 are how many ? 
 
 Jf, How many apples are on Mary's table? How 
 many on Willie's ? How many on both ? • 4 apples and 
 4 apples more are how many ? 
 
 How many are 
 
 1 and 7 ? 2 and 6 ? 2 and 3 ? 5 and 3 ? 4 and 4 ? 
 
 2 and 1 ? 3 and 5 ? 2 and 4 ? 6 and 2 ? 3 and 4 ? 
 
MENTAL AND WRITTEN ARITHMETIC. 35 
 
 5. If Mary should give Elizabeth her peach, how 
 many peaches would she and Willie have left? 8 
 peaches less 1 peach are how many ? 
 
 6, If WiUie should give Harry 3 pears, how many 
 would he and Mary have left ? 8 pears less 3 pears are 
 how many? 
 
 7. If Mary should give Jane 2 oranges, how many 
 would she and WiUie have left? 8 oranges less 2 
 oranges are how many ? 
 
 8, If Willie should give his 4 apples to his mother, 
 how many would he and Mary have left ? 8 apples less 
 4 apples are how many ? 
 
 How many are 
 8 less 2 ? 8 less 4 ? 8 less 6 ? 8 less 3 ? 
 
 8 less 5? 8 less 7? 8 less 1 ? 7 less 4? 
 
 LESSON XX. 
 
 Add the numbers in each of the following columns 
 by 5's, in the manner explained on page 27, for adding 
 by 6's : 
 
 3 
 
 3 
 
 4 
 
 3 4 4 2 3 1 
 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 3 
 
 4 3 2 4 12 
 
 
 2 
 
 3 
 
 4 
 
 2 
 
 1 
 
 4 
 
 5 2 2 14 5 
 
 
 2 
 
 3 
 
 4 
 
 2 
 
 4 
 
 1 
 
 2 4 4 5 3 4 
 
 
 2 
 
 3 
 
 4 
 
 4 
 
 2 
 
 3 
 
 12 3 3 4 3 
 
 Add the following by 6's 
 
 - 
 
 2 
 
 3 
 
 4 
 
 5 
 
 3 
 
 4 
 
 5 2 2 3 5 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 4 
 
 5 
 
 3 
 
 4 5 4 14 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 2 
 
 4 
 
 2 
 
 2 12 3 2 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 3 
 
 5 
 
 4 
 
 4 3 3 4 4 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5 
 
 3 
 
 2 
 
 5 4 2 3 2 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5 
 
 4 
 
 3 
 
 4 3 5 4 11 
 
 2 
 
 3 
 
 4 
 
 5 
 
36 FIRST LESSONS IN 
 
 * 
 
 WRiTTEi^ Exercises. 
 
 1, Walter and Albert went hunting. Walter shot 5 
 squirrels, and Albert 3 ; how many did both kill ? They 
 lost 2 of the squirrels ; how many were left ? Walter 
 killed 6 pigeons, and Albert 2 ; how many did both 
 kill? They gave away 4 of the pigeons; how many 
 were left ? 
 
 2, Anna, Amelia, and Willie went to gather flowers. 
 Anna picked 4 lilies, and Willie gave her 4 more ; how 
 many had she in all ? She lost 3 ; how many had she 
 left? Amelia picked 7 lilies, and Willie gave her 1; 
 how many had she then ? She gave her mother 5 lilies ; 
 how many had she at last ? 
 
 LESSON XXL 
 
 Write the following in Equations, and read them : 
 
 7 and 1 are 8, 3 and 5 are 8, 7 less 6 equal 1 ; 
 
 6 and 2 are 8, 2 and 6 are 8, 8 less 5 equal 3, 
 
 5 and 3 are 8, 1 and 7 are 8 ; 8 less 7 equal 1, 
 4 and 4 are 8, 7 less 3 equal 4, 8 less 4 equal 4, 
 4 and 3 are 7 ; 7 less 5 equal 2, 8 less 6 equal 2, 
 
 6 and 1 are 7, 7 less 1 equal 6, 8 less 3 equal 5, 
 
 2 and 5 are 7, 7 less 2 equal 5, 8 less 1 equal 7, 
 
 3 and 4 are 7, 7 less 4 equal 3, 8 less 2 equal 6. 
 Write four Equations from each of the following 
 
 Groups of Three Numbers. 
 
 1, 7, and 8 ; 2, 6, and 8 ; 3, 5, and 8 ; 2, 5, and 7 
 3, 4, and 7. 
 
 Write three others from each of the following 
 
 JEquations. 
 
 3 + 6 = 8; 4 + 3 = 7; 5 + 3 = 8; 8-5=3; 
 1 + 7 = 8; 8-3 = 6; 7-3 = 4; 3 + 5=7. 
 
mental and written arithmetic, 37 
 Exercises for the Slate akd Board, 
 
 Addition, 
 
 
 
 
 
 
 
 I. 
 
 
 
 
 
 
 
 1 
 
 2 
 
 2 
 
 3 
 
 2 
 
 5 
 
 2 
 
 2 
 
 1 
 
 5 
 
 1 
 
 6 
 
 4 
 
 3 
 
 3 
 
 4 
 
 3 
 
 2 
 
 2 
 
 1 
 
 6 
 
 5 
 
 1 
 
 6 
 
 1 
 
 2 
 
 4 
 
 2 
 
 2 
 
 2 
 
 4 
 
 1 
 
 5 
 II. 
 
 1 
 
 2 
 
 2 
 
 1 
 
 1 
 
 2 
 
 3 
 
 2 
 
 3 
 
 4 
 
 4 
 
 4 
 
 2 
 
 3 
 
 3 
 
 3 
 
 4 
 
 1 
 
 2 
 
 1 
 
 3 
 
 2 
 
 3 
 
 1 
 
 2 
 
 4 
 
 3 
 
 2 
 
 1 
 
 1 
 
 4 
 
 1 
 
 4 
 
 3 
 
 3 
 
 1 
 
 3 
 
 1 
 
 1 
 
 1 
 
 2 
 
 3 
 
 2 
 
 3 
 
 4 
 
 
 
 
 
 
 Subtraction. 
 
 
 - 
 
 
 
 
 6 
 
 4 
 
 5 
 
 3 
 
 6 
 3 
 
 6 
 2 
 
 7 
 3 
 
 8 8 7 
 3 5 3 
 
 8 
 
 2 
 
 8 
 4 
 
 7 
 4 
 
 8 
 6 
 
 7 
 5 
 
 LESSON XXI L 
 
 BQZrA.TIOJ\rS. 
 
 2 + ? = 8 5 + ? = 7 7-2 = ? 4 + ? =r 8 
 
 v+3==8 8-2 = ? 7-?=4 3 + ?=:8 
 
 3+?=8 8 ~ ? = 5 ?-4=3 ?+4=8 
 
 2 + 5 = ? ? - 4 = 4 7-3 = ? 7-4 = ? 
 
 3+?=7 8-5=? 7-5=? ?-3=5 
 
 ?+4=7 ?+6=8 6+2=? ?+2=8 
 
 Add the numbers in the following columns by 7's, in 
 the manner explained on page 27, for adding by 6's : 
 
 4 
 
 5 
 
 6 
 
 5 
 
 4 
 
 2 
 
 5 
 
 4 
 
 5 
 
 3 
 
 5 
 
 4 
 
 5 
 
 5 
 
 5 
 
 3 
 
 5 
 
 5 
 
 4 
 
 1 
 
 6 
 
 6 
 
 2 
 
 6 
 
 6 
 
 4 
 
 4 
 
 3 
 
 4 
 
 2 
 
 4 
 
 4 
 
 6 
 
 6 
 
 6 
 
 6 
 
 4 
 
 3 
 
 3 
 
 3 
 
 4 
 
 5 
 
 3 
 
 6 
 
 5 
 
 4 
 
 6 
 
 6 
 
 5 
 
 5 
 
 6 
 
 5 
 
 5 
 
 4 
 
 3 
 
 6 
 
 2 
 
 6 
 
 5 
 
 6 
 
 5 
 
 5 
 
 1 
 
 2 
 
 4 
 
38 
 
 FISST LESSONS 7iV 
 
 LESSON XXIII. 
 
 1. Learn this Addition 
 and Subtraction Table. 
 
 2. If we write the Equa- 
 tion 3 + 4 = 7 in the form 
 of an Example in Addition, 
 as shown at the 
 right, the order 3 ) p^^^^^ 
 of the numbers 4 f 
 is not changed. 7 Sum. 
 Hence we may 
 use the method shown on 
 page 23 also To find any 
 
 Number in an Example in Addition, when missing ; 
 thus: 
 
 
 _L 
 
 2 
 
 3 
 
 £ 
 
 A 
 
 6 
 
 7_ 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4- 
 
 5 
 
 6 
 
 7 
 
 8 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 
 
 
 5 
 
 6 
 
 7 
 
 8 
 
 
 
 
 
 6 
 
 7 
 
 8 
 
 
 
 
 
 
 7 
 
 ± 
 
 
 
 « 
 
 ^ 
 
 ^ 
 
 
 Names. 
 
 Methods of Finding. 
 
 Subtract the Second from 
 
 the Third. 
 Subtract the First from the 
 
 Third. 
 Add the First and Second. 
 
 Exercises eor the Slate and Board. 
 
 5 
 3 
 
 9 
 
 ? 
 
 2 
 7 
 
 I. 
 ? 
 5 
 
 4 
 
 7 
 
 4 
 
 9 
 
 8 8 
 
 ? 
 3 
 
 7 
 
 6 8 
 
 II. 
 ? 
 
 2 
 6 
 
 ? ? 
 
 i 1 
 
 8 4 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 39 
 
 LESSON XXIV. 
 
 If we take the Equation 8 — 3=5, and write it in the 
 
 form of an Example in Subtraction, 
 as shown at the right, the order of 
 the numbers is not changed. Hence 
 we may use the method shown on 
 page 24 also To find a missing 
 numher in an Example in Subtraction ; thus : 
 
 8 Minuend, 
 3 Subtrahend. 
 
 ^ 5 Difference, 
 
 Missing Numbers. 
 
 Names. 
 
 
 Methods of Finding. 
 
 ElEST. 
 
 MlKUEI^D. 
 
 
 Add the Second and 
 Third. 
 
 SECOi^D. 
 
 SUBTRAHEN^D, 
 
 
 Subtract the Third 
 from the Eirst. 
 
 Thikd. 
 
 Difference. 
 
 
 Subtract the Second 
 from the First. 
 
 Exercises for the 
 
 Slate and Board. 
 
 7 7 8 
 
 1. 
 
 8 8 ? 
 
 ? 
 
 ? ? ? ? ? 
 
 3 4 3 
 
 5 4 5 
 
 2 
 
 4 3 6 3 2 
 
 ? ? 
 
 ? ? 
 
 II. 
 
 7867? 778 ??86 
 ? ? 3 ? 5 ? ? ?. 3 4 ? . ? 
 
 In the same manner as we added by 7's on page 37, 
 add each column by 8's in the following 
 
 Exercises for the Slate and Board. 
 
 7 
 
 7 
 
 6 
 
 5 
 
 4 
 
 6 
 
 7 
 
 6 
 
 6 
 
 5 
 
 4 
 
 5 
 
 5 
 
 3 
 
 6 
 
 5 
 
 7 
 
 6 
 
 5 
 
 5 
 
 6 
 
 5 
 
 3 
 
 4 
 
 3 
 
 3 
 
 3 
 
 4 
 
 4 
 
 3 
 
 2 
 
 4 
 
 3 
 
 3 
 
 6 
 
 5 
 
 
 
 6 
 
 7 
 
 3 
 
 6 
 
 5 
 
 3 
 
 5 
 
 4 
 
 7 
 
 4 
 
 7 
 
 5 
 
 6 
 
 2 
 
 7 
 
 3 
 
 4 
 
 7 
 
 4 
 
 5 
 
 4 
 
 7 
 
 3 
 
40 
 
 FIRST LESSONS IN 
 
 LESSON XXV. 
 
 1, In this picture, how many beehives in each of the 
 two rows ? How many in both rows ? You see one 
 hive standing apart from the two rows. If you count 
 this with the others, how many are there ? We make a 
 figure Nine thus: 9. Make one. One and how 
 many more make 9 ? 
 
 2, How many birds do you see oyer this house ? How 
 many are about to light on the tree ? How many birds 
 in all ? 7 birds and 2 more are how many ? 
 
 |. S, How many squirrels have climbed the tree ? How 
 many others are running towards the tree ? 4 squirrels 
 and 2 more are how many ? How many squirrels are 
 on the fence, running away from the tree ? How many 
 squirrels in all ? 
 
MENTAL AND WRITTEN ARITHMETIC, 41 
 
 Jf, How many windows in the npper story, in the side 
 of the house ? How many in the lower story, in the 
 side of the house ? How many windows are 3 and 2 
 more ? How many windows do you see in the and of 
 the house ? 5 and 4 more are how many ? If a door 
 should be put in place of one of the windows, how many 
 would be left ? 1 window from 9 leaves how many? 
 
 5, If the boy shoot two of the squirrels, how many 
 will be left ? 2 from 9 leave how many ? 
 
 6, If 3 hives be carried away, how many will remain ? 
 3 from 9 leave how many ? 
 
 7, K the boy shoot 4 of the birds, how many will be 
 left ? 4 from 9 leave how many ? 
 
 Weittek Exercises. 
 
 1, Harry and Edward went fishing. Harry caught 2 
 sunfish, and Edward 7 ; how many did both catch ? 2 
 and 7 more are how many ? 3 of the fishes were lost 
 fi-om the basket ; how many were left ? 3 from 9 leave 
 how many? 
 
 2, Harry caught 3 perch, and Edward 6 ; how many 
 did both boys catch ? 3 and 6 are how many ? They 
 gave away 5 perch ; how many were left ? 5 perch from 
 9 leave how many ? 
 
 3, Harry caught 5 bass, and Edward 4 ; how many 
 were caught in all? 5 and 4 are how many? They 
 gave away 2 bass ; how many were left ? 2 from 9 leave 
 how many ? They also lost 2 bass. 4 bass from 9 leave 
 how many ? 
 
 Jf, Harry caught 6 eels, and lost 4 of them; how 
 many had he left ? He then caught 2 more ; how many 
 had he at last ? Edward caught 5 eels ; how many eels 
 had Harry and Edward, to carry home ? 4 eels and 5 
 eels are how many ? 
 
42 FIRST LESSONS IN 
 
 LESSON XXVI. 
 
 EXEEOISES FOR THE SlATE A.^I> BoAED. 
 
 AddUion. 
 
 I. 
 
 6 4 523235537262 
 23344142231214 
 
 II. 
 
 3262543323343 
 231/2111242111 
 3112222121234 
 1212112312111 
 
 
 
 
 
 
 Subtraction, 
 
 
 
 
 
 9 
 3 
 
 7 
 3 
 
 8 
 3 
 
 7 
 3 
 
 9 
 
 3 
 
 8 7 9 8 
 4 4 5 2 
 
 7 9 
 5.^6 
 
 8 
 5 
 
 9 
 
 7 
 
 9 
 4 
 
 JEquations, 
 
 7-f?=9 ?+2 = 9 9-3 = ? 9-2 = ? 
 
 9 -_ 5 = ? ?-3r=6 9-?=5 9 - ? = 3 
 
 4+?=9 9-?=2 ?-5=4 ?-4=5 
 
 54-?=9 3 + ? = 9 9 - ? =:= 4 9-? = 6 
 
 Examples in JLddition, 
 
 3735??4212?2??4?? 
 
 6? ?? 24? 7? ? 3? ^ ^ 'i I ^ 
 9989998?"989998998 
 
 Examples in Subtraction. 
 
 9998 9.98??9?9??9?8 
 62??4??78?6?54?2? 
 
 3?73? 5 4213324 4 472 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 43 
 
 LESSON XXVII. 
 
 1. Frank had 2 apples on a fniit-dish. He took 2 
 apples from the dish and gave them to a beggar. How 
 many apples were left on the dish? 2 apples from 2 
 apples leave how many apples ? We use a figure written 
 thus, 0^ to stand for Nothing. Its name is Zero^ 
 or Naughty or Cipher. Each of these words means 
 KoTHiKG ; and the figure, 0^ also means Nothing, 
 
 2, In Arithmetic we use no figures but these : 0, 1, 
 2, 3, 4, 5, 6, 7, 8, 9. 
 
 Write the following in Equations; and from each 
 Equation so written write three others in the manner 
 explained in Lesson XVIII. 
 
 5 and 4 are 9, 
 
 6 and 3 are 9 ; 
 
 9 less 3 equal 6, 
 9 less 6 equal 3 ; 
 4 
 
 7 and 2 are 9, 
 2 and 7 are 9 ; 
 9 less 5 equal 4, 
 9 less 4 equal 5 ; 
 
 4 and 5 are 9, 
 3 and 6 are 9 ; 
 9 less 2 equal 7, 
 9 less 7 equal 2. 
 
44 
 
 FIRST LESSONS IN 
 
 
 Learn and recite 
 
 tte annexed Addi- 
 
 tion 
 
 and Subtrac- 
 
 tion Table : 
 
 
 
 
 Add the nunabers 
 
 in 
 
 the 
 
 following ( 
 
 30l- 
 
 umns 
 
 by9's 
 
 
 
 6 
 
 3 
 
 7 6 
 
 4 
 
 4 
 
 5 
 
 6 
 
 8 6 
 
 5 
 
 2 
 
 7 
 
 7 
 
 2 4 
 
 6 
 
 5 
 
 3 
 
 5 
 
 5 8 
 
 6 
 
 7 
 
 5 
 
 4 
 
 5 5 
 
 5 
 
 3 
 
 3 
 
 6 
 
 6 3 
 
 8 
 
 7 
 
 7 
 
 5 
 
 3 4 
 
 % 
 
 8 
 
 
 J_ 
 
 Z^ 
 
 A 
 
 £ 
 
 i 
 
 1. 
 
 L 
 
 8^ 
 
 1 
 
 2 
 
 3 
 
 4. 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 3 
 
 4. 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 
 
 
 7 
 
 8 
 
 9 
 
 
 
 
 
 
 
 8 
 
 9 
 
 
 
 ^ 
 
 „, 
 
 ee* 
 
 _ 
 
 ^ 
 
 ^ 
 
 LESSON XXVIII. 
 
 1, In the cut on the next page, in the first row, how 
 many cubes in the greater part? How many in the 
 other part ? How many, in all, in that row ? 9 cubes 
 and 1 more are how many ? 
 
 2, In the second row, how many cubes in the greater 
 part ? How many in the other part ? How many, in 
 all, in the second row ? 8 cubes and 2 more are how 
 
 many 
 
 ? 2 and 8 more ? 
 
 S, In the third row, how many cubes in the greater 
 part ? In the other part ? In the third row ? 7 cubes 
 and 3 more are how many ? 3 and 7 more ? 
 
 ^. In the fourth row, how many cubes in the greater 
 part? In the other? In the fourth row? 6 cubes 
 and 4 more are how many ? 4 and 6 more ? 
 
 5. In the fifth row, how many cubes in each part? 
 5 cubes and 5 more are how many ? 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 45 
 
 mm. 
 
 3/4-/5 
 
 /a/4- /6 /e , 
 
 nETHROW 
 FOURTH BOW 
 TEQBD HOW 
 SECOND ROW 
 FIRST ROW 
 
 6. If we take 1 cube from the first row, how many 
 will be left? If, instead, we take away '9 cubes, how 
 many will remain ? 1 cube from Ten cubes leaves how 
 many ? 9 cubes from Ten" cubes leave how many ? 
 
 7. How many cubes will be left in the second row, if 
 we take away 2 cubes ? If we take 8 ? 2 cubes from 
 Te]^ leave how many ? 8 from Tek how many ? 
 
 8. How many cubes will be left in the third row, if 
 we take away 3 ? If we take 7 ? 3 cubes from Teit 
 leave how many ? 7 from TEiq" how many ? 
 
 9. Four cubes taken from the fourth row will leave 
 how many ? 6 taken away leave how many ? 4 cubes 
 from Tek leave how many ? 6 from Te^ how many ? 
 
 10. Five cubes taken from the fifth row will leave 
 how many ? 5 cubes taken from Ten leave how many ? 
 
 Learn the following 
 
 TA-HILE: 
 
 9 and 1 are Ten^, 
 8 and 2 are Ten", 
 7 and 3 are Teist, 
 6 and 4 are Ten, 
 5 and 5 are Ten, 
 4 and 6 are Ten, 
 3 and 7 are Ten, 
 2 and 8 are Ten, 
 1 and 9 are Ten ; 
 
 1 from Ten leaves 9, 
 
 2 from Ten leave 8, 
 
 3 from Ten leave 7, 
 
 4 from Ten leave 6, 
 
 5 from Ten leave 5, 
 
 6 from Ten leave 4, 
 
 7 from Ten leave 3, 
 
 8 from Ten leave 2, 
 
 9 from Ten leave 1. 
 
46 FIRST LESSONS IN 
 
 ^0!««Ki 
 
 LESSON XXIX. 
 
 L Walter arranged his set of Alphabet Blocks on a 
 sheet of paper, in the order shown above. He then pro- 
 ceeded to count them, printing the numbers on the 
 paper in words, just above the blocks. You see that 
 there were Twenty-six blocks. 
 
 2, He counted them again, and wrote the numbers 
 just below the blocks, in figures. For the blocks A, B, 
 0, D, E, F, G, H, I, he counted and wrote 1, 2, 3, 4, 5, 
 6, 7, 8, 9. For the block J he counted Ten, as before ; 
 but, not finding any single figure for Tee", he printed 
 the ivord TEiT. Counting the blocks K, L, M, N, 0, P, 
 Q, E, S, T, he wrote below them 1, 2, 3, 4, 5, 6, 7, 8, 9, 
 Teit. 
 
 For the blocks IT, V, W, X, Y, Z, he counted and 
 wrote 1, 2, 3, 4, 5, 6. 
 
 Finding that the two sets of numbers were unlike 
 beyond Tek, and not knowing how to write a number 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 47 
 
 greater than Nine, in figures, he was at first puzzled. 
 Finally, he wrote and gave to his teacher the following 
 
 TEN 
 
 and 1 
 and 2 
 and 3 
 and 4 
 and 5 
 and 6 
 and 7 
 and 8 
 and 9 
 
 TABLE. 
 
 2 TENS 
 
 are Eleven, 
 are Twelve, 
 are Thirteen, 
 are Fourteen, 
 are Fifteen, 
 are Sixteen, 
 are Seventeen, 
 are Eighteen, 
 are Nineteen. 
 
 are Twenty, 
 and 1 are Twenty-one, 
 and 2 are Twenty-two, 
 and 3 are Twenty-three, 
 and 4 are Twenty-four, 
 and 5 are Twenty-five, 
 and 6 are Twenty-six. 
 
 LESSON XXX. 
 
 myinnMEiMniEMMiiuMWM^M 
 
 flRST TEN-CROU P 
 
 EaEaEiEOiimmmmBxaamvm^m'm 
 
 «3 W 
 H O 
 
 SECOND TEN-GROUP 26 
 
 TWEKTY-SIX. 
 
 SIX ONES 
 
 Mary counted and numbered Tei^ of her Alphabet 
 Blocks, and, naming them a Ten-group^ placed them 
 at her left hand. Forming a second Ten-group, she 
 placed it with the first. The remaining Six blocks she 
 numbered and placed at the right. 
 
 Said she to her teacher : " I have Six single blocks, 
 standing at the right. I will write a figure 6, to stand 
 for these. Since I have Two Ten-groups, I will writs a 
 figure 2, to stand for them ; remembering that this 2 
 
48 
 
 FIRST LESSONS IN 
 
 does not stand for 2 blocTcs^ but for 2 Ten-groups ^ each 
 having Ten blocks. 
 
 " Since the 2 Ten-groups are placed at the left of the 
 6 single blocks, I will write the figure 2, which stands for 
 the 2 Ten-groups, at the left of the /^?^re 6, which 
 stands for the 6 single blocks ; thus, 20." 
 
 " That is correct/^ answered the teacher. " Two Tens 
 are named Twenty. . Your figures, 26, are to be read 
 Twentg-six, For an]^ number of Ones, or single blocks 
 which are not more than 9, you write one figure. Then, 
 if you have any number of Tens which are not more 
 than 9, you write one figure for them ; placing it at the 
 left of the figure which stands for the Ones» If there 
 are no Ones, you write a Cipher , thus, 0, at the right of 
 the figure standing for the Tens. 78 equals 7 Tens and 
 8 Ones ; 90 equals 9 Tens and no (0) Ones." 
 
 Some numbers are named as shown in the following 
 tabz:e: 
 
 Thirty, 
 ^ Forty, 
 
 3 Tens 
 
 4 Tens 
 
 5 Tens | Fifty, 
 
 6 Tens | Sixty, 
 
 7 Tens ^ Seventy, 
 
 8 Tens '^ Eighty, 
 
 9 Tens Ninety. 
 
 Thirty-one, etc. 
 . ^ Forty-one, etc. 
 
 3 Tens and 1 One 
 
 4 Tens and 1 One ^ ^ ._.,... 
 
 5 Tens and 1 One | Fifty-one, etc. 
 
 6 Tens and 1 One | Sixty-one, etc. 
 
 7 Tens and 1 One |^ Sevenfcy-one,etc. 
 
 8 Tens and 1 One ® Eighty-one, etc. 
 
 9 Tens and 1 One Ninety-one, etc. 
 
 Some numbers greater than 9 are written thus : 
 
 TABLE. 
 
 By Words. By Figures. 
 
 Ten, 10; 
 
 Eleven, 11 : 
 
 Twelve, 12 : 
 
 Thirteen, 13: 
 
 By Words. By Figures. 
 
 Fourteen, 14 ; 
 
 Fifteen, 15 
 
 Sixteen, 16 
 
 Seventeen, 17 
 
 By Words. By Figures. 
 
 Eighteen, 18 
 Nineteen, 19 
 Twenty, 20 
 Twenty-one, 21 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 49 
 
 By Words. By Figures. By Words, By Figures. By Words. By Figures. 
 
 Twenty-two, 22; Thirty-one, 31; Seventy, 70 
 
 Twenty-three, 23 ; Forty, 40 ; Eighty, 80 
 
 Twenty-four, 24; Fifty, 50; Ninety, 90 
 
 Thirty, 30; Sixty, 60; Ninety-nine, 99. 
 
 LESSON XXXI. 
 
 TVrlie the following J^umbers hi JP'lgures . 
 
 Twenty-seven ; 
 Thirty-three ; 
 Fifteen ; 
 Thirty-nine ; 
 Fourteen ; 
 Forty- five; 
 Thirteen ; 
 
 Fifty-three ; 
 Eleven ; 
 Sixty-eight ; 
 Twenty ; 
 Seventy-seven ; 
 Thirty; 
 Eighty-six ; 
 
 'Forty; 
 Thirty-four ; 
 Seventeen ; 
 Nineteen ; 
 Ninety-three ; 
 Fifty; 
 Ninety-nine. 
 
 Bead aloud the following Numbers : 
 
 17 41 50 64 55 36 69 58 
 
 27 44 13 12 91 10 9 84 
 
 37 49 72 85 11 93 47 39 
 
 Exercises for the Slate and Board. 
 
 Addition. 
 
 60 
 80 
 99 
 
 3 
 
 4 
 
 2 
 
 5 
 
 2 
 
 2 
 
 3 
 
 2 
 
 4 
 
 3 
 
 4 
 
 6 
 
 4 
 
 4 
 
 2 
 
 3 
 
 1 
 
 2 
 
 1 
 
 5 
 
 4 
 
 3 
 
 1 
 
 3 
 
 1 
 
 5 
 
 3 
 
 3 
 
 5 
 
 4 
 
 5 
 
 7 
 
 2 
 
 4 
 
 3 
 
 6 
 
 2 
 
 3 
 
 1 
 
 
 
 
 
 Subtraction. 
 
 
 
 
 
 9 
 
 10 
 
 8 
 
 10 
 
 10 10 9 
 
 10 
 
 10 
 
 9 
 
 9 
 
 2 
 
 % 
 
 4 
 
 5 
 
 3 6 5 
 
 4 
 
 J_ 
 
 3 
 
 4 
 
 JEquations. 
 
 5 + 5 = ? 5 + ? = 10 10-5 = ? ?-5 = 5 
 
 ?+4 = 10 6 + ? = 10 10-?=4 ?-3 = l. 
 
 10-?=:3 ?--7 = 3 ?+2 = 10 ?-2 = 8 
 
50 
 
 MBST LESSONS IN 
 
 LESSON XXXII. 
 
 1. Learn and re- 
 cite the annexed 
 Table : 
 
 2, 2 Ones and 3 
 Ones are 5 Ones; 
 and 2 Tens and 3 
 Tens are 5 Tens, or 
 50. 3 Ones taken 
 from 7 Ones leave 
 4 Ones ; and 3 Tens 
 taken from 7 J'e/xs 
 leave 4 I'e^^, or 40. 
 
 In every case, 
 Tens are added to 
 jT^/i^, or subtracted from Tens, in the same manner as 
 Ones are added to or subtracted from Ones. 
 
 
 1 
 
 2_ 
 
 ^ 
 
 £^ 
 
 ± 
 
 6 
 
 7 
 
 8 
 
 ^ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 To 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 3 
 
 4. 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 
 
 
 
 7 
 
 8 
 
 9 
 
 10 
 
 il 
 
 
 
 
 
 
 8 
 
 9 
 
 10 
 
 II 
 
 . 
 
 
 
 
 
 
 9 
 
 10 
 
 11 
 
 ^ 
 
 ,_. 
 
 
 
 
 
 
 Exercises for the Slate and Board. 
 
 JLddition, 
 
 I. 
 
 11 
 12 
 
 21 
 13 
 
 32 
 23 
 
 43 
 34 
 
 49 
 30 
 
 54 
 22 
 
 37 
 52 
 
 65 
 31 
 
 46 
 23 
 
 81 
 17 
 
 76 
 13 
 
 21 
 14 
 32 
 
 15 
 32 
 40 
 
 30 
 24 
 15 
 
 51 
 16 
 21 
 
 11 
 23 
 45 
 
 n. 
 
 33 
 
 23 
 11 
 
 12 
 23 
 34 
 
 42 
 13 
 24 
 
 52 
 24 
 12 
 
 16 
 63 
 20 
 
 71 
 14 
 14 
 
 
 
 
 
 Subtraction. 
 
 
 
 • 
 
 
 22 
 
 11 
 
 33 
 22 
 
 45 
 33 
 
 52 
 20 
 
 58 
 33 
 
 65 
 41 
 
 87 
 32 
 
 76 
 45 
 
 69 
 35 
 
 88 
 73 
 
 99 
 33 
 
MENTAL AND WRITTEN ARITHMETIC. 51 
 
 LESSON XXXIII. 
 
 Mental Exeecises. 
 
 9 + 3 = ? 
 
 ? + 4 = ll 
 
 8 + 3 = ? 
 
 ? + 5 = ll 
 
 8 + ? = ii 
 
 6 + ? = 11 
 
 3 + ? = 11 
 
 7 + ? = ll 
 
 6 + 5 = ? ■ 
 
 ? + 8 = ll 
 
 ? + 7 = 11 
 
 ? + 6 = 11 
 
 3 + ? = 11 
 
 5 + ? =11 
 
 ? + 2 = ll 
 
 4 + ? = 11 
 
 ? + 3 = 11 
 
 11 - 2 = ? 
 
 ? + 9 = ll 
 
 7 + 4 = ? 
 
 11 - 4 = ? 
 
 11-6 = ? 
 
 11 - 3 = ? 
 
 11-5 = ? 
 
 11-8 = ? 
 
 11 - 1 = ? 
 
 11-7 = ? ' 
 
 11-9 = ? 
 
 10 - 3 = ? 
 
 10 - 5 = ? 
 
 10 - 6 = ? 
 
 10 - 4 = ? 
 
 Exercises for the Slate ai^d Board. 
 
 1, Frank had 4 rabbits, and his father gave him 7 
 more ; how many had he in all ? 
 
 2, A farmer liad 16 sheep in one pasture, and 22 in 
 another ; how many had he in both ? He sold 25 sheep ; 
 how many had he left ? 
 
 3, In a school there were 43 boys, and 35 girls ; how 
 many pupils in all? 15 pupils left; how many re- 
 mained ? 
 
