3. Divide £25 15s 10d equally among 8 persons.

In this example we find that 8 is contained in £25, 3 times and £1 over. Now this £1 has yet to be divided by 8, as well as the 15s and 10d. Then by multiplying the £1 by 20 and adding the 15s gives 35s, which contains 8, 4 times and 3s over. Multiplying the 3s by 12 and adding in the 10d, gives 46d, which contains 8, 5 times and 6d over. The 6d being reduced, gives 24 farthings which contains 8, 3 times. Therefore, each of the denominations has been divided by 8.

OPERATION.

8)£25 15s 10d(£3

24

£ 1

20
8)35s(4s

32

3s

12

8)46(5d
40

6d

4

8)24far.(3far. Ans. £3 4s 53d.

§ 76. Therefore, a denominate number may be divided into any number of equal parts by dividing each of its denominations by the divisor.

RULE.

I. Set down the number to be divided in the order of its denominations from the highest to the lowest, and write the divisor on the left.

II. Find how often the divisor is contained in the figures of the highest denomination.

III. Reduce the remainder, if there be any, to the next lower denomination, and add the figures of the dividend expressing that denomination, and then divide the sum by the divisor.

IV. Proceed in the same way for all the denominations to the last, and if there be a remainder place the divisor under it, as in division of simple numbers. Each of the quotients will be of the same denomination as its dividend, and the several quotients connected together will be the entire quotient sought.

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 4s 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 6s 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 34d.

1. Bought 65 yards of cloth for which I paid £72 14s 44d: 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.

Albany, July 1, 1837.

Mr. James Sears,

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.

2. A gentleman purchased of a silversmith, 2 dozen silver spoons each weighing 3oz. 4pwt. 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 67b. 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. 7d.

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?

Widow's portion

£600 9s 8d.

Ans. Each child's portion £1050 16s 11d.

OF VULGAR FRACTIONS.

(Before proceeding farther let the pupil study carefully from $ 42 to Denominate Numbers.)

$77. There are five kinds of Vulgar Fractions, Proper, Improper, Simple, Compound, and Mixed.

A PROPER FRACTION is one in which the numerator is less than the denominator. The value of every proper fraction is less than 1. (See § 44.)

The following are proper fractions: