(133.) To Divide one Duodecimal Polynomial by another. The method of doing this is shown in the following

EXAMPLE.

To find the breadth of a surface whose area is 40sq. ft. 3' 4" and length 16ft. 8in.

16f. 8')40f. 3′ 4′′ (2ft. 5′

33f. 4'

6f. 11' 4"

6f. 11' 4"

0 0 0

We divide the left hand term, 16ft., of the divisor into the left hand term, 40ft., of the dividend, and obtain the quotient term 2ft. We multiply the entire divisor by the 2ft., and subtract the product 33f. 4' from the corresponding part of the dividend.

To the remainder, 6f. 11', we subjoin the next term, 4" of the dividend; we now divide 16ft. into 6f. 11' which is 83', and obtain the quotient term 5'; we multiply the entire divisor by the 5', and find that the operation is completed.

The proper index for any quotient term may always be known, by considering that the left hand term of the divisor

the quotient term must produce a term with the same index as that into which the division is made; thus 16ft. divided into 83' gives 5', because, 16ft. x 5' produces 80' (131).

Without employing Duodecimals, the given polynomials might be taken in feet and fractions of a foot.

9. The area of a surface is 228 sq. ft. 8' 2'', and its length

is 30ft. 10in.; what is its breadth ?

10. The area of a surface is 510 sq.ft. breadth is 6ft. 4in., what is its length?

Ans. 7ft. 5in. 10' 8", and its

Ans. Soft. 8in.

ALIQUOT PARTS.

(134.) An ALIQUOT PART of a quantity is an exact half, third, or fourth, and so on, of the quantity.

Thus 10s. is an aliquot part of 1£, being of 1£.

It is often convenient to regard the lower orders of units in a polynomial multiplier as aliquot parts of one or more of the higher units.

EXAMPLE.

To find the value of 2lb. 5oz. 12dwt. of silver ware, at $45.12 per lb.

$45.125

2

The value of 2lb. is 90.250 ; twice the value of 1lb.

$111.3082, value of the whole.

For the aliquot parts we say, 4oz. is of a lb., 1oz. is of 4oz., 10 dut. is of an oz., 2dwt. is of 10 dwt.—The values of these several parts, added to the value of 2lb., make up the value of the whole quantity.

For convenience, the aliquot parts should be so apportioned, when practicable, that none of the divisors shall exceed 12.

No general Rule can be given for calculating by aliquot parts; but practice will soon render the method easy.

EXERCISES.

1. Find the value of 3lb. 4oz. 17dwt. of jewelry, at $50.50 per lb.

Aliquot parts may be taken for the oz. and dwt. thus; 4oz. is of a lb., 10dwt. is of 4oz., 5dwt. is 183

2dwt. is of 10dwt.

of 10dwt., and

Ans. $171.909'.

2. Find the sum that should be paid for 13cwt. 2qr. 14lb. of soap, at $3,621 per cwt.

Ans. 49.39'.

3. Find the sum that should be paid for 3 T. 10cwt. 3qr. of iron, at $30.37 per ton. Ans. $107.45'. 4. A farmer sold 125bu. 3pk. 1qt. of wheat, at $0.87 per bushel. What did the whole amount to?

Ans. $110.058. 5. A merchant sold 10yd. 3qr. 2na. of silk, at $1.50 per yard. What did the whole amount to?

Ans. $16.3125. 6. A townsman bought a lot of ground containing 3A. 2R. 25 P., at $75 per acre. What did he pay for the lot?

7. An agriculturist sold 19cwt. 3qr. $7.50 per hundredweight. What did the

Ans. $274.218'. 21lb. of hemp, at

hemp amount to? Ans. $149.53'.

8. A farmer bought 12bu. 2pk. 5qt. 1pt. of clover seed, at $10 per bushel. What did the whole amount to?

Ans. $126.718'.

9. Allowing an acre of ground to produce 30 bushels of wheat, what would be the produce of a field containing 204. 1R. 24P.? Ans. 612 bushels.