 ^. Willie found a hen's nest with 15 eggs, and Walter 
 found one with 13 ; how many eggs did both find? 
 
 5. In one flock of pigeons were 54, and in another 
 35 ; how many pigeons in both flocks ? 
 
 Addition, 
 
 52 
 23 
 
 34 
 
 15 
 
 18 
 21 
 
 26 
 -4i 
 
 57 84 
 31 13 
 
 Subtraction, 
 
 45 
 23 
 
 74 
 22 
 
 56 
 23 
 
 64 
 14 
 
 74 
 31 
 
 98 
 35 
 
 76 
 43 
 
 69 
 25 
 
 48 87 
 15 36 
 
 36 
 
 14 
 
 66 
 33 
 
 52 
 40 
 
 75 
 25 
 
53 
 
 FIRST LESSONS IN 
 
 LESSON XXXIV. 
 
 A/k\/ih,v/,tf,u\J:if^'h///i 
 
 1, In the upper row of these cubes, how many cubes 
 are in the greater part ? How many in the other part ? 
 In both parts ? 
 
 2, In the lower row, how many cubes are in the 
 greater part ? In the other part ? In both ? 
 
 S, We wish to find how many cubes there are in all; 
 that is, find the Sum of 18 and 17. 
 
 1st. We first add the 7 cubes in the lower row, and the 
 8 in the upper row. Since 7 and 3 are 10, we take 3 of 
 the 8 cubes, and, adding them to the 7, have 10 cubes. 
 Since 3 taken from 8 leave 5, we have 5 of the 8 cubes 
 left. 
 
 As shown in the following cut, 
 
 we separate the 8 cubes into 3 cubes and 5 cubes, and 
 then place the 3 cubes at the end of the row of 7 cubes. 
 Counting them together, we have 10 cubes. 
 
 Thus we have in all a group of 10 cubes, and 5 single 
 cubes ; which are 15 cubes. 
 
 2d. We next arrange the cubes as shown in the cut 
 at the top of the opposite page. The Ten-group, which 
 we have formed, we carry to the left and place with 
 
MENTAL AND WRITTEN ARITHMETIC. 53 
 
 the 2 other Ten-groups. Counting the 2 Tens with the 
 1 Ten CARRIED, we find they are 3 Tens. Thus, we 
 have, in all, 3 Tens and 5 Ones; which are 35 cubes. 
 
 In like manner, we add 18 and 17 by 
 figures. We write the numbers as shown addition. 
 at the right. Taking 3 of the 8 Ones, 18 ] p^rts. 
 we add them to the 7 Ones to make 10, ^ i 
 and then add the remaining 5 Ones to 35 Sum. 
 the 10, and thus obtain 15 as the Sum 
 of 8 and 7. We then carry to the left the 1 Ten of 
 the 15, and, addmg it with the 1 Ten of the 17 and the 
 1 Ten of the 18, obtain 3 Tens. Thus we find that the 
 Sum of 18 and 17 is 3 Tens and 5 Ones, or 35. 
 
 In like manner we add any hvo or more numhers, 
 when their Sum is not greater than 99. We carry to 
 THE left all the Tens formed by adding the figures in 
 the right-hand column. 
 
 TBSTIJV-G THB SUM. 
 
 After adding numbers we sometimes doubt the cor- 
 rectness of our work. In such cases it is well to add 
 the figures a second time, commencing at the top and 
 adding to the bottom. If we obtain the same result as 
 in the first instance, it is presumed that we have found 
 the true Sum. This second addition is named Test' 
 ing the Sum. 
 
54 
 
 FIRST LESSONS IN 
 
 LESSON XXXV, 
 
 This Table should 
 be learned so thor- 
 oughly that any 
 part of it can be 
 given without hesi- 
 tation. Give special 
 attention to that 
 part of the Table 
 which includes 
 numbers greater 
 than 10. No fur- 
 ther Table for Ad- 
 dition or Subtrac- 
 tion will be needed 
 if this be mastered. 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 \1. 
 
 \1 
 
 
 ^grf^^s 
 
 i?g=^I^^g 
 
 
 ImmI 
 
 lgs=s^ 
 
 1 
 
 ^^m 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 3 
 4 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 15 
 
 7 
 8 
 
 8 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 9 
 
 10 
 
 II 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 9 
 
 10 
 
 II 
 
 12 
 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 Exercises eor the Slate akd Board. 
 
 Addition, 
 
 25 
 
 65 
 
 16 
 34 
 
 37 
 
 13 
 
 48 
 12 
 
 56 
 35 
 
 27 
 16 
 
 57 
 26 
 
 46 
 37 
 
 54 
 
 29 
 
 
 
 
 
 
 II. 
 
 
 
 
 
 8 
 
 23 
 
 39 
 
 63 
 
 18 
 
 19 
 
 24 
 
 11 
 
 62 
 
 42 
 
 7 
 
 50 
 
 23 
 
 14 
 
 25 
 
 44 
 
 18 
 
 45 
 
 18 
 
 19 
 
 5 
 
 17 
 
 28 
 
 19 
 
 34 
 
 23 
 
 HI. 
 
 35 
 
 37 
 
 16 
 
 27 
 
 9 
 
 14 
 
 16 
 
 12 
 
 34 
 
 12 
 
 11 
 
 14 
 
 13 
 
 19 
 
 6 
 
 35 
 
 13 
 
 14 
 
 13 
 
 14 
 
 14 
 
 31 
 
 27 
 
 33 
 
 8 
 
 13 
 
 47 
 
 17 
 
 28 
 
 36 
 
 42 
 
 14 
 
 18 
 
 22 
 
 7 
 
 28 
 
 14 
 
 33 
 
 13 
 
 27 
 
 19 
 
 28 
 
 30 
 
 11 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 55 
 
 LESSON XXXVI. 
 
 Add by Ti. 
 
 Add by 8'e. 
 
 AddbyO's. , 
 
 Add by lO'a. 
 
 Add. 
 
 6 3 1 
 
 7 6 
 
 7 8 
 
 8 7 
 
 15 18 
 
 5 6 6 
 
 .5 7 
 
 8 7 
 
 9 8 
 
 34 14 
 
 4 3 5 
 
 6 5 
 
 6 5 
 
 5 6 
 
 10 33 
 
 5 4 4 
 
 3 7 
 
 3 6 
 
 6 5 
 
 37 17 
 
 16 5 
 
 4 7 
 
 4 1 
 
 3 4 
 
 14 35 
 
 Written Exercises. 
 
 1. How many bushels of apples in 3 piles, the first 
 containing 26 bushels, the second 35 bushels, and the 
 third 29 bushels? 
 
 2, A poultry dealer sold some turkeys for 46 dollars, 
 some geese for 23 dollars, and some chickens for 27 dol- . 
 lars ; how many dollars did he receive in all ? 
 
 S. A farmer sold 38 bushels of wheat, 26 bushels of 
 corn, 17 bushels of rye, and 16 bushels of barley; how 
 many bushels of grain did he sell in all ? 
 
 Jf, In 3 days, Albert picked 23 quarts of strawberries, 
 Harry 29 quarts, Alfred 34 quarts, and Walter 10 quarts; 
 how many quarts did the 4 boys pick ? 
 
 5, How many pigeons are twenty-eight pigeons, 
 thirty-four pigeons, and twenty-nine pigeons ? 
 
 6, Frank rode twenty-nine miles Monday, thirty 
 Tuesday, and eighteen Wednesday; how many miles 
 did he ride during the three days? 
 
 7, Willie had 47 peaches in his basket, Frederick 16 
 in his, and Henry 28 in his ; how many peaches had the 
 three boys ? 
 
 8, Homer had 45 oranges in his basket, and Horatio 
 39 in his ; how many oranges had both boys ? 
 
 9, Nellie spelled 39 words, and Mary 48 ; how many 
 did both spell? 
 
56 
 
 FIRST LESSONS IN 
 
 LESSON XXXVIL 
 
 1.— MINUEND. 
 
 ■■iHiHiE3EHE3fli 
 
 ■ 
 
 
 ■• 'mM 
 
 WHnifliEiOI^M 
 
 
 lilil|ii'iii::^::^vr:i:.^;i^^^ 
 
 iii 
 
 SUBTRAHEND. 
 
 
 1 
 
 IT 
 
 E 
 
 :n' I 
 
 ■ 
 
 Iii 
 
 1 
 
 i 
 
 i!!i:'- 
 
 
 J! 
 
 I 
 
 The riglit-liand figure in the Subtrahend greater than 
 the figure above it in the Minuend, 
 
 Example. From 35 subtract 18. subtraction. 
 
 Writing the 18 under the 35, we find 35 Minuend, 
 that 8 cannot be subtracted from 5. 18 Subtrahend, 
 
 First : Subtraction hij Objects, 
 
 The above cut is in 2 Parts. In Part 1. how 
 many Ten-groups are in the Minuend ? How many 
 single cubes ? 3 Tens and 5 Ones are how many ? 
 How many Ten-groups are in the Subtrahend ? How 
 many single cubes ? 1 Ten and 8 Ones are how many ? 
 
MENTAL AND WRITTEN ARITHMETIC, 57 
 
 This Minuend, then, having 35 cubes, and Subtrahend 
 having 18 cubes, answer to the Minuend and Subtrahend 
 in the given Example. Hence the Difference between 
 35 cubes and 18 cubes will be the answer to the given 
 Example. 
 
 Since 8 cubes cannot be subtracted from 5 cubes, we 
 take 1 of the 3 Ten-groups in the Minuend and caery 
 it TO THE RIGHT, and place it with the 5 cubes, as 10 
 separate cubes. 
 
 This Minuend, as re-arranged, is shown' in Part 11. of 
 the cut. Since the Minuend and Subtrahend in Part 11. 
 are equal to those in Part I., the Difference in Part II. 
 and in Part I. must be the same. We will find the Dif- 
 ference in Part II. 
 
 We separate the Ten-group, which stands with the 5 
 cubes, into two parts, one having 8 cubes and the other 
 2. Since there are 8 cubes in the Subtrahend, we sub- 
 tract 8 cubes from the 10 cubes. There are then left, 
 in the Minuend, 2 of the 10 cubes and also the 5 cubes. 
 Placing 5 cubes and 2 cubes in the Difference, we have 
 7 cubes as the Difference between the 8 cubes of the 
 Subtrahend and the 10 cubes and 5 cubes of the 
 Minuend. 
 
 In Part I. we had 1 Ten to subtract from 3 Tens ; but 
 in Part IL, having carried to the right 1 Ten in the 
 Minuend, we have only 2 Tens remaining at the left. 
 Subtracting 1 Ten from 2 Tens, and obtaining 1 Ten 
 for the Difference, we place 1 Ten-group in the Differ- 
 ence. Thus we find that 18 cubes taken from 35 cubes 
 leave 17 cubes. 
 
 Second : Subtraction by Figures, 
 
 Wliat we have done with the cubes we will now do 
 with the figures. As shown in the margin, we write 
 
58 FIRST LESSONS IN 
 
 the 18 under the 35. We then re-arrange the Minuend, 
 by first taking 1 of the 3 Tens and 
 
 CAERYIKG it TO THE RIGHT and writ- Subtraction. 
 
 ing it as 10 Ones, over the 5 Ones, !. ^^ ,^. 
 and then drawing a Hne through the f I Minuend, 
 3, to show that it is not to be used, ^ SuUrahend. 
 and writing the 2 remaining Tens ^ ^ Difference, 
 over the former 3. 
 
 First, we subtract 8 from 5 and 10 taken together. 
 Subtracting, 8 from 10 leave 2 ; and this 2 and the 5, 
 added together, give 7 Ones for the Difference. Sub- 
 tracting 1 Ten from 2 Tens, we have 1 Ten for the 
 Difference in Tens. Thus we obtain for our Difference 
 1 Ten and 7 Ones ; or 17. 
 
 In every Example like this, the work is performed in 
 the manner just explained. It is not necessary, how- 
 ever, to change the figures of the Minuend. In this 
 Exam^^le, we write the numbers as 
 shown in the margin. Subtracting, subtraction. 
 
 we say, not 8 from 5, but 8 from 10 35 Minuend, 
 leave 2 ; 2 and 5 are 7 ; and then 18 SuUrahend, 
 write 7 in the Difference. Finally, 17 Difference. 
 we say, not 1 Ten from 3 Tens, but 1 
 Ten from 2 Tens leaves 1 Ten ; and write 1 Ten in the 
 Difference. 
 
 
 Examples 
 
 POR • 
 
 THE Slate xi^jy Board. 
 
 
 29 
 13 
 
 54 
 29 
 
 35 
 
 18 
 
 72 
 36 
 
 I. 
 
 91 
 43 
 
 II. 
 
 57 
 29 
 
 74 
 68 
 
 33 
 16 
 
 96 
 48 
 
 65 
 43 
 
 82 
 37 
 
 23 
 
 15 
 
 44 
 
 28 
 
 63 
 
 27 
 
 56 
 
 28 
 
 71 
 58 
 
 62 
 33 
 
 78 
 39 
 
MENTAL AND WRITTEN ARITHMETIC. 69 
 
 LESSON XXXVIII. 
 su:bt^a ctiojv. 
 
 Testing tlie Difference, 
 
 If the Difference between two numbers be added to 
 the less, the Sum will equal the greater. If this Differ- 
 ence be subtracted from the greater number, the result 
 will equal the less. Hence, we may Test -our Difference 
 in Subtraction by either of two methods. 
 
 First Method: Testing by Addition, 
 
 Add the Difference to the Subtrahend, If the Sum 
 equals the Minuend, it is presumed that we have found 
 the true Difference. 
 
 Second Method: Testing hy Stibtraction. 
 
 Subtract the Difference from the Minuend. If the re- 
 sult equals the first Subtrahend it is presumed that the 
 first Dfference is correct. 
 
 Find and test the Difference in each of the following 
 Examples for the Slate akd Boaed. 
 
 Testing by First Method, 
 
 83 
 
 52 
 
 70 93 80 47 
 
 90 
 
 75 65 
 
 64 
 
 38 
 
 36 79 39 18 
 
 Testing lyy Second, MethoA. 
 
 45 
 
 25 47 
 
 33 
 
 79 
 
 43 53 39 62 
 
 71 
 
 80 91 
 
 19 
 
 49 
 
 28 14 19 53 
 
 AdeUHan at Sight. 
 
 52 
 
 65 74 
 
 5 
 
 7 9 
 
 6 4 8 4 5 9 
 
 3 
 
 5 5 6 
 
 4 
 
 4 4 
 
 4 4 4 5 5 5 
 
 5 
 
 6 8 3 
 
60 FIRST LESSONS IJf 
 
 LESSON XXXIX. 
 
 
 
 
 
 Addition, 
 
 
 
 
 
 u 
 
 19 
 
 24 
 
 18 
 
 37 
 
 19 
 
 23 
 
 13 
 
 12 
 
 15 
 
 23 
 
 n 
 
 32 
 
 14 
 
 25 
 
 11 
 
 41 
 
 29 
 
 16 
 
 16 
 
 15 
 
 13 
 
 13 
 
 13 
 
 23 
 
 13 
 
 29 
 
 17 
 
 28 
 
 17 
 
 29 
 
 28 
 
 17 
 
 35 
 
 32 
 
 29 
 
 
 
 
 
 Subtraction. 
 
 
 
 
 
 83 
 
 94 
 
 64 
 
 77 
 
 85 
 
 97 
 
 54 
 
 78 
 
 91 
 
 57 
 
 31 
 
 24 
 
 59 
 
 25 
 
 88 
 
 27 
 
 59 
 
 42 
 
 Written Exercises. 
 
 Addition and Subtraction. 
 
 L Edwin had a set of 26 Alphabet Blocks on his 
 desk, and Susan had the same number on her desk ; 
 how many blocks did both have ? 
 
 Edwin put 17 of his blocks in the box; how many 
 were left on his desk ? 
 
 Susan put 19 of her blocks in the box ; how many 
 were left on her desk ? 
 
 How many did both put in their boxes ? 
 
 How many remained on both desks ? 
 
 2. Mr. Newton had 55 sheep, and Mr. Lawton had 
 42 ; how many sheep did both have ? 
 
 How many more sheep had Mr. Newton than Mr. 
 Lawton ? 
 
 Mr. Newton sold 29 sheep to Mr. Lawton ; how many 
 had Mr. Lawton then ? 
 
 How many had Mr. Newton left ? 
 
 How many more had Mr. Lawton than Mr. Newton ? 
 
 
 
 
 
 Addition at Sight. 
 
 
 
 
 
 7 
 5 
 
 9 
 
 6 
 
 7 
 6 
 
 8 
 6 
 
 6 5 9 7 4 
 
 g Y 7 7 7 
 
 8 
 
 7 
 
 6 
 9 
 
 6 
 
 7 
 
 6 
 8 
 
MENTAL AND WRITTEN ARITHMETIC, 61 
 
 LESSON XL 
 Examples for the Slate and Board. 
 
 Addition. 
 
 18 
 
 27 
 
 13 
 
 16 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 13 
 
 13 
 
 28 
 
 23 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 17 
 
 14 
 
 12 
 
 11 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 18 
 
 25 
 
 17 
 
 35 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 19 
 
 16 
 
 25 
 
 13 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 Mental Exercises. 
 
 1, One of 2 hens has 7 chickens, and the other 9 ; 
 how many chickens have both hens ? 
 
 2, Charles has 9 marbles, and Frank 6 ; how many 
 marbles have both boys? 
 
 5, Flora picked' 8 quarts of strawberries, and Ella 9; 
 how many quarts did both pick ? 
 
 Jf. Edward shot 11 squirrels, and Henry 6 ; how many 
 more did Edward shoot than Henry ? 
 How many did both boys shoot? 
 
 6, Harry bought 12 peaches, and gave 5 of them to 
 his sister ; how many had he left ? 
 
 6, Walter was 18 years old, and "Willie 9; how many 
 years was Walter older than Willie ? 
 
 7. On a tree were 17 pigeons, and 9 of them flew 
 away ; how many were left on the tree ? 
 
 Addition at Sight* 
 
 2 
 8 
 
 7 
 4 
 
 3 
 
 8 
 
 5 
 
 7 
 
 8 7 4 8 5 
 5 7 8 6 8 
 
 Subtraction at Sight. 
 
 8 
 
 7 
 
 9 
 
 5 
 
 8 
 8 
 
 3 
 9 
 
 4 
 
 2 
 
 5 
 3 
 
 5 
 _3 
 
 6 
 3 
 
 6 6 7 7 8 
 2 4 2 4 3 
 
 7 
 3 
 
 7 
 5 
 
 8 
 4 
 
 8 
 6 
 
62 
 
 FIRST LESSONS IN 
 
 One iluNDKED and Fifty- Six; 
 
 or, in figures, 
 
 156, 
 
 LESSON XL/. 
 
 A maker of Alphabet Blocks prepared 6 sets for let- 
 tering, and requested his little daughter to count them 
 and tell him how many there were. She counted them, 
 wrote on them, and arranged them as shown in the 
 above cut. 
 
 Said she to her father: "Because I cannot count 
 more than Ten, I have counted the blocks in groups, 
 each having Ten blocks. I have placed each Ten in a 
 row, writing the numbers on the blocks as T counted 
 them, and finally arranged the groups side by side. 
 After making as many Tens as I could, I placed the 
 remaining blocks at the right of these, and counted 
 them 1, 2, 3, 4, 5, 6. 
 
 " I have tried to count my Ten-groups, but find there 
 are more than Ten of them. "When, at first, I was 
 counting the single Uoclcs, I counted them in groups of 
 Ten single Uoclcs, because I could not count more than 
 Ten. Now, also, when I am counting my Ten-groupSy 
 because I cannot count beyond Ten, I will, in the same 
 
MENTAL AND WRITTEN ARITHMETIC, 63 
 
 manner, count my Ten-groups into larger groups, each 
 having Ten Ten-groups, As I count these Ten-groups 
 I number them, and write on the ends of them 1, 2, 3, 
 4, 5, 6, 7, 8, 9, Ten. These Ten Ten -groups I separate 
 from the others, and remove them a little to the left, 
 and call them 1 large group. I count the other Ten- 
 groups, and write on them 1, 2, 3, 4, 5. 
 
 " Since there are six single blocks at the right, I will 
 write a figure 6, to stand for them ; thus, 6. Since 
 there SiYefive Ten-groups standing by themselves, at the 
 left of the six blocks, I will write a figure 5 to stand for 
 them ; placing it at the left of the figure 6. Since there 
 is one large group, at the left of the 5 Ten-groups, I will 
 write a figure 1 to stand for this ; placing it at the left 
 of the figure 5, 
 
 " My figures will then be placed in the same order as 
 the groups and single blocks ; thus, 1 5 6.^' 
 
 Her father said : " Your large group is named One 
 Hundred ; your 5 Ten-groups are named Fifty ; and 
 your figures, 156, are read One Hundred and Fifty-sixP 
 
 Numbers greater than 99 are named and written thus : 
 
 TAJiljE. 
 
 10 Tens are One Hundred ; written, 100. 
 20 Tens are Two Hundred ; written, 200. 
 30 Tens are Three Hundred ; written, 300. 
 40 Tens are Four Hundred ; written, 400. 
 60 Tens are Five Hundred ; written, 500. 
 60 Tens are Six Hundred ; written, 600. 
 70 Tens are Seven Hundred ; written, 700. 
 80 Tens are Eight Hundred ; written, 800. 
 90 Tens are Nine Hundred ; written, 900. 
 
 101 is read One Hundred and One. 
 570 is read Five Hundred and Seventy. 
 999 is read Nine Hundred and Ninety-Nine. 
 
64 FIRST LESSONS IN 
 
 LESSON XLII. 
 
 Write, in figures, the following : 
 
 Five Hundred and Twenty- Nine Hundred and Ninety- 
 three ; nine ; 
 
 Four Hundred and Eighty- Seven Hundred and Eleven; 
 seven ; 
 
 Six Hundred and Ninety; Five Hundred and Seven; 
 
 Eight Hundred and Forty ; Seven Hundred ; 
 
 Seven Hundred and Ten ; Three Hundred and Eight; 
 
 Addition. 
 
 Example. Find the Sum of 456 and 231. 
 
 ExPLA:srATiON. We add the Ones and Tens in the 
 same manner as if there were no Hun- 
 dreds, and finding the Sum to be 87 addition. 
 write it under the columns. Adding 456 | p^^^^ 
 2 Hundreds and 4 Hundreds in the ^^ 3 
 same manner as we added Ones, and 687 Sum, 
 Tens, and writing the Sum, 6 Hundreds, 
 we have 687 as the entire Sum of 456 and 231. 
 
 Exercises for the Slate ai^d Board. 
 I. 
 
 147 
 236 
 
 238 
 354 
 
 546 
 234 
 
 329 
 453 
 
 176 
 215 
 
 415 
 127 
 
 824 
 159 
 
 213 
 158 
 
 215 
 326 
 144 
 
 423 
 135 
 
 218 
 
 154 
 218 
 323 
 
 II. 
 231 
 154 
 
 208 
 
 119 
 218 
 351 
 
 235 
 144 
 216 
 
 310 
 
 207 
 156 
 
 407 
 200 
 126 
 
 Siihtraction, 
 
 We subtract Hundreds from Hundreds in the same 
 manner as we do Tens from Tens, or Ones from Ones. 
 
 Exercises for the Slate akd Board. 
 549 763 452 985 691 826 398 745 
 321 245 147 756 243 518 289 245 
 
MENTAL AND WRITTEN ARITRMETia 65 
 
 LESSON XLin. 
 
 Carrying every 10 Tens to the Left as 1 Hundred. 
 
 Example. — Find the Sum of 574 and 253. 
 
 ExPLAi^ATiOiq^. — Adding the Ones, we write their 
 Sum, 7. Adding the Tens, their Sum is 
 12 Tens; or 10 Tens and 2 Tens. We addition. 
 write the 2 Tens under the column of 574 ) p^^^^^ 
 Tens. Since the 10 Tens are 1 Hun- ?^^ ) 
 dred, we carry them to the left as 1 Hun- 827 Sum, 
 dred, and add this Hundred with the 
 other Hundreds, in the same manner as heretofore, in 
 adding Tens and Ones, we have carried to the left every 
 10 Ones, from the column of Ones, and added them as 
 1 Ten with the other Tens at the left. Adding 2 Hun- 
 dreds, 5 Hundreds, and the I Hmidred which we carried 
 to the left, we write 8 under the column of Hundreds. 
 
 Thus we find the Sum of 574 and 253 to be 827. 
 
 Exercises eor the Slate akd Board. 
 
 235 
 
 162 
 
 421 
 
 L 
 
 324 
 
 142 
 
 170 
 
 214 
 
 362 
 
 142 
 
 231 
 
 235 
 
 132 
 
 250 
 
 218 
 
 152 
 
 123 
 
 231 
 
 354 
 
 162 
 
 261 
 
 134 
 
 221 
 
 451 
 
 154 
 
 152 
 
 215 
 
 315 
 
 IL 
 
 173 
 
 219 ' 
 
 123 
 
 324 
 
 218 
 
 237 
 
 143 
 
 267 
 
 326 
 
 102 
 
 257 
 
 153 
 
 172 
 
 123 
 
 374 
 
 184 
 
 184 
 
 391 
 
 164 
 
 247 
 
 154 
 
 132 
 
 125 
 
 212 
 
 115 
 III 
 
 173 
 
 231 
 
 141 
 
 431 
 
 173 
 
 214 
 
 365 
 
 143 
 
 237 
 
 378 
 
 215 
 
 132 
 
 254 
 
 175 
 
 138 
 
 375 
 
 114 
 
 152 
 
 171 
 
 269 
 
 135 
 
 143 
 
 243 
 
 114 
 
 345 
 
 125 
 
 124 
 
 171 
 
 156 
 
 257 
 
 156 
 
 252 
 
 103 
 
 261 
 
 453 
 
 183 
 
66 FIRST LESSONS IN 
 
 LESSON XLIV. 
 
 Carrying 1 Hundred to the Right as 10 Tens. 
 
 Example. Subtract 379 from 652. 
 
 Expla:n-atio:n'. Carrying to the right 1 of the 5 Tens 
 in the Minuend, and subtracting 9 
 Ones from 10 Ones and 2 Ones, solution. 
 
 we have 3 Ones left. Hence we f^^ Minuend. 
 
 ., o;i^i, 1 ^rk 379 Subtrahend. 
 
 write 3 under the column oi Ones. — -- 
 
 Having carried to the right 1 of ^^^ Difference. 
 the 5 Tens in the Minuend, there are only 4 Tens left. 
 Since we can not subtract 7 Tens from 4 Tens, we carry 
 to the right 1 of the 6 Hundreds in the Minuend, and, 
 calling it 10 Tens, subtract the 7 Tens from the 10 Tens 
 and 4 Tens, saying: 7 Tens from 10 Tens leave 3 Tens; 
 3 Tens and 4 Tens are 7 Tens. We then write 7 Tens. 
 
 Having carried to the right 1 of the 6 Hundreds in 
 the Minuend, we now subtract the 3 Hundreds in the 
 Subtrahend from the 5 Hundreds left in the Minuend, 
 and write 2 Hundreds in the Difference. 
 
 Hence the Difference between 379 and 652 is 273. 
 
 
 EXEKCISES : 
 
 FOE THE Slate 
 
 A.^jy BOAED. 
 
 
 576 
 234 
 
 765 
 284 
 
 648 
 
 257 
 
 I. 
 
 429 946 
 156 493 
 
 II. 
 
 328 
 254 
 
 724 
 251 
 
 514 
 127 
 
 842 
 
 257 
 
 578 
 259 
 
 421 
 156 
 
 627 724 
 253 468 
 
 III. 
 
 218 
 104 
 
 327 
 139 
 
 876 
 588 
 
 711 
 524 
 
 526 
 389 
 
 410 
 215 
 
 674 836 
 
 285 647 
 
 325 
 147 
 
 939 
 
 756 
 
 575 
 197 
 
MENTAL AND WRITTEN ARITHMETIC. 67 
 
 LESSON XLV. 
 
 Exercises for the Slate ais^d Board. 
 
 Addition, 
 
 158 
 
 189 
 
 176 
 
 184 176 
 
 154 
 
 187 
 
 197 
 
 176 
 
 147 
 
 124 
 
 117 145 
 
 238 
 
 176 
 
 137 
 
 167 
 
 135 
 
 189 
 
 269 179 
 
 197 
 
 198 
 
 167 
 
 134 
 
 178 
 
 165 
 
 175 186 
 
 286 
 
 184 
 
 147 
 
 179 
 
 186 
 
 248 
 
 236 179 , 
 II. 
 
 124 
 
 153 
 
 157 
 
 123 
 
 134 
 
 125 
 
 136 137 
 
 148 
 
 129 
 
 148 
 
 123 
 
 134 
 
 135 
 
 116 117 
 
 128 
 
 119 
 
 128 
 
 123 
 
 134 
 
 125 
 
 126 147 
 
 118 
 
 139 
 
 118 
 
 123 
 
 134 
 
 115 
 
 116 127 
 
 138 
 
 159 
 
 138 
 
 123 
 
 134 
 
 135 
 
 136 117 
 
 158. 
 
 119 
 
 158 
 
 123 
 
 134 
 
 115 
 
 126 137 
 
 138 
 
 139 
 
 118 
 
 123 
 
 134 
 
 125 
 
 136 117 
 
 128 
 
 149 
 
 121 
 
 
 
 Addition at Sight, 
 
 
 
 
 9 7 4 
 
 8 2 
 
 6 9 9 
 
 9 9 
 
 5 9 9 
 
 9 9 9 
 
 9 9 
 
 9 4 8 
 
 2 6 
 
 9 7 3 
 
 
 
 
 Subtraction, 
 
 
 
 
 524 
 
 743 
 
 621 
 
 I. 
 
 812 546 
 
 954 
 
 726 
 
 314 
 
 173 
 
 429 
 
 238 
 
 734 268 
 II. 
 
 267 
 
 459 
 
 198 
 
 957 
 
 742 
 
 525 
 
 326 813 
 
 640 
 
 435 
 
 293 
 
 243 
 
 529 
 
 389 
 
 178 508 
 
 239 
 
 276 
 
 108 
 
 
 
 Subtraction at Sight 
 
 , 
 
 
 
 9 1 
 
 ) 8 
 
 9 8 
 
 10 9 10 
 
 9 
 
 8 10 
 
 10 
 
 2 4 3 
 
 3 5 
 
 _5 6 _8 
 
 5 
 
 6 _6 
 
 4 
 
68 
 
 FIRST LESSONS IJST 
 
 ONE 'j.kJ.<j^i:,.i^±j ONE HUNDRED AND ELBYBN; 
 
 or, written in figures, 
 
 1,111. 
 
 LESSON XLVI. 
 
 A mechanic sawed out a basketful of Alphabet Blocks 
 and gave them to his little son to count. 
 
 The boy first counted them and arranged them as 
 shown in the above cut. Then, turning to his father, 
 he said: '^I cannot count beyond Ten. I first counted 
 the blocks in groups having Ten blocks in each. Each 
 of these groups I call a Ten-groujp. After making as 
 many Ten-groups as I could I had 1 single block left, 
 which I have placed by itself, at the right hand. 
 
 " I next counted these Ten-groups in the same man- 
 ner as I counted the single blocks ; putting Ten Ten- 
 
HENTAL AND WRITTEN ABITIIMETIC. G9 
 
 groups together, and placing them one Ten-group above 
 another ; thus making of every Ten Ten-groups 1 larger 
 group. 
 
 " After making as many as I could of these larger 
 groups, I had 1 Ten-group remaining, which I have 
 placed at the left of the single block." 
 
 " Each of these larger groups is named a Hundred ; 
 and the 1 Ten-group and the- single block, taken to- 
 gether, are named eleven," remarked the- father. 
 
 " Then," said the son, " I will call each of these larger 
 groups a Hui^DRED-GROUP. I then counted my Hun- 
 dred-groups, placing Ten of them side by side, to make 
 1 very large group. I had 1 Hundred-group remaining, 
 which I placed at the left of the 1 Ten-group. At the 
 left of this Hundred-group I placed the 1 very large 
 group." 
 
 Said the father: "This largest group is named a 
 Thousand. Can you tell me how many blocks there 
 are, and then write the number h^ figures V^ 
 
 The son, after reflecting a moment, promptly replied : 
 "There are Oxe Thousan^d Oke Hundred and 
 Eleven blocks. I have one single block, standing alone, 
 and also one group of each kind standing by itself. I 
 will write a figure 1 to stand for the one block, and also 
 a figure 1 to stand for each one group of the different 
 kinds; writing the figures in the same order as the 
 groups are placed ; thus, 1111." 
 
 The boy answered correctly. And always in writing 
 a number having Thousands, we write a figure showing 
 the number of Thousands, placing it at the left of the 
 figure written for Hundreds. 
 
 Eead the following numbers : 
 
 1111; 1119; 1110; 1211; 1201; 1200; 1000; 5009; 
 5900; 5990; 7080; 5017; 1001; 9999. 
 
W FIRST LESSONS IN 
 
 LESSON XLVIL 
 
 JVO TA no AT ;;ij[r^ JSrUMB'UA TIOJV. 
 
 Writing numbers in figures is named Notation. 
 
 When numbers are written in figures, reading them 
 in luords is named Numeration* 
 
 Eead, or Numerate, the following numbers : 
 
 1560, 1506, 1056, 156, 1500, 1050, 1005, 7089, 9090, 
 9017, 1921, 5786, 1870. 
 
 Write in figures the following numbers : 
 
 1, Three Thousand Five Hundred and Seventy-six ; 
 
 2, One Thousand Two Hundred and Ten ; 
 S, Two Thousand One Hundred and Three ; 
 ^. Four Thousand and Thirty ; 
 
 5. Five Thousand and Thirteen ; 
 
 6. Seven Thousand and Three ; 
 
 7. One Thousand Two Hundred. 
 
 A.ddition, 
 
 In adding Hundreds and Thousands, we carry to the 
 left every 10 Hundreds, calling them 1 Thousand, and 
 add this Thousand with those in the column of Thou- 
 sands. 
 
 Exercises eor the Slate ai^d Board. 
 
 1542 
 
 1020 
 
 2153 
 
 1450 
 
 1000 
 
 2157 
 
 1740 
 
 2176 
 
 1507 
 
 1208 
 
 1234 
 
 2000 
 
 1528 
 
 2031 
 
 1628 
 
 2418 
 
 1759 
 
 2357 
 
 2060 
 
 1S72 
 
 1507 
 
 3473 
 
 1256 
 
 1080 
 
 1567 
 
 1007 
 
 2143 
 
 1423 
 
 1024 
 
 2569 
 
 3417 
 
 1728 
 
 3000 
 
 1726 
 
 2158 
 
 
 
 A.ddition at Sight, 
 
 
 
 8 5 
 
 9 6 
 
 7 9 
 
 7 7 
 
 8 
 
 8 8 
 
 8 9 
 
 5 9 
 
 6 8 
 
 6 7 
 
 7 8 
 
 6 
 
 8 9 
 
 7 9 
 
MENTAL AND WRITTEN ABITHMETIC, 71 
 
 LESSON XLVIII. 
 su:B2'^actiojv. 
 
 In Subtraction, whenever the Hundreds in the Sub- 
 trahend are more than those above them in the Minuend, 
 we carry to the right 1 of the Thousands in the Minuend, 
 calhng it 10 Hundreds, in the same manner as we have 
 heretofore carried to the right 1 Hundred, calhng it 10 
 Tens. 
 
 EXEBCISES FOR THE SlATE AifD BOAED. 
 
 5497 
 2145 
 
 7421 
 5176 
 
 9354 
 
 5927 
 
 I. 
 
 3257 
 1476 
 
 8542 
 4716 
 
 6215 
 4389 
 
 9666 
 5999 
 
 4571 
 
 2786 
 
 5274 
 1548 
 
 7345 
 5456 
 
 II. 
 
 9876 
 4894 
 
 Addition. 
 
 6721 
 3834 
 
 2925 
 1673 
 
 8007 
 5002 
 
 1439 1247 1020 1300 1172 1526 1000 1210 
 
 1256 1072 1513 1040 1094 1017 1200 1030 
 
 1073 1364 1327 1000 1207 1432 2030 1140 
 
 1142 1289 1156 1708 1165 1157 1005 1327 
 
 1381 1070 1208 1159 1082 1215 1234 1456 
 
 1574 1528 1097 1023 1521 1000 1357 1579 
 
 Addition at Sight, 
 
 The right-hand figure in the Sum will be unchanged 
 so long as the right-hand figures in the numbers to be 
 added remain unchanged, however we vary the Tens. 
 Thus ; 3 and 5 are 8 ; 13 and 5 are 18 ; 13 and 15 are 
 28. Changing the Tens in the numbers to be added 
 changes the Tens in the Sum, but not the Ones, 
 
 1 11 11 2 12 12 1 11 11 2 12 12 
 
 2 2 12 2 2 12 3 3 13 3 3 13 
 
72 FIBST LESSONS IN 
 
 LESSON XL/X. 
 
 ^BZATIOjY ^BTyTjEJBJSr ^D^ITIOJV ^jV!2> 
 SU'BT^A CTIOJV, 
 
 Example 1. What is the Sum of 347 and 456 ? 
 Adding as in other cases the Sum is 
 
 803. ADDITION. 
 
 , Example 2. The Sum of two 347) Two 
 numbers is 803, and one of them is ^^ ) Numbers. 
 456 ; what is the other number ? 803 Sum. 
 
 Explanation^. We notice 
 that the Minuend is like the subtraction. 
 
 Sum just obtained by Addi- Sum, 803 Minuend. 
 tion ; and also that the Sub- ^One l^^g SuUrahend. 
 
 trahend is one of the two Number.) 
 
 numbers just added. By 
 
 subtracting we shall obtain, for the Difference, the other 
 
 of the two numbers added. 
 
 Since we cannot take 6 Ones from 3 Ones, we seek 
 in the column of Tens 1 Ten to carry to the right, but 
 find none. We will go back to our work in Addition 
 and find the reason for this. 
 
 Adding 6 Ones and 7 Ones, we had for the Sum 13 
 Ones ; or 10 Ones and 3 Ones. We wrote the 3 Ones, 
 and carried the 10 Ones, as 1 Ten, to the left. Adding 
 the 5 Tens, 4 Tens, and the 1 Ten brought from the 
 column of Ones, we had 10 Tens. These 10 Tens we 
 carried to the left, as 1 Hundred, and wrote no Tens, 
 One of these Tens came from the column of Ones. 
 Thus, in adding 456 and 347, we carried 10 Ones not only 
 to the column of Tens, but afterwards, with the 9 Tens, 
 to the column of Hundreds. Now, when we come to 
 Subtract 456 from this Sum, 803, to obtain the other 
 number, we cannot Subtract until after seeking our 10 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 73 
 
 Ones, and also our 9 Tens, in the column of Hundreds, 
 wliere loe left them, and carrying them lack to the right. 
 
 We take 1 of the 8 Hundreds, and, carrying it to the 
 column of Tens, separate it into 9 Tens and 1 Ten ; and 
 leaving the 9 Tens in the column of Tens,/rom which 
 ive formerly carried them, we then carry the 1 Ten to 
 the column of Ones, from which 
 it had been carried, and write it as solution. 
 
 10 Ones. Subtracting 6 Ones from 
 10 Ones and 3 Ones, 5 Tens from 
 9 Tens, and 4 Hundreds from 7 
 Hundreds, the Difference is 347, 
 the other number. 
 
 7 9-10 J Minuend 
 $ 3 j re-arranged. 
 4 5 6 Subtrahend, 
 
 3 4 7 Difference, 
 
 INFERENCES, 
 
 I. — If ^e unite two numbers by Addition", we may 
 disunite them by Subtractiok. 
 
 II. — If, in adding two Numbers, we carey to the 
 LEFT, in Subtracting either ISTumber from the Sum to 
 find the other, we must carry to the right the same 
 Numbers Avhich we carried to the left in adding. 
 
 Exercises for the Slate A]srD Board. 
 
 ■ Addition: 
 
 f 4526 
 1375 
 
 
 f 2746 
 5273 
 
 f 3254 
 1647 
 
 f 3127 
 4375 
 
 1. 
 
 Subtraction : 
 
 5901 
 1375 
 
 2. 
 
 3. 
 
 8019 
 5273 
 
 4.. 
 
 4901 
 1647 
 
 7502 
 . 4375 
 
 Addition: 
 
 ri542 
 3486 
 
 
 r 6849 
 2053 
 
 ' 1872 
 4029 
 
 r 3529 
 4490 
 
 3. 
 
 Subtraction : 
 
 5028 
 1543 
 
 6. 
 
 8902 
 6849 
 
 8.- 
 
 5901 
 
 , 1872 
 
 8019 
 .3529 
 
74 MRST LESSONS IN 
 
 LESSON L. 
 
 By turning now to page 68, you will see that tlie 
 Thousand-group of blocks is in the form of a cube; 
 being 10 blocks in hight, in width, and in length. It 
 contains One Thousand blocks. 
 
 This same mechanic made Alphabet Blocks in large 
 quantities, and used to pile them up in cubical Thou- 
 sand-groups. Then, wrapping a strong paper around 
 each group, he tied up the Thousand blocks firmly in a 
 package. 
 
 Having a yery large number of such packages on 
 hand one day, he requested his little son and one of the 
 workmen to count them, with the blocks previously 
 counted by the boy. 
 
 They counted the packages in Ten-groups; then 
 counted the Ten-groups by Tens, to make Hundred- 
 groups ; then counted these Hundred-groups by Tens, 
 to make Thousand-groups. 
 
 They found that there were just as many pacTcages 
 now as there were single UocJcs formerly counted by the 
 boy ; as shown on page 68. They were piled up and 
 arranged in the same manner. At the right was 1 sin- 
 gle package ; at the left of this 1 Ten-group ; next 1 
 Hundred-group; and last, at the left, 1 Thousand- 
 group. 
 
 They now placed their single package at the left of 
 the blocks previously counted by the boy, close by the 
 side of his Thousand-group of blocks, which they tie(|^ 
 up as one package. At the left of these they placed their 
 Ten-group of packages; at the left of this their Hun- 
 dred-group of packages ; and last their Thousand-group 
 of packages. 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 75 
 
 The blocks and groups then stood thus : 
 
 Cube op 
 Packages. 
 
 Packages. 
 
 Blocks. 
 
 P^ 
 
 ^ 
 
 
 
 P. 
 
 
 
 ? ^ 
 
 p 
 
 
 
 P 
 
 , 
 
 
 One - 
 sand-grc 
 Package 
 
 o 
 
 cL 
 
 _J! 
 
 Q 
 
 p^ 
 
 
 One 
 dred-gr 
 
 One 
 5n-grou 
 
 Two 
 
 ackages 
 
 One 
 dred-gr 
 
 One 
 m-grou 
 
 One 
 Block. 
 
 
 
 
 Ah 
 
 
 H 
 
 
 Said the boy : " The blocks not in packages stand at 
 the right, and are One Hundred and Eleven. We will 
 write the number of these thus: 111.'' 
 
 The man replied: "Our packages, like our blocks, 
 are Cubes. And since we have counted and arranged 
 them in the same manner and order as we did the 
 blocks, we will also tvrite the figures showing their num- 
 ber in the same manner and order as we did those show- 
 ing the number of single blocks. We have 2 packages, 
 1 Ten-group of packages, and 1 Hundred-group of pack- 
 ages; or. One Hundred and Twelve packages. Since 
 these stand at the left of our 111 blocks, we will write 
 the figures 112 at the left of 111 ; placing a mark like a 
 comma between the two groups of figures; thus, 112,111. 
 Each of these 112 packages contains a Thousand blocks. 
 Hence the figures 112,111 are read: One Hundred and 
 Twelve Thousand One Hundred and Eleven. 
 
 "Since we have 1 very large cube, containing One 
 Thousand packages, or 10 Hundred-groups of packages, 
 which stands at the left of our 112 packages, we will 
 write for this a figure 1, placing it at the left of 112, 
 with a point after it; thus, 1,112,111. 
 
 " This very large cube of packages is named a MJitf 
 lion. A Thousand is a large cube containing a Thou- 
 6 
 
76 FIRST LESSONS IN 
 
 sand Hocks, A Million is a very large cube containing 
 a Thousand TJiousandsr 
 
 If, now, we had more Millions, as, for instance, Three 
 Hundred and Sixty-five Millions, instead of One Mil- 
 lion, we would write them by placing the figures 365 
 at the left of 112,111 ; thus, 365,112,111. We see that, 
 commencing at the right, we have separated the figures 
 of this number into groups each having three figures. 
 Each group of figures is named a Period. The point, 
 made like a comma, separating the Periods from each 
 other, is named a Period-point. 
 
 The manner of separating a number having nine 
 figures into Periods, and naming the Periods and the 
 Subdivisions in each, is shown in the following 
 
 
 
 
 TABZD 
 
 '• 
 
 
 
 
 
 
 
 THREE PERIODS. 
 
 
 
 
 
 Numbered, 
 
 
 3d. 
 
 
 
 2d. 
 
 
 
 1st. 
 
 
 
 Millions. 
 
 Thousands. 
 
 
 Ones. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~> 
 
 1 
 
 
 
 
 
 ^ 
 
 
 x& 
 
 
 
 
 
 
 
 
 
 
 {d 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 x& 
 
 
 C3 
 
 o 
 
 OQ 
 
 
 
 
 
 Subdivided 
 
 1 
 
 1 
 
 
 s 
 
 
 m 
 
 
 
 INTO AND 
 
 Named, 
 
 
 
 g 
 
 1 
 
 O 
 
 1 
 
 O 
 
 
 a;* 
 
 ca 
 
 S 
 
 
 w 
 
 H 
 
 a 
 
 m 
 
 H 
 
 H. 
 
 w 
 
 H 
 
 o 
 
 
 3 
 
 6 
 
 3 > 
 
 , 1 
 
 1 
 
 2 , 
 
 . 1 
 
 J 
 
 1 
 
 All the Periods are built up in the same manner, each 
 from its own cube. Since the cube from which the 
 right-hand Period is built up is One^ or One block, this 
 Period is named the Period of Ones. The middle 
 Period, having a Thousand for its cube, is named the 
 Period of Hiousands. For a like reason the left- 
 hand Period is named the Period of Millions. 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 77 
 
 LESSON LI. 
 
 JVOTATIOJV AJ\r2) J\rirM£:'RATIOJV. 
 
 
 TABIjE, 
 
 
 
 1 One is written 1 
 
 10 Ones 
 
 are 1 Ten; written 10 
 
 10 Tens 
 
 " 1 Hundred; * 
 
 100 
 
 10 Hundreds 
 
 " 1 Thousand; 
 
 1,000 
 
 10 Thousands 
 
 " 1 Ten-thousand; ' ' 
 
 10,000 
 
 10 Ten-thousands 
 
 " 1 Hundred-thousand; * 
 
 100,000 
 
 10 Hundred-thousands 
 
 " 1 Million ; 
 
 1,000,000 
 
 10 Millions 
 
 " 1 Ten-million; 
 
 10,000,000 
 
 10 Ten-millions 
 
 " 1 Hundred-million ' 
 
 ' 100,000,000 
 
 To write numbers in figures : Commence at the left 
 and ivfite the periods, one after another, in the same 
 order as the words are written. 
 
 Write 171 JFigures the following : 
 
 L Five Thousand ; Four Hundred ; Seven Thousand 
 and Twenty; Nine Thousand and Seven; Thirteen 
 Thousand, and Eleven. 
 
 2, One Hundred and Three Thousand, One Hundred 
 and Seventeen ; Three Hundred Thousand, and Twenty- 
 five ; Five Millions, One Hundred and Eleven Thousand, 
 and Seventeen. 
 
 To read, or numerate, a number written in figures : 
 
 1st. Beginning at the right, separate the nvmber into 
 periods ; 
 
 2d, Beginning at the right, name all the periods; 
 
 3d. Beginning at the left, read the periods in order, 
 giving the name of every period except that at the right, 
 
 JV^umerate the following JV^iimbers : 
 
 137256984 ; 40576402 ; 170510001 ; 103520407 ; 
 
 19030040 ; 21005000 ; 100301200 ; 31001501. 
 
T8 
 
 FIRST LESSONS IN 
 
 LESSON L/L 
 
 First numerate the numbers in each of the following 
 Exercises, and then find their Sum. 
 
 EXEKCISES FOR THE SlATE AlJfD BOARD. 
 
 35467 
 
 127508 
 
 I. 
 1234567 
 
 10154125 
 
 12536421 
 
 12243 
 
 105623 
 
 2514360 
 
 12253326 
 
 12037058 
 
 10516 
 
 100510 
 
 1602507 
 
 15107908 
 
 10250870 
 
 27638 
 
 209018 
 
 1019015 
 
 11015017 
 
 20900500 
 
 32507 
 
 234126 
 
 II. 
 3509215 
 
 27514310 
 
 12141871 
 
 10325 
 
 576387 
 
 5280317 
 
 41310080 
 
 27423612 
 
 47018 
 
 154218 
 
 4052001 
 
 60040085 
 
 12751423 
 
 53106 
 
 627324 
 
 6003200 
 
 21005002 
 
 20205602 
 
 61007 
 
 542987 
 
 1100005 
 
 13241576 
 
 10030050 
 
 27589 
 
 173238 
 
 7020050 
 
 11376529 
 
 10400800 
 
 Notation, Nutneration, and A.ddltion, 
 
 In each of the following Examples, first write the 
 numbers, then numerate them, ^nd finally add them. 
 
 Exercises for the Slate a:n^d Board. 
 
 I. Twenty thousand, One Hundred and Five; Thir- 
 teen Thousand, and Fifteen ; Seventeen Thousand, and 
 Nine. 
 
 II. One Hundred and Twenty-five Thousand, Three 
 Hundred and Eleven ; Three Hundred and Seven 
 Thousand, Five Hundred and Four; Five Hundred and 
 Eleven Thousand, and Fifteen. 
 
 III. Three Million, Five Hundred and Twenty-five 
 Thousand, One Hundred and Twenty-Seven ; Five Mil- 
 lion, Three Hundred and Seven Thousand, Seven Hun- 
 dred and Eight; Nine Million, Five Thousand, and Six. 
 
MENTAL AND WRITTEN ARITHMETIC, 79 
 
 
 LESSON Llll. 
 
 
 
 SZrSTlici.CTIOJV. 
 
 
 867,542 
 573,814 
 
 5,473,207 
 1,632,154 
 
 I. 
 
 63,521,365 
 21,507,193 
 
 750,319,476 
 473,207,528 
 
 953,541 
 684,703 
 
 2,507,318 
 1,498,173 
 
 II. 
 58,390,504 
 28,279,371 
 
 972,548,765 
 481,395,976 
 
 Addition and Subtraction. 
 
 Writtei^ Exercises. 
 
 1, Three grain dealers purchased wheat as follows: 
 A purchased 59,487 bushels ; B, 47,376 ; and C, 39,894. 
 
 (a.) How many bushels did the three men purchase ? 
 {b.) How many bushels did A and B together pur- 
 chase more than C ? 
 
 2, A father divided his property among his children 
 as follows: He gave to Charles,17,510 dollars; to Henry, 
 21,437 dollars ; to William, 25,196 dollars ; to Amelia, 
 13,087 dollars ; to Sarah, 15,193 dollars ; and to Susan, 
 11,981 dollars. 
 
 {a.) How many dollars did he give to his three sons ? 
 (J.) How many dollars to his three daughters? 
 {c.) How many dollars, in all, to his children ? 
 {d.) How many more dollars were given to his sons 
 than to his daughters ? 
 
 3, In the year 1850, the population of the city of New 
 York was 515,547; Boston, 136,881; Philadelphia, 
 340,045, 
 
 ' («.) What was the total population of the three cities? 
 (J.) How much did the population of New York ex- 
 ceed that of both Philadelphia and Boston ? 
 
80 FIRST LESSONS IN 
 
 LESSON LIV. 
 
 JVUMB^ATIOJV ciJY^ ^^^ITIOJSr. 
 
 First read, or numerate, the numbers, and then find 
 their Sum, in each of the following 
 
 Exercises for the Slate Aiq"D Board. 
 
 33,507 
 
 234,126 
 
 3,509,215 
 
 27,514,310 
 
 12,418,712 
 
 10,325 
 
 576,387 
 
 5,280,317 
 
 41,310,080 
 
 27,236,129 
 
 47,018 
 
 154,218 
 
 4,052,001 
 
 60,040,085 
 
 12,514,237 
 
 53,106 
 
 627,324 
 
 6,003,200 
 
 21,005,002 
 
 20,056,025 
 
 61,007 
 
 542,987 
 
 1,100,005 
 
 13,241,576 
 
 10,300,500 
 
 27,589 
 
 173,238 
 
 7,020,050 
 
 11,376,529 
 
 10,008,006 
 
 Notation and Addition, 
 
 In each of the following Examples, first write the 
 numbers, then numerate them, and finally find the Sum. 
 
 Examples for the Slate ai^t> Board. 
 
 I. Twenty Thousand, One Hundred and five ; Thir-. 
 teen Thousand, and fifteen ; Seventeen Thousand, and 
 Nine. 
 
 II. One Hundred and Twenty-five Thousand, Three 
 Hundred and Eleven ; Three Hundred and Seven Thou- 
 sand, Five Hundred and Four; Five Hundred and 
 Eleven Thousand, and Fifteen. 
 
 III. Three Million, Five Hundred and Twenty-four 
 Thousand, One Hundred and Twenty-seven ; Five Mil- 
 lion, Three Hundred and Seven Thousand, Seven Hun- 
 dred and Eight ; Nine Million, Five Thousand, and Six. 
 
 IV. Two Hundred and Seventy-six Million, Three 
 Hundred and Seven Thousand, One Hundred and Nine- 
 teen ; One Hundred and Two Million, Forty-five Thou- 
 sand, and Twelve; Five Hundred Million, Eight Thou- 
 sand, and Nine. 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 81 
 
 /■ / r ^ 
 
 k 
 
 fc 
 
 t h^ 'fc. \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ■™ 
 
 ^ 
 
 
 ^ 
 
 ^ 
 
 
 
 _ 
 
 ^ 
 
 4 
 4 
 8 
 
 5 
 ]0 
 
 6 
 _6 
 12 
 
 7 8 
 _7 _8 
 14 16 
 
 9 
 _9 
 18 
 
 f r ^ ^ ^_ \ 
 
 fc 
 
 ^ 1 
 
 ^ '^ \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 iO 
 
 12 
 
 14 
 
 16 
 
 j^ 
 
 ' LESSON LV. 
 
 These first 
 two horizontal 
 rojN^s of cubes 
 were begun at 
 the left with 
 one cube in 
 each, and 
 lengthened by 
 adding another 
 cube to each 
 from time to 
 time, till each 
 row had 9 
 cubes. The 
 cubes in the 
 upper row are numbered. 
 
 Below the figure 1 written on the first cube in the 
 upper row we have written two figure l^s, and have 
 written their Sum below, to show how many cubes there 
 were in both rows when there was 1 cube in each row. 
 
 Under the 2 written in the upper row we have written 
 two figure 2's, and below them their Sum, to show that 
 there were 4 cubes in both rows when there were 2 cubes 
 in each row. 
 
 In the same manner we have found the number of 
 cubes in the 2 rows at each point from their commence- 
 ment till their completion. 
 
 In the second cut we have written these Sums on the 
 cubes in the lower row. We see that when there were 
 3 cubes in each row there were 6 cubes in both rows ; 
 when there were 4 cubes in each row there were 8 in 
 both rows. 
 
82 FIRST LESSONS IN 
 
 When the numbers added to form a Sum are equals 
 Tve name the Addition Gn^aded Addition^ since 
 the Sum is composed of equal or graded parts. 
 
 Since we found these Sums by adding two I's, two 
 2's, two 3's, &c., it is plain that the Sums show how 
 many 2 times 1, 2 times 2, 2 times 3, &c., are. Hence, 
 our 
 
 GMADED ADDITION TABLE. 
 
 2 Times 
 
 1 are 2, 4 are 8, 7 are 14, 
 
 2 are 4, 5 are 10, 8 are 16, 
 
 3 are 6, 6 are 12, 9 are 18. 
 
 When we subtract one number from another not once 
 only, but as many times as possible, we name this Sub- 
 traction Graded Subtraction. 
 
 We may also consider these cubes as piled up in ver- 
 tical columns, each having 2 cubes. 
 
 From the first column, or 2 cubes at the left, we can 
 Subtract 2 cubes once. From the first 2 columns, or 4 
 cubes at the left, we can Subtract 2 cubes 2 times ; from 
 6 cubes 3 times; from 8 cubes 4 times; and so on. 
 Hence we can make the following, or First Form of 
 
 G BAD ED STTBTBACTION TABLE, 
 
 3 can be Subtracted from 
 
 2 1 Time, 8 4 Times, 14 7 Times, 
 
 4 2 Times, 10 5 Times, 16 8 Times, 
 
 6 3 Times, 12 6 Times, 18 9 Times. 
 
 Since 2 cubes are contained in any number of cubes 
 as many times as they can be Subtracted from that num- 
 ber of cubes, it is evident that we may have the follow- 
 ing, more common, or Second Form of 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 83 
 
 GRADED SUBTRACTION TABLE. 
 
 3 are contained in 
 
 2 1 Time, 8 4 Times, 14 7 Times, 
 
 4 2 Times, 10 5 Times, 16 8 Times, 
 
 6 3 Times, 12 6 Times, 18 9 Times. 
 
 We can read these Tables from the cubes as a 
 
 GRADED ADDITION AND SUBTRACTION TABLE. 
 
 f 
 
 r 
 
 f ~^ 
 
 ^ 
 
 fc 
 
 it \ 
 
 r f^ ^ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 ki 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 To read this as a Graded Addition Table, we com- 
 mence with the 2 at the bottom, at the left; then 
 take a number in the upper row, as 3 ; and lastly take, 
 in the lower row, the number below that taken in the 
 upper row, which in this case would be 6 ; saying : 2 
 times S are 6, In like manner we proceed, saying : 2 
 times If. are 8 ; 2 times 5 are 10 ; and so on. 
 
 To read it as a Graded Subtraction Table, we com- 
 mence with the 2 at the left, as before, but take the two 
 other numbers in a contrary order ; reading first the 2 
 at the left ; then a number in the lower row, as 6 ; and 
 lastly the number 3 above this ; saying : 2 are in 6, S 
 times. Thus we go on, saying : 2 are in 8, 4 times j 2 
 are in 10, 5 times ; and so on. 
 
 This form of the Tables, as represented ly the cubes, 
 is the more convenient. 
 
 These Tables should be perfectly learned, and often 
 recited. 
 
84 FIRST LESSONS IN 
 
 LESSON LVI. 
 
 Writtei^ Mejsttal Exercises. 
 
 L Two horses make 1 span. How many horses are 
 there in 2 spans ? 2 times 2 horses are how many ? 
 
 2, One wagon has 4 wheels. How many wheels have 
 2 wagons ? 2 times 4 wheels are how many wheels ? 
 
 <^. Fanny found 2 birds' nests, each having 6 eggs. 
 How many eggs were there in all ? 2 times 6 eggs are 
 how many eggs ? 
 
 Jf. In each of 2 windows there are 8 panes of glass. 
 How many panes are there in both windows ? 2 times 
 8 panes are how many ? 
 
 5, Willie had two rose-bushes with 3 roses on each. 
 How many roses had he on both bushes ? 2 times 3 
 roses are how many roses ? 
 
 6, Counting my thumbs as fingers, I have 5 fingers 
 on each hand. How many fingers have I on both 
 hands ? 2 times 5 fingers are how many ? 
 
 7, Frank had 7 peaches, and Mary also had 7. How 
 many had both ? 2 times 7 peaches are how many ? 
 
 8, Two hens have each 9 chickens. How many 
 chickens have both hens ? 2 times 9 are how many ? 
 
 How many times can we Subtract 
 
 2 wagon-wheels from 8 2 horses from 4 horses ? 
 
 wheels ? 2 fingers from 10 fingers ? 
 
 2 birds' eggs from 12 eggs ? 2 roses from 6 roses ? 
 
 2 chickens from 18 chick- 2 window-panes from 16 
 
 ens ? panes ? 
 
 2 peaches from 14 peaches ? 
 
 9, How many cents, in all, are 2 times 4 cents anq 
 2 times 3 cents ? 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 85 
 
 f t k- r- 
 
 L. 
 
 fc 
 
 ^ ! 
 
 fe-- '^. \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 _ 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 4 
 
 4 
 
 _4 
 
 12 
 
 5 
 
 5 
 
 _5 
 
 15 
 
 6 
 '6 
 _6 
 18 
 
 7 8 9 
 
 7 8 9 
 
 _7 _8 _9 
 
 21 24 27 
 
 LESSON LVIL 
 
 In this cut, 
 the 2 horizon- 
 tal rows of 
 cubes shown 
 in the last cut 
 are seen placed 
 over another 
 row of cubes. 
 Below these 
 cubes numbers 
 are written in 
 vertical col- 
 umns. The three 4's, and the 12 written below them, 
 which is their Sum, show that when there were 4 cubes 
 in each of the 3 horizontal rows there were 12 cubes in 
 the 3 rows. So of the other columns of figures and 
 their Sums. These Sums are written on the cubes in 
 the lower row. 
 
 The new part added to the Table is read thus as a 
 part of the 
 
 GBADEJ) ADDITION TABLE, 
 
 3 Times 
 
 4 are 12, 
 
 5 are 15, 
 
 6 are 18, 
 It is read thus as a part of the 
 
 *GItADED SUBTRACTION TABLE, 
 
 3 can be Subtracted from 
 
 1 Time, 12 4 Times, 21 
 
 2 Times, 15 5 Times, 24 
 
 3 Times, 18 6 Times, 27 
 Eepeat the entire Table ; first as a Graded Addition 
 
 Table, and then as a Graded Subtraction Table. 
 
 1 are 3, 
 
 2 are 6, 
 
 3 are 9, 
 
 3 
 
 6 
 9 
 
 7 are 21, 
 
 8 are 24, 
 
 9 axe 27. 
 
 7 Times, 
 
 8 Times, 
 
 9 Times. 
 
8G FIRST LESSONS IN 
 
 LESSON LVIII. 
 
 Writtei^ Mei^tal Exercises. 
 
 1, Charles found 3 birds' nests, each having 4 eggs. 
 How many eggs were there in the 3 nests ? 
 
 2, Emma attended school 5 days each week for 3 
 weeks. How many days did she attend in 3 weeks ? 
 
 3, In a school are 3 classes in reading, with 9 pupils 
 in each class. How many pupils in the 3 classes? 
 
 -4. Each of 3 boys has 8 cents. How many cents have 
 the 3 boys ? 3 times 8 are how many ? 
 
 5, Three boys went fishing, and each caught 7 fish. 
 How many fish did the 3 boys catch ? 
 
 6, A board fence was 6 boards high. How many 
 boards were there in 3 lengths of the fence ? 
 
 7. A father had 12 apples, and gave them to his sons, 
 giving each boy 3 apples. How many sons were there ? 
 3 apples are in 12 apples how many times ? 
 
 8, A company of girls picked 18 quarts of berries, 
 each girl picking 3 quarts. How many girls were there ? 
 3 are in 18 how many times ? 
 
 9, Florence had 15 roses in her flower-garden, and 
 there were 3 roses on each bush. How many rose-bushes 
 were there ? 3 are in 15 how many times ? 
 
 10. A beggar received 24 cents from a company of 
 boys, each boy giving him 3 cents. How many boys 
 were there ? 3 are in 24 how many times ? . 
 
 11. Willie had 27 cents in 3-cent pieces. How many 
 3-cent pieces had he ? 3 are in 27 how many times? 
 
 12. A mother had 21 flowers, and made bouquets of 
 them for her children, putting 3 flowers in each bou- 
 quet. How many children had she ? 3 are in 21 how 
 many times ? 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 
 87 
 
 LESSON LIX. 
 
 MUZ TITZICA TIOjV. 
 
 Example. A tailor cut from a roll of clotli enoiigli 
 for 7 pairs of boys' pants, using 2 yards for each pair. 
 How many yards did he use ? 
 
 ExPLAN"ATiON". He first measured 2 yards at one end 
 of the roll, and then hj folding the cloth first one way 
 and then the other, until he had 7 thicknesses, he thus 
 measured 7 times 2 yards. 
 
 In thus obtaining 7 times 2 yards, he folded the 2 
 yards many times. Taking 2 yards 7 times, and finding 
 how many yards we have, is sometimes named Jfulti-^ 
 plication. Multiplication me'dixs folding many times, 
 or taking many times. We have taken 2 yards 7 times. 
 
