D.APPLE.ON&CO.,KewYork. [ /r APPLETOUS' MATHEMATICAL &EEJES. f PEIMAEY ARITHMETIC. BT G. P. QUACKENBOS, LL. D., AUTHOR or "AN ENGLISH GRAMMAR;" *' FIRST LESSONS IN COMPOSITION ; " "ADVANCED COURSE OP COMPOSITION AND RHETORIC;" "A NATURAL PHILOSO- phy;" "illustrated school history op the united states;" "primary history op the united states ; " etc. UPON THE BASIS OP THBW0EK8 OP GEO. R. PERKINS, LL.D. KEW YORK: D. APPLETON AND COMPANY, 1, 8, AND 6 BOND STEEET. '^~''' ' "preface, g^^' This little book, the first of our Series of Arithmetics, presup- poses no knowledge of numbers whatever. It tries to give a cor- rect idea of their value, both absolute and relative, by treating them concretely as well as abstractly from the very outset, pictorial illustrations being freely used for that purpose. We have not sought in its pages to go far, but to go thoroughly. It is therefore confined to the four fundamental operations, a very brief view of fractions, the most important tables of moneys, weights, and meas- ures, and easy exercises under them. These subjects, it is believed, are so treated, as to make the young pupil think for himself, and to lay the right kind of foundation for a mathematical course. One thing is taught at a time, and great care has been taken to present every thing in its proper place. It will be observed that through- out the book slate exercises go hand in hand with mental opera- tions. In the author^s opinion, they are indispensable, from the first, for inspiring the beginner with that interest in the subject which is essential to success. This book imposes no labor on the teacher, in the way of ex- plaining orally, furnishing examples, &c. It is thought to contain in itself all that is needed. The author would only ask those who use it to be sure that every lesson is thoroughly mastered, before proceeding to the next. Each principle presented, each table, should be perfectly understood and memorized before leaving it, even though it be necessary to repeat the lesson again and again. Time saved by allowing a single lesson to pass unmastered, will be a tenfold loss in the end. The Roman Notation is not explained in this volume ; but by means of the numerals used in the lesson-headings the pupil may readily be made to understand it. Entered, according to Act of Congress, in the year 1868, by D. APPLETOJrA CO., In the Clerk's OflOice of the District Court of the United States for the Southern District of New York. auL. \ PRIMARY ARITHMETIC. LESSON I. Touch your head. How many heads have you ? One. Touch your eyes. How many eyes have you ? " Two. Touch the joints of your fore-finger. How many are there ? Thkee. Touch the fingers of your left hand. How many are there ? -^ Fouk. Touch the fingers and thumb of your left hand. How many do they make ? Five. When we say one, two, tueee, four, five, what do we do ? Answer, We count. Count five ; count four ; count three ; count two. What are one, two, three, four, five, called ? Answer. Numbers. Which is the smallest of these numbers ? Which is the greatest of these numbers ? 54 I \ 92 A- ;couNTmG. LSSSON II. Count the birds on this branch. How many are there ? Five. If another ah'ghts on it, how many will there be ? Six, If another joins -them, how many will that make ? Seven. If another comes, how many ? Eight. ^One more will make how many? Nine. And one more, how many ? Ten. Connt six ; count seven ; eight ; nine ; ten. "Which is greater, ten or five ? Two or six ? What do we use, to express numbers ? Answer. Ten characters, called Figures. Learn the names of these ten figures, and how to make them neatly on your slate. NAITGHT ONB TWO THBKB POUB FIVE BIX BEVEN EIGHT NINB O / 2 -/ ^ d 7 ^ ^ COUNTING. — NUMERATION. — NOTATION. LESSON III. Eleven // Twelve /2 Thirteen /S Count these rabbits. How many are there ? How do we express ten with figures ? Answer. By putting a naught after a one (10). Learn the numbers, going up from ten : — Fourteen / -4 Seventeen / y FifteeV /S Eighteen '^^ Sixteen / d Nineteen / p Twenty ^ . . 20 Kead these numbers : 13 ; 7 ; 19 ; 11 ; 9 ; 2 ; 20; 12; 5; 16; 3; 10; 4; 18; 6; 15; 8; 17. Write in figures, six ; ten ; three ; eighteen ; eight ; twenty ; two ; twelve ; fourteen ; five. What is the reading of numbers called? Answer. Nnmeration. What is the writing of numbers called? Answer. "FTotation. COUNTING. LESSON IV. Now we have some marks in diflPerent rows. Count those in each row. The number of marks is given after each row ; then come the written figures that represent this number, then the printed figures. I I I I I J mill iiiiiii iiiiiiii iiiiiiiii iiiiiiiiii iiiiiniiii . . . . iiiiiiiiiiii . . . . iiiiiiiiiiiii . . . iiiiiiiiiiiiii . . . iiiiiiiiiiiiiii . . Ml I II I I I I I I I I I I . . One Two Three Four , Five Six Seven Eight !N'lNE Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen N'iNETEEN Twenty 7 // /^ /^ /^ ^7 y/ ^0 Which is greater, twenty or ten ? 10 11 12 13 14 15 16 17 18 19 20 11 or 18 ? COUNTING. LESSON V. Learn the twenties : — Twenty-one 21 I Twenty-four 24 Twenty-two 22 Twenty-five 25 Twenty-three 23 | Twenty-six 26 In the twenties, which figure remains the same, and which does not ? Answer, The left-hand figure remains the same — 2. The right-hand figure changes. What comes after 29 ? Answer, Thirty (30). Learn the thirties : — Twenty-seven 27 Twenty-eight 28 Twenty-nine 29 Thirty-one 31 Thirty-four 34 Thirty-seven 37 Thirty-two 32 Thirty-five 35 Thirty-eight 38 Thirty-three 33 Thirty-six 36 Thirty-nine 89 What is the left-hand figure in all the thirties ? Which figure changes ? What comes after 39 ? Answer, Forty (40). Learn the forties : — Forty-one 41 Forty-four 44 Forty-seven 47 Forty-two 42 Forty-five 45 Forty-eight 48 Forty-three 43 Forty-six 46 Forty-nine 49 Fifty .... 50 Which is greater, fifty or forty? 49 or 39? 28 or 38? 26 or 17? 9 or 15 ? 15 or 30? Count from 1 to 60. Count from 20 to 1 backwards ; twenty^ nineteen^ eighteen^ &c. 8 OOUiniNG. LESSON VI. Learn the numbers, going up from fifty : — Fifty 60 Sixty-seven 67 Eighty-four 84 Fifty-one 51 Sixty-eight 68 Eighty-five 85 Fifty-two 52 Sixty-nine 69 Eighty-six 86 Fifty-three 53 Seventy 70 Eighty-seven 87 Fifty-four 54 Seventy-one 71 Eighty-eight 88 Fifty-five 55 Seventy-two 72 Eighty-nine 89 Fifty-six 56 Seventy-three 73 Ninety 90 Fifty-seven 57 Seventy-four 74 Ninety-one 91 Fifty-eight 58 Seventy-five 75 Ninety-two 92 Fifty-nine 59 Seventy-six 76 Ninety-three 93 Sixty 60 Seventy-seven 77 Ninety-four 94 Sixty-one 61 Seventy-eight 78 Ninety-five 95 Sixty-two 62 Seventy-nine 79 Ninety-six 96 Sixty-three 63 Eighty 80 Ninety-seven 97 Sixty-four 64 Eighty-one 81 Ninety-eight 98 Sixty-five 65 Eighty-two 82 Ninety-nine 99 Sixty-six 66 Eighty-three 83 One hundred 100 How many figures does 100 contain ? What is the greatest number that can be ex- pressed with two figures ? Answer, 99. What is the greatest number that can be ex- pressed with one figure ? Ansvjer. 9. Name in order the numbers between 1 and 100 that contain naught. Ans. 10 , 20, 30, 40, 50, 60, 70, 80, 90, 100. COUNTING. LESSON VII. Here is a factory ten stories high. How many windows are there in each story ? Count the windows in the two lowest stories. How many do you find? How many rows of 10 each make 20 ? How many tens in 20 ? Count the windows in the three lowest stories. How many tens in thirty ? Add another row of windows, and how many do you find ? How many tens in forty ? Count the windows in the five lowest rows. How many tens in fifty ? How many tens in 60 ? How many tens in 70 ? How many tens in eighty ? How many tens in 90 ? Count all the windows. How many are there ? How many rows of 10 each make 100 ? How many tens in one hundred ? 10 UNITS. TENS. LESSON VIII. Learn this table : — 20 is 2 tens. 30 is 3 tens. 40 is 4 tens. 50 is 5 tens. 60 is 6 tens. 70 is 7 tens. 80 is 8 tens. 90 is 9 tens. Which figure shows the number of tens ? Answer. The second, or left-hand figure. What is the second place called ? Answer, The place of tens. What is the first, or right-hand place called ? Answer. The place of units. Write down 22. What figures do you use ? Have these twos the same value ? Answer. No. The 2 in the second place is ten times as great as the 2 in the first place. Why is this ? Answer. Because the 2 in the second place denotes two tens^ or twenty ; the 2 in the first place denotes two units^ or two. What, then, is the effect of removing a figure from the first place to the second ? Answer. Its value is increased ten times ; 20 is ten times as great as 2. Write down the numbers from 1 to 100. UNITS. TENS. 11 LESSON IX. How do you write six tens ? Ans, 6 in the second place denotes six tens. As there are no units, I put a after it — 60. How do you write six tens, one unit ? Arts. 6 in the second place denotes six tens ; 1 in the first place denotes one unit — 61. Read this number. Arts, Sixty-one. How must a column of numbers be written ? Ans. So that units may stand under units, and tens under tens. Write the following : — Four tens, three units ; nme tens, nine units ; three tens, seven units ; eight tens, two units ; five tens, two units ; two tens, five units ; one ten ; seven tens. Kead the numbers just written. Mention the tens and units in the following numbers, and then read them : — 80 {eight tens^ eighty) ; 75 {seven tens^ five units^ seventy-five) ; 69; 93; 41; 28; 7; 50; 17; 32; 84; 22; 61; 14 ; 57 ; 46 ; 70 ; 11 ; 98 ; 23. Which is greater, a ten or a unit ? How can you turn three into thirty ? Count from 100 to 75 backwards. Count from 50 to 20 backwards. Write down in columns the numbers in order, going backwards from 100 to 1. 12 HUNDREDS. LESSON X. How much do 10 tens make ? Ans. One hundred — 100. Which figure shows the number of hundreds ? Ans. The third figure — 1. How, then, do we write two hundred ? Ans. By putting a 2 in the third place — 200. Learn the hundreds : — One hundred 100 Two hundred 200 Three hundred 300 Four hundred 400 Five hundred 500 Six hundred 600 Seven hundred TOO Eight hundred 800 Nine hundred . . . 900 How many times greater is 1 in the third place than 1 in the second place ? Ans, Ten times ; 100 is ten times 10. What is the effect, then, of removing a figure one place to the left ? Arts, Its value is increased ten times. Name the places, going from right to left. Atis. Units, tens, hundreds. Count from 100 to 200. Thus \—One lirni- dred cmd one^ one hundred and two^ &c. Write the numbers from 100 to 200. Thus : 101 104 107 110 113 116 102 105 108 111 114 117 103 106 109 112 115 118, &c. NUMERATION. NOTATION. 13 J LESSON XI. What is Numeration ? Answer. The art of reading numbers. What is Notation ? Answer, The art of writing numbers. EXERCISE IN NUMERATION. Read the following numbers : — 198 912 15 853 4 909 611 1 268 309 200 88 82 79 41 470 224 625 Y03 816 526 6 36 64 434 349 62 518 147 517 EXERCISE IN NOTATION. Write these numbers in figures : — Eighty-three ; Fourteen ; Two hundred and sixty ; Nine hundred and forty-eight ; One hun- dred and nine ; Twelve ; Ninety-one ; Seven hundred and fifty-seven ; Thirty-four. Three hundred and fifteen ; Seventy-two ; Six hundred and twenty-four ; One hundred ; Four hundred and seven ; Eight hundred and eleven ; Eighty-eight ; Five hundred and forty ; Fifty ; Seven hundred and twenty ; Ninety-nine. Counting.— Count from 800 to 900. 14 ADDITION. LESSON XII. Look at the picture. You see one diimney in one part of the roof, and one chimney in another part; how many chimneys do you see in all? Count them. One and one are — how many ?. One sheep is feeding by itself, and two sheep are feeding together ; how many sheep are there in all ? One and two are — . One boy and three girls are walking. How many in all are walking ? One and three are — . One dog is lying down, four dogs are running ; how many dogs are there altogether ? One and four are — . ADDITION. 15 There is one window on one side of the house, and five windows on another. How many win- dows are there altogether ? One and five are — . There is one girl jumping the rope, and six girls not jumping it. How many girls are there in all ? One and six are — . On the lawn we see one boy and seven girls ; counting the boy and girls together, how many are on the lawn? One and seven are — . One tree stands on one side of the house, and eight trees on the other. How many trees do you count altogether ? One and eight are — . One chicken by itself, and nine chickens in a group ; how many chickens in all ? One and nine are — . One hen and ten chickens ; counting hen and chickens together, how many do you find ? One and ten are — . What you have just answered, will now be put in a table. Learn it perfectly. 1 and 1 are 2. 1 and 6 are 7. 1 and 2 are 3. 1 and 7 are 8. 1 and 3 are 4. 1 and 8 are 9. 1 and 4 are 5. 1 and 9 are 10. 1 and 5 are 6. 1 and 10 are 11. When you put numbers together, to find ho^;^ much they make, what is the process called ? A71S, Addition. 16 ADDITION. LESSON ZIII. What is Addition? Ans. Addition is the process of uniting two or more numbers in one. When you unite two or more numbers in one, what do you find? Ans, I find their Sum. Two men are riding ; one is walking by their side ; how many men in all ? Add 2 and 1. Jane has two kittens ; her sister also has two. How many kittens have both ? Add 2 and 2. Jane had two dollars, and her father gave her three more. How many had she then ? Learn the Tables so as to say them forward and backward, in order or skipping about. 2 and 1 are 3. 2 and 2 are 4. 2 and 3 are 5. 2 and 4 are 6. 2 and 5 are 7. 2 and 6 are 8. 2 and 7 are 9. 2 and 8 are 10. 2 and 9 are 11. 2 and 10 are 12. ADDITION. 17 LESSON XIV. 3 and 1 are 4. 3 and 2 are 5. 3 and 3 are 6. 3 and 4 are 7. 3 and 5 are 8. 3 and 6 are 9. 3 and 7 are 10. 3 and 8 are 11. 3 and 9 are 12. 3 and 10 are 13. [Let the pupil in all cases add each number as a whole, and not by counting or taking one unit at a time. Say three and three are six; not three, four, Jive, six.] 1. A regiment inarched three miles before dinner, and five miles after dinner. How far did it march in all ? Add 3 and 5. 2. If you have 3 marbles in one pocket, and 7 in another, how many have you in both ? 3. Three horses are in the stable, and six are in the field. How many horses in all ? 4. Euth bought three books; she had ten books before. How many has she now ? Now you have a few examples in addition for your slate. Copy the figures in each column, add them, and write the sum below the line. 1127859 10 12331321 1 10 9 8 74 65 7 1 1 1 2 10 21 1 1 1 2 1 12 11 2 18 ADDITION. LESSON XV. "1^ /\ /\ /\ J/ >/ \/ \/ How many are 1 and 4 ? How many are 4 and 1 ? How many are 2 and 4 ? 4 and 2 ? How many are 3 and 4 ? 4 and 3 ? When you are adding numbers, does it make any difference whicli you take first ? Here are four eggs in one bird's -nest, four in another ; count how many there are in both. Add 4 and 4. Four kittens are running one way, and five the other ; how many in all ? 4 and 5 make — „ Four rats and six rats make how many rats ? How many are 4 and 6 ? 4 and 1 are 5. 4 and 6 are 10. 4 and 2 are 6. 4 and Y are 11. 4 and 3 are 7. 4 and 8 are 12. 4 and 4 are 8. 4 and 9 are 13. 4 and 5 are 9. 4 and 10 are 14. QUESTIOJ^S IN ADDITIOK. 19 liSSSON XVI. 1. Harry gets four good marks on Monday, / and seven on Tuesday ; how many does lie get both days ? How many do 4 and 7 make ? 2. I have four fingers on each hand ; how many have I on both ? What is the sum of 4 and 4? 3. If four cars leave in one train, and ten in another, how many leave in both ? Add 4 and 10. 4. Ida had one pencil, then she bought three, and the next day her father gave her seven more ; how many had she then in all? 1, 3, and 7, make how many ? 5. A toyman sold two dolls in the morning, two in the afternoon, and nine in the evening ; how many did he sell altogether ? 2 and 2 added to 9 make how many ? 