10. A merchant sold 15yd. 3qr. 3na. of silk, at 1.37 per yard, and 5yd. 2qr. of lace, at $2.50 per yard. What did the whole amount to ? Ans. $35.661'.

11. Find what would be the expense of putting up 230r. 23yd. of fencing, at the rate of $0.75 per rod.

Ans. $172.875.

12. Allowing an acre of meadow ground to produce 2T. 16cwt. 2qr. of hay, what would be the produce of a meadow containing 12A. 2R. 10P.? Ans. 35 T. 9cwt. 3 qr.

MISCELLANEOUS EXERCISES

ON MONOMIALS AND POLYNOMIALS, CURRENCIES, DUODECIMALS, AND ALIQUOT PARTS.

1. What will 2lb. 8oz. 13dwt. of silver ware amount to at the rate of $0.314 per dwt.? Ans. $204.0625.

2. What will 2bu. 3pk. 3qt. of strawberries amount to at the rate of $0.12 per quart? Ans. $11.375. 3. What will 2hhd. 40gal. 3qt. of beer amount to, if retailed at $0.03 per pint? Ans. $35.70. 4. What will 3bar. 16gal. 3qt. of brandy amount to, if retailed at $0.061 per gill? Ans. $222.50.

5. Find the expense of putting up 1m. 3fur. 20r. of fencing, at the rate of $0.75 per rod. Ans. $345.00. 6. At the rate of 3 gills per day, what quantity of brandy will a toper drink in a year of 365 days?

Ans. 34gal. 1pt. 3gr. 7. At the rate of 5 pints per day, what quantity of milk will a family consume in a year of 365 days?

Ans. 228gal. Oqt. 1pt. 8. At the rate of 2006. per acre, what quantity of plaster will be required to sow a field containing 95 acres?

Ans. 8T. 9cwt. 2qr. 16lb.

9. A farmer sold corn to the amount of $100, which he laid out for wheat at 75 cents per bushel. How much wheat did he purchase? Ans. 133hu. 1pk. 2qt. 11pt.

How

10. A merchant invested the profits of five years' business, amounting to $7349.314, in land at $24.12 per acre. much land did he purchase? Ans. 304A. 2R. 21.44P.

11. What will 4cwt. 3qr. 1976. of hemp amount to, at $6.87 per hundredweight?

The 3qr. 1976. may be reduced to a decimal of a cut. (118... 2), and the price per cwt. be then multiplied by the whole quantity in cwt.; or the calculation may be made by means of aliquot parts. Ans. $33.818'.

12. Required the sum that should be paid for 13yd. 1qr. 3na. of lace, at $0.93 per yard. Ans. $12.597'. 13. Required the sum that should be paid for 10 T. 13cwt. 2qr. 23lb. of coal, at $5.37 per ton.

Ans. $57.431'.

ton, and sold What profit Ans. $11.928'.

14. An iron-monger bought iron at $45 per 13cwt. 2qr. 15lb. of the same at $62 per ton. was made on the quantity sold?

15. A merchant bought in New York 135 yd. 3qr. of linen, at 2s. 3d. per yard, and 74yd. 21⁄2gr. of silk at 6s. 9d. per yard. Required the whole amount in Federal Money.

Ans. $101.144'.

16. A farmer sold in Philadelphia 400bu. 3pk. of wheat, at 6s. 11d. per bushel, and 1751 bu. of oats, at 3 shillings per bushel. Required the whole amount in Federal Money. Ans. $439.644'.

17. A lot of broadcloth imported from Liverpool amounted to 125£. 16s. 10d. sterling. Required the amount in Federal Money, according to the legal value of the pound sterling in the United States. Ans. $609.07'.

18. A farmer sold 3 T. 16cwt. 3qr. 21lb. of hemp, at $5 per cwt., and invested the proceeds in land at $37 per acre. What quantity of land did the farmer purchase?

Ans. 10A. 1R. 1.28P