 We name the ^ yards the Multiplicand. Multi- 
 plicand means something to he folded or tahen many 
 times. 
 
 We name 7 the Multiplier. The Multiplier tells 
 how many times the Multiplicand is to be taken. 
 
88 
 
 FIRST LESSONS IN 
 
 There are two Solutions for this Exam- 
 ple. First: We write a figure 2 seven 
 times, and, adding the 2's, write the Sum, 
 14, below. . This method of performing 
 the work we name Graded Addition. 
 Second : We write the 2 once only, and 
 then writing a figure 7 under it, to show 
 how many times the 2 is to he taken, draw 
 a line under the 7, and saying : " 7 times 
 2 are H,'' write 14 below the line. 
 This method of performing the work 
 we name Multiplication, The result 
 
 First Solution. 
 By Addition. 
 2' 
 
 ^ Equal 
 
 2 1 ^"^ 
 2 I Graded 
 
 2 Parts. 
 
 _2^ 
 
 14 Sum. 
 
 is 14 in both cases ; but the 14 ob- 
 
 Second Solution. 
 By Multiplication. 
 
 2 Multiplicand* 
 7 Multiplier. 
 14 Product. 
 
 tained by the first Solution is named 
 the Sum, and that obtained by the 
 second Solution is named the Product. Product 
 means something produced. H is produced by multi- 
 plying 2 by 7. 
 
 EXEKCISES POR THE SlATE 
 
 AND Board. 
 
 
 
 Graded Addition 
 
 
 , 
 
 3 Equal 
 
 or Graded ■ 
 
 Parts. 
 
 '4 20 
 
 4 20 
 
 .4 20 
 
 24 
 24 
 24 
 
 100 
 100 
 100 
 
 124 3,000 
 124 3,000 
 124 3,000 
 
 3,124 
 3,124 
 3,124 
 
 - 
 
 Multiplication, 
 
 
 
 Multiplicands: 4 
 Multipliers : 3 
 
 20 
 _3 
 
 X. 
 
 24 100 
 3 3 
 
 124 3.000 
 3 3 
 
 3,124 
 3 
 
 51,323 
 3 
 
 93,423 
 
 2 
 
 
 II. 
 
 83,213 
 3 
 
 50,706 
 
 2 
 
 90,708 
 3 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 89 
 
 FIRST SOLUTION. 
 
 By Addition. 
 
 5 + 5 + 5 = 15. 
 
 SECOND SOLUTION. 
 
 _ By Multiplication. 
 
 5 X 3 = 15 
 
 LESSON LX. 
 
 Example. How many are 3 times 5 ? 
 
 ExPLAN^ATiON^. In the first Solution we write 5 three 
 times, and, adding, write the Sum, 
 15 ; thus, 5 + 5 + 5 = 15. In the 
 second Solution we write 5 once 
 only ; and, since it is to be taken S 
 times and added, we write 3 at the 
 right of 5, and, turning the Sign 
 Plus thus, X , place it hetiveen the 
 5 and 3. The whole stands thus : 
 5 X 3 = 15 ; and is read : 5 mul- 
 tiplied by 3 equal 15. When the Sign Plus is thus 
 turned and used it does not show that 5 and 3 are to be 
 added together, but shows that 5 are to be taken 3 times 
 and added, to find the Sum; or, 
 which means the same, it shows 
 that 5 are to be multiplied by 3. 
 For this reason, when the Sign Plus 
 is thus turned and used we name 
 it the Sign of Multiplica- 
 tion. The Sign of Addition is a 
 Vertical Cross. The Sign of Mul- 
 tiplication is an Inclined Cross, 
 
 We can write 
 
 6 multiplied by 3 equal 18 ; or 6 x 
 9 multiplied by 3 equal 27 ; or 9 x 
 
 Write, learn, and recite, the following 
 ta:bJjE, 
 
 3 = 18; 
 
 3 = 27. 
 
 X 2 = 
 
 x 2 = 2 
 
 X 2 = 4 
 
 X 2 = 6 
 
 X 2 = 8 
 
 2 = 10 
 2 = 12 
 2 = 14 
 2 =: 16 
 2 = 18 
 
 3 = 
 3 = 3 
 3 = 6 
 3 = 9 
 3 = 12 
 
 5 X 
 
 6 X 
 
 7 X 
 
 8 X 
 
 9 X 
 
 = 15 
 = 18 
 = 21 
 = 24 
 
 3 = 27 
 
90 FIRST LESSONS IN 
 
 LESSON LX/. 
 
 Example A. Multiply 879 by 3. 
 
 ExPLAi^ATiojsr. First ; we multiply 9 Ones by 3, and 
 write the Product, 27 Ones. We name this a Partial 
 Product^ because it forms 
 only a part of the true Pro- solution. 
 
 duct. Second; we multiply 7 879 Multiplicand, 
 
 Tens by 3, and, having 21 Tens, . § Multiplier. 
 
 or 2 Hundreds and 1 Ten, for 27 1 p / • ? 
 
 our second Partial Product, 21 [p^^^J^^^^^ 
 write 1 Ten in the column of ^^ ^ 
 Tens, and 2 Hundreds at the 2,637 Product, 
 left, in the column of Hundreds. 
 
 Third ; we multiply 8 Hundreds by 3, and, having 24 
 Hundreds, or 2 Thousands and 4 Hundreds for our 
 third Partial Product, write 4 Hundreds in the column 
 of Hundreds, and 2 Thousands at the left. Now we 
 have taken the 8 Hundreds 3 times, the 7 Tens 3 times, 
 and the 9 Ones 3 times. If we add these three Partial 
 Products we shall have 3 times 879 for the final Product. 
 
 First, we bring down the 7 Ones into the Product. 
 Next, we add 1 Ten and 2 Tens, and write their Sum, 3 
 Tens, in the Product. Adding 4 Hundreds and 2 Hun- 
 dreds, we write their Sum, 6 Hundreds, in the Product. 
 Finally, we write 2 Thousands at the left of 6 Hundreds, 
 and have 2,637 for our final Product. 
 
 EXEKCISES FOR THE SlATE AKD BoARD. 
 
 57,489 
 3 
 
 95,847 
 3 
 
 I. 
 
 48,796 
 3 
 
 II. 
 
 85,798 
 3 
 
 79,846 
 3 
 
 64,587 
 3 
 
 84,758 
 3 
 
 89,798 
 3 
 
 94,876 
 3 
 
 58,764 
 3 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 91 
 
 LESSON LXIL 
 
 Example B. Multiply 52,819 by 3. 
 
 ExPLAi5"ATioi^. The work in this Solution is placed 
 in nearly the same order as 
 that in the Solution of Ex- 
 ample A, The second and 
 third Partial Products are, 
 however, placed in the same 
 horizontal line, since they do 
 not interfere with each other; 
 so also the fourth and fifth. 
 
 52,819 
 3 
 
 271 
 24-3 
 15-6 
 
 Multiplicand. 
 Multiplier. 
 
 Partial 
 Products. 
 
 158,457 Product. 
 
 Exercises for the Slate akd Board. 
 
 I. 
 
 584,319 769,815 715,394 428,173 936,284 
 
 152,873 
 3 
 
 231,312 
 3 
 
 II. 
 987,546 
 3 
 
 514,271 
 3 
 
 Example C. Multiply 50,608 by 3. 
 
 ExPLAi^ATioi^. Since no in the Multi- 
 plicand gives rise to any Partial Product, 
 we write in place of a Partial Product, in 
 such cases. 
 
 859,897 
 3 
 
 SOLUTION. 
 
 50,608 
 3 
 
 24 
 18-0 
 15-0 
 
 151,824 
 
 Exercises for the Slate ai^d Board. 
 
 I. 
 
 20,708 69,008 304,500 50,980 100,709 
 
 2 
 
 2 
 
 2 
 
 2 
 
 20,030 
 3 
 
 58,090 
 3 
 
 II. 
 600,708 
 3 
 
 12,500 
 3 
 
 907,008 
 3 
 
92 
 
 FIRST LESSONS IN 
 
 4 Multiplicand. 
 3 Multiplier. 
 
 12 Product. 
 
 LESSON LXIII. 
 
 These 12 apples are arranged in two sets of rows ; one 
 set running lengthwise of the table, and the other 
 crosswise. 
 
 FiKST : If we consider the rows pour m a row. 
 
 as running lengthwise, there are 3 
 rows. In each row are 4 apples. 
 In 3 rows there are 3 times 4 ap- 
 ples; which are 12 apples. The 
 Multiplicand is 4, the number of apples in each row ; 
 the Multiplier is 3, the number of rows ; and the Pro- 
 duct is 12, the number of apples on the table. 
 
 Second : If we consider the rows 
 as running crosswise, there are 4 
 rows. In each row are 3 apples, 
 giving 3 for the Multiplicand. In 
 4 rows there are 4 times 3 npples ; 
 which are 12 apples ; giving 4 for a 
 Multiplier, and 12 for the Product. 
 
 We notice that the two numbers multiplied together 
 are the same in both cases ; and also the Products ; but 
 
 THREE IN A ROW. 
 
 3 Multiplicand. 
 
 4 Multiplier. 
 
 12 Product. 
 
MENTAL AND WRITTEN ARITHMETIC, 93 
 
 the Multiplicand and Multiplier of the first change 
 places in the second. 
 
 . In each case the Multiplicmid shows the number of 
 apples in a row, the Multiplier the number of roivs, and 
 the Product the number of apples arranged, 
 
 Inferen'CE. I7i all cases where the Product represents 
 MATERIAL THiJS'GS, as apples, peachcs, oranges, 1st, these 
 things can be so arranged as to stand in two sets of 
 roios ; 2d, the number of thiiigs in one row of either set 
 mag be tahen as the Multiplicand, and the number of 
 things in one row of the other set as Multiplier ; and, 
 3d, the number of things arranged will be the Product, 
 
 Example. A lady gave 4 apples to each of 3 boys, 
 and 3 apples to each of 4 girls ; how many apples did 
 she give to the 3 boys, and how many to the 4 girls ? 
 
 Explain ATioN. The arrangement of apples in the 
 foregoing cut will give us both answers to this Example. 
 * In the first case there are 4 apples in each row, and 3 
 rows ; or one row for each boy. 4 x 3 = 12. Hence 
 12 apples were given to the 3 boys. 
 
 In the second case there are 3 apples in each row, and 
 4 rows ; one row for each girl. 3 x 4 = 12. Hence 
 12 apples were given to the 4 girls. 
 
 3 and 4 multiplied together malce ov produce 12. For 
 this reason 3 and 4 are named the Factors of 12. 
 Factor means Maker or Producer, 12 is the thing made, 
 OY produced, and hence is named the Product* Pro- 
 duct means something made or produced. 
 
 From the foregoing Explanations we infer this 
 
 FRINCIPIjE in 3rULTirZICATION, 
 
 If two Numbers are to be midtiplied together either 
 may be used as the Multiplicand, and the other as the 
 Multiplier. 
 
94 
 
 FIRST LESSOJSS IN 
 
 LESSON LXIV. 
 
 By the use of the Principle stated in the preceding 
 Lesson, we can change the Multiplication Tables already 
 learned into Tables haying 1, 2 and 3 as Multiplicands, 
 and 1, 2, 3, 4, 5, 6, 7, 8 and 9, as Multipliers ; thus : 
 
 
 
 FinST TABZE. 
 
 SECOND TABZE, 
 
 
 
 
 3 Times 1 are 2, 
 
 and 1 Time 3 is 2 ; 
 
 
 
 
 2 Times 2 are 4, 
 
 and 2 Times 2 are 4 ; 
 
 
 
 
 2 Times 3 are 6, 
 
 and 3 Times 2 are 6 ; 
 
 
 
 
 2 Times 4 are 8, 
 
 and 4 Times 2 are 8 ; 
 
 
 
 
 2 Times 5 are 10, 
 
 and 5 Times 2 are 10 ; 
 
 
 
 
 2 Times 6 are 12, 
 
 and 6 Times 2 are 12 ; 
 
 
 
 
 2 Times 7 are 14, 
 
 and 7 Times 2 are 14 ; 
 
 
 
 
 2 Times 8 are 16, 
 
 and 8 Times 2 are 16 ; 
 
 
 
 
 2 Times 9 are 18, 
 
 and 9 Times 2 are 18. 
 
 
 
 
 3 Times 1 are 3,' 
 
 and 1 Time 3 is 3 ; 
 
 - 
 
 
 
 3 Times 2 are 6, 
 
 and 2 Times 3 are 6 ; 
 
 
 
 
 3 Times 3 are 9, 
 
 and 3 Times 3 are 9 ; 
 
 
 
 
 3 Times 4 are 12, 
 
 and 4 Times 3 are 12 ; 
 
 
 
 
 3 Times 5 are 15, 
 
 and 5 Times 3 are 15 ; 
 
 
 
 
 3 Times 6 are 18, 
 
 and 6 Times 3 are 18 ; 
 
 
 
 
 3 Times 7 are 21, 
 
 and 7 Times 3 are 21 ; 
 
 
 
 
 3 Times 8 are 24, 
 
 and 8 Times 3 are 24 ; 
 
 
 
 
 3 Times 9 are 27, 
 
 and 9 Times 3 are 27. 
 
 
 
 Learn, and often recite, the above Tables. 
 
 
 
 
 Mektal Exercises. 
 
 
 5 
 
 X 
 
 2 =: ? 6x2 = 
 
 ? 7x2=? 2x6 
 
 = ? 
 
 2 
 
 X 
 
 8 = ? 8x2 = 
 
 ? 2x8=? 7x3 
 
 = ? 
 
 3 
 
 X 
 
 4 --= ? 3x6 = 
 
 ? 3x8=? 8x3 
 
 = ? 
 
 2 
 
 X 
 
 5 = ? 2x7 = 
 
 ? 2x9=? 9x2 
 
 = ? 
 
 3 
 
 X 
 
 9 = ? 3x5 = 
 
 ? 9x3=? 3x7 
 
 = ? 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 95 
 
 LESSON LXV. 
 
 Exercises for the Slate aij^d Board. 
 I. 
 
 2,132 
 4 
 
 30,201 
 4 
 
 3,213 
 > 5 
 
 1,323 
 6 
 
 II. 
 
 2,131 
 
 7 
 
 3,231 
 
 8 
 
 2,333 
 9 
 
 21,032 
 5 
 
 23,103 
 6 
 
 31,232 
 
 7 
 
 13,233 
 
 8 
 
 33,232 
 9 
 
 Multiply 31,213,210 by 4 
 Multiply 23,131,032 by 4 
 Multiply 12,302,313 by 4 
 Multiply 30,131,231 by 4 
 Multiply 21,312,103 by 4 
 
 ; by 5 
 
 ; by 6; 
 
 ; by 5 
 
 ; by 6; 
 
 ; by 5 
 
 . by 6; 
 
 ; by 5 
 
 by 6; 
 
 ; by 5. 
 
 byO; 
 
 by 7; 
 by 7; 
 by 7; 
 by 7; 
 by 7; 
 
 by 8. 
 by 8. 
 by 8. 
 by 9. 
 by 9. 
 
 LESSON LXVI. 
 
 Exercises for the Slate akd Board. 
 
 Multiply 31,020,230 by 3 ; by 4 ; by 5 ; by 6 ; by 7. 
 
 Multiply 12,130,302 by 3 ; by 4 ; by 5 ; by 6 ; by 7. 
 
 Multiply 31,213,023 by 3 ; by 4 ; by 5 ; by 6 ; by 7. 
 
 Multiply 21,302,013 by 3 ; by 4 ; by 5 ; by 6 ; by 8. 
 
 Multiply 30,130,231 by 3 ; by 4; by 5; by 6; by 8. 
 
 Multiplication at Sight, 
 
 4 X 
 
 2 X 
 
 8 X 
 
 2 X 
 
 7 X 
 
 X 2 
 
 X 4 
 
 X 8 
 
 X 6 
 
 X 2 
 
 .3 
 5 
 
 8 
 2 
 9 
 
96 
 
 MBST LESSONS IN 
 
 LESSON LXVII. 
 
 Example. — Harry's mother gave 12 apples to her 
 children, giving 4 apples to each. How many children 
 had she ? 
 
 Explaintatiok. — 1st: 
 Subtracting 4 apples from 
 12 apples, we have 8 ap- 
 ples left. These 4 apples 
 are arranged in the 1st row 
 in the picture. 2d: Sub- 
 tracting 4 apples from 8 
 apples, we have 4 apples 
 left. The 4 apples last 
 subtracted are arranged in 
 
 the 2d roio. 3d : Subtracting 4 apples from 4 apples, we 
 find none remaining. These 4 apples are arranged in 
 the 3d row. 
 
 FIRST SOLUTION. 
 
 By Graded Subtraction. 
 
 12 ajjples. Minuend, 
 
 1st SuUraliend. 
 
 2d 
 
 3d 
 No apples left. 
 
MENTAL AND WRITTEN ARITHMETIC. 97 
 
 In this manner we divide 12 apples into 3 groups, 
 each having J^. ap2)les. Since ^ apples were given to 
 each child, the number of groups and the nurnber of 
 children must have been the same. Hence there were 
 3 children. 
 
 Testing the Result, 
 
 If 12 apples can be divided into 3 tbst. 
 
 groups, each having 4 apples, then 3 ^^ Graded AddUim, 
 times 4 apples must equal 12 apples. -z^^ ^ Apples. 
 Therefore, we write three 4's and add - q^ a « 
 them. Finding the Sum to be 12, we — 
 
 conclude that there must have been 3 ^^^^ ^^ 
 groups, with 4 apples in each. Hence there were 3 
 children. 
 
 Secokd. — We can perform our work by another 
 method, and find the same result. We 
 write 4 apples, and, drawing a line second solution. 
 below, write 12 apples below this line. ^ Groups. 
 We then ask : Hotv many times can 4 _Z Apples. 
 apples be subtracted from 12 apples ? 12 " 
 or, Hoio many times are 4 apples con- 
 tained in 12 apples ? Finding that 4 are in 12 3 times, 
 we write 3 above the 4 apples, to show how many groups 
 there are. Since 12 apples can be divided into 3 groups, 
 each having 4 apples, Harry's mother must have had 3 
 children. 
 
 Testing the Result, 
 
 If there are in 12 apples 3 groups, each having 4 
 apples, then there are in all 3 times 4 apples; which 
 are 12 apples. Since 3 and 4, in the Solution, stand 
 over 12 in the same manner as the Multiplicand and 
 Multiplier stand over their Product, we will multiply 
 them together. 
 
98 FIRST LESSONS IN 
 
 Using 3 as Multiplier, the Product is 12 apples. 
 Hence, in 12 apples there are 3 groups, each having 4 
 apples. Therefore, 3 children received 12 apples. 
 
 The method which we have just used, in the Second 
 Solution, is named Division. 12 is named the 
 Dividend, 4 the Divisor, and 3 the Quotient. 
 
 L DivisiOi?" means dividing. We have divided 12 
 apples. 
 
 2, Dividend means something to le divided. 12 is 
 the number to he divided. 
 
 S. Divisor means divider. We have used 4 as a 
 divider of 12. 
 
 Jf. QuoTiEi^T means how man^ times. 3 shows how 
 many times 4 are contained in 12. 
 
 Third. There is still another method of Solution. 
 1st: Since 12 apples are to be 
 
 T.TT •i-.r» ckJ O* THIRD SOLUTION. 
 
 divided, we write 12. 2d : Since -. o _^_ 4. _ q 
 each child is to receive 4 apples, we — _ . 
 
 write 4 at the right of 12. 3d: 
 Since 4 apples are to be subtracted 
 from 12 apples, we write the Sign 
 Minus between 12 and 4. 4th : 
 Since 4 is to be subtracted, not once 
 only, but as many times as possible, 
 we write a dot above the centre of 
 the Sign Minus, and another below 
 it, to show this. 5th : Since we wish to show what 
 number of times this result equals, we write the Sign of 
 Equality after the 4. 6th : Since we find that the num- 
 ber of times 4 can be subtracted from 12 equals 3 times, 
 we write 3 at the right of the Sign of Equality. 7th : 
 We read this expression thus : 12 divided by ^ equal 8. 
 
 The Sign thus made by changing the Sign Minus is 
 named the Sign of Division. 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 99 
 
 LESSON LXVIII. 
 
 By Graded Subtraction^ wg find how many times 
 one number can be subtracted from another, or is con- 
 tained in another. 
 
 Division is a short method of performing Graded 
 Subtraction, 
 
 The Graded Subtraction Tables on pages 83 and 85 
 can be thus read as a 
 
 DIVISION TABLE, 
 
 2 are in 
 
 3 are in 
 
 2 Once, 
 4 2 Times, 
 6 3 Times, 
 8 4 Times, 
 10 5 Times, 
 
 12 6 Times, 
 14 7 Times, 
 16 8 Times, 
 18 9 Times. 
 
 3 Once, 
 
 6 2 Times, 
 
 9 3 Times, 
 
 12 4 Times, 
 
 15 5 Times, 
 
 18 6 Times, 
 21 7 Times, 
 24 8 Times, 
 27 9 Times. 
 
 Mental Exercises. 
 
 14~2 = ? 9-^3 = ? 4-^2 = ? 6-^3 = ? 
 
 18 -^3 = ? 21 -^3 = ? 12 -^2 = ? 8-4-2 = ? 
 
 12-f-3 = ? 10 -^2 = ? 15-4-3:==? 16~2 = ? 
 
 6-^2 = ? 18-4-2 = ? 24~3 = ? 27-4-3 = ? 
 
 1, If 2 boys can sit in one seat at school, how many 
 seats will be required for 12 boys ? 
 
 Explanation. Since 1 seat is required for 2 boys, 
 the number of seats required for 12 boys is equal to the 
 number of times 2 boys are contained in 12 boys. 2 are 
 in 12 6 times. Therefore 6 seats are required for 12 
 boys. 
 
 2, 18 apples were divided among a company of boys. 
 Each boy had 2 apples. How many boys were there ? 
 
 3, 2 horses make 1 span. How many spans will 16 
 horses make ? 
 
ioo 
 
 FIRST LESSONS IN 
 
 LESSON LXIX. 
 
 THB Tyro GITJEJJSr JSrUM'SB'RS STAJVDIJVG 
 
 It often happens that the things for which the Divi- 
 dend and the Divisor stand are unlike in hind. 
 
 Example. Harry's mother gave 12 peaches to her 3 
 children, Albert, Harry and Walter, giving each the 
 same number. How many peaches did each receive ? 
 
 Explain" ATioi^. The Dividend 
 is 12 peaches ; and the number 
 S, which stands for hoys, is the 
 Divisor. 1st: We take 3 peaches 
 from 12 peaches, and place them 
 on the table, in a row, headed 
 " 1st 3.'' Each boy has one of 
 these 3 peaches. 2d: Finding, 
 by Subtraction, that 3 peaches 
 are contained in 12 peaches ^ 
 times, we place 4 ^^ws on the 
 table. 
 
 
 SOLUTIOir. 
 
 
 B7J Subtraction. 
 
 12 Peaches. 
 
 
 3 
 
 i6 
 
 1st 
 
 9 
 
 cc 
 
 
 3 
 
 (C 
 
 Sd 
 
 6 
 
 <6 
 
 
 3 
 
 iC 
 
 Sd 
 
 3 
 
 i( 
 
 
 _3 
 
 <e 
 
 J4h 
 
 No Peaches 
 
 ', 
 
MENTAL AND WRIT'L'PN Au'lTBTlfMTXC, :»01 
 
 Each boy has 1 peach in each row, or 4 peaches 
 in all. 
 
 But we can consider the row5 running crosswise of 
 these. Then each boy's peaches will stand in a row. 
 There will be 3 rows ; one row for each boy, with 4 
 peaches. 
 
 In this Solution, 1st: We divide the Dividend, 12 
 peaches, into parts, each having 3 peaches, and find that 
 there are 4 parts, or rows. 2d : We then consider the 
 rows running crosswise of the first set, and find there 
 are 3 parts, or rows, with 4 peaches in each. 
 
 We divide by 3 in the same manner 
 as we would if it stood iov peaches. In solution. 
 the end, however, the Divisor, 3, is ^y^^^^- 
 made to stand for the number of parts, ^ Quottent. 
 or rows, into which we divide the Divi- — 
 dend; and the Quotient, 4, shows the ^^ Dividend, 
 size of each part of the Dividend. Hence, 
 
 I^rinciple 1, 
 
 When the Divisor shows the kumber of parts into 
 which the Dividend is divided, the Quotient shows the 
 SIZE OF EACH PART of the Dividend. 
 
 Principle 2, 
 
 When the Divisor shozvs the size of each of the 
 PARTS into which the Dividend is divided, the Quotient 
 
 €h0WS THE NUMBER OF THE PARTS. 
 I^nHple 3, 
 
 If we regard both the Divisor and Quotient in any 
 Example in Division, one of them tcill show the num- 
 ber of parts into which the Dividend is divided, and 
 the other the size of each part. 
 
K)3 'WR^T -LESSONS IN 
 
 LESSON LXX, 
 
 MUZTITI.ICATIOJ\r. 
 
 Example 1. Willie's mother gave 5 apples to each of 
 her 3 children. How many apples did she give in all ? 
 
 ExPLAiq-ATiON". 1st. Our Product, 
 15, shows tlie whole number of ap- solution. 
 
 pies given. 2d. Our Multiplier, 3, 5 Multiplicand. 
 shows the number of parts, or groups, _^ Multiplier, 
 into which the 15 apples were di- 15 Product, 
 vided. 3d. Our Multiplicand, 5, 
 shows the size of each of the parts, or groups, into which 
 the 15 apples were divided, in giving them to the 3 
 children. 
 
 0irisioj\r. 
 
 Example 2. Willie's mother divided 15 apples among 
 her 3 children, giving each the same number. How 
 many apples did she give each child ? 
 
 T-. -rrr . . . w ^ ,1 80LUTI0K. 
 
 Explanation^. Writmg 15 as the k n t' t 
 
 Dividend, and 3 as Divisor, and pro- 3 Divisor 
 
 ceeding as in the Solution on page ^ Dividend. 
 101, the Quotient is 5 apples. 
 
 Examining these two Solutions, we observe : 
 
 1st. 15 f which shoivs the i^umber of thiitgs di- 
 vided, is the Product in MultijMcation and the Divi- 
 dend in Division. 
 
 2d. 3f tuhich shows the kumber of parts into 
 which 15 things are divided, is the Multiplier in Mul- 
 tiplication and the Divisor in Division. 
 
 3d. 89 which shows the size of each of the parts 
 into which 15 things are divided, is the Multiplicand 
 in Multiplication and the Quotient in Division. 
 
 J4h. The SAME THREE NUMBERS occur in both Solutions, 
 
MENTAL AND WRITTEN ARITHMETIC. 103 
 
 INFERENCES, 
 
 1st. The Pkoduct in Multiplication may he taken as 
 
 DlVIDEJSTD. 
 
 ^d. The Multiplier in Multiplication may he taken 
 as DiYisoR. 
 
 Sd. If the Product he taken as Dividend, and the Mul- 
 tijMer as Divisor, the Quotieot: in Division will he the 
 same as the Multiplica^^d in Multiplication. 
 
 Jfih. If the Divisor and Quotient be multiplied 
 
 TOGETHER, THE RESULT WILL EQUAL THE DIVIDEKD. 
 Principle 4. 
 
 Division consists in finding a kumber which, wheit 
 
 MULTIPLIED BY OKE OF TWO GIVEN* NUMBERS, wHl 
 
 give a Product equal to the other given numher. 
 
 TESTING TME QUOTIENT. 
 
 After obtaining our Quotient, we can always test it 
 by Diference Jf. Multiplying it by the Divisor, if the 
 result equals the Dividend we presume that our Quo- 
 tient is the true one. 
 
 LESSON LXXI. 
 
 Find and test the Quotients in the following 
 Exercises for the Slate and Board. 
 Quotients. 6 ???????? ? 
 Divisors. _5_3^^_2_3_2^^_3 
 
 Dividends. 12 18 14 15 18 21 16 27 10 24 
 
 Multiplication, 
 
 Example 1. Find the Product solution. 
 
 arising from multiplying 34 by 2. 34 Multiplicand. 
 
 __ TTT 1 i • 1 ^ Jjluitvplier. 
 
 Explanation. We multiply — ^ 
 
 first the 4 Ones, and then the 3 ^^ Product. 
 
 Tens, by 2, as in former Examples, 
 
 and, writing the results in the Product, have 68. 
 
104 FIRST LESSONS IN 
 
 Division, 
 
 Example 2. Find the Quotient arising from dividing 
 68 by 2. 
 
 ExPLAKATiOK. From "Principle solution. 
 
 4/^ we find that Division consists, in 34 Quotient. 
 this Example, in finding a number _^ Divisor. 
 which, when multiplied by 2, will G8 Dividend. 
 give a result equal to 68. 68 Test Product. 
 
 1st. We find how many Tens mul- 
 tiplied by 2 will give the 6 Tens of the Dividend. Since 
 3 Tens multiplied by 2 give 6 Tens, we write 3 Tens in 
 the Quotient. 
 
 2d. Finding that 4 Ones multiplied by 2 give 8 Ones, 
 we write 4 Ones in the Quotient. 
 
 3d. Testing our Quotient^ we multiply 34 by 2, and 
 write the result 68, as a Test Product^ below the Divi- 
 dend. 
 
 Find and Test the Quotients in the following 
 
 Exercises eor the Slate an^d Board. 
 
 Quotients. 
 Divisors. 
 
 ? 
 2 
 
 ? 
 2 
 
 X. 
 
 ? 
 
 2 
 
 ? 
 
 2 
 
 ? 
 
 2 
 
 ? 
 
 2 
 
 ? 
 2 
 
 Dividends. 
 
 42 
 
 64 
 
 46 
 
 68 
 
 48 
 
 88 
 
 68 
 
 Test Producti 
 
 !. 42 
 
 
 IL 
 
 .... 
 
 
 .... 
 
 
 Divisors. 
 
 ? 
 3 
 
 V 
 
 3 
 
 ? 
 
 3 
 
 ? 
 3 
 
 ? 
 3 
 
 ? 
 3 
 
 ? 
 •3 
 
 Dividends. 
 
 63 
 
 66 
 
 39 
 III. 
 
 69 
 
 93 
 
 99 
 
 369 
 
 468 -- 2 
 639 --3 
 
 684 -f 
 306^ 
 
 -2 
 -3 
 
 2,468 
 9,630 
 
 -f- 2 
 -v-3 
 
 
 68,468 
 60,936 
 
 -^ 2 
 -i-3 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 105 
 
 LESSON LXXII. 
 
 M772. TI'PI.ICA^TIOJV. 
 
 Example A. Multiply 376 by 
 
 376 
 
 __2 
 
 12 
 14 
 6 
 
 752' 
 
 SOLUTIOiy. 
 
 Multiplicand. 
 Multiplier. 
 
 Partial 
 Products. 
 
 Product. 
 
 2. 
 
 ExPLAN^ATiOK. Multiplying 6 
 Ones, then 7 Tens, and lastly 3 
 Hundreds, separately by 2, and 
 writing the Partial Products and 
 adding them, we have 752 for our 
 final Product. 
 
 ^irisiojv-. 
 
 Example B. Divide 752 by 2. 
 
 ExPLAKATiois". 1st. Our Hundreds 
 in the Quotient cannot be more than 
 3, since 4 Hundreds multiplied by 2 
 would give 8 Hundreds. Hence we 
 write 3 Hundreds in the Quotient. 
 Multiplying 3 Hundreds by 2, we 
 write the Product, 6 Hundreds, under 
 the 7 Hundreds. Drawing a line below 
 the 6, we subtract 6 from 7, and have 
 1 Hundred remaining. 
 
 2d. We now bring down the 5 Tens of the Dividend ; 
 and, writing 5 at the right of the 1 Hundred, have 1 
 Hundred and 5 Tens, or 15 Tens, for a new Partial 
 Dividend. Proceeding to divide this by 2, we see that 
 we cannot have more than 7 Tens in the Quotient. 
 Writing 7 Tens in the Quotient, and multiplying them 
 by 2, we write the Product, 14 Tens, under the 15 Tens. 
 Subtracting, we have 1 Ten remaining. 
 
 3d. Lastly, we bring down the 2 Ones of the Divi- 
 dend ; and, writing 2 at the right of the 1 Ten, have 1 
 Ten and 2 Ones, or 12 Ones. Dividing 12 Ones by 2, 
 
 376 Quotient. 
 2 Divisor, 
 
 752 Dividend. 
 6_ 
 
 15 
 14 
 
 12 
 12 
 
106 FIRST LESSONS IN . 
 
 we write 6 Ones in the Quotient. Multiplying 6 Ones 
 by 2, and subtracting, there is no Remainder, Hence 
 our entire Quotient is 376. 
 
 TIi:STING THE QXTOTIENT. 
 
 To I'est our Quotient we would multiply it by the 
 Divisor. The work would be the same as in Example 
 Ar 
 
 From our work in both Examples, we obserye, 1st : 
 The Partial Products in both are 6 Hundreds, 14 Tens, 
 and 12 Ones; but they stand in contrary order. In 
 Example A, we added these, and obtained 752. In 
 Example B, we subtracted them from 752, in a reverse 
 order, and had no Remainder, 2d : In Example A, we 
 multiplied together two numbers, 376 and 2, and obtained 
 752 for a Product ; and in Example B we divided 
 this Product by one of the numbers, 2, and obtained, for 
 a Quotient, the other number, 376. Hence, we draw the 
 following 
 
 INFERENCE, 
 
 DiYisiOK is precisely the reyekse of Multiplica- 
 
 TIOIS". 
 
 Principle 5. 
 
 If the Product of two numbers be diyided by oke of 
 them, the Quotient will be the other. 
 
 Find and Test the Quotients in the following 
 
 Exercises eor the Slate ajstd Board. 
 
 I. 
 
 Quotients, ?????? 
 
 Divisors, _2 _3 _2 __3 _2 3 
 
 Divide7ids, 312 435 538 768 756 867. 
 
 II. 
 5,871 -^ 3 9,786 -:- 2 7,467 ~ 3 9,536 -r- 2 
 3,976 -^ 2 8,568 -^ 3 5,796 -^ 2 4,125 -r- 3 
 7,641 -T- 3 5,876 -^ 2 8,085 ~ 3 1,974 -^• 2 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 107 
 
 LESSON LXXIII. 
 
 The three numbers will be precisely alike, when ob- 
 tained, in both Examples of each Pair of the following 
 
 EXEKCISES FOR THE SlATE AND BOARD. 
 
 Multiplication 
 
 1. \ 
 
 Division 
 
 59,758 
 
 2 
 
 l 119,516 
 
 789,675 
 3 
 
 2,369,025 
 
 S.\ 
 
 376,586 
 2 
 
 l 753,172 
 
 By Principle 5 we change our Multiplication Tables 
 into two sets of Division Tables ; thus 
 
 2 lare 2, 
 
 3 2 
 
 1 Time, 
 
 It 2 
 
 9 Times, 
 
 2 2 are 4, 
 
 2 4 
 
 2 Times, 
 
 2 4 
 
 2 Times, 
 
 2 3 are 6, 
 
 2 6 
 
 3 Times, 
 
 3 6 
 
 2 Times, 
 
 2^4 are 8, 
 
 2g 8 
 
 4 Times, 
 
 4 8 
 
 2 Times, 
 
 2 1 5 are 10, 
 
 2X10 
 
 5 Times, 
 
 5I1O 
 
 2 Times, 
 
 2 "^ 6 are 12, 
 
 2«12 
 
 6 Times, 
 
 6|12 
 
 2 Times, 
 
 2 7 are 14, 
 
 2 14 
 
 7 Times, 
 
 7 14 
 
 2 Times, 
 
 2 8 are 16, 
 
 2 16 
 
 8 Times, 
 
 8 16 
 
 2 Times, 
 
 2 9 are 18, 
 
 2 18 
 
 9 Times, 
 
 9 18 
 
 2 Times. 
 
 3 1 are 3, 
 
 3 3 
 
 1 Time, 
 
 1|3 
 
 3 Times, 
 
 3 2 are 6, 
 
 3 6 
 
 2 Times, 
 
 2 6 
 
 3 Times, 
 
 3 3 are 9, 
 
 3 9 
 
 3 Times, 
 
 3 9 
 
 3 Times, 
 
 3 ^ 4 are 12, 
 
 3gl2 
 
 4 Times, 
 
 4 12 
 
 3 Times, 
 
 3 1 5 are 15, 
 
 3^15 
 
 5 Times, 
 
 5ll5 
 
 3 Times, 
 
 3 "^ 6 are 18, 
 
 3«18 
 
 6 Times, 
 
 6|18 
 
 3 Times, 
 
 3 7 are 21, 
 
 3 21 
 
 7 Times, 
 
 7 21 
 
 3 Times, 
 
 3 8 are 24, 
 
 3 24 
 
 8 Times, 
 
 8 24 
 
 3 Times, 
 
 3 9 are 27, 
 8 
 
 3 27 
 
 9 Times, 
 
 9 27 
 
 3 Times. 
 
108 FIRST LESSONS IN 
 
 LESSON LXXIV. 
 
 The equation 5 x 3 = 15 formed by Multiplication^ 
 we name an JEquation hy Multiplication. 
 
 Since the equation 15 -r- 3 = 5 is formed by Division^ 
 we wiU name it an JEquation by Dlmsion. 
 
 From three numbers such that no two of them are 
 equal, and the greatest equals the Product of the two 
 others, we can form two Equations hy Multiplication^ 
 by the Principle given on page 93, and also two Equor 
 tions by Division^ by Principle 5 in Division. 
 
 To Form two Equations by Multiplication: 
 
 KULE. 
 
 I. Write the two smaller numbers^ with the Sign x 
 between them, for the First Member of an Equation. 
 
 II. Write the greatest number for the Second Member, 
 placing the Sign = between the Members. 
 
 III. Form the second Equation from the first, by 
 changing the places of the two smaller numbers. 
 
 To Form two Equations by Division: 
 
 EULE. 
 
 I. Write the greatest number, and after it one of the 
 other numbers, placing the Sign -r- between them, for the 
 First Member of an Equation. 
 
 n. Write the remaining number for the Second Mem- 
 ber, placing the Sign = between the Members. 
 
 in. F(yrm the second Equation by Division from the 
 first, by changing the places of the two smaller numbers. 
 
 Form 4 Equations from each of these sets of numbers : 
 2, 7 and 14; 6, 3 and 18 ; 2, 8 and 16; 3, 7 and 21 ; 
 9, 2 and 18; 3, 8 and 24 ; 6, 2 and 12 ; 9, 3 and 27 ; 
 5, 3 and 15 ; 4, 3 and 12 ; 2, 5 and 10 ; 4, 2 and 8. 
 
MENTAL AND WRITTEN ARITHMETIC. 109 
 
 LESSON LXXV. 
 
 In the Equation 2 x 3 = 6, if 6 were erased, it could 
 be found by 7nulti2Jlying together 2 and 3 ; if 2 were 
 erased, we could find it by dividing 6 by 3; and if 3 
 were erased, it could be found by dividing 6 by 2. 
 Hence, 
 
 To Replace a Number in an Equation by 
 Multiplication: 
 
 KULE. 
 
 L If the Second Member he missing, find it ly Multi- 
 plying together the tioo numbers in the First Member. 
 
 II. If either number in the First Member be missing, 
 find it by Dividin^g the secokd member by the other 
 NUMBER in the First Member. 
 
 Exercises for the Slate and Board. 
 
 3x? = 18?x5 = 15 8x2 = ? 6x3 = ? 
 ?x3 = 12 7x?==14 7x3 = ? 7x?=21 
 3 X 8 = ? 3 X ? = 27 ? X 6 = 18 2 X ? = 16 
 
 As we see at the right, 
 i\iQ three numbers mi]iQ^ Quotient. 7 Multiplicand. 
 Solution of an Example Divisor. _3 Multiplier. 
 in Multiplication and Dividend. 21 Product. 
 one in Division are the 
 same, and stand in the same order. Hence, 
 
 To FIND ANY' NUMBER IN AN EXAMPLE IN MULTI- 
 PLICATION OR Division, when missing: 
 
 EULE. 
 
 I. If EITHER OF THE FIRST TWO, Or Smaller numbers, 
 be missing, find it by dividing the third, or greatest, 
 by the smaller number tohich is given. - 
 
 II. If the THIRD, or greatest number, be missing, find it 
 by multiplying together the two others. 
 
110 
 
 FIRST LESSONS IN 
 
 Exercises por the Slate ai^d Board. 
 5 ? 8 ? ? 3 3 ? - ? 9 
 
 10 14 16 18 12 15 18 21 24 27 
 
 LESSON LXXVI. 
 
 We now place 
 the 3 liorizon- 
 tal rows of 
 cubes, on page 
 85, over an- 
 other row. 
 The portion 
 now added to 
 the Table is 
 formed in the same manner as the portion then made. 
 
 From the Multiplication Table given below, and also 
 from each one hereafter given, make a second Table by 
 changing the places of the figures in the first and second 
 columns; thus: 1 time 4 is 4; 2 times 4 are 8; 3 times 
 4 are 12 ; &c., &c. 
 
 / / r 
 
 W~ 
 
 L. 
 
 m^ 
 
 - \ 
 
 r f" ^ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 J4 
 
 8 
 
 12 
 
 _I6_ 
 
 20 
 
 24 
 
 28 
 
 32 
 
 ^ 
 
 TABLE, 
 
 
 DIVISION 
 
 TABLES, 
 
 4t 1 are 4, 
 
 4 4 
 
 1 Time, 
 
 1 1 4 4 Times, 
 
 4 2 are 8, 
 
 4 8 
 
 2 Times, 
 
 2' 8 4 Times, 
 
 4 3 are 12, 
 
 4 12 
 
 3 Times, 
 
 3 12 4 Times, 
 
 4 « 4 are 16, 
 
 4|16 
 
 4 Times, 
 
 4 16 4 Times, 
 
 4| 5 are 20, 
 
 4 120 
 
 5 Times, 
 
 5I2O 4 Times, 
 
 4^ 6 are 24, 
 
 4''24 
 
 6 Times, 
 
 6|24 4 Times, I 
 
 4 7 are 28, 
 
 , 4 28 
 
 7 Times, 
 
 7 28 4 Times, 
 
 4 8 are 32, 
 
 4 32 
 
 8 Times, 
 
 8 32 4 Times, 
 
 4 9 are 36. 
 
 4 36 
 
 9 Times, 
 
 9 36 4 Times. 
 
MENTAL AND WRITTEN ARITHMETIC, 111 
 
 The same 3 columns of figures occur in all the 3 Ta- 
 bles on the preceding page. 
 
 Mental Exercises. 
 
 I. 
 
 9x4 = ? 5x?=2Q 7x?==28 8x?=32 
 
 6x? = 18 ?x4 = 24 ?x4r=:28 9x? = 36 
 
 ?x8 = 32 4x?=28 6x?=24 ?x4 = 36 
 
 II. 
 
 16-T-4 = ? 21-4-7 = ? 28 -f-? =7 ? -r 6 = 4 
 
 21 ^9 = r^ 20-f-4 = ? 16-T-4 = ? ?-4-4 = 4 
 
 24-v-4 = ? ?-T-4 = 9 ?-f-8 = 4 36-r-?=4 
 
 
 LESSON LXXVII. 
 
 
 
 MUL TI^ZICA TIOJV. 
 
 
 587,936 
 4 
 
 I. 
 759,864 434,234 
 4 9 
 
 II. 
 
 342,344 
 8 
 
 241,243 
 
 7 
 
 423,132 214,324 
 6 5 
 
 879,567 
 4 
 
 III. 
 Multiply 414,342 by 4; by 5; by 6; by 7; by 8; 
 Multiply 340,424 by 4 ; by 5 ; by 6 ; by 7 ; by 8. 
 
 ^irisiojsr. 
 I. 
 
 4 3 4 8 
 
 7,849,896 8,626,587 4,787,104 2,738,752 
 
 II. 
 7 6 5 4 
 
 1,688,701 2,538,726 7,171,060 9,276,872 
 
112 
 
 FIB8T LESSONS IN 
 
 LXXVIII. 
 
 LESSON 
 
 It should be 
 noticed that 
 the first part 
 of the Tables 
 given below, 
 extending t o 
 the line com- 
 mencing " 5 
 times 4 are 
 20/' has been 
 given in the preceding Tables. The new part is printed 
 in the Italic type. 
 
 MTTLTIPJLICATION 
 TABLE, 
 
 f 
 
 r 
 
 L-^K" fc^ 
 
 fc 
 
 c— 1 
 
 ^ ^ ^ 
 
 1 
 
 2 
 
 3 
 
 4. 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 2i 
 
 24 
 
 27 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 js 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 [ 
 
 5 
 
 1 
 
 are 
 
 5, 
 
 5 
 
 2 
 
 are 
 
 10, 
 
 5 
 
 3 
 
 are 
 
 15, 
 
 ^t 
 
 4 
 
 are 
 
 20, 
 
 ^1 
 
 5 
 
 are 
 
 25, 
 
 6^ 
 
 (5 
 
 are 
 
 SO, 
 
 5 
 
 7 
 
 are 
 
 85, 
 
 5 
 
 8 
 
 are 
 
 JfO, 
 
 6 
 
 9 
 
 are 
 
 J^. 
 
 5 
 
 5 
 5 
 5j 
 
 5 
 10 
 15 
 20 
 
 DIVISION TABZBS, 
 
 1 Time, l| 
 
 2"* 
 3 
 4 
 
 5 
 10 
 15 
 20 
 
 5 I 25 
 5"" SO 
 
 S5 
 
 JfO 
 
 5 $25 
 6tS0 
 
 7 S5 
 
 8 Jfi 
 
 9 JfB 
 
 5 Times, 
 5 Times, 
 5 Times, 
 5 Times, 
 5 Times, 
 5 Times, 
 5 Times, 
 5 Times, 
 5 Times. 
 
 2 Times, 
 
 3 Times, 
 
 4 Times, 
 
 5 Times, 
 
 6 Times, 
 
 7 Times, 
 
 8 Times, 
 
 9 Times, 
 
 "Write one more Equation by Multiplication and two 
 by Division from each of the following 
 
 Equations. 
 
 x7 = 35 6x5 = 30 
 x8 = 32 9x4 = 36 
 
 MEi^TAL EXEKCISES. 
 I. 
 
 X ? = 36 5 X ? = 30 
 X ? := 40 9 X ? = 27 
 X 5 == 25 ? X 9 =3 45 
 
 24 
 40 
 
 = ? 
 = 36 
 = 40 
 
 4 
 
 8 
 
 ? 
 
 4 
 
 X 
 
 7 
 
 r:: 
 
 28 
 
 9 
 
 X 
 
 5 
 
 = 
 
 45 
 
 5 
 
 X 
 
 9 
 
 ^ 
 
 ? 
 
 5 
 
 X 
 
 ? 
 
 = 
 
 45 
 
 7 
 
 X 
 
 p 
 
 = 
 
 35 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 113 
 
 II. 
 
 35 ^5 = ? ?^7 
 30 ^ 5 = ? 40 ^ 8 
 
 = 3 
 
 = ? 
 
 36 ^ 
 
 28 i- 
 
 4 = 
 
 • 7 = 
 
 ? 36 -f- 9 = ? 
 ? ? -^ 9 = 5 
 
 5 ? ? 9 
 
 7 7 4 5 
 
 ? 
 9 
 
 III. 
 
 7 
 ? 
 
 5 
 8 
 
 5 ? 9 
 
 ? 5 ? 
 
 ? 35 28 ? 
 
 45 
 
 35 
 
 ? 
 
 30 40 45 
 
 LESSON LXXIX. 
 
 Exercises for the Slate akd Board. 
 
 MuUipUcation. 
 
 
 
 1. Multiply 874,596 by 2; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 2. Multiply 746,789 by 2 ; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 3. Multiply 876,598 by 2; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 A. Multiply 715,829 by 2; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 5. Multiply 769,854 by 2; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 I>ivUU>n, 
 
 
 
 L Divide 114,840 by 2; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 2. Divide 689,040 by 2 ; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 3. Divide 803,880 by 2 ; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 Jf. Divide 344,520 by 2; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 5. Divide 107,640 by 2; 
 
 by 3; 
 
 by 4; 
 
 by 5. 
 
 Multiplication at Sight 
 
 , 
 
 
 4x4 4x5 6x5 
 
 4x7 
 
 6x4 
 
 7x4 
 
 5x5 4x6 5x4 
 
 7x5 
 
 5x6 
 
 3x9 
 
 3x8 5x8 8x4 
 
 9x3 
 
 8x3 
 
 4x9 
 
 Division at Sight. 
 
 
 
 6^2 5-4-5 8-f-4 
 
 10-4-2 
 
 10-4-5 
 
 14 -=-3 
 
 8-=-2 20-f-5 9-f-3 
 
 25 -^ 5 
 
 12-4-2 
 
 1^-4-6 
 
 15-4-5 16 -T- 4 16-1-8 
 
 12 H- 3 
 
 12-4-4 
 
 18^2 
 
 20-4-4 15 -^ 3 14 -^ 7 
 
 16-4-2 
 
 18-4-3 
 
 21-4-3 
 
114 
 
 FIRST LESSONS IN 
 
 LESSON LXXX. 
 
 The four 
 Tables given 
 below should 
 be thoroughly 
 learned. They 
 can all be read 
 from the Table 
 of cubes shown 
 at the right. 
 The new part 
 now added is 
 printed in Ita- 
 lic type. 
 MULTIBTj tcation 
 
 TABL£J. 
 
 f 
 
 f- ^ ^ 
 
 L 
 
 1= 
 
 ^- \ 
 
 ^ ^ \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 4 
 
 8 
 
 12 
 
 i6 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 6 
 
 12 
 
 ii 
 
 24 
 
 30 
 
 3£ 
 
 42 
 
 48 
 
 54^ 
 
 DIVISION TABIBS, 
 
 6,1 5 are 30, 
 ^^^are.^^, 
 6 1l^x^Ji2, 
 6 8 are 4^^, 
 6 Pare 54. 
 
 6 1 are 6, 6 6 1 Time, 1 "J 
 
 6 2 are 12, 6 12 2 Times, 2*" 
 
 6 3 are 18, 6 18 3 Times, 3 
 
 6 * 4 are 24, 6 1 24 4 Times, 4 
 
 6 1^ 30 5 Times, 5 1 
 
 e'^Se ^ Times, 
 
 6 42 7 Times, 
 
 6 JfB ^ Times, 
 
 6 ^5Jf, 9 Times, 
 Write three other Equations from each of the follow- 
 ing 
 
 8 
 9 
 
 6 ©Times, 
 
 12 6 Times, 
 
 18 6 Times, 
 
 24 6 Times, 
 
 30 6 Times, 
 
 36 6 Times, 
 
 6 Times, 
 
 6 Times, 
 
 6 Times. 
 
 JfS 
 54 
 
 X 7 = 42 
 X 8 =40 
 
 Equations. 
 
 6x8 = 48 6x9 = 54 5x6 = 30 
 7x5 = 35 9x4 = 36 9x5 = 45 
 
 ? X 7 = 42 
 9 X ? = 54 
 
 Mei^tal Exekcises. 
 I. 
 
 6x8 = ? 7 X ? = 42 
 ? X 8 = 48 ? X 6 = 48 
 
 6 X ? = 36 
 ? X 6 = 54 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 115 
 
 II. 
 
 36-T-6 = ? 54-T-? = 6 25 -^5 = ? 30-^? = 5 
 
 
 
 
 
 III. 
 
 
 
 
 
 
 ? 
 
 ? 
 
 7 
 
 8 
 
 e 
 
 ? 
 
 8 
 
 ? 
 
 5 
 
 9 
 
 6 
 
 e 
 
 ? 
 
 ? 
 
 9 
 
 6 
 
 6 
 
 9 
 
 ? 
 
 6 
 
 36 
 
 42 
 
 42 
 
 48 
 
 48 
 
 54 
 
 ? 
 
 54 
 
 35 
 
 ? 
 
 LESSON LXXXI. 
 
 MirZ TJTZICil TIOJV. 
 
 1, Multiply 875,964 by 2 
 
 ; by 3^ 
 
 by 4; 
 
 by 5 
 
 , bye. 
 
 2, Multiply 587,958 by 2 
 
 ; by 3 
 
 . by 4; 
 
 by 5 
 
 bye. 
 
 S. Multiply 978,547 by 2 
 
 ; by 3 
 
 ; by 4; 
 
 by 5 
 
 ; bye. 
 
 ^. Multiply 849,756 by 2 
 
 ; by 3 
 
 ; by 4; 
 
 by 5 
 
 ; bye. 
 
 5. Multiply 354,623 by 5 
 
 ; by 6 
 
 ; by 7; 
 
 by 8 
 
 , by 9. 
 
 6, Multiply 560,365 by 5 
 
 ; by 6 
 
 ; by 7; 
 
 by 8 
 
 ; by 9. 
 
 7. Multiply 465,324 by 5 
 
 ; by 6 
 
 dsion. 
 
 ; by 7; 
 
 by 8 
 
 ; by 9. 
 
 1, Divide 3,232,740 by 2 ; 
 
 by 3; 
 
 by 4; 
 
 by 5 
 
 ; bye. 
 
 2. Divide 4,765,140 by 2 ; 
 
 by 3; 
 
 by 4; 
 
 by 5 
 
 ; bye. 
 
 S, Divide 1,854,240 by 2 ; 
 
 by 3; 
 
 by 4; 
 
 by 5 
 
 ; bye. 
 
 Jf. Divide 1,683,840 by 2 ; 
 
 by 3; 
 
 by 4; 
 
 by 5 
 
 ; bye. 
 
 J. Divide 2,918,520 by 2 ; 
 
 by 3. 
 
 by 4; 
 
 by 5 
 
 ; bye. 
 
 MultiplU 
 
 nation at 
 
 Sight. 
 
 
 
 4x5 6x4 8xi 
 
 5 3 > 
 
 c 9 5 
 
 X 8 
 
 7x5 
 
 6x6 7x6 6x 
 
 9 5 > 
 
 < 6 9 
 
 X 5 
 
 6x8 
 
 6x7 8x6 4x 
 
 5 5 > 
 
 < 9 9 
 
 X 6 
 
 6x5 
 
 Division at Sight, 
 
 42 -^ 6 36 -f- 6 45 -f- 9 54 -^ 6 48 -^ 8 40 -=- 8 
 
 35 -T- 7 54 -^ 9 48 -^ 6 42-7 30-4-6 45-^-5 
 
 36 -J- 9 35 -^ 5 30 -^ 5 32 ~ 4 40 -^ 5 25-5 
 
116 
 
 FIRST LESSONS IN 
 
 LESSON LXXXII. 
 
 > >- r *" 
 
 t. 
 
 fc 
 
 ^_ ' 
 
 = ^ \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 ID 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 48 
 
 54 
 
 7 
 
 14 
 
 21 
 
 28 
 
 35 
 
 42 
 
 49 
 
 56 
 
 63 
 
 MUZTIPZICA TION 
 TABLE. 
 
 7 1 are 7, 
 
 7 7 
 
 7 2 are 14, 
 
 7 14 
 
 7 3 are 21, 
 
 7 21 
 
 7 * 4 are 28, 
 
 7|28 
 
 7 1 5 are 35, 
 
 7^35 
 
 7^ 6 are 42, 
 
 7''42 
 
 7 l^XQJfiy 
 
 7 49 
 
 7 S^reSe, 
 
 7 56 
 
 7 9 are 63, 
 
 7 68 
 
 DIVISION 
 
 1 Time, 
 
 2 Times, 
 
 3 Times, 
 
 4 Times, 
 
 5 Times, 
 
 6 Times, 
 
 7 Times, 
 
 8 Times, 
 P Times, 
 
 TABIES, 
 
 2" 14 
 
 3 21 
 
 4 28 
 5 1 35 
 
 6 I 42 
 
 7 4^ 
 <9 ^^ 
 9 68 
 
 7 Times, 
 7 Times, 
 7 Times, 
 7 Times, 
 7 Times, 
 7 Times, 
 7 Times, 
 7 Times, 
 7 Times. 
 
 Write three others from each of the following 
 
 Equations. 
 
 7x8 = 56 8x6 = 48 6x9=54 
 35^7 = 5 42-^7 = 6 30-f-6=5 
 
 7 X 9 = 63 
 36 ~ 9 = 4 
 
 7 X ? = 42 
 ? X 7 = 63 
 
 Mektal Exercises. 
 I. 
 7x? = 56 7x? = 63 7x? = 49 
 ?x7 = 56 ?x9=63 9x7 = ? 
 
MENTAL AND WRITTEN ARITHMETIC. 117 
 
 11. 
 
 36 ^ 6 = ? 56 -^ ? = 8 42 -f- ? = 6 63 -^ ? = 9 
 
 49 ^7 = ? 63_^7^p ?,^7=7 p.^^.^^ 
 49_^?=:7 p_^g^7 5g^P^7 p^9^7 
 
 HI. 
 
 ? 9 7 9 ? ? 7 ? 6 
 
 _8_?9_?_7__7_?J_? 
 56 54 ? 63 56 49 63 48 42 
 
 LESSON LXXXIII, 
 Exercises for the Slate and Board. 
 
 
 Multiplication, 
 
 
 795,869 X 7 
 
 978,549 X 7 
 
 895,987 X 7 
 
 536,754 X 8 
 
 765,467 X 8 
 
 657,327 X 8 
 
 534,276 X 9 
 
 374,657 X 9 
 
 534,756 X 9 
 
 978,679 X 7 
 
 576,754 X 8 
 
 Division, 
 
 321,567 X 9 
 
 5,984,678 -^ 7 
 
 9,854,789 -^ 7 
 
 6,897,583 -- 7 
 
 5,416,040 H- 8 . 
 
 5,174,016 -4- 8 
 
 4,996,504 -V- 8 
 
 6,395,148 -T- 9 
 
 3,784,878 h- 9 
 
 1,918,575 -f- 9 
 
 9,546,327 -f- 7 
 
 3,715,616 -4- 8 
 
 5,982,039 -V- 9 
 
 Divide 544,320 by 
 
 4; by 5; 
 
 by 6; by 7. 
 
 Divide.5,987,520by 4; by 5; 
 
 by 6; by 7. 
 
 
 Multiplication, at Sight, 
 
 
 5x6 6x6 
 
 5x8 8x4 
 
 7x4 4x7 
 
 7x7 7x8 
 
 7x9 8x7 
 
 9x7 6x7 
 
 5x7 7x6 
 
 6x5 5x9 
 
 Division at Sight. 
 
 7x5 5x5 
 
 35 -T- 7 42 -^ 7 
 
 63 -f- 9 56 -^ 7 
 
 42-6 49 -f- 7 
 
 54 ^ 6 30 -f- 5 
 
 48 H- 6 36 ^ 6 
 
 30 -^ 6 48 -^ 8 
 
 56 -^ 8 63 ^ 7 
 
 25 -^ 5 54 -r^ 9 
 
 35 -^ 5 28 -f- 7 
 
118 
 
 FinsT lbsso:ns m 
 
 
 LESSON LXXXIV. 
 
 
 f r r fc. ■ 
 
 
 te 
 
 ^- \ 
 
 ^ P ^ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 5 
 
 iO 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 48 
 
 54 
 
 7 
 
 14 
 
 21 
 
 28 
 
 35 
 
 42 
 
 49 
 
 56 
 
 63 
 
 k 
 
 16 
 
 24 
 
 32 
 
 40 
 
 48 
 
 56 
 
 64 
 
 7Z^ 
 
 MUZTIPLICA TION 
 TABLE, 
 
 DIVISION TABLES, 
 
 8 1 are 8, 8 8 1 Time, 
 
 8 2 are 16, 8 16 2 Times, 
 
 8 3 are 24, 8 24 3 Times, 
 
 8^4 are 32, 8|32 4 Times, 
 
 8| 5 are 40, 8*^40 5 Times, 
 
 8^ 6 are 48, 8*48 6 Times, 
 
 8 7 are 56, 8 56 7 Times, 
 
 8 8 are 6 J/,, 8 GJf, 8 Times, 
 
 8 9 are 72, 8 72 9 Times, 
 
 Write three others from each of the following 
 
 Equations, 
 
 6 X 8 = 48 7 X 9 = 63 9 x 8 = 72 7 x 8 = 56 
 
 Mektal Exeecises. 
 I. 
 
 7x8 = ? 8x?=:56 ?x6=:48 8x? = 64 
 9x8=r? 9x?=:63 8x? = 72 ?x9 = 7'2 
 
 1-^8 8 Times, 
 
 2"'l6 8 Times, 
 
 3 24 8 Times, 
 
 4 32 8 Times, 
 5 1 40 8 Times, 
 
 6 I 48 8 Times, 
 
 7 56 8 Times, 
 
 8 64. ^ Times, 
 
 9 72 8 Times. 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 119 
 
 64 4- 8 = ? 56 ^ r 
 
 ? -^ 7 = 8 73 H- 8 
 
 II. 
 
 = 7 73 -4- 
 
 = ? ?-r 
 
 9 = 
 9 = 
 
 ? 
 8 
 
 ? 
 56 
 
 -T-8 = 8 
 -=-? = 8 
 
 7 ? 8 9 
 7 8?? 
 ? 56 64 73 
 
 III. 
 
 ? 8 
 9 9 
 
 63 ? 
 
 7 
 ? 
 
 56 
 
 
 ? 
 8 
 
 73 
 
 ? 8 
 8 8 
 
 64 ? 
 
 LESSON LXXXV. 
 Exercises for the Slate and Board. 
 
 JHultiplication, 
 
 785,968 X 8 978,786 x 8 
 
 874,569 X 8 678,587 x 9 
 
 854,867 X 9 758,675 x 9 
 
 1. Multiply 859,756 by 5 ; by 6 ; 
 
 2. Multiply 596,873 by 5 ; by 6 ; 
 S. Multiply 584,762 by 4 ; by 7 ; 
 Jf,. Multiply 378,578 by 4 ; by 7 ; 
 
 687,952 
 789,872 
 245,439 
 
 jyivision. 
 
 598,648 -^ 8 
 759,132 -r- 9 
 753,588 -T- 9 
 
 1. Divide 164,304 by 4; 
 
 2. Divide 332,472 by 4; 
 S. Divide 349,440 by 4 ; 
 ^. Divide 903,504 by 2 ; 
 5. Divide 910,680 by 3 ; 
 
 by 6; 
 by 6; 
 by 6; 
 by 3; 
 by 4; 
 
 8x9 7x8 
 
 Multiplieation at Sight. 
 
 8x8 7x9 
 
 879,587 X 8 
 768,576 X 9 
 795,879 X 8 
 
 by 7; by 8. 
 
 by 7; by 8. 
 
 by 8 ; by 9. 
 
 by 8; by 9. 
 
 987,688 -T- 8 
 653,445 -^ 9 
 579,568 -^ 8 
 
 by 7 
 by 7 
 by 7 
 by 4 
 by 5 
 
 by 8. 
 by 8. 
 by 8. 
 by 8. 
 by 8. 
 
 9x8 6x8 
 
 Division at Sight. 
 
 63 -^ 9 64-4-8 72 4- 8 56 -^ 8 72 4- 9 48-4-8 
 1.9-7 56 4- 7 48 4- 6 63 4- 7 40 4- 5 45 -^ 9 
 
120 
 
 FIRST LESSONS IN 
 
 
 LESSON LXXXVI. 
 
 
 f r ^ 
 
 ^- t_ 1 
 
 fe fct- %,. ^ 
 
 fe^ \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3 
 
 6 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 6 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 48 
 
 54 
 
 7 
 
 14 
 
 21 
 
 28 
 
 35 
 
 42 
 
 49 
 
 56 
 
 63 
 
 8 
 
 16 
 
 24 
 
 32 
 
 40 
 
 48 
 
 56 
 
 64 
 
 7Z 
 
 m 
 
 18 
 
 27 
 
 36 
 
 45 
 
 54 
 
 63 
 
 72 
 
 M 
 
 TjLBLE. 
 
 
 DIVISION TABLES. 
 