6. If I give four dollars for a vest, and eight dollars for a coat, how much do both cost me ? Add the following numbers on your slate. Be sure to set units under units, tens under tens. Add twenty-three and thirty-four. Add one hundred, twelve, and three hundred and forty- three. Add forty-four to four hundred and one. Add four and eleven. Add seventy-four and two hundred and two. Add ninety-four and three hundred. 20 ADDITION. LESSON XVII. How many are one and five ? ->$• ">HHHHc How many are five and one ? •>r-/Hc">Hr '/<- How many are 2 and 5 ? 5 and 2 ? How many are 3 and 5 ? 5 and 3 ? Five lilies and four lilies are how many lilies? Five boys are sailing in one boat, and five in another. Count how many there are in both. Five tops are lying on the floor, and six are spinning ; how many do you see in all ? How many are 5 and 7? Mill Illllll How many are 5 and 8? ||||| II Mill I How many are 5 and 9? Mill I Mill I II How many are 5 and 10? Mill I Mill MM Which is the greater, five or ten ? 5 and 1 are 6. 5 and 6 are 11. 5 and 2 are 7, 5 and 7 are 12. 5 and 3 are 8. 5 and 8 are 13. 5 and 4 are 9. 5 and 9 are 14. 5 and 5 are 10. 5 and 10 are 15. QUESTIONS IN ADDITION. 21 LESSON XVIII. 1. Five years ago Mary was nine years old ; how old is she now ? How much are 5 and 9 ? 2. We have five toes on each foot ; how many toes have we in all ? What is the sum of 5 and 5 ? 3. Horace has five marbles in one bag, and ten in another ; how many has he in both ? Add 5 and 10. 4. Five pigs are in the orchard, and seven in the sty. How many pigs are there altogether ? 6. Louise has three cents, Mary two, and George eight ; how many cents have all three ? 6. Susan has four books ; if her brother gives her one more, and her sister two, how many books will she have? Add 4, 1, and 2. 7. Eichard caught five fish, and Ben six ; how many did both catch ? How many are 5 and 6 ? 8. The first story of a house contains two rooms, the second three, and the third three ; how many rooms does the whole house contain? Here are some examples for your slate. Eead each of the numbers added, and also the answers. (10) (11) (12) (13) (14) (15) 4: 403 927 8 5 765 14 515 20 610 372 121 51 30 32 41 112 102 ' 22 ADDITION. LESSON XIZ \/ \/ \f \f^\^^, 7 r*/ C/ \"i **/ r' How many are one and six ? How many are six and one ? ->r*>Hr"-/Hwr '/c How many are 2 and 6 ? 6 and 2 ? How many are 3 and 6 ? 6 and 3 ? How many are 4 and 6 ? 6 and 4 ? How many are 5 and 6 ? 6 and 5 ? Six flies and six flies are how many flies ? Six sqnirrels are eating, seven are running; connt how many there are in all. If you put six forks and eight forks together, how many will yon have ? How many are 6 and 9? |||||! MINMII How many are 6 and 10? |1|||| 111 Hi HI | 6 and 1 are 7. 6 and 6 are 12. 6 and 2 are 8. 6 and 7 are 13. 6 and 3 are 9. 6 and 8 are 14. 6 and 4 are 10. 6 and 9 are 15. 6 and 5 are 11. 6 and 10 are 16. QUESTIONS IN ADDITION, 23 IiESSON XX. 1. Troy is six miles above Albany, and Lan- singburg is three miles farther ; how far is Lan- singburg from Albany ? 6 and 3 make how many ? 2. If I have 6 dollars in my pocket, and 9 in my purse, how mnch have I in all ? Add 6 and 9. 3. Three dogs, and three dogs, and three dogs, make how many ? How many are 3, 3, and 3 ? 4. Two ships, four schooners, and ten brigs, are in port ; how many vessels does that make in all ? What is the sum of 2, 4, and 10 ? 5. A baker sold six loaves of wheat bread, and two of rye ; how many loaves did he sell in all ? 6. Mary gave a poor man six cents, and Kate gave him five ; how much did he receive from both ? How much are 6 and 5 ? 7. If six horses are in the stable, and I put in eight more, how many will be there then ? 8. Six goats and one goat make how many ? 9. Six shoes in one drawer, and six in another, make how many in all ? Add the following numbers on your slate : — (10) (") (12) (13) (14) (IS) 80 92 43 66 730 80 31 25 653 42 10 605 26 41 203 11 256 113 137 798 24 .UDDmON". ^/ \f \f \f \f \/ \f */ c*/ v**/ r*/ c"/ r^ x"*/ \* LBSSON XXI. How many are 1 and 7 ? How many are 7 and is / x* / r* -"> v-* r*/ r*/ ;• "> x- How many are 2 and 7 ? 7 and 2 ? How many are 3 and 7 ? 7 and 3 ? How many are 4 and 7 ? 7 and 4 ? How many are 5 and 7 ? 7 and 5 ? How many are 6 and 7 ? 7 and 6 ? Mary has a bunch of seven cherries, and buys seven more ; how many has she then i Count and see. Seven boys and eight girls make how many children in all ? How many are 7 and 8 ? ^ W W W ^ W W Seven wine glasses in one %I%l%I\i%l%i _ ^<^^ ^d iiiiie in another make how many in all ? How many are 7 and 9 ? How many are 7 and 10 ? 7 and 1 are 8. 7 and 6 are 13. 7 and 2 are 9. 7 and 7 are 14. 7 and 3 are 10. 7 and 8 are 15. 7 and 4 are 11. 7 and 9 are 16. 7 and 5 are 12. 7 and 10 are 17. QUESTIONS IN ADDITION. 25 LESSON XXII. 1. To get home, I shall have to go seven miles by steamboat, and five by stage ; how far will 1 have to travel ? What is the sum of 7 and 5 ? 2. A farmer had seven rooms in his house, and built a wing with three rooms more ; how many rooms had he then ? How many are 7 and 3 ? 3. A gardener set out three lilacs, four roses, and ten verbenas ; how many plants did he set out in all? Add 3, 4, and 10. 4. Mary dressed five dolls for the fair, Augusta two, and Helen seven ; how many did all three dress ? How many are 5, 2, and 7 ? 6. If my black hen has seven chickens, and my white hen nine, how many have both ? 6. How far will a horse go in 2 hours, if he travels 7 miles the first hour, and 6 the second ? 7. How many are 6 and 4 ? 6 and 9 ? 4 and 7? 7and4? 7and2? 3andl0? 2and8? 1, 6, and 1 ? 4, 3, and 8 ? 6, 1, and 6 ? 1 and 7? 6 and 3? Here are some examples for your slate. Make your figures neatly. Kead each number, and tell whether the upper or lower one is the greater. (8) (9) (10) (11) (12) (13) 163 329 48 856 775 507 425 670 731 132 11 182 26 ADDITION. IiESSON XXIII. How many are 1 and 8 ? •>(* "vc-JC—X—X—X- How many are 8 and 1 ? "/ (— > H c"/ c"/ r*/ How many are 2 and 8 ? How many are 3 and 8 ? How many are 4 and 8 ? How many are 5 and 8 ? How many are 6 and 8 ? How many are 7 and 8 2 c'/r'/C'/C/C 8 and 2? 8 and 3 ? 8 and 4? 8 and 5? 8 and 6? 8 and 73 Eight doves in one group, and eight in another, make how many in all ? How many are 8 and 8 ? Eight rounds in one ladder, and nine rounds in another, make how many rounds in all ? What is the sum of 8 and 9 ? Eight spokes are in one wheel, and ten in another ; how many are in both? 8 and 10 are — . 8 and 1 are 9. 8 and 6 are 14. ^ 8 and 2 are 10. 8 and 7 are 15. 8 and 3 are 11. 8 and 8 are 16. , , 8 and 4 are 12. 8 and 9 are 17. 8 and 5 are 13. 8 and 10 are 18. QUESTIONS IN ADDITION, 27 LESSON XZIV. 1. A lady bought eight cents' worth of needles, and six cents' worth of thread ; how much did she lay out ? How many are 8 and 6 ? 2. Eight birds are sitting on a bush ; if ten more alight by their side, how many will there be ? What is the sum of 8 and 10 ? 3. If I spend six dollars for clothes, two dol- lars for books, and three dollars for groceries, how much do I spend in all ? 6, 2, and 3, are — . 4. Sarah has three rabbits, Kuth five, and Herman seven. If Sarah and Euth give theirs to Herman, how many will he have ? Add 3, 5, and 7. 5. If a dairy-maid makes eight pounds of butter one day, and four the next, how many does she make in all ? How many are 8 and 4 ? 6. How many boys will two benches hold, if one holds eight and the other nine? Add 8 and 9. 7. Add on your slate seventeen and eighty. 8. Add twenty to two hundred and twelve. Kead these numbers ; add them. (9) (10) (11) (12) (13) (U) (15) 63 57 91 40 21 35 Y2 82 81 88 88 - 85 84 85 145 157 28 ADDITION. LSSSON XXV. How many are one and nine ? f ttftftttt How many are nine and one ? ttttttttt t How many are 2 and 9 ? 9 and 2 ? How many are 3 and 9 ? 9 and 3 ? How many are 4 and 9 ? 9 and 4 ? How many are 5 and 9 ? 9 and 5 ? How many are 6 and 9 ? 9 and 6 ? How many are 7 and 9 ? 9 and 7 ? How many are 8 and 9 ? 9 and 8 ? 4- ^ Nine stars in one other, make how many Count and see. stars in all ? If there are nine cannon- balls in one pile, and ten in another, how many balls are there in both ? What is the sum of 9 and 10 J A jeweller has nine watches in his safe, and five on his counter ; how many has he in all ? 9 and 1 are 10. 9 and 6 are 15. 9 and 2 are 11. 9 and 7 are 16. 9 and 3 are 12. 9 and 8 are 17. 9 and 4 are 13. 9 and 9 are 18. 9 and 5 are 14. 9 and 10 are 19. QUESTIONS IN ADDITION. 29 LESSON XXVI. 1. If Harry has nine dollars in the bank, and four in his pocket, how much has he in all ? 2. A milkman sold one customer three quarts of milk, another six quarts, and a third seven ; how many quarts did he sell all three? What is the sum of 3, 6, and 7 ? 3. Blanche has two canary birds, Eose seven, and Alice three ; if Eose and Alice give theirs to Blanche, how many will she then have ? 4. If one field yields nine bushels of potatoes, and another ten, how much do both yield ? 5. If Frank gets nine good marks in the morn- ing, and five in the afternoon, how many does he get during the day ? How many are 9 and 5 ? 6. Nine currants on one string, and nine on another, make how many currants in all ? 9 and 9 are how many ? 7. There are nine eggs in one hen's nest, and six in another ; if we put the eggs together, how many will there be ? What is the sum of 9 and 6 ? 8. What is the sum of 1, 8, and 2 ? Eead the following numbers. Add them. (9) (10) (11) (12) (13) (14) 510 601 17 82 140 91 67 95 700 33 806 45 412 203 41 62 42 52 30 ADDITION. IiBSSON ZXVII. 10 1 1 10 How many are one and ten ? How many are ten and one ? How many are 2 and 10 ? 10 and 2 ? How many are 3 and 10 ? 10 and 3 ? How many are 4 and 10 ? 10 and 4 ? How many are 5 and 10 ? 10 and 5 ? How many are 6 and 10 ? 10 and 6 ? How many are 7 and 10 ? 10 and 7 ? How many are 8 and 10 ? 10 and 8 ? How many are 9 and 10 ? 10 and 9 ? How many are two tens ? Ans. Twenty. 10 How many are ten and ten ? 15 You have eight fingers and two thumbs ; I have the same. How many fingers and thumbs have we together? What is the sum of 8, 2, and 10? A dime is worth ten cents; a half-dime is worth five cents. How many cents are both together worth ? How many are 10 and 5 ? What is the sum of 7, 3, and 8 ? What is the sum of 6, 4, and 3 ? 10 and 1 are 11. 10 and 2 are 12. 10 and 3 are 13. 10 and 4 are 14. 10 and 5 are 15. 10 and 6 are 16 10 and 7 are 17. 10 and 8 are 18. 10 and 9 are 19 10 and 10 are 20. THOUSANDS. 31 LESSON ZXVIII. What is the greatest number that can be expressed with one figure? Ans. Nine (9). If you add 1 to 9, what do you get ? Ans. Ten (10). What is the greatest number that can be expressed with two figures ? Ans. Ninety-nine (99). If you add 1 to 99, what do you get ? Ans. One hundred (100). What is the greatest number that can be expressed with three figures ? Ans. Nine hundred and ninety-nine (999). If you add 1 to 999, what do you get ? Ans. One thousand (1000). How many figures are used in writing 1000 ? What figure shows the number of thousands ? Ans. The fourth figure — 1. How, then, do we write two thousand? Ans. By putting a 2 in the fourth place. Learn the thousands : — One thousand 1000 Two thousand 2000 Three thousand 3000 Four thousand 4000 Nine thousanc . Five thousand 5000 Six thousand 6000 Seven thousand 7000 Eight thousand 800.0 . . . 9000 32 EXERCISE IN NOTATION. LESSON XXIX. Learn the even hundreds between 1000 and 2000 :— 1100, one thousand one hundred, or eleven hundred. 1200, one thousand two hundred, or twelve hundred, 1300, one thousand three hundred, or thirteen hundred. 1400, one thousand four hundred, or fourteen hundred. 1500, one thousand five hundred, or fifteen hundred. 1600, one thousand six hundred, or sixteen hundred. IVOO, one thousand seven hundred, or seventeen hundred. 1800, one thousand eight hundred, or eighteen hundred. 1900, one thousand nine hundred, or nineteen hundred. 2000, two thousand. Write on your slate the numbers between 1000 and 1100, and then read them. Thus : 1001, one thousand and one. 1002, one thousand and two, &c "Write one thousand and ten. 1010. Write two thousand and ten. 2010. Write 1 thousand 2 hundred and ten. 1210. Write 8 thousand 2 hundred and ten. 8210. Write one thousand four hundred. 1400. Write six thousand four hundred. 6400. Name the places, going from right to left. Ans, Units, Tens, Hundreds, Thousands. Eead and then add the following numbers : — 1024 1031 1059 1400 1099 2024 3031 4000 5000 1600 NOTATION, NUMERATION. 33 lilSSSON XXX. What place denotes thousands ? Ans. The fourth place. How do you write five thousands, nine hun- dreds, seven tens, one unit ? Ans, 5 in the fourth place denotes five thou- sands ; 9 in the third place, nine hundreds ; 7 in the second place, seven tens ; 1 in the first place, one unit. Thus we get 5971. Eead this number. Ans. Five thousand nine hundred and seventy- one. Write the following, placing units under units, tens under tens, &c. : Four thousands, 8 hun- dreds, 6 tens, 2 units ; five thousands, 3 hun- dreds, 1 ten, 1 unit ; nine thousands, 2 hundreds, 8 tens, 7 units ; seven thousands, 4 hundreds. Eead the numbers just written. Mention the thousands, hundreds, tens, and units, in the following numbers : — 4629 ; 9780 ; 5111 ; 1236 ; 7405 ; 2293 ; 529 ; 18. When a naught stands in the third or second place, must you say no hundreds, no tens f Ans. No ; it is passed over in reading. Thus, 9007 is read nine thousand and seven. Eead 7012; 6099; 3003; 8064; 1011; 5002. 34: NOTATION, NUMERATION, COUNTING. LESSON XXXI. Write the following numbers with figures : — One thousand six hundred and fifty-three. Five thousand nine hundred and eighteen. Seven thousand eight hundred and ninety. Three thousand five hundred and seven. Eight thousand and forty-nine. Eead the following numbers : — 1321 ; 2906 ; 6870; 5012; 1864; 9990; 1007; 8700; 3401. When we count, one^ two^ three^ four^ &e., how do we get each number ? Ans, By adding 1 to the number going before. Now in stead of adding 1, add 2 each time : 1 7 13 19 25 31 37 3 9 15 21 27 33 39 5 11 17 23 29 35 41 and so on to 99. In adding units, if the sum is 9, what do we do ? Arts. We write the 9 in the units' place. If the sum is 10, what do we do ? Ans, 10 consists of units, 1 ten. 6 Hence we write in the first or units' place, _^ and 1 in the second or tens' place. 10 Whenever the sum consists of two figures, how must we place them ? Ans, Set the right-hand figure in the units' place, the left-hand figure in the tens' place. ADDITION. 36 LESSON XXXII. Add the following numbers : — (1) (2) (3) (4) (5) (6) (7) 4 5 8 7 9 4 9 3 4 13 111 2 3 4 2 2 11 12 3 4 5 6 7 26 Now suppose we have to add 26 and 17. 17 Begin at the right. 7 and 6 are 13 — three 43 units and 1 ten. We write 3 in the units' place, and as there are other tens we add the 1 ten in with them, making 4, which we write in the tens' place. Answer, 43. "When the sum of a column consists of two figures, which figure must we write under the column added ? Ans, The right-hand figure. "What must we do with the left-hand figure ? Ans. We must add the left-hand figure to the next column. What if thel*e are no more columns to be added ? Ans. Then we must write the left-hand figure in the sum, in the next place on the left. What do you write down, and what do you add to the next column, if the sum of a column is 96? Ifitis71; 80; 52; 11; 48; 84:; 17? 36 ADDING COLUMNS ALOUD. IiSSSON XXXIII. Look at tins example. Learn to add eacli column aloud, in the following manner: — 1959 1st column. — Two^ jive^ six^ eighty sev- 1032 enteen — write down 7, and add 1 to the c^c^^o ^6xt column. 1342 ^^ column. — One, jwe, seven, ten, Jlf- teen — write down 5, and add 1 to the next ^^^"^ column. 3d column. — One, four, six, nine, eighteen — write down 8, and add 1 to the next column. 