PROOF OF MULTIPLICATION. Divide the product by the multiplier, and if the quotient is equal to the multiplicand, the work may be considered right. PROOF OF DIVISION. Multiply the quotient by the divisor, and if the product is equal to the dividend, the work may be considered right. Q. How may a denominate number be divided? How do you set down the number to be divided? How do you then divide? When there is a remainder what do you do with it? Of what denomination will each of the quotients be? How do you prove multiplication? How do you prove division? NOTE. When the divisor does not exceed 12 the division may be made after the manner of short division in simple numbers. 2. Divide £25 15s 4d by 8. OPERATION. 8) £25 15s 4d We first say 8 into 25, 3 times and £1 or 20s over. Then after adding the 15s, we say, 8 into 35, 4 times and 3s over. Then reducing the 3s to pence and adding in the 4d, we say 8 into 40, 5 times. £3 48 5d Q. When the divisor does not exceed 12, how may the division be performed? NOTE. When the divisor is a composite number, and exceeds 12, the work may be shortened by dividing by the factors in succession, as in division of simple numbers. EXAMPLES. 1. Divide £28 2s 4d by the composite number 21. Here the factors are 3 and 7. OPERATION. 7)£28 28 4d £4 Os 4d OPERATION. 3)£4 Os 4d £1 68 91d. Hence, the answer sought is £1 6s 9}d. Q. When the divisor is a composite number, how may the division be performed? 2. Divide £57 3s 4d by 35=5×7. 3. Divide £85 4s by 72. 4. Divide £31 2s 101d by 99. APPLICATIONS. Ans. £1 12s 8d. Ans. £ S d. Ans. 6s 31d. 1. Bought 65 yards of cloth for which I paid £72 148 4d: what did it cost per yard? Ans. £1 2s 44d. 2. Bought 64 gallons of brandy for £30 8s: what did it cost per gallon? Ans. 9s 6d. 3. Bought 144 reams of paper for £96: what did it cost me per ream? Ans. S d. 4. Sixty-three barrels of sugar contain 7 T. 16cwt. 3qr. 217b.: how much is there in each barrel? Ans. 2cwt. 1qr. 27lb. 5. A farmer has a granary containing 232 bushels 3 pecks 7 quarts of wheat, and he wishes to put it in 105 bags: how much will each bag contain? Ans. bu. qt. 6. One hundred and seventy-six men consumed in a week 13cwt. 3qr. 1lb. 6oz. of bread: how much did each man consume? Ans. 8lb. 12oz. 2dr. APPLICATIONS IN THE FOUR RULES. Mr. James Sears, Albany, July 1, 1837. Bought of Albert Titus. 3lb. of green tea at 7s 6d per pound, 27yd. of muslin at 1s 6d per yard, 4cwt. of sugar at £2 2s 8d per cwt. 2hhd. of molasses at 2s 6d per gallon, 67b. of raisins at 1s 7d per pound. Received payment, Albert Titus. £27 18s 2d. 2. A gentleman purchased of a silversmith, 2 dozen silver spoons each weighing 3oz. 4put. 1gr.; 2 dozen of tea spoons, each weighing 15pwt. 16gr.; 3 tankards each weighing 22oz. 14pwt. He sold him old silver to the amount of 6lb. 10oz. 3pwt.; how much remained to be paid for? Ans. 6lb. 9oz. 12pwt. 3. What will be the cost of 22 tons of hay, at £2 1s 10d per ton? Ans. £46 0s 4d. 4. If two hogsheads of wine cost £67 4s: what does it cost per gallon? Ans. s d. 5. If 4cwt. of sugar cost £14: what is it per pound? Ans. 74d. 6. A man paid £67 4s for a pile of wood containing 64 cords; he sold 30 cords for £29 16s: for how much must he sell the remainder per cord so as not to lose? Ans. £ S. 7. If 78cwt. 3qr. 10lb. of sugar be equally divided among 5 men, what will be each one's share? Ans. 15cwt. 3qr. 2lb. 8. A printer uses one sheet of paper for every 16 pages of an octavo book: how much paper will be necessary to print 500 copies of a book containing 336 pages, allowing 2 quires of waste paper in each ream? Ans. 24 reams 5 quires 12 sheets. 9. A farmer wishes to divide 108 acres into 8 equal fields: how much will there be in each field? Ans. A. R. 10. Out of a pipe of wine, a merchant draws 12 bottles, each containing 1 pint 3 gills: he then fills six 5 gallon demijohns; then he draws off 3 dozen bottles, each containing 1 quart 2 gills: how much remained in the cask? Ans. 82gal. 1pt. 11. A man lends his neighbor £135 6s 8d and takes in part payment 4 cows at £5 8s apiece, also a horse worth £50: how much remained due? Ans. £ S. d. 12. A farmer has 6T. 8cwt. 2qr. 14lb. of hay to be removed in 6 equal loads: how much must be carried at each load? Ans. 1T. 1cwt. 1qr. 21lb. 13. A person at his death left landed estate to the amount of £2000, and personal property to the amount of £2803 17s 4d. He directed that his widow should receive one eighth of the whole, and that the residue should be equally divided among his four children. What was the widow's and each child's portion? £600 9s 8d. Widow's portion OF VULGAR FRACTIONS. Before proceeding farther let the pupil study carefully from $ 42 to Denominate Numbers.) 1. If a unit be each part called? 1 SECTION I. divided into two equal parts, what is How do you express one of the parts?, 2. If a unit be divided into three equal parts, what is each part called? How do you express one of the parts? How do you express two of them? How do you express three of them? 3. If a unit be divided into four equal parts, how do you express one of the parts? Two of the parts? Three of the parts? Four of the parts? 4. How many halves are there in one thing? many fourths or quarters are there? is a half than a quarter? How How much greater 5. If a unit be divided into five equal parts, what is each part called? How do you express three of the parts? Four of them? Five of them? 6. If a unit be divided into six equal parts, what is each part called? How do you express one-sixth? How do you express two of the parts? Three of them? How do you express six of them? 7. How many thirds are there in a unit? How many sixths are there? How much greater then is one-third than one-sixth? Two 8. If a unit be divided into seven equal parts, what is each part called? How do you express one part? parts? Four parts? Six parts? Seven parts? 9. If a unit be each part called? divided into eight equal parts, what is How do you express four of the parts? Five of them? Six of them? Seven of them? Eight of them? 10. How many fourths or quarters are there in a unit? How many eighths are there? How much greater, then, is a quarter than an eighth? How many eighths are equal to two quarters? How many to three quarters ? 11. If a unit be divided into nine equal parts, what is each part called? How do you express one part? Two parts? Four parts? Six parts? Nine parts? 12. If a unit be divided into ten equal parts, how do you express one of the parts? How do you express two of them? Ten of them? How 13. How many fifths are there in one unit? many tenths are there? How much greater is a fifth than a tenth? How many tenths are equal to two-fifths? How many tenths are equal to four-fifths? To five-fifths? 14. If a unit be divided into eleven equal parts, what is each part called? How do you express one of the |