 9 1 are 9, 
 
 9 9 
 
 1 Time, 
 
 
 l| 9 9 Times, 
 
 9 2 are 18, 
 
 9 18 
 
 2 Times, 
 
 
 2"* 18 9 Times, 
 
 9 3 are 27, 
 
 9 27 
 
 3 Times, 
 
 
 3 27 9 Times, 
 
 9 «> 4 are 36, 
 
 9-36 
 
 4 Times, 
 
 
 4-S 36 9 Times, 
 
 9| 5 are 45, 
 
 9 145 
 
 5 Times, 
 
 
 5 1 45 9 Times, 
 
 9^ 6 are 54, 
 
 9 54 
 
 6 Times, 
 
 
 6 54 9 Times, 
 
 9 7 are 63, 
 
 9 63 
 
 7 Times, 
 
 
 7 63 9 Times, 
 
 9 8 are 72, 
 
 9 72 
 
 8 Times, 
 
 
 8 72 9 Times, 
 
 9 QaieSL 
 
 
 9 are in 
 
 81 
 
 9 Times. 
 
 
 Mektal Exercises 
 
 
 8 X ? = 56 
 
 ? X 9 = 
 
 : 63 8 X 
 
 ? = 
 
 72 9x9 = ? 
 
 ? X 9 = 72 
 
 7 X ? = 
 
 63 9 X 
 
 7 = 
 
 ? 9x8=? 
 
 8x8 = ? 
 
 9 X ? = 
 
 81 ? X 
 
 8 = 
 
 56 ? X 9 = 81 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 121 
 
 II. 
 
 72 4-8 = ? 81-^-9 = ? ?-^8 = 8 ?h-9 = 9 
 
 ? 
 8 
 
 III. 
 
 9 ? 
 9 _9 
 
 ? 
 
 8 
 
 ? 
 
 ? 
 
 8 
 
 63 72 63 72 
 
 81 
 
 72 81 64 
 
 ft ^' ^ ■ 
 
 k-. 
 
 fc 
 
 ^ ■ t 
 
 t^ ) 
 
 ^ \ 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 3 
 
 
 9 
 
 12 
 
 15 
 
 18 
 
 21 
 
 24 
 
 27 
 
 4 
 
 
 
 16 
 
 20 
 
 24 
 
 28 
 
 32 
 
 36 
 
 5 
 
 
 
 
 25 
 
 30 
 
 35 
 
 40 
 
 45 
 
 6 
 
 
 
 
 
 36 
 
 42 
 
 48 
 
 54 
 
 7 
 
 
 
 
 
 
 49 
 
 56 
 
 63 
 
 8 
 
 
 
 
 
 
 
 64 
 
 72 
 
 J 9 
 
 
 
 
 
 
 
 _ 
 
 8l[ 
 
 By comparing the above Table with that given on the 
 preceding page, it will be seen that the numbers omitted 
 in forming this Table from that are the same as those 
 retained. Hence they were unnecessary. 
 
 In reading this Table we find our Multiplier, or Di- 
 visor, in the upper horizontal row whenever it is greater 
 than the Multiplicand, or Quotient. 
 6 
 
122 FIRST LESSONS IN 
 
 LESSON LXXXVII. 
 
 Example A. Multiply 964 by 3. 
 
 ExpLA:N^ATio:Nr. 1st. The first Solution is iu the form 
 heretofore used, the Partial Products being written. 
 
 2d. In the second Solution the Par- 
 tial Products are not written out in first solution. 
 full. We first multiply 4 Ones by 3, 964 
 
 and, obtaining 12 Ones, or 1 Ten and 3 
 
 2 Ones, for a Partial Product, write 
 
 the 2 Ones, and retain the i Ten in 
 
 the mind. N^ext, we multiply the 6 
 
 Tens by 3, and, obtaining 18 Tens for 2,892 
 
 a Partial Product, we add to these the 
 
 1 Ten retained in the mind, and have second solution. 
 
 19 Tens, or 1 Hundred and 9 Tens. 964 
 
 We write the 9 Tens in the Product, r 
 
 and retain the 1 Hundred in the mind. 2,892 
 
 Finally, multiplying 9 Hundreds by 3, 
 we have 27 Hundreds for a Partial Product, to which 
 we add the 1 Hundred retained, and write the Sum, 28 
 Hundreds, in the Product. Our final Product is 2,892 ; 
 the same as in the first Solution. 
 
 These 2 methods of Multiplication differ in this only ; 
 in the first the Partial Products are written, while in 
 the second only the final Product is written. The writ- 
 ten work in the second is much shorter than in the first. 
 
 DEFINITIONS, 
 
 1, JLong Multiplication is the method of Mul- 
 tiplication used where we obtain the final Product by 
 writing out the Partial Products and finding their 
 Sum. 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 123 
 
 2, Short Multiplication is the method of MuU 
 tijjUcation used where, after writing the Multiplicand 
 and Multiplier, we write the final Product only, perform- 
 ing the rest of the work mejitally. 
 
 Find the Products by Short Multiplication in these 
 
 Exercises for the Slate ai^^d Board. 
 
 879,567 X 2 543,763 x 3 825,046 x 4 
 
 342,354 X 5 243,652 x 6 105,824 x 9 
 
 243,054 X 6 540,324 x 8 134,564 x 9 
 
 536,726 X 6 947,540 x 8 ' 302,132 x 9 
 
 LESSON LXXXVIII. 
 
 Example B. Divide 2,892 by 3. 
 
 ExPLAKATioi^r. 1st. The first Solution is full. 
 
 2d. In the second Solution we shorten the work. 
 Finding that 28 Hundreds divided 
 by 3 give 9 Hundreds, we write 9 
 Hundreds in the Quotient. We then 
 multiply 9 Hundreds by 3 ; and, 
 instead of writi7ig the Product, 27 
 Hundreds, we retain it in the mind, 
 and subtracting it from 28 Hun- 
 dreds mentally, find 1 Hundred re- 
 maining. Next, we mentally unite 
 the 9 Tens with the 1 Hundred left, 
 and retained in the mind, and have 
 1 Hundred and 9 Tens, or 19 Tens, 
 for a new Partial Dividend, which 
 we retain in the mind. Finding 
 that 19 Tens divided by 3 give 6 
 Tens, we write 6 Tens in the Quo- 
 tient. Multiplying 6 Tens by 3 and obtaining 18 Tens 
 for a Product, we subtract 18 Tens from 19 Tens men-- 
 9 
 
 FIRST SOLUTION. 
 
 964 Quotient. 
 3 Divisor. 
 
 2,892 Dividend. 
 27 
 
 19 
 
 18 
 
 
 12 
 
 12 
 
 
 SECOND SOLUTION. 
 
 964 
 3 
 
 2,892 
 
124 FIRST LESSONS IN 
 
 tally ; and, finding 1 Ten remaining, we retain this in 
 tlie mind. Finally, uniting the 2 Ones of the Dividend 
 with the 1 Ten retained, we have for our last Partial 
 Dividend 1 Ten and 2 Ones, or 12. We do not write 
 this ; but, dividing by 3 mentally, we write the result, 
 4, in the Quotient. We have 964 for our final Quotient. 
 The method used in the second Solution is much 
 shorter than that in the first, since in the second we 
 perform the work mentally , without writing the Partial 
 Dividends and Products, 
 
 DEFINITIONS, 
 
 1, Iiong Division is the method of Division used 
 where we obtain the Quotient by writing the Partial 
 Dividends and Products, 
 
 2, Short Division is the method of Division used 
 where we obtain the Quotient without writing the Par- 
 tial Dividends and Products. 
 
 Perform the work by Short Division in the following 
 
 Exercises for the Slate ai^b Board. 
 
 789,156 -V- 2 
 
 
 I. 
 591,732 ~ 2 
 
 
 769,518 -- 2 
 
 768,927 -f- 3 
 
 
 857,421 -r- 3 
 
 
 257,613 -f- 3 
 
 736,928 -^ 4 
 
 
 175,628 -^ 4 
 
 
 279,536 -- 4 
 
 623,715 -V- 5 
 
 
 859,235 -^ 5 
 
 
 973,265 -^ 5 
 
 157,326 -^ 6 
 
 
 312,714 -f- 6 
 
 
 517,896 -r- 6 
 
 548,765 -^ 7 
 
 
 227,353 -f- 7 
 
 
 257,327 -T- 7 
 
 264,976 ^ 8 
 
 
 512,760 -^ 8 
 
 
 512,064 ~ 8 
 
 325,719 -4- 9 
 
 
 289,881 -T- 9 
 
 
 123,453 -f- 9 
 
 Divide 7,206,480 by 
 
 II. 
 
 2 ; by 3 ; 
 
 by 4; 
 
 by 5; by 6. 
 
 Divide 14,405,760 
 
 by 
 
 2; by3; 
 
 by 4; 
 
 by 5; by 6. 
 
 Divide 5,040,720 
 
 by 
 
 2; by3; 
 
 by 4; 
 
 by 5; by 6. 
 
 Divide 833,280 by 
 
 4; by5; 
 
 by 6; 
 
 by 7; by 8. 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 125 
 
 LESSON LXXXIX, 
 
 MUZ TIP Lie A TION 
 
 
 
 
 
 
 TABLE 
 
 
 
 
 DIVISION TABIES, 
 
 10 1 is 
 
 10, 
 
 10 
 
 10 
 
 1 Time, 
 
 ll 
 
 10 
 
 10 Times, 
 
 10 2 are 
 
 20, 
 
 10 
 
 20 
 
 2 Times, 
 
 2*^ 
 
 20 
 
 10 Times, 
 
 ^ 10 3 are 
 
 30, 
 
 10 
 
 30 
 
 3 Times, 
 
 3 
 
 30 
 
 10 Times, 
 
 10 4 are 
 
 40, 
 
 10 
 
 40 
 
 4 Times, 
 
 4 
 
 40 
 
 10 Times, 
 
 10 1 are 
 
 50, 
 
 10- 
 
 50 
 
 5 Times, 
 
 5.1 
 
 50 
 
 10 Times, 
 
 10 1 6 are 
 
 60, 
 
 10 1 
 
 60 
 
 6 Times, 
 
 6 1 
 
 60 
 
 10 Times, 
 
 10 7 are 
 
 70, 
 
 10 
 
 70 
 
 7 Times, 
 
 7 
 
 70 
 
 10 Times, 
 
 10 8 are 
 
 80, 
 
 10 
 
 80 
 
 8 Times, 
 
 8 
 
 80 
 
 10 Times, 
 
 10 9 are 
 
 90, 
 
 10 
 
 90 
 
 9 Times, 
 
 9 
 
 90 
 
 10 Times, 
 
 10 10 are 
 
 100. 
 
 10 
 
 100 10 Times. 
 
 10 
 
 100 
 
 10 Times. 
 
 Multiplicands. 
 
 10 
 
 I. 
 10 
 
 10 
 
 
 7 
 
 II. 
 8 9 
 
 Multijoliers, 
 
 
 7 
 
 8 
 
 9 
 
 
 10 
 
 10 10 
 
 Products. 
 
 
 70 
 
 80 
 
 90 
 
 
 70 
 
 80 90 
 
 In these two sets of Examples, the numbers multiplied 
 together are the same, and hence the Products must he 
 the same in both sets. 
 
 From the Principle on page 93, we see that if two 
 numbers are to be multiplied together either may be used 
 as the Multiplicand^ and the other as the Multiplier. 
 Hence we use the Multiplicands of the first set as Mul- 
 tipliers in the second set. 
 
 The Products are the same in both sets; and are 
 formed in the second set by annexing a to each Multi- 
 plicand. 
 
 Example. Multiply 25 by 10. 
 
 Explanation". Writing the 10 
 under the 25 so that the 1 shall 
 stand under the 5, we bring down 
 
 25 Multiplicand. 
 10 Multiplier. 
 
 250 Product. 
 
126 FIRST LESSONS IN 
 
 the 25 into the Product, and then annex a cipher, to 
 obtain the final Product. By annexing the cipher the 
 6 Ones are made 5 Tens, which are 10 times as many as 
 5 Ones ; and the 2 Tens are made 2 Hundreds, which 
 are 10 times as many as 2 Tens. Therefore, since 250 
 is 10 times as large as 25, it is the true Product. Hence 
 we have this 
 
 INFERENCE, 
 
 Annexing a cipher at the right of a number multiplies 
 it hy 10, 
 
 Multiply each of the following numbers by 10 : 
 125; 896; 2,570; 37,896; 543,768; 5,020; 500. 
 
 LESSON XC. 
 
 In each of the first 4 Equations, 
 at the right, 2 numbers are mul- tactors. products. 
 tiplied together to make a third 2x3 =6 
 number. 2x5 =10 
 
 The numbers thus multiplied 3x3 ~ -, ^ 
 
 together to make a Product are %^\ 5 — qa 
 named Factors. Factor means ^ ^ 3 >< ^ Z 42 
 maker ; and Product means some- 
 thing made, or produced. 
 
 Thus, 2 and 5 are the Factors of 10, their Product ; 
 and 2, 3 and 5 are the Factors of 30, their Product. 
 
 The numbers 6, 10, 9, 15, 30, and 42, given above, are 
 composed of the numbers multiplied together to make 
 them, and are therefore named Composite NuTn- 
 bers. 
 
 The numbers multiplied together to form another 
 number are named its Component Factors. 
 
MENTAL AND WRITTEN ARITHMETIC. 127 
 
 nEFINITIONS. 
 
 1. A Factor is one of the numbers multiplied 
 together to produce another number. 
 
 2, A Prime Factor is any Factor which cannot 
 itself be produced by multiplying together other Factors. 
 
 S. A Prime Number is any number which cannot 
 be produced by multiplying together other numbers. 
 
 4. A Composite Number is any number which 
 can be produced by multiplying together other numbers. 
 
 5, A Com2^07ient Factor is one of the Factors 
 multiplied together to produce a Composite IN^umber. 
 
 6, An Fven Number is any number having 2 as 
 one of its Prime Factors. 
 
 7. An Odd Number is any number not having 2 
 as one of its Prime Factors. 
 
 Eemakks. 
 
 1, Every number whose right-hand figure is zero, or 
 is exactly divisible by 2, is an Even Number. 
 
 2. Every Even Number, except 2, is Composite. 
 
 S. Every number whose right-hand figure is an Odd 
 Number is itself an Odd Number. 
 
 Of the following numbers what ones are Prime, and 
 what ones Composite ? What Even, and what Odd ? 
 
 2, 4, 9, 11, 8, 16, 21, 25, 17, 63, 164. 
 6, 7, 6, 10, 13, 14, 15, 19, 22, 27, 120. 
 
 Exercises for the Slate akd Board. 
 
 Multiplication. 
 
 Multiply 7,859,634 by 5 ; by 6 ; by 7 ; by 8; by 9. 
 Multiply 9,473,857 by 4 ; by 6 ; by 7 ; by 8 ; by 9. 
 Multiply 5,837,918 by 3 ; by 5 ; by 7; by 8; by 9. 
 
 Division, 
 
 Divide 6,053,040 by 5 ; by 6 ; by 7; by 8 ; by 9. 
 Divide 6,562,584 by 3 ; by 4 ; by 7 ; by 8 ; by 9. 
 
128 FIRST LESSONS IN 
 
 LESSON XGI. 
 
 We have heretofore seen that any Product can be 
 divided by either of the numbers multiplied together 
 to produce it. In the same manner any Product formed 
 of any number of Factors can be divided by any of its 
 Factors, or by all of them in succession. 
 
 We will take the Prime Factors 13, 7, and 3, and find 
 their Product, and then divide this Product by its Prime 
 Factors, 13, 7, and 3, and see what we shall have for a 
 Quotient. 
 
 \.—Multiplicati(m. 
 
 
 ^.—Division. 
 
 1st Factor, 13 
 
 13 
 
 2d Quotient. 
 
 2d Factor, 7 
 
 _7_ 
 
 2d Divisor, 
 
 91 
 
 91 
 
 1st Quotient, 
 
 3d Factor, 3 
 
 3 
 
 1st Divisor, 
 
 Product, 273 
 
 273 
 
 1st Dividend, 
 
 1st, MuLTiPLiCATioisr. Multiplying 13 and 7 to- 
 gether, and then multiplying their Product, 91, by 3y 
 the third Factor, we obtain 273 for a final Product. 
 
 2d, Divisi02!C. Dividing this Product, 273, by 3^ the 
 last MultijMer, we obtain 91 for our first Quotient. 
 
 DividiJig 91 hy 7^ another of the 8 Factors 7nultipUed 
 together, we obtain, for a Quotient, 13, the other of the 
 8 Factors. 
 
 If, now, we should divide our last Quotient, 13, by 
 13 f the final Quotient would be 1. 
 
 We have divided the number 273 by each of its Prime 
 Factors in succession, and obtained 1 for a final Quo- 
 tient. Thus it is evident that, in whatever order we 
 multiply the numbers together, we may divide the Pro- 
 
MENTAL AND WRITTEN ARITHMETIC. 129 
 
 duct by all the numbers in succession, in a contrary 
 order* 
 
 Hence we have the following 
 
 rinciple in ZHvision. A, 
 
 Any Composite Number can he divided ly any one of 
 its Component Prime Factors, or dy all of them m sue- 
 cessio7iy using each Quotient for a neio Dividend. 
 
 Since any given Composite Number can be formed by 
 multiplying together its Prime Factor Sy and cannot be 
 formed by multiplying together any other Prime Factors, 
 it is evident that it cannot be exactly divided by any 
 other Prirrie Factor. Hence 
 
 Principle in Division. D. 
 
 No Composite Number can be exactly divided by any 
 Prime Number which is not one of its Prime Factors. 
 Hence, 
 
 To FIND THE Prime Factors of any Number: 
 
 EULE. 
 
 Divide the Number by any Prime Number that tvill 
 exactly divide it ; then divide the Quotient by any Prime 
 Number that will exactly divide it ; and so continue to 
 divide until a Quotient is obtained that is itself a Prime 
 Number. The several Divisors and the final Quotient 
 will be the Prime Factors of the given Prime Number. 
 
 Find and write all the Prime Factors of each of the 
 following Composite ISTumbers : 
 
 16 
 
 24 
 
 30 
 
 40 
 
 50 
 
 60 
 
 75 
 
 85 
 
 100 
 
 18 
 
 25 
 
 32 
 
 42 
 
 54 
 
 64 
 
 80 
 
 90 
 
 112 
 
 20 
 
 27 
 
 35 
 
 45 
 
 55 
 
 70 
 
 81 
 
 95 
 
 120 
 
 21 
 
 28 
 
 36 
 
 48 
 
 56 
 
 72 
 
 84 
 
 96 
 
 128 
 
130 
 
 FIRST LESSONS IN 
 
 365 Multiplicand. 
 9 1st Multiplier. 
 
 3,285 
 
 2 M Multiplier. 
 
 6,570 Product. 
 
 LESSON XGII. 
 
 Example, Multiply 365 by 18. 
 
 ExPLAKATioiq-. 1st. We separate the Multiplier, 18, 
 into the Component Factors 2 and 9. 
 
 2d. We multiply 365 by 9, one 
 of the Component Factors of 18, 
 thus taking 365 9 times, 
 
 3d. We multiply this Product, 
 3,285, by 2, the other Component 
 Factor of 18 ; thus taking 9 times 
 365 2 times. Since 2 times 9 
 times are 18 times, we have thus taken 365 18 times ; 
 or multiplied 365 by 18. 
 
 Instead of 2 and 9, as the Component Factors of 18, we 
 might have taken 3 and 6, or 2, 3 and 3. Multiplying by 
 these, we should have obtained the same Product. Hence, 
 
 To Multiply by any Composite Number: 
 
 EULE. 
 
 I. Separate the Multiplier into any number of Com^ 
 ponent Factors. 
 
 II. Multiply the Multiplicand ly one of these Factors ; 
 then this Product hy another, and so on till all the Com- 
 ponent Factors have leen used as Multipliers. The final 
 Product will be the true Product. 
 
 By using Component Factors as Multipliers, 
 Multiply 23,546 by 12 
 Multiply 37,985 by 24 
 Multiply 85,896 by 42 
 Multiply 98,583 by 56 
 Multiply 56,874 by 75 
 
 by 14 
 
 by 15 
 
 by 18. 
 
 by 27 
 
 by 35 
 
 ; by 36. 
 
 by 45^ 
 
 by 48 
 
 by 49. 
 
 by 63; 
 
 by 64 
 
 by 70. 
 
 by 81, 
 
 by 84^ 
 
 by 96. 
 
MENTAL AND WRITTEN ARITHMETIC. 131 
 
 LESSON XGIIL 
 
 ^iriDIJVG ^r A COM'POSITB jyUMl^B^. 
 
 l.—MuUiplication. 2.— Division. 
 
 Multiplicand, 376 376 Final Quotient. 
 
 1st Multiplier. 7 7 2d Divisor. 
 
 1st Product. 2,632 2,632 1st Quotient. 
 
 2d Multiplier. 2 2 1st Divisor. 
 
 Final Product. 5,264 5,264 Dividend. 
 
 Above we have multiplied 376 by 14, by using 7 and 
 2, the Component Factors of 14, as Multipliers. The 
 Product is 5,264. If, now, this Product be divided by 
 14, our Multiplier, the Quotient will be the Multipli- 
 cand, 376. But, above, we have obtained this result by 
 dividing by 2 and 7, the Component Factors of 14. 
 
 We notice that in our Division we retraced all our 
 work of Multiplication. Hence, 
 
 To Divide by ant Composite Number: 
 
 EULE. 
 
 I. Separate the Divisor into any number of Component 
 Factors. 
 
 II. Divide the Dividend ly any one of these Factors ; 
 then divide the Quotient by another Factor, and so on 
 till all the Factors have been used as Divisors. 
 
 The final Quotient will be the true Quotient. 
 By using Component Factors as Divisors, 
 
 Divide 40,320 by 12 ; 
 
 by 14; 
 
 by 15; 
 
 by 16. 
 
 Divide 816,480 by 24 ; 
 
 by 27; 
 
 by 35; 
 
 by 36. 
 
 Divide 4,445,280 by 42 ; 
 
 by 45; 
 
 by 48; 
 
 by 49. 
 
 The figures 1, 2, 3, 4, 5, 6, 7, 8, 9, always 
 Member. They are therefore called Numeral Fig' 
 tireSf or Nuinerals. Zero (0) never itself _ expresses 
 Number, but shows the absence of Number. 
 
132 FIRST LESSONS IN 
 
 LESSON XCIV. 
 
 SISTIJ^G OJ^ / WITH CI^H^^S A.J\r- 
 
 Example. Multiply 365 by 100. 
 
 ExpLAKATio:^-. 10 and 10 may be regarded as the 
 Component Factors of 100. Hence we can multiply by 
 100 by multiplying by 10 and 10. But we may multi- 
 ply by 10 by annexing one cipher at the right of the 
 Multiplicand. Hence, to multiply by 10 twice, that is, 
 by 100, we must annex tico ciphers. Therefore, 365 x 
 100 = 36,500. The ciphers at the right of 1, in 100, 
 or 1,000, or 10,000, or any other number, show how 
 many times 10 is a factor in the number. Hence, 
 
 To Multiply by any Number consisting of 1 
 WITH Ciphers annexed at the right: 
 
 EULE. 
 
 Annex at the right of the Multiplicand as many ci- 
 phers as there are in the Multiplier, TJie result will he 
 the true Product, 
 Multiply 785 by 1,000 ; by 10,000 ; by 100,000. 
 
 Multiplying by any Number consisting of one 
 Numeral Fig ure with Ciphers annexed : 
 
 The Component Factors of 500 may be taken as 5 and 
 100; since these numbers, multiplied together, make 
 500. Hence, we can multiply by 500 by multiplying by 
 its Factors, 5 and 100. It is also plain that we can mul- 
 tiply by 5,000 by multiplying by its Factors, 5 and 
 1,000. Therefore, to multiply by 5,000, we can first 
 multiply by 5, and then multiply the Product thus ob- 
 tained by 1,000, by annexing three ciphers. Hence, 
 
MENTAL AND WRITTEN ABITH:^ETIC, 133 
 
 To Multiply by a Number consisting of a sin- 
 gle Numeral Fig ure, with Ciphers annexed : 
 
 KULE. 
 
 I. MuUijjly the Multiplicand hy the left-hand figure 
 of the Multi^jlier. 
 
 II. Annex at the right of this Product as many 
 Ciphers as there are standing at the right in the Multi- 
 plier, Tlie result will be the true Product. 
 
 Exercises for the Slate ai^d Board. 
 7,854 X 70 3,586 x 700 734 x 5,000 
 
 9,873 X 90- 8,962 x 800 548 x 7,000 
 
 5,420 X 80 5,423 x 900 820 x 9,000 
 
 In the number 555, the figures do not each express 
 the same vahie. The 5 at the right stands for 5 Ones ; 
 the 5 at the left of this for 5 Tens, or 50 ; and the 5 at 
 the left for 5 Hundreds, or 500. Hence, the value of 
 each figure 5 is determined by its place or locality. 
 
 DEFINITIONS. 
 
 1. The Simple Value of any Numeral Figure is 
 the value which it expresses when standing in the 2^l(^oe 
 of Ones. 
 
 2. The Local Value of any Numeral Figure is the 
 value which it expresses when standing in any place. 
 
 Eemarks. 
 
 1. The Simple . Value of a Numeral Figure is always 
 the same. 
 
 2. The Local Value of a Numeral Figure changes as 
 often as the place of the figure is changed. 
 
 3. When a Numeral Figure stands in the place of 
 Ones, its Simple and Local Value are the same. 
 
 4. The figure has no value, either Simple or Local. 
 
134 
 
 MBST LESSONS IN 
 
 LESSON XCIV. 
 
 Example A. Multiply solution. 
 549 by 375. 
 
 Explanation. 1st. We 
 multiply 549 by 5, or take 2,745 
 it 5 times, and write the 38,430 
 Partial Product, 2,745. 2d. ^^^^^^^ 
 We multiply 549 by 70, or 205,875 
 take it 70 times, and write 
 
 the Partial Product, 38,430. 3d. We multiply 549 by 
 300, or take it 300 times, and write the Partial Product, 
 164,700. 4th. We add the 3 Partial Products, and take 
 their Sum as the final Product. 
 
 Thus we have taken 549, our Multiplicand, 300 times, 
 and 70 times, and 5 times, or 875 times. Hence, we 
 have the true Product. 
 
 549 Multiplicand. 
 375 Multiplier, 
 
 5 Times 5Jf9. 
 
 70 Times BJfi. 
 
 800 Times 5Jfi, 
 
 875 Times 5J^9. 
 
 249 
 _J05 
 
 1,245 
 
 _74^00 
 
 75,945 
 
 Example B. Multiply 249 by 305. 
 
 Explanation. We first take 249 5 times, 
 then 800 times, and, writing the Partial 
 Products, add them. 
 
 We do not have any Partial Product 
 arising from multiplying by in the Mul- 
 tiplier ; since taking 249 no (0) times is not taking it at 
 all. The other Partial Products are written as in the 
 preceding Solution. 
 
 For each Numeral Figure of the Multiplier there is a 
 corresponding Partial Product, obtained by multiplying 
 the whole Multiplicand by this Figure. Hence, 
 
 To Multiply one Number by another: 
 
 EULE. 
 
 I. Write the Multiplier under the Multiplicands 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 135 
 
 II. Commencing at the right, midtij^ly the whole MuU 
 tijMcand hy each Numeral Figure of the Multiplier, 
 regarding the Local Value of each Figure, and write the 
 Partial Products, 
 
 III. Add the Partial Products, and tahe their Sum as 
 the final Product. 
 
 EXEKCISES FOR THE SlATE AKD BoARD. 
 
 Multiply 3,582 by 125 ; by 342 ; by 976 ; by 748. 
 Multiply 7,643 by 502 ; by 430 ; by 900 ; by 708. 
 Multiply 9,536 by 739 ; by 608 ; by 968 ; by 374. 
 
 Multiply 5,894 by 5,423 ; by 4,672 ; by 6,702 ; by 3,064. 
 Multiply 3,698 by 5,008 ; by 7,041 ; by 8,302 ; by 7,006. 
 
 Multiply 7,874 by 25,376 ; by 30,708 ; by 50,023. 
 Multiply 657 by 46,002 ; by 50,007 ; by 10,101. 
 
 LESSON XCV. 
 
 Example. Multiply 347 by 
 235. 
 
 ExPLAKATiON". The first of 
 these two Solutions is in the 
 form heretofore used. 
 
 The second Solution differs 
 from the first only in having 
 ciphers omitted at the right of 
 the Partial Products. 
 
 We observe that the second Partial Product, 10,41, is 
 found by multiplying 347 by 3 ; and the third, 69,4, by 
 multiplying 347 by 2. But, since the figure 3 in the 
 Multiplier stood for 30, or 3 x 10, 10,41 must still be 
 
 1st solution. 
 
 347 
 _235 
 
 1,735 
 10,410 
 69,400 
 
 81,545 
 
 2d solution. 
 
 347 
 235 
 
 1,735 
 
 10,41 
 69,4 
 
 81,545 
 
136 FIRST LESSONS IN 
 
 multiplied by 10. It has been written one place to the 
 lefty so as to leave room for a cipher at the right , on mul- 
 tiplying it by 10. So, also, the Partial Product 69,4 has 
 been written two places to the left, so as to leave room 
 for two ciphers at the right, on multiplying by 100. 
 
 In the second Solution the right-hand figure of each 
 Partial Product is written directly helow the correspond- 
 ing figure of the Multiplier, Hence, when the ciphers 
 at the right of the Partial Products are omitted, 
 
 To Write the Partial Products: 
 
 EULE. 
 
 Multiply the Multiplicand hy each Numeral Figure of 
 the Multiplier, and write the Partial Products so that 
 the EIGHT-HAND FIGURE in each shall stand directly 
 
 BELOW THE CORRESPOKDIKG FIGURE OF THE MULTI- 
 PLIER. 
 
 If there are any ciphers at the right of the first Nu- 
 meral Figure of the Multiplier, write the same number 
 of ciphers at the right of the first Partial Product 
 
 The second Partial Product, 10,41, is read 10,410; 
 and the third, 69,4, is read 69,400. Hence, 
 
 To Bead the Partial Products : 
 
 EULE. 
 
 Read each Partial Product the same as if the omit- 
 ted CIPHERS WERE WRITTEN. 
 
 Exercises for the Slate and Board. 
 
 Multiply 6,852 by .237 ; by 543 ; by 678 ; by 906. 
 
 Multiply 3,826 by 460 ; by 307 ; by 500; by 1,060, 
 
 Multiply 6,978 by 1,502 ; by 2,007; by 2,030; by 9,706. 
 
 Multiply 8,059 by 6,025; by 3,006; by 9,408; by 3,500. 
 
 Multiply 7,684 by 2,643 ; by 5,489; by 3,762; by 7,563. 
 
IIENTAL AND WRITTEN ARITHMETIC, 137 
 
 LESSON XGVI, 
 
 Example. Find the Product arising from multiply- 
 ing together 468 and 12. 
 
 FIRST SOLUTION. 
 
 12 MiiUiplicand. 
 468 MuUiplier. 
 
 96 = 8 ti7nes 12. 
 
 72 = 60 times 12. 
 
 4,8 = iOO times 12. 
 
 5^ = ^68 times. 12. 
 
 SECOND SOLUTION. 
 
 4jS8 Multiplicand, 
 12 Multiplier. 
 
 96 = 12 times 8. 
 
 72 = 12 times 60. 
 
 4,8_ = 12 times JfiO. 
 
 5,616 =. 12 times 1^68. 
 
 ExPLAN'ATiOi^r. As the Product will be the same 
 whichever of the two numbers be taken as Multiplier, 
 in the first Solution we have made 468 Multiplier, and 
 in the second Solution 12. The first Solution is in the 
 usual form. 
 
 In the second Solution we first multiplied 8 by 12; 
 then 6, or 60, by 12; and finally 4, or 400, by 12. Thus 
 we have taken 400, and 60, and 8, 12 times ; or have 
 taken 468 12 times. The Partial Products stand in tlie 
 same order, and are the same, in hotli Solutions. 
 
 In the second Solution, we find it easier to obtain the 
 Product of 8 multiplied by 12 by multiplying 12 hy 8, 
 the Product being the same in both cases. So, also, to 
 find the Product of 60 multiplied by 12, we multiply 12 
 by 60. In the same manner, we find the Product of 400 
 X 12 by multiplying 12 by 400. 
 
 The peculiarity in the second Solution consists in 
 multiplying first the right-hand figure of the Multipli- 
 cand hy the entire Multiplier, then the next figure of the 
 Multiplica7id hy the entire Multiplier, and so on until 
 all the figures of the Multiplicand have been multiplied 
 by the Multiplier. But, though we consider X'^ the 
 
138 
 
 FIRST LESSONS IN 
 
 Multiplier, we obtain the Partial Products by using the 
 figures of the Multiplicand as Multipliers, and 12 as 
 Multiplicand. 
 