4:TH column. — One, iAJOo, four, seven, eight, nine — write it down. Answer, 9857. Add the following in the same way, not on the slate, but aloud. (1) (2) (3) (4) (S) 112 1270 1492 1666 374 22 321 2101 1022 311 332 2011 2121 232 1002 232 3322 1060 2432 2123 211 1346 2714 4212 5463 909 8270 9488 9564 9273 Commencing with 2, give the numbers formed by adding 2 each time, up to 100. Thus : — 2 6 10 14 18 22 26 4 8 12 16 20 24 28, &c. ADDING THREE. 37 LESSON XZXIV. Add the following numbers. When the left- hand figure of any sum is to be added to the next column, write it in the space under the column. (1) (2) (3) (4) 1228 3222 1252 2122 1654 222 2202 1422 2343 2222 1942 1465 3421 3965 2749 4358 How much are 3 and 3 ? 3 and 13 ? How much are 3 and 4? 3 and 34 ? How much are 3 and 5 ? 3 and 45 ? How much are 3 and 6 ? 3 and 66 ? How much are 3 and 7? 3 and 87? Which is greater, 900 or 700 ? 900 or 1000 ? Which is greater, 1 hundred or 1 thousand ? Commencing with 3, give the numbers formed by adding 3 each time : — 3 18 33 48 6 21 36 51 9 24 39 54 12 27 42 57 15 30 45 60 If I have 27 roses, and buy 3 more, how many will I then have ? If some one gives me three more, and I raise 3 from slips, how many will I then have in all ? 63 78 93 66 81 96 69 84 99 72 87 102 75 90 105 38 ADDING FOUR. LESSON XXXV. How many are 4 and 4 ? 4 and 14 ? 4 and 24? 4and34? 4and44? 4and54? 4and64? How many are 4 and 5 ? 4 and 75 ? 4 and 85 ? How many are 4 and 6 ? 4 and 86? 4 and 96? How many are 4 and 7? 4 and 47? 4 and 67? Robert has 8 fish in his basket and 4 on the grass ; how many has he in all ? How many are 8 and 4? 18 and 4? 28 and 4? 48 and 4? How many are 4 and 9? 4and89? 34and9? Commencing witli 4, give the numbers formed J by adding 4 each time. Thus:- — 4 20 36 52 68 84 8 24 40 56 72 88 12 28 44 60 76 92 16 32 48 64 80 96 ADDING FIVE. 39 LESSON XXXVI. How many are 5 and 5 ? 5 and 15 ? 5 and 25 ? 5 and 45 ? How many are 5 and 6 ? 5 and 26 ? 5 and 36 ? 5 and 46? 5 and 56 ? 55 and 6 ? 5 and 66 ? 65 and 6 ? How many are 5 and 7 ? 5 and 77 ? How many are 5 and 8 ? 5 and 88 ? How many are 5 and 9 ? 5 and 29 ? 25 and 9? 5 and 49? 45 and 9? 5 and 59 ? 5and69? Commencing with 5, give the numbers formed by adding 5 each time. Thus : — 5 20 35 50 65 80 95 10 25 40 55 70 85 100 15 30 45 60 75 90 105 What do these numbers end in ? 1. John had twenty cents ; his mother gave him five, and his father five more. How many had he then? How many are 20, 5, and 5 ? 2. Forty-eight apples in one basket, and five in another, make how many apples in all ? 3. A traveller rode thirty-nine miles one day, and five miles the next. How far did he ride both days ? Add 39 and 5. 4. I have just found five pins, said Eli, and I had seventeen before. How many have I now ? 40 ADDITION. LESSON XZXVII. How many are 6 and 6? 6 and 16 ? 46 and 6 ? How many are 6 and 7 ? 6 and 27 ? 27 and 6 ? How many are 6 and 8 ? 6 and 88 ? 38 and 6 ? 58 and 6 ? How many are 6 and 9 ? 6 and 99? 29 and 6? How many are 7 and 7 ? 7 and 47 ? 87 and 7? How many are 7 and 8? 7 and 58? 28 and 7 ? How many are 7 and 9 ? 69 and 7 ? How many are 8 and 8 ? 8 and 78 ? 18 and 8? How many are 8 and 9 ? 8 and 99? 68 and 9 ? 69 and 8 ? How many are 9 and 9 ? 9 and 59? 79 and 9? How many are 10 and 2 ? 40 and 2 ? 6 and 10 ? 6 and 60 ? 10 and 8 ? 90 and 8 ? 10 and 9 ? 9 and 100 ? 30 and 3 ? 1. There are twenty-nine girls in one class, and eight in another ; how many in both ? 2. How many apple-trees have I on my farm, if there are 27 in one field and 6 in another ? 3. A half dollar is worth fifty cents, and a dime ten cents. How much are both worth ? 4. Nine pigeons join a flock of thirty-eight. How many does the flock then contain ? 5. Sixteen roses in one bouquet, and eight in another, make how many in both ? EXAMPLES IN ADDITION. 41 LESSON XXXVIII. Eead the following numbers in turn. Add aloud (not on the slate), as shown on page 36. (1) (2) (3) (4) 546 145 834 689 8063 2083 1045 1989 789 1978 626 3754 4261 939 3997 1965 825 1987 1698 876 94:84 7132 8200 9273 Some examples for the slate follow. Be sure to set units under units, tens under tens, &c. 1. Add ninety-nine ; eighty-seven ; fifty-six ; twenty-four ; and one hundred and eight. 2. Add four hundred and fifty ; nineteen ; eight thousand six hundred and nine ; seven hun- dred and twenty-three ; and eighty-four. 3. Add two thousand four hundred and twenty-one ; nine hundred and eighty-seven ; eighteen hundred and ninety-nine ; three thou- sand and sixteen ; nine hundred and ninety-nine. 4. Add three thousand and eight ; nine hun- dred and ninety ; seventeen hundred and fifty-six ; five hundred and ninety-seven; two thousand eight hundred and eighty-eight. What is Numeration? What is Notation? What is Addition ? 42 SUBTEACTION. LESSON XXZIZ. Two pigeons were on the ground. One has flown away ; how many are left ? Two cows were standing near a pond. One of them has gone into the water ; how many are left on the bank ? Take 1 from 2, and what is left What is this process called ? Ans, Subtraction. What is Subtraction ? Ans, Subtraction is the process of taking one number from another. When we count one, two, three, four, &c., how do we get each number ? Ans, By adding 1 to the number going before. When we count backward, ten, nine, eight, &c., how do we get each number ? Ans, By subtracting 1 each time from the number going before. Count backward from 100 to 1. from 1 leaves 0. from 2 leaves 1. from 3 leaves 2. from 4 leaves 3. from 5 leaves 4. from from from from from 6 leaves 5. 7 leaves 6. 8 leaves 7. 9 leaves 8. 10 leaves 9. SXJBTKACTION. 43 LESSON XL. What is Subtraction ? When we take one number from another, what is the result called ? Arts, The Remainder. Three boys were riding. Two got out to gather berries ; how many were left in the wagon ? Four dogs ran barking after the wagon. Two have turned back ; how many are lef^ there ? Five cows were standing in a field. Two have lain down ; how many remain standing ? 2 from 2 leaves 0. 2 from 3 leaves 1. 2 from 4 leaves 2. 2 from 5 leaves 3. 2 from 6 leaves 4. 2 from 7 leaves 5. 2 from 8 leaves 6. 2 from 9 leaves 7, 2 from 10 leaves 8. 2 from 11 leaves 9. 44 QUESTIONS IN SUBTRACTION. LESSON XLI. Which of two numbers must we take from the other, to find their difference ? Ans, We must take the less from the greater. 1. Susan has six apples, and gives away two ; how many has she left ? 2. If John has ten marbles, and loses one, how many has he then ? 3. A man having five sheep, sold one ; how many remained unsold ? 4. Mary gives a beggar two cakes. She had nine before ; how many has she left ? 5. If there are two boys in a class of seven children, how many girls are in the class ? 6. 2 from 4 leaves how many ? 2 from 24 ? 2 from 54 ? 2 from 74 2 7. Two from 6, how many ? 2 from 96 ? 8. Two from 8, how many ? 2 from 48 ? 2 from 78 ? 2 from 38 ? Commencing with 50, give the numbers ob- tained by subtracting 2 each time. Thus : 60, 48, 46, 44, 42, 40, 38, 36, 34, 32, 30, &c. EXAMPLES EOB THE SLATE, (1) (2) (3) (4) (5) From 98 891 437 1652 7234 Take 12 120 121 212 2112 Remainder SUBTRACTION. 45 LESSON ZLII. What is Subtraction ? What is the result obtained by Subtraction called ? There are 4 panes of glass in this barn win- dow. If 3 are broken, how many are unbroken ? There were five sheep in the barn-yard, but three have gone into the field. How many remain ? Count them in the picture. Nine chickens were on the ground ; 3 have gone to roost. How many are left on the ground ? from from from 3 leaves 0. 4 leaves 1. 5 leaves 2. from 6 leaves 3. from 7 leaves 4. 3 from 8 leaves 5. 3 from 9 leaves 6. 3 from 10 leaves 7. 3 from 11 leaves 8. 3 from 12 leaves 9. 46 SUBTRACTION". LESSON XLIII. Take four oxen from four oxen, and how many- are left ? Cover four over, and count the rest. Take four eggs from a nest con- taining five, and how many remain ? Cover 4, and count the rest. Here is a knife with six blades. If 4 of them are closed, how many are open ? 