 By the method used in the second Solution, obtain 
 the Products in the following 
 
 EXEECISES FOR THE SlATE AND BOARD. 
 
 Multiply 473 by 13; by 25 ; by 36 ; by 47; by 58. 
 
 Multiply 568 by 45; by 73 ; by 84; by 39 ; by 96. 
 
 Multiply 2,437 by 123 ; by 234 ; by 543 ; by 736. 
 
 Multiply 5,283 by 542 ; by 745 ; by 827 ; by 976. 
 
 LESSON XCVIL 
 
 Example A. Multiply 236 by 12. 
 Example B. Divide 2,832 by 12. 
 
 Solution op Ex. A. 
 
 Multiplicand. 236 
 MultijMer. 12 
 
 12 times 6 = ^2 
 12 times SO = 36 
 12 times 200 =_2^_ 
 
 2^32 
 
 Solution op Ex. B. 
 
 236 Quotient, 
 12 Divisor. 
 
 ^,832 Dividend. 
 
 2,4 1= 12 times 200. 
 
 43 
 
 36_ = 12 times 
 
 72 
 
 72 = 12 times 
 
 6. 
 
 Expla:n'ATIOIT. In the Solution of Example A, the 
 Product is obtained by the method used in the last 
 Lesson. We find the Product to be 2,832 ; the same as 
 the Dividend in Example B. Our Multiplicand is 236, 
 our Multiplier 12, and our Product 2,832. 
 
 In Example B we are required to divide this Product, 
 2,832, by the Multiplier, 12. Hence, according to Prin- 
 
MENTAL AND WRITTEN ARITHMETIC. 139 
 
 ciple 5 in Diyision, on page 106, our Quotient must be 
 the same as our Multiplicand, 236. We will obtain it 
 by the Solution. 
 
 Writing the Divisor above the Dividend, with a line 
 between them, we see that we are to find a Multiplicand 
 which when multiplied by 12 will give 2,882 for a 
 Product. 
 
 1st. We first ask: What is the greatest number of 
 Hundreds which, when written in the Multiplicand (or 
 Quotient) and multiplied by 12, will give a Partial 
 Product not exceeding 28 Hundreds ? Finding this 
 number of Hundreds to be 2, we write 2 Hundreds in 
 the Multiplicand (or Quotient). Multiplying 2 Hundreds 
 by 12, or 12 by 200, and subtracting the Partial Product, 
 24 Hundreds, from 28 Hundreds, 4 Hundreds are left. 
 
 2d. Bringing down the 3 Tens from 2,832, and writ- 
 ing them at the right of 4 Hundreds, we have 4 Hun- 
 dreds and 3 Tens, or 43 Tens, for our next Partial Divi- 
 dend. We now ask : What is the greatest number of 
 Tens which, when written in the Multiplicand and 
 multiplied by 12, will give a Partial Product not exceed- 
 ing 43 Tens. Finding this number of Tens to be 3, we 
 write 3 Tens in the Multiplicand (or Quotient), and 
 multiplying the 3 Tens by 12 (or 12 by 30), and sub- 
 tracting the Partial Product, 36 Tens, from 43 Tens, we 
 write the Eemainder, 7 Tens. 
 
 3d. Finally, bringing down the 2 Ones from 2,832, 
 and writing them at the right of our 7 Tens, we have 7 
 Tens and 2 Ones, or 72 Ones, for a Partial Dividend. 
 Finding that 6 Ones, when written in the Multiplicand 
 (or Quotient) and multiplied by 12, will give 72 Ones, 
 we write 6 Ones as the last figure in the MultipHcand 
 (or Quotient). Multiplying and subtracting, as before, 
 10 
 
140 FIRST LESSONS IN 
 
 we have no Remainder. Therefore^ our Multiplicand, or 
 Quotient, is 236. Hence, 
 
 To Divide one Number by another : 
 
 EULE. 
 
 I. Write the Divisor above the Dividend, separating 
 them ly a horizontal line, 
 
 II. Commencing at the left, take as a Partial Dividend 
 such a part of the entire Dividend as, without regarding 
 its Local Value, ivill contain the Divisor at least okce^ 
 and NOT MOKE than nine times ; and, determining the 
 first figure of the Quotient ^ write it in its place, giving it 
 its proper Local Value, 
 
 III. Multip)ly this Divisor ly the Quotient figure, and 
 subtract the result from the Partial Dividend, 
 
 IV. Write the next figure of the Dividend at the right 
 of the Remainder ; and, using the number thus formed 
 as a new Partial Dividend, determine the second figure 
 of the Quotient in the same manner as the first was ob- 
 tained ; and multiply and subtract as before. 
 
 Proceed in this manner till all the figures of the Quo- 
 tient are obtained, and the worh is completed. 
 
 Kemarks. 
 
 1. If any Partial Product is greater than the Partial 
 Dividend under which it is written, the corresponding 
 Quotient figure is too large, and must be diminished. 
 
 2. If any Remainder is greater than the Divisor, the 
 corresponding Quotient figure is too small, and must be 
 increased. 
 
 3. Wheneyer any Partial Dividend is less than the 
 Divisor, the corresponding Quotient figure is ; and 
 the next Partial Dividend is formed directly from this, 
 by writing the next figure of the Dividend at the right 
 of it. 
 
3£ENTAL AND WRITTEN ARITHMETIC. 141 
 
 EXEECISES FOR THE SlATE AND BOARD. 
 
 Divide 
 
 33,760 
 
 by 12; 
 
 by 13; 
 
 by 14; 
 
 by 15. 
 
 Divide 1,970,640 
 
 by 23; 
 
 by 34; 
 
 by 45; 
 
 by 56. 
 
 Divide 
 
 375,480 
 
 by 149 ; 
 
 by 398 ; 
 
 by 447 ; 
 
 by 745. 
 
 Divide 
 
 134,488 
 
 by 347; 
 
 by 494 ; 
 
 by 741 ; 
 
 by 1,739. 
 
 LESSOlV XCVIII. 
 
 Example. Divide 1,589 by 58. 
 
 ExPLA:NrATiO]sr. We divide as here- solution. 
 
 tofore, and obtain 27 for a Quotient; ^^ Quotient. 
 
 but, on subtracting the last Partial ^ Divisor. 
 
 Product, we have 23 for a Re- 1,^^^ Dividend. 
 
 mainder. This last Remainder is ^^^^ 
 
 named the Final Remainder. 429 
 
 It is never called the Difference, as _^^ 
 
 in Subtraction. 23 Remainder. 
 
 DEFINITION. 
 
 The Final Hemainder in Division is that part 
 of the Dividend left, still undivided, after obtaining all 
 the figures of the Quotient. 
 
 Exercises for the Slate akd Board. 
 
 Divide 23,578 by 35 ; by 59 ; by 68 ; by 79. 
 Divide 376,982 by 143 ; by 256 ; by 374 ; by 578. 
 Divide 438,796 by 527 ; by 743 ; by 963 ; by 829. 
 Divide 25,762,159 by 159 ; by 1,524 ; by 3,284. 
 
 Divide 57,349,284 by 6,023 ; by 7,009 ; by 8,219. 
 
 Divide 29,513,784 by 374 ; by 183 ; by 921 ; by 548. 
 Divide 73,182,546 by 1,316 ; by 589 ; by 918 ; by 3,600. 
 Divide 10,020,010 by 101 ; by 1,010 ; by 10 ; by 100. 
 Divide 10,000,000 by 10; by 100; by 1,000; by 10,000. 
 
14a FIRST LESSONS IN 
 
 LESSON XO/X. 
 
 We multiply by 10, 100, 1,000, &c., by annexiDg at 
 the right of the Multiplicand as many ciphers as stand 
 at the right in the Multiplier. 
 
 According to Principle 5, on page 106, if the Product 
 be divided by the Multiplier the Quotient will be the 
 Multiplicand. But it is evident that we can obtain the 
 Multiplicand, or Quotient, in such cases, by dropping at 
 the right of the Product, or Dividend, the same ciphers 
 which we have just annexed. Hence, 
 
 To DIVIDE BY 10, 100, 1,000, &C., WHEN THE DIVI- 
 DEND HAS AS MANY CIPHERS AT THE RIGHT AS 
 ARE FOUND IN THE DiVISOR : 
 
 KULE. 
 
 Drop at the right of the Dividend as many ciphers as 
 stand at the right in the Divisor, The figures remaining 
 will he the Quotient. 
 
 Exercises eor the Slate and Board. 
 Divide 57,830,000 by 100 ; by 1,000 ; by 10,000. 
 Divide 21,700,000 by 10 ; by 10,000 ; by 100,000. 
 
 Divide 65,430,000 by 10 ; by 1,000 ; by 10,000. 
 
 Example 1. Divide 54,768 by 100. 
 
 EXPLAKATION^ 1. Before solution. 
 
 dividing, we separate our 54,700 4-68. 
 
 Dividend into tico parts, 54^^100=547 Quotient. 
 
 ,54,700 and 68. l^ext we Fi7ial Rem., 68. 
 
 divide the first part, 54,700 
 
 by 100, by dropping two ciphers, and have 547 for a 
 
 Quotient. It is plain that 68 will not contain 100 even 
 
 once. Hence it will be our Final Eemainder. Our 
 
 Quotient is 547, and our Final Eemainder 68. 
 
MENTAL AND WRITTEN ARITHMETIC, 
 
 143 
 
 54,768 = 54,000 + 768 
 54,000 +-1,000=54 QuoH. 
 Final Remainder, 768. 
 
 Example 2. Divide 54,768 
 by 1,000. 
 
 ExPLAis^ATiOK. Eemoving 
 the three right-hand figures, 
 768, and supplying their 
 
 places with ciphers, we separate the Dividend into two 
 parts, 54,000 and 758. Dividing 54,000 by 1,000, by 
 rejecting the three ciphers at the right, we obtain 54 for 
 our Quotient. The other part of the Dividend, 768, 
 being less than the Divisor, will not contain it even 
 once, and hence will be our Final Kemainder. • 
 
 On examination, we find that when we divide by 100, 
 1,000, &c., the figures removed at the right of the Divi- 
 dend form our Final Kemainder, and the figures at the 
 left of these, in the Dividend, form our Quotient. Hence, 
 
 To Divide by 10, 100, 1,000, &c.: 
 
 EULE. 
 
 Remove at the right of the Dividend as many figures 
 as there are ciphers in the Divisor, and take the number 
 composed of the figures so removed for the Final Re- 
 mainder, and the number composed of the figures at the 
 left of these for the Quotient. 
 
 Example 3. Di- 
 vide 7,958 by 400. 
 
 Explanation". 
 The first Solution is 
 in the form hereto- 
 fore given. We see 
 that the ciphers in 
 the Divisor appear 
 in the Partial Prod- 
 ucts, but do not affect the work so as to change either 
 the Quotient or the Final Remainder. 
 
 1st solution. 
 
 19 QuoH. 
 _400 DivW, 
 
 7958 Divided, 
 400_ 
 
 3958 
 3600 
 
 358 Fin. Rem, 
 
 2d solutiok 
 
 19 QuoH. 
 
 4 1 GO Div'r. 
 
 79|58 Divided. 
 4_ 
 
 39 
 36_ 
 
 358 Fin. Rem. 
 
144 
 
 FIRST LESSONS IN 
 
 The figures 58, at the right in the Dividend, appear 
 in the Final Eemainder unchanged. Hence, in the sec- 
 ond Solution the work is shortened, by cutting off the 
 ciphers at the right in the Divisor, and the figures 58 at 
 the right in the Dividend, and then annexing the 58 to 
 the second Eemainder. Therefore, 
 
 To Divide by a Number with Ciphers at the 
 Right: 
 
 KULE. 
 
 I. Cut off the ciphers at the right of the Divisor, and an 
 equal rjLumber of figures at the right of the Dividend, 
 
 II. Divide the Dividend, thus changed, by the changed 
 Divisor, and use the Quotient thus obtained as the true 
 Quotient. 
 
 III. Annex to the last Eemainder the figures cut off from 
 the Dividend, and use the number thus formed as the Final 
 Remainder, 
 
 EXEKCISES FOR THE SlATE Al^D BOAKD. 
 
 Divide 78,546 by 10 
 
 ; by 100 
 
 ; by 1,000. 
 
 Divide 57,968 by 100 
 
 ; by 1,000 
 
 ; by 10,000. 
 
 Divide 30,102 by 10^ 
 
 by 100 , 
 
 by 1,000. 
 
 Divide 73,001 by 100 
 
 ; by 1,000 
 
 ; by 10,000. 
 
 Divide 237,849 by 60^ 
 
 by 700 
 
 by 8,000. 
 
 Divide 546,789 by 80 
 
 by 900 , 
 
 by 70,000. 
 
 Divide 107,050 by 90; 
 
 by 500; 
 
 by 8,000. 
 
 Divide 700,520 by 30^ 
 
 by 650 
 
 by 3,500. 
 
 Divide 672,518 by 230 ^ 
 
 by 3,100; 
 
 by 30,100. 
 
 Divide 127,950 by 100 
 
 by 7,200 ; 
 
 by 50,001. 
 
 Divide 718,312 by 101 
 
 by 2,001 ; 
 
 by 10,001. 
 
MENTAL AND WRITTEN ARITHMETIC. 145 
 
 LESSON C. 
 
 Example. ist solution. 2d solution. 
 
 Multiply 15 by Multiplicand, 15 = 3x5 
 
 14 Multiplier. 14= 2x7 
 
 ExPLAiq-A- 60 3x5x2x7 
 
 Tioisr. 1st. The 1^ _^ 
 
 first Solution Product. 210 14 
 
 is in the usual __^ 
 
 form, and the Product is 210. 2d. In the 70 
 
 second Solution, we factor 15 into 3 and 3 
 
 5, and 14 into 2 and 7. Writing the 210 FrodH, 
 Factors of both Multiplicand and Mul- 
 tiplier in a line, with the Sign x between, for the Fac- 
 tors of the Product, we multiply them together, and 
 obtain 210 for a Product. Hence, 
 
 General Principle in Multiplication. 
 
 The Product is composed of the Factors of the Multipli- 
 cand and Multiplier, and IS'O othehs. 
 
 Since the Product in Multiplication becomes the Divi- 
 dend in Division, the Multiplier the Divisor^ and the 
 Multiplicand the Quotient, therefore, in Division, when 
 there is no Remainder, we have 
 
 General Principles in Division. 
 
 1. The Dividend is composed of the Fdctors of the Divi- 
 sor and Quotient, and KG others. 
 
 2. If the Factors of the Divisor he rejected from the 
 Dividend, the remaining Factors will be those of the Quo- 
 tient. Hence, 
 
 To Multiply by any Number: 
 
 EULE. 
 
 Connect the Factors of the Multiplier with those of the 
 Multiplicand by the Sign x . The Product of these Fac- 
 tors will be the true Product 
 
 7 
 
146 FIRST LESSONS IN 
 
 Hence, also, when there is no Remainder^ 
 To Divide by any Number: 
 
 KULE. 
 
 From the Dividend remove Factors equal to those of the 
 Divisor, The Product of tJie remaining Factors will he 
 the true Quotient, 
 
 We saw, on page 98, that the Sign of Division, -^, is 
 made from the Sign Minus, — . 
 
 We may use the Sign Minus in still another manner 
 to show that one number is to be divided by another. 
 We may show that 12 are to be divided hy 3, in the 
 manner seen at the right. 
 
 1st : We write the Divi- Dividend, 12 _ 
 dend, 12. 2d: We write Divisor, 3 " ^ V^^^^^^^- 
 the Sign Minus below the 
 
 Dividend. 3d: We write the Divisor below the Sign 
 Minus. 
 
 If we make this Expression the First Member of an 
 
 Equation, the Second Member will be the Quotient. 
 
 12 
 The Expression — = 4 is read : " 12 divided ly 3 equal 
 
 ^" The Quotient, 4, is called the Value of the Ex- 
 
 . 12 
 pression — 
 
 Find the Value of each Expression in the following 
 
 Exercises for the Slate ai^d Board. 
 I. 
 
 15 = ? 
 
 5 
 
 27 _ 
 3 
 
 ? 
 
 43 _ 
 
 7 ~ 
 
 11. 
 
 ? 
 
 72 _p 
 8 
 
 63 _p 
 
 7 
 
 936 
 
 675 
 
 
 959 
 
 
 1792 
 
 3753 
 
 3 
 
 5 
 
 
 7 
 
 
 8 
 
 9 
 
MENTAL AND WRITTEN ARITHMETIC. 147 
 
 Example. Divide 378 by 42. 
 
 ExPLAi^ATiON. From the second General Principle in 
 Division we see that the Quotient consists of those Fac- 
 tors of the Dividend which are not in the Divisor. 
 Hence, by the method 
 
 shown on page 129 we solution. 
 
 factor 378 into 2x3 378=2 x 3 x 7 x 9. 
 X 7 X 9, and 42 into 42=2 x 3 x 7. Hence, 
 2x3x7. Writing 378^ 2 x3 x 7 x _9^^ Quotient. 
 the Factors of the Di- 42 2x3x7 
 vidend above those of 
 
 the Divisor, for Division, we reject from the Dividend 
 the Factors 2 x 3 x 7, of the Divisor, as directed by 
 the preceding Rule, and have the remaining Factor, 9, 
 for our Quotient. 
 
 Obtain the Quotients by factoring in the following 
 
 EXEECISES FOR THE SlATE Al^D BOARD. 
 
 132 195 2205 2940 3888 864j0 
 66 39 441 420 432 1728 
 
 LESSON 01. 
 
 Since our Quotient always consists of those Factors of 
 the Dividend remaining after taking away the Factors 
 of the Divisor, it is plain that (Principle 3) putting any 
 neiu Factor into the Dividend does in effect put that 
 Factor into the Quotient ; and that (Principle 4) remov- 
 ing from the Dividend any Factor already there in effect 
 removes it from the Quotient, 
 
 It is also evident that (Principle 5) if any new Factor 
 be put into the Divisor, when we take the Factors of the 
 
148 FIRST LESSONS IN 
 
 Divisor from the Dividend we must also take this new 
 Factor from the Dividend^ and thus in effect take it from 
 the Quotient, or divide the Quotient by it; and that 
 (Principle 6) if any one of the Factors of the Divisor 
 be removed from the Divisor before taking the Factors 
 of the Divisor from the Dividend, the Factor so removed 
 will not be taken from the Dividend, as it should be, 
 but will leave the corresponding Factor of the Dividend 
 in the Quotient, and will thus, in effect, multiply the 
 Quotient by this Factor. 
 
 It is also clear that (Principle 7) if any new Factor 
 be put into both Dividend and Divisor, the one in the 
 Divisor will cause the removal of the one in the Divi- 
 dend, and hence the Quotient will not be affected. And 
 (Principle 8) if the same Factor be removed from both 
 the Dividend and Divisor, the final effect will be the 
 same as if it had remained in both till all the Factors 
 of the Divisor were taken from the Dividend. Hence 
 the Quotient will not be affected. 
 
 Therefore, we shall have as 
 
 General Frinciples of Division : 
 
 3. Multiplying tJie Dividend hy any Factor in effect 
 multiplies the. Quotient hy that Factor, 
 
 4. Dividing the Dividend hy any Factor in effect di- 
 vides the Quotient hy that Factor, 
 
 5. Multiplying the Divisor hy any Factor in effect di- 
 vides the Quotient hy that Factor, 
 
 6. Dividing the Divisor hy any Factor in effect multi- 
 plies the Quotient hy that Factor. 
 
 7. Multiplying hoth Dividend and Divisor hy the same 
 Factor does not affect the Quotient, 
 
 8. Dividing hoth Dividend and Divisor hy the same 
 Factor does not affect the Quotient, 
 
MENTAL AND WRITTEN ARITH3IETIC. 149 
 
 Dividing both Dividend and Divisor by the same 
 Factor is the same as rejecting that Factor from both. 
 
 DEFINITION, 
 
 Cancellation is rejecting equal Factors from both 
 Dividend and Divisor. 
 
 Example. Divide 60 by 15. 
 
 SOLUTION. 
 
 Dividend. 60 $x0x4 .^^.^ 
 
 — = — — — 4. Quotient. 
 
 Divisor. 15 ? x ^ 
 
 ExPLAiq-ATiON". Factoring the Divisor into 3 and 5, 
 and the Dividend into 3, 5 and 4, and rejecting the 
 Factors 3 and 5 from both, by Principle 8, we have 4 
 for the Quotient. 
 
 By Cancellation find the Quotients in the following 
 
 Exercises for the Slate akd Board. 
 
 1728 
 144* 
 
 When the Divisor is contained in the Dividend with- 
 out a Eemainder, the Divisor is named an Exact 
 Divisor. 
 
 Example. Divide 30 by 42. 
 
 SOLUTION. 
 
 Dividend. 30 ^ x $ x 5 5 . 
 
 zzz — r^: Jjin^ 
 
 Divisor. 42 ^ x $ x 7 7 
 
 ExPLANATioiq-. Factoring and canceling, we find no 
 Factor 7 in the Dividend. Hence, 
 
 General Principle in Division, 
 
 9. The Dividend cannot he exactly divided hy the 
 Divisor wlien any Factor of the Divisor is not found in 
 it. 
 
 48 
 
 70 
 
 168 
 
 210 
 
 385 
 
 735 
 
 12' 
 
 14' 
 
 U' 
 
 30' 
 
 35' 
 
 105' 
 
150 
 
 FIRST LESSONS IN 
 
 1 
 
 LESSON GIL 
 
 Example. Clifford's mother divided a watermelon 
 between him and his sister. What did each receive ? 
 
 ExPLANATiOiq-. Writing the Dividend 
 and Di^dsor as in former cases, we find solution. 
 
 that we can not so factor the Dividend Dividend, 1 
 as to obtain a Factor 2. Hence, accord- Divisor, 2 
 ing to Principle 9, 1 can not be exactly 
 divided by 2. 
 
 But since there are 2 children, and there is only 1 
 melon, it is evident that the melon must be divided into 
 2 equal 'parts, and 1 'part given to each child. 
 
 When any single thing is divided into 2 equal parts, 
 these parts are named halves. One of these is called 
 one Half, and is written |. 
 
 If an apple be cut into S equal parts, these parts are 
 named Thirds. One part is named one Third, and 
 written ^, Two Thirds are written f . 
 
MENTAL AND WRITTEN ARITHMETIC. 151 
 
 If a pear be divided into ^ equal parts, these parts are ' 
 named Fou7^ths. One Fourth is written |; two 
 Fourths are written | ; and three Fourths J. 
 
 If 5 oranges are to be divided between 2 children, we 
 can give each child 2 oranges, that is 4 oranges to the 
 2 children, and then divide the fifth orange into 2 
 Halves, and give 1 Half-orange to each child. Each 
 child would then have 2 oranges and 1 Half-orange ; 
 which are written ^| oranges. 
 
 When a watermelon is divided into, 2 equal parts, or 
 an apple into 3 equal parts, the melon or apple is cut or 
 fractured, and one of the parts is 2i fragment, or Frac- 
 tion of the entire thing. Hence, one Half, one Third, 
 one Fourth, two Thirds, three Fourths, or ^, I, \, |, |, 
 are named Fractions. 
 
 DEFINITIONS, 
 
 1. An Integral Unit is a single entire thing. 
 
 2. A Fractional Unit is one of the equal parts 
 into which an Integral Unit is divided. 
 
 S. An Integral Number ^ or Integer^ is an 
 Integral Unit or collection of Integral Units. 
 
 4. A Fractional l^umber^ or Fraction^ is a 
 Fractional Unit, or a collection of Fractional Units. 
 
 5. A Mixed Number is a number consisting of 
 loth an Integer and a Fraction. 
 
 Eemakks. 
 
 1. Integral means ivhole, or entire. Hence, an Integer 
 is frequently called a Whole Number. 
 
 2. An Integral Unit is commonly called simply a 
 Unit ; or, sometimes, a If nit One. 
 
152 
 
 FIRST LESSONS IN 
 
 LESSON CIIL 
 
 Suppose we wish to divide 3 apples equally between 
 2 persons. 1st : It is evident that we can divide each 
 apple into 2 equal parts, and give each person 1 part 
 from each apple. He would have as many parts as there 
 were apples ; or 3 parts. He would receive 3 Halves ; 
 or |. 2d : If we chose, however, we might divide the 3 
 apples between the 2 persons by at first giving each 1 
 apple, or 2 apples to both, and then cutting the third 
 apple into 2 equal parts and giving 1 part to eaeh. 
 Each person would thus have 1 entire apjole, and 1 Half- 
 apple ; or 1^ apples. It is plain that each person would 
 receive the same in both cases ; hence, | are the same as 
 1^. This must be evident. For, since 2 of the 3 Halves 
 will make one apple, 3 Halves are the same as 1 \, 
 
 If 4 apples were to be divided among 3 persons, we 
 might cut each apple into 3 pieces, and give each person 
 a piece from each apple. Each person would then have 
 4 pieces, or |. And, since 3 Thirds make 1 apple, he 
 would have the same as 1 apple and one Third of an 
 apple; or>l| apples. 
 
MENTAL AND WRITTEN ARITHMETIC. 153 
 
 In the expressions, | and |, the Dividends, 3 apples 
 and 4 apples, show the number of things divided ; and, 
 since each person has 1 piece from each thing divided, 
 ^3 and 4 also show the number of parts each person re- 
 ceives. The Divisors, 2 and 3, show the name or kind 
 of the parts received by each person. Name means the 
 same as Denomination. 
 
 The Expressions | and | are Fractions. Hence, 
 
 DEFINITIONS. 
 
 6. The DiviDEKD in a Fraction is named the Nu" 
 meratoVf because it tells the Number of parts in the 
 Fraction, 
 
 7. The Divisor in a Fraction is named the Denom- 
 inatOTf because it tells the ^N^ame, or Dekomikatiok, 
 of the parts in the Fraction, 
 
 8. The Numerator and Dekomikator, taJcen to- 
 gether, are named the Terms of the Fraction, 
 
 9. The LINE {Sign Minus) loritten between the Terms 
 of a Fraction is named the Dividing -line ^ because 
 it shows thdt the Numerator is to be divided by the 
 Denominator, 
 
 10. The Value of a Fraction is the Quotient aris- 
 ing from dividing the Numerator by the Denominator, 
 
 11. A Minor Fraction is a Fraction whose value 
 is LESS THAN the Unit One. 
 
 12. A Major Fraction is a Fraction whose value 
 EQUALS OR EXCEEDS the Unit One ; that is, whose value 
 is greater than that of any Minor Fraction. 
 
 Eemark. 
 Minor means less, or smaller; and Major means 
 greater, 
 
 13. Like Fractions are Fractions having like or 
 equal Denominators. 
 
154 FIRST LESSONS IN 
 
 14 Tinlike Fractions are Fractions having uk« 
 
 liIKE OR UifEQUAL DeKOMIJS^ATORS. 
 
 Kemark. 
 f and 4 are Like Fractions, | and /y Unlike. 
 15. To Reduce a Fraction is to change its 
 
 FORM without CHAKGIi^G ITS VALUE. 
 
 Example 1. How many apples in ^^- apples? 
 
 ExPLAKATioiT. Since 2 Halves make one apple, we 
 shall have as many apples as 2 Halves are contained 
 times in 12 Halves ; which are 6 times. Hence ^^ = 6. 
 
 Example 2. How many apples in -Y- apples ? 
 
 ExPLAN^ATiO]sr. In 16 Thirds there are as many Ones 
 as 3 Thirds are contained times in 16 Thirds. 3 are in 
 16 5 times, with 1 for a Remainder. Hence, 16 Thirds^ 
 or J36, are equal to 5 Units and 1 Third; or 5|. Hence, 
 
 To Reduce a Major Fraction to an Integer, 
 OR Mixed Number: 
 
 Eule. 
 
 Divide the Numerator of the Fraction ly the Denomi- 
 nator; and if there is a Remainder use it for the Numer- 
 ator of a Fraction, with the Divisor for Denominator, 
 and annex this Fraction to the Quotient. 
 
 Eeduce the Fractions to Integers or Mixed Numbers 
 in these 
 
 Exercises for the Slate akd Board. 
 I. 
 
 11 
 
 17 39 148 
 
 739 
 
 90 360 
 
 540 
 
 2' 
 
 5' 8' 6 ' 
 
 7' 
 II. 
 
 9' 6' 
 
 90 ■ 
 
 84 
 
 795 1189 
 
 1973 
 
 19000 
 
 7354 
 
 21' 
 
 25 ' 32 ' 
 
 176 ' 
 
 1700 ' 
 
 679 
 
MENTAL AND WRITTEN ARITHMETIC, 155 
 
 LISSON CIV. 
 
 Example 1. In 5 apples how many Thirds ? 
 
 ExPLAi^ATio^sr. Since there are 3 Thirds in 1 apple, 
 in 5 apples there are 5 times 3 Thirds ; which are 15 
 Thirds; or J/* C)r, since there are 3 times as many 
 Thirds as there are apples, we may find the number of 
 Thirds by multiplying the number of apples by 3. 3 
 times 5 are 15. Hence, there are 15 Thirds; or -*/-. 
 Therefore, 
 
 To Reduce an Integer to the form of a 
 Fraction: 
 
 EULE. 
 
 Multiply the Integer dy the Denominator of the re- 
 quired Fraction, and under this Product, used as the 
 Numerator of the result, write the required Denominator, 
 
 Exercises for the Slate and Board. 
 Eeduce 7 to Thirds ; 13 to Fifths ; 37 to Ninths. 
 Keduce 123 to 25ths ; 527 to 75fchs ; 317 to llths. 
 
 Example 2. In 8| apples how many Thirds ? 
 
 Explanatio:n". Keducing 8, or 8 apples, to Thirds, by 
 the preceding Eule, we have (3 times 8 are) 24 Thirds. 
 We have also 2 other Thirds (|). Adding 24 Thirds 
 and 2 Thirds, we have for a result 26 Thirds ; or ^f. 
 Hence, 
 
 To Reduce a Mixed Number to the form of a 
 Fraction: 
 
 Eule. 
 Multiply the Integer ly the Denominator of the FraC' 
 tion, and to this Product add the Numerator, Under 
 this Sum, tised as a Numerator, write the Denominator 
 of the give7i Fraction for a Denominator. 
 11 
 
156 first lessons in 
 
 Exercises for the Slate akd Board. 
 
 I. 
 
 5|; 111; 19«3; 23i ; 265,\; IS?/^; 378^. 
 
 II. 
 18311; 72311; 618i||; 372|if; 958?i|. 
 
 If James has 5 oranges and John 3 oranges, we find 
 how many they both have by adding together 5 and 3 
 and obtaining their Sum, 8. So if they have things of 
 any other kind, and the things which they both have 
 are of the same hindy we find how many they both have 
 by adding the numbers showing how many each has. 
 
 Example 3. Frank has | of a watermelon, and Harry 
 has f of it. What have both ? 
 
 ExPLAKATioiT. Since Frank had 3 pieces and Harry 
 2 pieces, and both had pieces of the same hind, or size, 
 we add 3 pieces and 2 pieces, and have 5 pieces, or |. 
 Hence, 
 
 To Add Like Fractions: 
 
 Eule. 
 Find tJie Sum of the Numerators of the Fractions, 
 and, using this for the Numerator of the result, write the 
 common Denominator for a Denominator, 
 
 Exercises for the Slate akd Board. 
 
 5.3^, A . 1^. 'l5 + l?^. 11 + ^ = ? 
 9 9 * 17 17 * 25 25 * 32 32 
 
 II. 
 
 35 35 '56 56 * 85 85 * 125 125 
 
 Since Subtraction is the reverse of Addition, from our 
 Rule for the Addition of Fractions we must have 
 
MENTAL AND WRITTEN ARITHMETIC. 
 
 157 
 
 To Subtract a Fraction from a Like Fraction: 
 
 KULE. 
 
 Subtract the Numerator of the SuUrahend from that 
 of the Minuend, and, using the Difference as the Numer- 
 ator of the result, write the common Denominator for a 
 Denominator. 
 
 Exercises for the Slate and Board. 
 
 7 4 
 --- = ? 
 
 8 8 
 
 13 13 
 
 39 
 
 27 
 
 18 
 
 27 
 
 ==::? 
 
 49 
 53 
 
 53 
 
 LESSON GV. 
 
 Example. Walter's mother gave him ^ of a cake, and 
 f of another cake of the same size. What had he in 
 all? 
 
 rmST CAKE. 
 
 SECOND CAKE. 
 
 Explanation. In the 
 cut, at the right, we see 
 the first cake cut into 
 Halves, and the Half 
 given to Walter placed 
 below it. 
 
 We see also the second 
 cake cut into Thirds, 
 and the ^ Thirds given 
 to Walter placed below. 
 
 We observe that the 1 
 Half and the 2 Thirds 
 are not parts of the same 
 
 hind, or size, and hence cannot he counted together, or 
 added. We must cut the 1 Half into smaller parts, and 
 
158 
 
 FIRST LESSONS IN 
 
 also the 2 Thirds into smaller parts^ in such manner 
 that all the parts shall be of the same size. 
 
 Cutting the 1 Half 
 into 3 equal parts, as 
 shown at the right, we 
 see that there would be 
 6 such parts in the 
 whole cake. Hence i 
 of the cake is the same 
 as § of the cake. 
 
 Cutting each of the 2 Thirds into 2 equal parts, they 
 make 4 parts. There would be 6 such parts in the 
 second cake. Hence the 4 parts are Sixths ; and it fol- 
 lows that f of this cake are the same as % of it. 
 
 Therefore Walter had | of the first cake, and % of the 
 second cake. Since these parts are all of the same size, 
 they can be added by the Eule in 
 the last Lesson. Adding them 
 according to the Eule, | and | 
 are |. This is a Major Fraction. 
 Keducing it to a Mixed ^NTumber 
 according to the Eule on page 
 154, we have 1^. This is shown 
 in the cut at the right, by placing 
 the 3 Sixths and 4 Sixths together. 
 6 of the 7 Sixths make a whole cake, and the remaining 
 
 1 Sixth is placed on the top of this cake. Hence Walter 
 received 1 cake and 1 Sixth of a cake. 
 
 Thus we have first changed h into |, and | into |, 
 and then added them and obtained |, or IJ. 
 
 The Numerator of { shows that there is 1 part in the 
 Fraction ; and the Denominator, 2, shows that there are 
 
 2 such parts in one calce. So, also, in any Fraction, the 
 Numerator shows the number of parts in the Fraction, 
 
 ^^Whs.o^^ 
 
MENTAL AND WRITTEN ARITHMETIC, 159 
 
 and the Denominator shows the number of such parts in 
 a Unit, Hence, when we cut the 1 part in the Numer- 
 ator of ^ into 3 parts, there will be S times as many 
 such parts as there are Halves in the cake ; or 3 
 times 2 parts, which are 6 parts. Hence, cutting our ^ 
 cake into |, the Numerator and Denominator of | are 
 leach S times as large as the corresponding terms of i. 
 ^hat is, we change | to f hy multijjlying loth its terms 
 ly S, This does not change the value of the Fraction. 
 This agrees with the 7th Principle of Division. 
 
 When we cut each of the 2 parts in | into 2 parts, 
 changing the 2 Thirds to 4 Sixths, or | to |, we make 
 both terms of | tivice as large, or multiply both terms by 
 2, This has not changed the value of the Fraction. 
 
 By multiplying both terms of each Fraction by the 
 Denominator of the other, we have reduced the Unlike 
 Fractions i and | to the Like Fractions § and |. 
 
 In the same manner we may reduce ^, | and |, to 
 Like Fractions by multiplying both terms of i by the 
 Denominators 3 and 4 ; both terms of | by the Denom- 
 inators 2 and 4 ; and both terms of | by 2 and 3. 
 
 If we have any number of Unlike Fractions, we may 
 proceed in the same manner. Hence, 
 
 To Reduce Unlike to Like Fractions: 
 
 EULE. 
 
 Multiply both terms of each Fraction by each of the 
 other Denominators successively. 
 
 ' Eemark. — It is necessary to obtain the Denominator 
 of only the first reduced Fraction, since all the other 
 Denominators are like it. 
 
160 
 
 FIRST LESSONS IJST 
 
 Eeduce Unlike to Like Fractions in the following 
 
 EXEKCISES FOR THE SlATE AKD BoARD. 
 
 I. 
 
 I and I ; 
 
 and 4; 
 
 3? 5 and 7y y 
 
 II. 
 
 h % I and y\ ; i f and ^3 ; 
 
 3 4 pnrl 8 
 3> lT> TT '^'^^ T7« 
 
 LESSON GVI. 
 
 Example. Add 5f and ^. 
 
 ExPLAi^ATiOK. Eeducing | 
 and f to Like Fracticns, we 
 have ^f and ^ ?. Adding these, 
 we have |f, or l^J. Having 
 
 2 
 
 3 
 
 1 4 
 
 2T 
 
 + ^f 
 
 and f 
 
 32 
 2T? 
 
 or m. 
 
 5 + 4 + 1=10. Hence 
 
 of and 4f 
 
 the Sum of the Fractions, we 
 add to this the Integers 5 and 4. The Sum of 5, 4 and 
 1 is 10. Writing the Fractional part of the Sum after 
 this, we have 10^ {. Hence, 
 
 To Add Mixed Numbers: 
 
 EULE. 
 
 Add the Fractions, and if their Sum is a Major Frac- 
 tion reduce it to an Integer or Mixed Number, Add the 
 Integral part of this result ivith the given Integers, and 
 to this Sum annex the Fractional part. 
 
 Exercises for the Slate akd Board. 
 I. 
 
 3I + 3I-? 7| + lli=? 12H + 9J-|=? Uf + 2J^T=? 
 
 II- 
 4f + 5i=? 6f + 8|=? 10J+9|=? 16f + 13/yrr:? 
 
MENTAL AND WRITTEN ARITHMETIC. 161 
 
 Example. From 8| sub- 
 tract 4|. SOLUTION. 
 
 ExPLANATioif. Reducing \=~hy ^^^ I=t% 
 \ and I to the Like Frac- 8/5 = 7f§. Hence 
 tions j% and f^, we find that 8|-4|==7f §-4yV 
 y^^ cannot be subtracted ff--y%= j|, and 7— 4=3. 
 from j\, since it exceeds it. Hence 8| — 4|=3}|. 
 Therefore we take one of 
 
 the 8 Units, and, calling it ||, add it to the ^5, and have 
 f f. Our Minuend is then 7f f, and ojir Subtrahend 4y%. 
 Subtracting y^^ from ff we have j| for the Fractional 
 part of our Eemainder. Subtracting 4 Ones from 7 
 Ones, we have 3 Ones left. Uniting both parts of our 
 Remainder, we have 3} | for the true Remainder. Hence, 
 
 To Subtract a Mixed Number, or a Fraction^ 
 FROM A Mixed Number or an Integer : 
 
 Rule. 
 
 I. Reduce to Like Fractions the Fraction in the Sub- 
 trahend, and also such part of the Minuend, {including 
 the Fraction, if any,) as shall equal or exceed this. 
 
 II. Subtract the Fractional part of the Subtrahend 
 from that in the Minuend, and the Integer in the Sub- 
 trahend from that in the Minuend, and unite the two 
 partial Remainders into one Final Remainder. 
 
 Exercises for the Slate and Board. 
 
 I. 
 7|-5|=? 9|-5i=? ll|-5/y=? 10|-3f=? 
 
 II. 
 12J-5|=? 15~64=? 21/7-31=? 35-21/3=? 
 
162 FIRST LESSONS IN 
 
 LESSON evil. 
 
 MZrZTITZIC^TIOJV* oiJV^ ^lYISIOJST 01^ 
 I^^;ACTIOJVS. 
 
 In a Fraction the Numerator is a Dividend, and the 
 Denominator a Divisor. The Value of the Fraction is 
 the Quotient. Hence we may make any change, in the 
 Fraction, which does not change the Quotient. But, 
 according to the 8th Principle in Division, dividing 
 both Dividend and Divisor (Numerator and Denomina- 
 tor) by the same number does not alter the Quotient. 
 
 If we take the Fraction /^ and factor both terms, we 
 have f^. Dividing both terms by 3, by rejecting the 
 common Factor 3, we have |. Hence -f^ equal |. The 
 terms of | are smaller than those of y%, and hence more 
 convenient. We reduced y% to | by rejecting the Factors 
 common to loth terms, 
 
 DEFINITION. 
 
 A Fraction is expressed in its Lowest Terms when 
 there is no Factor common to both terms. Hence, 
 
 To Reduce a Fraction to Lowest Terms: 
 
 EULE. 
 
 Reject all the CoMMOiq' Factors from both Terms. 
 Exercises for the Slate akd Board. 
 
 18. 21. 24. 90. i3o. 216. 375. _1 44 
 
 ■g"? ^ 2 7 > 35? Ib^y f89? B^O? 600? ll ■2'S* 
 
 SOLUTION. 
 
 Example. Multiply ^ by 3. 
 
 EXPLAKATIOK. 1st. Ac- First Method. 
 
 cording to the 3d Prin- i ^g—i x 3 — 3—1 ^^^^ 
 ciple in Division, multi- 
 plying the Dividend multi- Second MethM. 
 plies the Quotient. Hence, ^x 3 = ^^-3 =2* ^^^' 
 
MENTAL AND WRITTEN ARITHMETIC. 163 
 
 we multiply the Numerator of \ by 3, and obtain | for 
 a Product ; or, in Lowest Terms, \. 
 
 2d. According to the 6th Principle in Division, divid- 
 ing the Divisor multiplies the Quotient. Hence we 
 multiply J by 3 by dividing the Denominator by 3, and 
 have \ for our Final Product, as before. Hence, 
 
 To Multiply a Fraction by an Integer: 
 
 EULE. 
 
 Multiply the Numerator of the Fraction ly the Integer, 
 and for a Denominator write the given Denominator ; or, 
 Divide the Denominator of the Fraction by the Integer, 
 and write the given Numerator over this for a Numer- 
 ator* 
 
 Exercises for the Slate aitd Board. 
 
 First Method, 
 
 J X 3 = ? I X 2 = ? /_ X 4 = ? 3^ X 11 = ? 
 
 Second Method. 
 
 § X 3 = ? Vo"? X 7 = ? 4^ X 9 = ? /g X 8 = ? 
 
 BOLTTTION. 
 
 First Method. 
 
 Example. Divide f by 3. 
 
 Explanation. According 
 to the 4th General Princi- ^^^^^^^^^ Quotient 
 pie in Division, dividing the ^ ' ^ ^' ^ 
 Dividend divides the Quotient. 
 
 Hence, we divide the Numerator of f by 3, and obtain 
 f for our Quotient. 
 
 According to the 5th General Principle in Division, 
 multiplying the Divisor divides the Quotient. Hence, 
 multiplying the Deno- 
 minator of f by 3, we h ave second Method. 
 
 -^j ; or, in Lowest Terms, fi~3=f X3=i/V=f- Quotient. 
 ^, as before. Hence, 
 
164 FIRST LESSONS IN 
 
 To Divide a Fraction by an Integer: 
 
 EULE. 
 
 Divide the Numerator of the Fraction ly the Integer^ 
 and under this Quotient tvrite the given Denominator 
 for a Denominator ; or, Multiply the Denominator of 
 the Fraction hy the Integer, and over this Product write 
 the given Numerator for a Numerator, 
 
 Eemakk. The result obtained by the Second Method 
 should be reduced to Lowest Terms. 
 
 Exercises eor the Slate akd Board. 
 
 First Method. 
 Second Method. 
 
 7y — — r 3^-v-O — r y^g -T- < — f 
 
 LESSON CVIII, 
 
 Example 1. Multiply 14 by 4. 
 
 Explain" ATioiq" T o solution. 
 
 multiply 14 by 4 is to 14 x 1=14-^7== V =2. Prod, 
 take one-seventh of 14. 
 
 To take one-seventh of 14 we must divide 14 by 7. The 
 result, written as a Fraction, is y. 
 
 Example 2. Mul- solution. 
 
 tiplyl4by|. 1=1x3. 
 
 ExPLANATioiq-. 14x3=42. 
 We have seen that 42 x 4=42^7=-\^=6. Product, 
 when our Multiplier 
 
 is composed of Factors we can obtain the Product by 
 multiplying by the several Factors in succession. Find- 
 ing that our Multiplier is composed of two Factors, we 
 
MENTAL AND WRITTEN ARITRMETIC. 165 
 
 obtain our Product by multiplying by its Factors, 3 and 
 4. 14 X 3 gives 42. And 42 x 4 is the same as 42 -^ 7, 
 which is Y"? ^^ ^• 
 
 Example 3. Mul- solution. 
 
 tiply|by-|. 1 = 1x3 
 
 Explanation". |x3=f''3 = '5^ 
 We multiply | by 3 ^f x 1 .=J,^-7=J#-x7=l|. Prod. 
 and ^, the Factors of 
 
 -|. We multiply | by 3 by multiplying the Numerator 
 by 3. We multiply this result by \ by dividing by 7 ; 
 which we do by multiplying the Denominator by 7. 
 This gives \ | for the Product. Examining these Ex- 
 amples and Solutions, we see that we multiply by a 
 Fraction by multiplying the Multiplicand by the Nu- 
 merator of the Multiplier, and dividing the result by 
 the Denominator of the Multiplier. Hence, 
 
 To Multiply by a Fraction: 
 
 EULE. 
 
 Multiply the Multiplicand hy the Numerator of the 
 Multiplier, and divide this result hy the Denominator. 
 
 Eemarks. 
 
 1. Mixed INTumbers can be multiplied together by the 
 above Rule, after reducing them to Major Fractions. 
 
 2. The Product should be reduced to Lowest Terms, 
 or to an Integer or Mixed Number. 
 
 Exercises for the Slate an^d Board. 
 
 
 Fractions. 
 
 
 8xi = ? 
 f X 1 = ? 
 
 12 X 1 = ? 15 X 1 = ? 
 4 X 1 = ? tV X 1 = ? 
 
 Mixed Numbers, 
 
 21 X 4 = ? 
 
 21 x If = ? 
 
 5| X 31 = V 71 X 81 = ? 
 
 9/t X 3? = ? 
 
166 FIRST LESSONS IN 
 
 LESSON CIX. 
 
 Example 1. Divide 5 by i. 
 
 EXPLAIN^ATION. It is plain solution. 
 
 that 4 is contained in 1 seven 5-h^=5 x 7==35. QuoH, 
 times. It must be contained 
 
 in 2 twice 7 times, and in 5 five times 7 times, or, which 
 is the same, 7 times 5 times; which are 35 times. 
 Hence we divide 5 by ^ by multiplying 5 by 7. 
 
 Example 2. Divide 5 by |. 
 
 Explanation. We solution. 
 
 factor our Divisor | 1 = ^x3 
 
 into 1 and 3. Divid- 5-^3=| 
 
 ing first by the Factor |-^|=| x 7=|^^ = V"- Q^oH. 
 
 3, we have |. Divid- 
 ing this result by the other Factor, 4, by multiplying 
 by 7, we have y for the result, or Quotient. 
 
 Example 3. Divide | by f . 
 
 Explanation. Factor- solution. 
 
 toring f into 4 and 5, we |=:X x 5 
 
 divide | by 5, by multi- |4-5=:|x5=^^o 
 
 plying the Denominator •/o"^t=2U x '^=^%^^=i0 
 by 5, and have for a result f i^lsV* ^''^^' 
 5%. We then divide ^% by 
 
 4, by multipljring by 7. This gives f J for the Quotient. 
 Eeducing f J to a Mixed Number, we have 1-^^ for our 
 final Quotient. 
 
 Examining these Solutions and Explanations, we see 
 that we have in each case divided by a Fraction by 
 dividing the Dividend by the Numerator of the Divisor, 
 and then multiplying this result by the Denominator 
 of the Divisor. Hence, 
 
MENTAL AND WRITTEN ARITHMETIC. 167 
 
 To Divide by a Fraction: 
 
 Rule. 
 
 Divide the Dividend hy the Numerator of the Divisor^ 
 and then multiply this result hy the Denominator of the 
 Divisor, 
 
 Eemakks. 
 
 1. The result should be reduced to Lowest Terms ; if 
 a Major Fraction, to an Integer or Mixed Number. 
 
 2. Mixed Numbers in the Dividend or Divisor may 
 be reduced to Major Fractions, and then the Division 
 be performed by the above Eule. 
 
 Exercises for the Slate akd Board. 
 
 Fractions, 
 
 9~| = ? 8-f-| = ? 15-^f=? l-r-|=? 
 
 4 • 3 — * g • ^ — • IT • 4 — ^ 25 • Id — • 
 
 Mixed Numbers, 
 
 2|- -^- 3| = ? 5| -V- 7| = ? 12| ~ 4^ = ? 3| -^ If = ? 
 
 To Teachers. — The Elementary Principles involved 
 in the Reduction of Denominate Numbers, the Addition 
 and Subtraction of Compound Numbers, and the Mul- 
 tiplication and Division of Compound by Simple Num- 
 bers, are the same which are used in Addition, Subtrac- 
 tion, Multiplication and Division of Simple Numbers, 
 and have already been fully set forth. In applying them 
 to Denominate and Compound Numbers, the slight 
 changes necessary to be made can be readily and easily 
 explained by Teachers. Therefore, the Examples here- 
 inafter given are not accompanied with explanations. 
 
168 FIRST LESSONS IN 
 
 DENOMINATE NUMBERS. 
 
 LESSON OX. 
 
 In measuring a quantity, we take some definite amount 
 of it for a Unit ; as a pint, a yard. We often have dif- 
 ferent Units for the same kind of quantity. 
 
 I^EFINITIONS. 
 
 1. A TTnit of Measure is the definite amount of 
 anything taken as a stakdaed of comparison in meas- 
 uring all quantities of that hind. 
 
 2. Denoinifiation is the name given to a Unit of 
 Measure ; as quart, ounce, shilling, 
 
 3. A Higher Denoi^nination is that one of two 
 Denominations whose Unit has the higher yalue. 
 
 4. A Lower Denomination is that one of two 
 Denominations whose Unit has the lower yalue. 
 
 5. A Denominate JS'umber is a number applied 
 to one or more Denominations ; as 4- gallons, 3 quarts, 
 
 6. A Simple Number is a number expressed either 
 in NO Denomination or only one. 
 
 7. A Compound NiiTnher is a number expressed 
 in more than one Denomination ; as 2 days 5 hours, 
 
 8. deduction of Denominate Numhe^^s is 
 changing the number and Denomination of a Denom- 
 inate Number without changing its Value. 
 
 9. Reduction Ascending is reducing a Denom- 
 inate Number to one of a Higher Denomination. 
 
 10. Meduction Descending is reducing a De- 
 nominate Number to one of a Lower Denomination. 
 
MENTAL AND WRITTEN ARITHMETIC. 169 
 
 LESSON CXI. 
 
 UJSriTBD ST A. TBS MOJSTBT. 
 
 Tfnited States Money ^ called also Federal 
 
 money f is the legal currency of the United States. 
 
 TABZE. 
 
 10 mills {m), are 1 cent. d. 
 
 10 cents are 1 dime. d. 
 
 10 dimes, or 100 ct., are 1 dollar. $. 
 
 10 dollars are 1 eagle. E, 
 
 Note. — The Table of Canada Money is the same as that of 
 United States Money. 
 
 Exercises in Reduction, 
 
 In 9 ct. how many mills ? 
 
 In 8 ct. 7 m. how many mills ? 
 
 In 7 d. how many cents ? How many mills ? 
 
 In 7 d. 8 ct. 9 m. how many mills ? 
 
 How many cents in 80 m. ? In 50 m. ? In 90 m. ? 
 
 How many cents and mills in 98 m. ? In 65 m. ? 
 
170 FIRST LESSONS IN 
 
 Addition, 
 
 What is the Sum of 4 ct. 5 m., and 3 ct. 2 m.? 
 What is the Sum of 6 ct. 8 m., and 5 ct. 7 m.? 
 What is the Sum of 8 d. 9 ct. 7 m., and 3 d. 8 ct. 6 m. i 
 
 Subtraction, 
 
 From 8 ct. 9 m. subtract 5 ct. 3 m.? 
 Prom 7 d. 8 ct 9 m. subtract 3d. 4 ct. 5 m. 
 From 5 d. 4 ct. 7 m. subtract 2 d. 9 ct. 3 m. 
 From 8 d. 3 ct. 4 m. subtract 3 d. 5 ct. 7 m. 
 
 LESSON CXII. 
 
 JEJSTGZISir MOJSTBT. 
 
 Bnglish Money ^ called also Sterling Money , 
 
 is the currency of Great Britain. 
 
 TABJjE. 
 
 4 farthings {far,) are 1 penny. d. 
 
 12 pence are 1 shilling. s, 
 
 20 shillings are 1 pound or sovereign. £, 
 
 21 shillings are 1 guinea. guin. 
 
 Exercises in Reduction. 
 
 How many farthings in 7d. ? In 9s. ? In 3s. 9d. ? 
 
 How many pence in 7s. ? In £3 ? In £6 7s. 9d. ? 
 
 How many farthings in £7 lis. 5d. ? In £9 7s. 8d. 
 3 far. ? 
 
 How many shillings in 48d. ? In 96 far. ? In 240 
 far.? 
 
 Keduce 987 far. to Higher Denominations. 
 
 Eeduce lis. and 1,765 far. to Higher Denominations. 
 
 In 21 pounds how many guineas ? 
 
MENTAL AND WRITTEN ARITHMETIC. 171 
 
 fFTi -^^^y?^ ^ — ■ — p 
 
 LESSON CXI/f. 
 
 liiqttid Measure^ called also Wine Measure^ 
 
 is used in measuring wines, oil^ molasses, milk, and 
 other liquids. 
 
 TABLE, 
 
 4 gills igi.) 
 
 2 pints 
 
 4 quarts 
 31^ gallons 
 
 2 barrels, or ) 
 63 gallons, ) 
 
 are 1 pint, 
 are 1 quart, 
 are 1 gallon, 
 are 1 barrel. 
 
 qt. 
 gal 
 
 are 1 hogshead. JiM. 
 
 Exercises for the Slate asd Board. 
 
 Meduction, 
 
 How many gills in 5 pt. ? In 7 qt. ? In 15 gal. ? 
 How many gills in 15 gal. 3 qt. 1 pt. 2 gi. ? 
 Eeduce 547 gi. to Higher Denominations- 
 
 Addition^ 
 
 To 19 gal. 3 qt. 1 pi 3 gi. add 9 gal. 2 qt. 1 pt. 2 gi. ? 
 To 20 gal. 2 qt 1 pt. 2 gi. add 10 gaL 3 qt. 1 pt 3 gL ? 
 12 
 
173 
 
 FTRST LESSONS IN 
 
 Subtraction, 
 
 From 27 gal". 3 qt. 1 pt. 3 gi. take 19 gal. 2 qt. 1 pt. 2 gi. 
 From 9 gal. qt. 1 pt. 1 gi. take 2 gal. 2 qt. pt. 3 gi. 
 
 Multiplication , 
 
 How mucli oil is there in 5 casks, each containing 12 
 gal. 2qt. 1 pt. 3gi.? 
 
 In 3 casks, each containing 10 gaL 1 qt. 1 pt.? 
 
 Division,. 
 
 If 18 gal. 3 qt. 1 pt. 2 gL of milk be equally divided 
 between two persons, what will each receive ? 
 
 LESSOIV CX/V. 
 
 Dry Measure is used in measuring grain, fruits, 
 roots,, seeds, salt, lime, charcoal, and various other ar- 
 ticles not fluid. 
 
 TABZE. 
 
 2 pints (pt.) are 1 quart. qf. 
 
 8 quarts are 1 peck. ph 
 
 4 pecks are 1 bushel. hu. 
 
 36 bushels (of coal) are 1 chaldron* chal. 
 
MENTAL AND WRITTEN ARITHMETIC. 173 
 
 EXEECISES FOR THE SlATE AKD BoAKD. 
 A ddition. 
 
 Add 5 bu. 3 pk. 7 qt. 1 pt. and 3 bu. 2 pk. 4 qt. 1 pt. 
 Add 7 bu. 1 pk. 5 qt. 1 pt. and 11 bu. 3 pk. 7 qt. 1 pt. 
 
 Subtraction, 
 
 From 31 bu. 3 pk. 7 qt. pt. take 27 bu. 3 pk. 4 qt. 1 pt. 
 From 25 bu. 3 pk. qt. pt. take 16 bu. 2 pk. 5 qt. 1 pt. 
 
 LESSON CXV. 
 
 Avoirdupois Weight is used for ail the ordinary 
 purposes of weighing, 
 
 TABLE, 
 
 are 1 ounce. oz. 
 
 are 1 pound. Ih 
 
 are 1 quarter. qr. 
 
 ^ ( hundred- ] 
 are 1 i _^,.^i.^^ j- 
 
 16 drams {dr,) 
 16 ounces 
 25 pounds 
 4 quarters, or \ 
 
 100 pounds, 
 
 I weight. 
 
 cwU 
 
 20 hundred weight are 1 ton. 
 
 T, 
 
174 FIRST LESSONS IN 
 
 EXEBCISES FOR THE SlATE A.^B BoARD. 
 JReduction* 
 
 How many drams in 3 qr. 18 lb. 13 oz. 11 dr. ? 
 Eeduce 1572 dr. to Higher Denominations. 
 
 jLddition, 
 
 Tolqr. 171b. 9 oz. 7 dr. add 1 qr. 15 lb. 13 oz. 14 dr. 
 To 3 qr. 21 lb. 11 oz. 13 dr. add 2 qr. 9 lb. 8 oz. 7 dr. 
 
 LESSON GXVI. 
 
 Troy Weight is used in weighing jewels, gold and 
 silyer. 
 
 TJLBZB. 
 
 24 grains (gr,) are 1 pennyweight, ptvt. 
 20 pennyweights are 1 ounce. oz. 
 
 12 ounces are 1 pound. lb. 
 
 Note. The Treacher will supply under this and the following 
 Tables all needed Exercises. 
 
 Apothecaries^ Weight is used by physicians in 
 compounding medicines ; but when medicines are 
 bought or sold Avoirdupois Weight is used. 
 
 TABLE. 
 
 20 grains {gr.) are 1 scruple, sc. or 3. 
 
 3 scruples ^re 1 dram. dr. or 3 . 
 
 8 drams are 1 ounce. oz. or § . 
 
 12 ounces are 1 pound, lb. or ft. 
 
MENTAL AND WRITTEN ABITMMETIC. 
 
 17b 
 
 LESSON CXVII. 
 
 Linear Measure — called also Long Measure 
 
 — is used in measuring lines, or distances. 
 
 12 inches (m.) 
 
 3 feet 
 
 51 yards, or 1Q\ it, 
 40 rods 
 
 TAJBZE, 
 
 are 1 foot ft 
 
 are 1 yard. yd. 
 
 are 1 rod, perch, or pole. rd. 
 
 are 1 furlong. fur. 
 
 8 furlongs, or 320 rd.,are 1 mile. mi. 
 
 SQUA^RB MBASU^B. 
 
 Square Measure is used for measuring surfaces ; 
 as of land, plastering, and paving. 
 
 A square foot is a square each of whose 4 sides is 1 
 foot, or 12 inches, in length. A square yard is a square 
 each of whose 4 sides is 1 yard, or 3 feet, in length. 
 
176 
 
 FIBST LESSONS IN 
 
 TABLE, 
 
 144 square inches {sq. in,) are 1 square foot. sq,ft 
 
 9 square feet are 1 square yard. sq. yd. 
 
 30| square yards are 1 square rod. sq, rd, 
 
 40 square rods are 1 rood. R, 
 
 i roods or) avelaove. A. 
 
 160 sq. rods, ) 
 
 640 acres are 1 square mile, sq, mi. 
 
 LESSON CXVIII. 
 
 Cubic Wen sure is used for measuring solids; as 
 timber, wood, and stone. 
 
 A cubic foot is a cube each of whose 12 edges is 1 
 foot, or 12 inches, in length. A cubic yard is a cube 
 measuring 1 yard, or 3 feet, on each edge. Each of its 
 6 faces is a square containing 9 square feet. A cubic 
 yard is shown in the above cut. 
 
MENTAL AND WRITTEN ARITHMETIC 
 
 177 
 
 TABLE. 
 
 1728 cubic inches are 1 cubic foot. cu,p, 
 
 27 cubic feet are 1 cubic yard. cu, yd, 
 
 42 cubic feet are 1 ton, shipping. t s. 
 
 24:| cubic feet are 1 perch, of ^tone. pch. 
 
 Wood Measure^ though part of Cubic Measure, 
 is sometimes embraced in a separate Table. 
 
 A pile of wood -8 ft long, 4 ft. wide, and 4 ft. high, 
 contains a cord. 
 
 TABLE, 
 
 16 cubic feet are 1 cord foot cd.fL 
 
 8 cord feet, or) ^re 1 cord. cd. 
 
 128 cubic feet, ) 
 
 EXEECISES FOR tHE SlATE AND BOARD. 
 
 In 1 cd. 5 cd. ft. 11 cu. ft. 187 cu. in., how many 
 cubic inches ? 
 
 Reduce 3 cu. yd. 5 cu. ft. 127 cu. in. to cubic inches. 
 To 3 cd. 7 cd. ft. 11 cu. ft. add 5 cd. 9 cd. ft. 8 cu. ft. 
 
178 
 
 FIRST LESSONS IN 
 
 LESSON CX/X. 
 
 TIM:^ MBASTJ'RB. 
 
 Time is Duration having a beginning and an end 
 Being definite, it can therefore be measured. 
 
 The Day and Year are the Natural Divisions of Tirne^ 
 since they are founded in Nature. 
 
 TABLB. 
 
 GO seconds {sec) 
 
 are 1 minute. 
 
 min. 
 
 60 minutes 
 
 are 1 hour. 
 
 h. 
 
 24 hoRirs 
 
 are 1 day. 
 
 da. 
 
 7 days 
 
 are 1 week. 
 
 wk. 
 
 365 days, or) 
 52 wk. 1 d.,) 
 
 are 1 common year. 
 
 yr. 
 
 366 days, or > 
 52 wk. 2 d.,f 
 
 are 1 leap year. 
 
 leap 1 
 
 12 calendar months 
 
 ; are 1 year. 
 
 yr. 
 
 100 years 
 
 are 1 century. 
 
 C. 
 
 Every fourth year in a century (except sometimes the 
 last) is a leap year; as 1804, 1808, 1812. 
 
3IENTAL AND WRITTEN ARITH3IETIC, 
 
 179 
 
 ^ly^isiojsr OjF tub tbah. 
 
 
 Months. Days. 
 
 Seasons. 
 
 January, 
 
 Jan., 
 
 1st month, has 31. 
 
 ^ 1 Winter. 
 
 February, 
 
 Feb., 
 
 2d month, has 28 or 2^ 
 
 March, 
 
 Mar., 
 
 3d month, has 31. 
 
 
 April, 
 
 Apr., 
 
 4th month, has 30. 
 
 >■ Spring. 
 
 May, 
 
 May, 
 
 5th month, has 31. 
 
 
 June, 
 
 June, 
 
 6th month, has 30. 
 
 ] 
 
 July, 
 
 July, 
 
 7th month, has 31. 
 
 \ Summer. 
 
 August, 
 
 Aug. 
 
 8th month, has 31. 
 
 J 
 
 September, Sept., 
 
 9th month, has 30. 
 
 1 
 
 October, 
 
 Oct., 
 
 10th month, has 31. 
 
 >■ Autumn. 
 
 November, 
 
 Nov., 
 
 11th month, has 30. 
 
 December, 
 
 Dec, 
 
 12th month, has 31. 
 
 Winter. 
 
 Note. — February has 29 days in none but leap years. 
 
 LESSON CXX, 
 
 Circular and Angular Measure is used in 
 measuring angles, and in the comparison of portions of 
 the circumferences of circles. 
 
 60 seconds ('') 
 60 minutes 
 30 degrees 
 
 90 degrees 
 
 360 degrees, or ^ 
 
 12 signs, or \ 
 
 4 quadrants, J 
 
 TABTjE, 
 
 are 1 minute, 
 are 1 degree. 
 1 sign. 
 
 are 
 
 are 
 
 j 1 quadrant, or ) quad. 
 \ right angle. f r. a, 
 
 j 1 circi 
 I of a ci 
 
 cumference ) 
 circle. j ' 
 
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