4 from 6 leaves how many ? Here are seven flowers on one stem, but four are broken ; how many are unbroken ? -X-X-X-X- -X--X-X-X- From eight stars take four stars ; how many are left ? A lady having nine wine glasses, broke four of them ; how many had she left ? Four from ten leaves how many ? iIitailJLiXiidL4L 4 from 4 leaves 0. 4 from 5 leaves 1. 4 from 6 leaves 2. 4 from 7 leaves 3. 4 from 8 leaves 4. 4 from 9 leaves 5. 4 from 10 leaves 6. 4 from 11 leaves 7. 4 from 12 leaves 8. 4 from 13 leaves 9. QUESTIONS IK SUBTKACTION. 47 LESSON XLIV. 1. Three from 7 leaves how many? 3 from 17? 3 from 37? 3 from 67? Three from' 8 leaves how many ? 3 from 88 ? 3 from 98 ? 3 from 28 ? 2. Three from 9 leaves how many ? 3 from 79? 3. Three from ten leaves how many ? Three from twenty ? 3 from 30 ? 3 from 40 ? 3 from 50 ? 4. Commencing with 60, give the numbers obtained by subtracting 3 each time. Thus : 60, 57, 64, 51, 48, 45, 42, 39, &c. 5. Twelve things make a dozen. If Jane buys a dozen needles, and gives away 3, how many does she keep ? If she loses 3 of these, how many has she then ? 6. A boy spends four cents ; how much has he left, if he had twelve cents at first ? 7. Amy is four years younger than Lucy, and Lucy is 13 ; how old is Amy ? 8. If there are eleven persons in an omnibus, and three get out, how many remain ? If four more get out, how many then are left ? Eead these numbers. Write them down on your slate. Subtract, and read the remainders. (9) (10) (11) (12) (13) From 647 983 1085 4293 7658 Take 434 423 1043 4131 3343 48 SUBTEACTION. LESSON XLV. Is 4: greater or less than 5 ? | | | | | | | 1 1 Can we subtract five from four ? Can we subtract five from five ? If we take five dolls away from five dolls, bow many are left ? Try and see. Whenever we subtract a number from itself, as 5 from 5, what is the remainder ? Ans, Nothing, or naught — 0. \jjf Njjt Five marks from ten marks leave I In I In ]^q^ many marks ? Cover five, and count the rest. Five marks from twenty marks leave how many marks ? Five from thirty leaves how many ? [jjj "^Hjj 5 from 40 ? 6 from 50 ? 5 from 60 % ^IHJ James had 6 books on his table ; 5 have fallen down. How many are left on the table ? 5 leaves 0. 6 leaves 1. 7 leaves 2. from 8 leaves 3. from 9 leaves 4. from from from from 10 leaves from 11 leaves from 12 leaves from 13 leaves from 14 leaves QUESTIONS IN SUBTEACTIOlSr. 49 LESSON XLVI. 1. Thirteen persons were dining together. Four got up and left ; how many remained ? 2. Maud is seven years old ; her brother, is five. Which is the older, and how much ? 3. A man who had fourteen cents in his pocket, gave away all but five to beggars. How much did he give away ? 4. Twelve persons were standing near a boiler which burst. Two were killed, and three were injured ; how many were unhurt ? 5. If I borrow eleven dollars, and pay back four, how many do I owe ? 6. Five from eight leaves how many ? 5 from 18 ? 5 from 28 ? 5 from 38 ? 5 from 48 ? 7. Five from nine leaves how many ? 5 from 69 ? 5 from 89 ? 5 from 79 ? 5 from 99 ? 8. K a boy who has thirty-seven hens, sells four, how many has he left ? 9. A certain boat holds 19 persons. If five are in it, how many empty seats are there ? 10. Five from 10 leaves how many ? 5 from 5 ? EXAMPLES FOR THE SLATE, (1) (2) (3) (4) (5) From 937 8165 4397 6028 5984 Take 425 5054 2154 4014 5234 50 SUBTKACTION'. LESSON XLVII. What is the smallest number that we can subtract 6 from ? How many does 6 from 6 leave ? Look at these seven mice. Six are running away; how many remain ? ^ Six from 7 leaves how many? 6 from 17? 6 from 27 ? 6 from 47 ? 6 from 67 ? This wheel has eight spokes. Six of them are whole ; how many are broken I Count and see. Six from 8 leaves how many ? 6 from 38 ? 6 from 58 ? 6 from 78 ? 6 from 98 ? Take six forks away from nine forks, and how many are left ? ^wC^c^r ^c /\ 7r^r^r^\ >^-»^-^ stars irom len stars leave how many stars ? 1 1 I I I I I I I I I Six marks from eleven marks leave how many marks ? 6 leaves 0. 7 leaves 1. 8 leaves 2. 9 leaves 3. 6 from 6 from 6 from 6 from 6 from 10 leaves 4. 6 from 11 leaves 5 6 from 12 leaves 6. 6 from 13 leaves 7 6 from 14 leaves 8 6 from 15 leaves 9. SUBTEACTION. 51 LESSON XLVIII. What is Subtraction? What is the answer to a question in Subtraction called ? What is the smallest number that we can subtract 7 from ? How many does seven from seven leave ? 7 from 27 ? 7 from 97 ? 7 from 57 ? 7 from 37 ? Eight fingers on two hands. If seven of them are held up straight, how many will be bent down ? How many does seven from eight leave ? 7 from 18 ? 7 from 88 ? 7 from 68 ? 7 from 48 ? 7 from 58? Here we have nine rin^s ' "n A n A A n A A A '" onastrbg. If you cover ^'J " ^ ^ " U U U 7 with your hand, how many will you see ? Seven from 9 leaves how many ? 7 from 79 ? Ten bees are flying round a hive. If 7 are on one side, how many are on the other ? Count and see. Seven from ten leaves how many ? 7 from 7 leaves 0. 7 from 8 leaves 1. 7 from 9 leaves 2. 7 from 10 leaves 3. 7 from 11 leaves 4. 7 from 12 leaves 7 from 13 leaves 7 from 14 leaves 7 from 15 leaves 7 from 16 leaves 62 QUESTIONS IN SUBTRACTION. IiESSON XLIX. 1. There were thirteen lights in a street. Six are blown ont ; how many are left burning ? 2. Henry broke 7 teeth ont of a comb. How many teeth were left, if there were 27 at first? 3. How many sngar-plums wonld you have to put with nine, in order to make fifteen ? 4. If a boy has eleven almonds, and eats all but seven, how many does he eat ? 5. A blind man received eight cents from one kind boy, and four from another. If he spent six cents on his way home, how many had he left? EXAMPLES FOB THE SLATE. 1. From ninety-seven subtract seventy-six. 2. Take six thousand and seventeen from eight thousand three hundred and forty-eight ? 3. From five thousand and seventy-two take three thousand and twenty-two. 4. A man having ninety-one dollars, bought a coat for twenty dollars ; how much had he left? 5. A ship with six hundred and sixty-nine passengers on board was wrecked. Four hundred and sixteen escaped ; how many were lost ? 6. A gale swept over a garden containing eighty-seven bean-poles. Seventy-six were blown down ; how many remained standing ? SUBTRACTION'. 63 LESSON L. What is the smallest number that we can take 8 from? How many does eight from eight leave? 8 from 108 ? 8 from 68 ? 8 from 18 ? 8 from 58 ? Nine marbles were placed in the centre of a ring. The first boy that shot, drove eight away from the centre ; how many were left there? 8 from 9 leaves how many ? 8 from 29 ? Ten trees were set out thus. If o o o o 8 afterwards died, how many re- © o mained ? Cover 8 over, and count o o o o the rest. 8 from 10 leaves — . Eleven spiders were on a wall, but eight have crawled down. How many remain there ? 8 from 11 leaves how many ? Take eight rings ^^ ^^^^ from twelve rings, and how many q q o o o o will be left? 8 from 8 from 8 from 10 leaves 8 from 11 8 leaves 0. 9 leaves 1. 2. leaves 3. 8 from 12 leaves 4. 8 from 13 leaves 8 from 14 leaves 8 from 15 leaves 8 from 16 leaves 8 from 17 leaves 54 SUBTRACTION. LESSON III. Here is a frame with wires stretched across it, and each wire runs through balls. Count the balls on the top wire. How many are there in all ? Now, if from these 10 balls, we push away 9 to the right, how many are left ? How many balls on the second wire ? Push away 9, and how many are left ? How many balls on the third wire ? Cover 9 with your finger, and how many do you see ? How many balls on the fourth wire ? Cover 9 over, and how many can you count ? How many balls on the fifth wire? Cover 9 over, and how many are left ? 9 from 9 leaves 0. 9 from 10 leaves 1. 9 from 11 leaves 2. 9 from 12 leaves 3. 9 from 13 leaves 4. 9 from 14 leaves 5. 9 from 15 leaves 6. 9 from 16 leaves 7. 9 from 17 leaves 8. 9 from 18 leaves 9. QUESTIONS IN SUBTEACTION, 55 LESSON LII. 1. A jar of lard weighed seventeen pounds. If the jar weighed eight pounds, what was the weight of the lard ? 2. I bought a dozen eggs, but eight of them were bad. How many were good ? 3. Sixteen boys went a skating. If nine of them fell on the ice, how many escaped falling ? 4. A farmer had eighteen cows. Four of them died, and he sold five ; how many were left? 5. Nine lemons fell from a tree on which thirteen were growing. How many remained ? 6. A fisherman who had seventeen lobsters, sold six in the morning and two in the afternoon. How many had he left ? 7. A boy having fourteen dollars in the bank, drew out all but eight. How much did he draw out? EXAMPLES FOR THE SLATE. (1) (2) (3) (4) (5) From 18 873 5268 9145 8460 Take 9 721 1045 8043 6250 (6) (7) (8) (9) (10) From 17 921 2397 5679 4396 Take 8 721 1163 4645 3291 56 SUBTE ACTION. LESSON LIII. From 652 Take 379 Look at this example. The lower number is the smaller ; hence it can be taken from the upper. Always begin to subtract at the right, 9 from 2. — We can not take 9 from 2, because 9 is greater than 2. So we add 10 to the 2, making it 12. 9 from 12 leaves 3. Write down 3 for the first figure of the remainder. Now, to balance the 10 units just added to 3 in the upper line, add 1 ten to the 7 tens in the lower line, — making 8. Subtract 8. 8 from 5. — ^We can not take 8 from 5. So add 10 to the 5, making it 15. 8 from 15 leaves 7. Write down 7 for the second figure of the re- mainder. To balance the 10 just added to the 5 tens in the upper line, add 1 hundred to the 3 hundreds in the lower line, — making 4. Subtract 4. 4 from 6 leaves 2. Write down 2 for the third figure of the remainder. Answer, 273. How, then, are we to proceed, when the lower figure is greater than the one above it ? Ans. When the lower figure is greater than the one above it^ we add 10 to the upper fi^gure^ sub- tract^ am^d then add 1 to the next lower figure. QUESTIONS IN SUBTRACTION^. 57 LESSON LIV. EXAMPLES FOR THE SLATE. (1) (2) (3) (4) (5) From 752 8364 9017 3429 8560 Take 479 2192 4108 3173 6936 (6) (7) (8) (9) (10) From 416 5832 7411 8267 6059 Take 273 1988 4143 3479 3421 11. There were seven hundred and live fish in a net. But, in drawing the net, ninety-six were lost ; how many fish were caught ? 12. Charles bought some meat for thirty-nine cents. He gave the butcher half a dollar (worth fifty cents). How much change did he get ? 13. A man who had four thousand and ten dollars, gave his son two thousand seven hundred and eight dollars. How much did he keep ? 14. A farmer sold eight hundred and twenty- five acres of land. How many had he left, if his- farm contained a thousand acres at first ? 15. Fanny had a present of a box containing 144 pens. If she used thirty-nine of them in a month, how many had she left ? 16. A drover sold 87 head of cattle out of a drove of 250. How many had he left ? 58 EXERCISE IN SUBTRACTION. LESSON LV. MENTAL EXERCISES. 1. A car started with 40 passengers. At tlie first station, 10 passengers got off and 3 got on. How many did it then contain ? 2. A rose-bush had 82 roses on it. The next day, 6 of these roses fell to pieces, and 9 new ones opened. How many were then on the bush ? 3. Leaving home with a hundred dollars in my purse, I spent five for a vest, five for a coat, and five for boots. How much had I left ? 4. Said Dick to Stephen, " Give me your nine chickens, and I shall have seventy-eight." How many chickens had Dick ? 5. A man rode 90 miles in 4 days. The first day he went 70 miles ; the second day, 10 ; the third day, 5. How far did he go the fourth day ? 6. Subtract the sum of 70, 10, and 5, from 90. 7. If Julia is twenty-three years old, and her brother is twenty-nine, what is the difference in their ages ? 8. Six, and five, and nine, and how many more, make twenty-eight ? 9. A man who had twenty cents, met five poor boys, and gave them each three cents. How many cents had he left ? 10. From 20 take the sum of 3, 3, 3, 3, and 3. MULTIPLICATION. 59 LESSON LVI. "What does once mean ? Arts, One time. What does injoice mean ? Ans, Two times. One bird, taken once, is one bird. Twice one bird is bow many birds ? Two birds, taken once, are two birds. Twice two birds are how many birds % "What is taking a number one or more times called ? Ans. Multiplying. Taking a number once is multiplying by 1. Taking a number twice is multiplying by 2. Taking a number 3 times is multiplying by 3. Multiplying is a short way of doing what ? Ans. Of adding a number to itself. Thus : — Twice 1 is 2. 1 and 1 are 2. Three times 1 is 3. 1 and 1 and 1 are 3. When we multiply, what is the result called ? Ans, The Product. When we say Twice one is twOy 2 is the product. What is Multiplication ? Ans. Multiplication is the process of taking a number a certain number of times. 60 MULTIPLICATION. Once 2 is 2. product ? LESSON LVII. One swan, taken once, is one swan. Once 1 is 1. If 1 is multiplied by 1, what is the product ? Two ducks, taken once, are two ducks. If 2 is multiplied by 1, what is the -/Hwr How many do three stars, taken once, make ? How much is once 3 ? If 3 is multiplied by 1, what is the product ? How many do four eggs, taken 2^. once, make ? How much is once 4 ? If 4 is multiplied by 1, what is the product ? Then, Once 1 is 1. Once 2 is 2. Once 3 is 3. Once 4 is 4. Once any number is the number itself. Learn the Table forward and backward. Once 1 is 1. Once 6 is 6 Once 2 is 2. Once 7 is 7 Odcc 3 is 3. Once 8 is 8 Once 4 is 4. Once 9 is 9 Once 5 is 5. Once 10 is 10 MULTIPLICATION. 61 LESSON LVIII. EXAMPLES FOB THE SLATE. (1) ^2) (3) (*) (S) Multiply 98 150 742 3018 6932 By 1 1 1 1 1 m • 1 • one aeer is how many deer ? How mucli is twice 1 ? Twice two trees are tow many trees ? How mucli is twice 2 ? \r \r x/ Twice three stars are how many stars ? "/{•■/(-"/r How much is twice 3 ? What is the same as multiplying a number by 2? Ans, Adding the number to itself. Twice 1 is 2. 1 and 1 are 2. Twice 2 is 4. 2 and 2 are 4. Twice 1 is 2. Twice 6 is 12. Twice 2 is 4. Twice 7 is 14. Twice 3 is 6. Twice 8 is 16. Twice 4 is 8. Twice 9 is 18. Twice 5 is 10. Twice 10 is 20. 62 MULTIPLICATION. LESSON LIX. 0, taken any number of times, is still 0. Twice is 0. 3 times is 0. 4 times is Go Now you have some examples for the felate. Read the numbers multiplied, and the products. (1) (2) (3) (*) (S) Multiply 34 121 620 4213 3004 By 2 2 2 2 2 6. How much is twice four hundred and three ? 7. Multiply two thousand one hundred and eleven by two. 8. If two thousand three hundred and forty one is multiplied by two, what is the product ? 9. How much is once seventy-six ? 10. Multiply four thousand and thirteen by 2, 11. What is the product of one and two thou- sand seven hundred ? 12. How much is twice five hundred and thirty-two ? 13. Multiply twelve hundred by two. 3 times 1 is 3. 3 times 2 is 6. 3 times 3 is 9. 3 times 4 is 12. 3 times 5 is 15. 3 times 6 is 18. 3 times 7 is 21. 3 times 8 is 24. 3 times 9 is 27. 3 times 10 is 30. MULTIPLICATION-. C3 LESSON liX. 1, If a cord of wood costs five dollars, what will three cords cost ? Model. — If 1 cord costs 5 dollars, 3 cords will cost 3 times 5 dollars, or 15 dollars. Answer, 15 dollars. Follow this model in all tlie mental exercises under Multiplication. 2, What will two books cost, at a dollar each ? 8. How much will a clerk, who gets eight dollars a week, earn in three weeks ? 4. At 10 dollars each, what will 2 desks cost ? 5. A man gave two beggars nine cents apiece ; how much did he give them both ? 6. If one woman can make three dresses in a week, how many dresses can three women make ? 7. I have two classes, each containing seven boys ; how many boys do both contain ? 8. If a barrel of flour lasts a family nine weeks, how long will three barrels last them ? 9. When oranges are worth two cents apiece, and lemons one cent, what will I have to pay for three oranges and two lemons ? A times 1 is 4. 4 times 2 is 8. 4 times 3 is 12. 4 times 4 is 16. 4 times 5 is 20. 4 times 6 is 24. 4 times 7 is 28. 4 times 8 is 32. 4 times 9 is 36. 4 times 10 is 40. 64 MULTIPLICATION. Ht LXI. If one girl can pick five quarts of berries in a morn- ing, how many quarts can four girls pick ? 2. There are four fingers on one hand ; how many are there on four hands ? 3. How many days are there in four weeks, there being seven days in one week ? 4. What will 4 combs cost, at 10 cents apiece ? 5. If one horse eats 8 quarts of oats in a day, how much will 4 horses eat at the same rate ? 6. A father gave each of his 4 sons 3 dollars ; how much did he give them in all ? 7. If one stage holds 9 men, how many men will 4 such stages hold ? Commencing with 5, give the numbers formed by adding 5 each time. Thus : 5, 10, 15, 20, 25, 30, &c. times 1 is 5. times 2 is 10. times 3 is 15. times 4 5 5 5 5 5 times 5 is 20. is 25. times times times times 6 is 30. 7 is 35. 8 is 40. 9 is 45. times 10 is 50. MITLTIPLICATIOIir. 65 liESSON LXII. How much is 5 times 4 ? How much is 4 times 5? Which is the greater? How much is 5 times 3 ? How much is 3 times 5 ? Which is the greater ? In finding the product of two numbers, does it matter which we multiply by ? 1. How many miles will a horse trot in 5 hours, if he trots 10 miles an hour ? 2. If Henry reads 5 books every week, how many will he read in 5 weeks ? 3. What are 5 pounds of pork worth, at 9 cents a pound ? 4. How many trees are there in 5 rows, con- taining 6 trees each ? 5. Louise writes 2 pages every day ; how many pages does she write in 5 days ? 6. There are 5 cages in the Museum, with 8 monkeys in each ; how many monkeys in all ? 7. Ned went 5 times to the station, and took 1 letter each time ; how many did he take in all ? 6 times 1 is 6. 6 times 2 is 12. 6 times 3 is 18. 6 times 4 is 24. 6 times 6 is 36. 6 times 7 is 42. 6 times 8 is 48. 6 times 9 is 54. 6 times 5 is 30. I 6 times 10 is 60. 66 MULTIPLICATION. LESSON LXIII. How mucli is 6 times 2 ? How mucJb is twice 6 ? Which is greater ? Which is greater, 6 times 4 or 5 times 5 ? Which is greater, 6 times 2 or 3 times 4 ? 1. Walking three miles an hour, how far will a man go in six hours ? 2. If one box of tea lasts a family six months, how long will six boxes last them ? 3. How many dollars are there in six bags, containing seven dollars each ? 4. One eagle is worth ten dollars. How many dollars are six eagles worth ? 5. How far will we get from Albany, sailing 6 hours in a boat that goes 9 miles an hour ? 6. Jane and her five sisters have each a canary bird. How many birds have they all ? 7. How many flowers are there in 6 nosegays, if each contains 8 flowers ? 8. Six pin-cushions, with five pins in each, will contain how many pins in all ? 7 times 1 is 7. 7 times 2 is 14. 7 times 3 is 21. 7 times 4 is 28. 7 times 5 is 35. 7 times 6 is 42. 7 times 7 is 49. 7 times 8 is 56. 7 times 9 is 63. 7 times 10 is 70. MULTIPLICATION". 67 LESSON LXIV. 1. How many panes of glass are there in 7 windows, containing 8 panes each ? 2. If a boy spends four cents a day, how mnch will he spend in a week, which contains 7 days ? 3. One fly has two wings ; how many wings have seven flies ? 4. What will 7 rings cost, at 9 dollars each ? 5. How many pages will Ella read in seven days, if she reads seven every day ? 6. How many dollars in seven eagles, allow- ing ten dollars to the eagle ? 7. How many boats pass a draw-bridge in 7 days, if five boats go through each day ? 8 times 1 is 8. 8 times 2 is 16. 8 times 3 is 24. 8 times 4 is 32. 8 times 5 is 40. 8 times 6 is 48. 8 times 7 is 56. 8 times 8 is 64. 8 times 9 is 72. 8 times 10 is 80. 68 MULTIPLICATION. LESSON LXV. 1. If these boys skate four miles an hour, how many miles will they skate in 8 hours ? 2. Twice every day Fred and Frank come with their sleds, to ride down hill. How many times do they come in eight days ? 3. If it takes 3 hours to cut a ton of ice from the pond, how long will it take to cut 8 tons ? 4. One day 8 sleigh-loads, of 9 persons each, came to the pond ; how many persons came ? 5. What will eight loads of evergreens cost, at six dollars a load ? 9 times 9 times 9 times 1 is 9. 2 is 18. 3 is 27. 9 times 4 is 36. 9 times 5 is 45. 9 times 9 times 9 times 9 times 6 is 54. 7 is 63. 8 is 72. 9 is 81. 9 times 10 is 90. MULTIPLICATION. 69 LESSON LXVI. (1) (2) (3) (4) (5) (6) Multiply 41 21 90 81 YO 621 By 5 6 7 8 9 4 205 2484 7. Which is greater, 9 times 4 or 7 times 5 ? 8. How many legs has one chair ? How many legs, then, will nine chairs have ? 9. How many fingers has one boy? How many fingers, then, will nine boys have ? 10. Ada's father gives her and her 8 brothers 6 cents each. How much does he give them all ? 11. I have five hens, and each hen has nine chickens. Hens and chickens, how many have 1? 12. How much will nine turkeys weigh, if their weight is eight pounds apiece ? 13. There are nine chains, each six feet long. What is the length of the whole nine ? 14. How many gallons in 9 ten-gallon casks ? Eepeat the even tens : 10, 20, 30, 40, 50, &c. 10 times 1 is 10. 10 times 2 is 20. 10 times 3 is 30. 10 times 4 is 40. 10 times 5 is 50. 10 times 6 is 60. 10 times 7 is 70. 10 times 8 is 80 10 times 9 is 90 10 times 10 is 100. ! 70 MULTIPLICATION. LESSON LXVII. 1 2 What is an easy way of multiply- 10 10 ing a number by 10 ? 10 20 Ans. Placing a naught after it? 1. Write down these numbers ; multiply them by 10 in the way just shown : — Eight hundred and forty-seven. Six hundred and seventy-nine. Two hundred and thirteen. Seven hundred and five. Thirty-three. Nineteen. Fifty-one. 2. Ten cents make a dime ; ten dimes make a dollar. How many cents, then, in a dollar? How many cents are seven dimes worth ? 3. If you can buy two apples for a cent, how many can you get for ten cents ? 4. Ann is five years old. If she lives to be ten times as old, what will be her age? 5. How many quarts will ten jars contain, if each holds three quarts ? 6. At 3 cents each, what will 10 oranges cost ? 7. There are nine inches in a quarter of a yard ; how many inches in ten quarters ? 8. If for ten days you get four good marks each day, how many do you get in all ? Review all the Tables, from page 60, till you can say them perfectly. MULTIPLYING BY ONE FIGUKE. 71 LESSON LXVIII. Multiply 734 By 6 \ Look at the above example. Always begin to multiply at the right. 6 times 4 is 24 — 4 units and 2 tens. Write the 4 units in the units' place, and add the 2 tens to the next product. 6 times 3 tens are 18 tens, and 2 tens are 20 tens — or 2 hundreds. Write in the tens' place, and add 2 hundreds to the next product. 6 times T hundreds are 42 hundreds, and 2 hundreds are 44 hundreds. Answer, 4404. When, in multiplying by a single figure, we get a product expressed by two figures, what must we do with them ? Ans. We must place the right-hand figure of the product under the figure rrmltiplied^ and add the left-hand figure to the next product. If 36 is your product, what do you do with each figure? If 63? If 72? Examples follow ; multiply as shown above, i Multiply 863 638 742 914 ^ By 4 5 7 9 3452 3190 5194 "8226 72 MTSCELLANBOFS EXAMPLES. IiESSON LXIZ. What] is Numeration ? What is Notation ? What is Addition ? What is Subtraction? What is Multiplication ? EXAMPLES FOB THS , SLATE. d of this Series arc pe k .'Hy gr- ^i , the ■ advance from step *to step is inductive and g d la^ ; h! g is anticipated, nothing equired to be supplied by the tt-^che^. The definitions are simple, the rules biief, the auii}-''^ uac ci Jibered with verbiage, the examples dravv?i frcm the pr. al r.at^ers of life tie arrangement th » mopt n.itural, .. .e iTiOih >dd taught the shortest possible. Every device is resorted to, to prevent the mere" TiiCchanical solving of examples; the pupil's mind is constantly kept on tbe alert, and his Arithmetic lesson is thus made an '• valuable nicntal* discipline. Teachers that want the best Books should examine the above. D..APPI-ETON & CO., New York.