11. Find the sum of .2 A., 3.1 R., and 4 P., in sq. yd.

Ans. 4840 sq. yd.

12. Find the difference between .75 A. and 1 R., in sq. yd. Ans. 2420 sq. yd.

Multiplication and Division of Monomials.

§ 190. In multiplication, the multiplier can be considered only as expressing repetitions of the multiplicand, or a part of the multiplicand.

§ 191. In Division, when the answer to be found is the number of times the dividend contains the divisor, or the part the dividend is of the divisor, these two terms must be taken in the same denomination of units.

For example, to find how many times 15 s. is contained in 3£.

Reducing, we find 3£=60 s.; then 15 s. in 60 s., 4 times. Or, 15 s.=15£=3£; and 3⁄4£ in 3£, 4 times.

13. How many yards of silk, at 5 shillings per yard, can be purchased for 9 £? Ans. 38 yd. 14. How many pounds of butter, at 9 pence per pound, can be purchased for 214 shillings? Ans. 28 lb. which contained 14 What part of the lot Ans. .195 of it.

15. A person having a lot of ground acres, sold 39 P. of it to his neighbor. did he sell?

16. A laborer who had 25 rods of ditching to execute, has accomplished 51 yards of it. What part of the whole work has he accomplished? Ans. .374' of it. 17. A farmer sowed 574 bushels of rye in a field, and found that it was at the rate of 3 pk. per acre. How many acres did he sow? Ans. 65 A.

18. An agriculturist bought, at one time, 2 T. of plaster, and at another, 15 cwt. How many acres of meadow can he sow with the whole, at the rate of 100 lb. per acre?

Ans. 62 A.

19. A wine merchant has 3 tuns and 1 pipe of wine, which he wishes to put into barrels of 31 gal. each. How many barrels will be required for the whole ? Ans. 28 bbl.

20. Bought at different times, in adjoining parcels, 3 A., 3 R., and 20 P. of ground. I wish to divide the whole into lots of 40 P. each; how many lots will there be?

Ans. 16 lots.

ADDITION OF POLYNOMIALS.

§ 192. Two or more polynomials in the same kind of measure, might be reduced to similar monomials, and, under the monomial form, might be added or subtracted.

In like manner, a polynomial might be multiplied or divided. But polynomials may be added, &c., under the polynomial form, such operations being usually called Compound Addition, Compound Subtraction, &c.

§ 193. For Polynomial or Compound Addition.

1. Set the polynomials with similar terms one under another, in separate columns.

2. Proceeding from right to left, add up each column of similar terms, and under each set its amount, if less than a unit of the next higher denomination.

3. If not less than such unit, divide the amount of the column by that number of its own denomination which makes a unit of the next higher; set the remainder, if any, under the column, and add the quotient to the next column of similar terms.

EXAMPLE.

To add together 13£ 7 s. 2 d., 49£ 18 s. 4 d., and 84£ 9 s. 5 d.

£ S. d.

13 7 2

49 18 4

84 9 5

147 14 11

Having set pounds under pounds, shillings under shillings, &c., we add up the column of d., and set down the amount 11 d., since it is less than 1 s., a unit of the next higher denomination.

Adding up the column of s., we find the amount to be 34 s., which, being more than 1£, we divide by 20, since 20s.=1£. Setting down the remainder 14 s. we add the quotient to the column of £, and find the whole sum to be

147£ 14s. 11 d.

By this Rule, an amount in a lower denomination is reduced,

when practicable, to units of the next higher denomination; and all units of the same denominaton are collected into one (§ 23).

sum.

EXERCISES.

1. Add together 125£, 13 s. 5 d., 19£, 4s. 10 d. 2 gr., and 12£, 16 s. 8 d. 3 qr.

2. Add together 23 lb. 8 oz. and 300 lb. 2 oz. 9 dwt. 13 gr. 3. Add together 3 T. 9 cwt. and 54 T. 7 cwt. 3 qr. 20 lb.

Ans. 157£, 15s. 0 d. 1 qr. 16 dwt., 36 lb. 5 oz. 8 dwt. 16 gr., Ans. 360 lb. 4oz. 14 dwt. 5 gr. 2 qr. 16 lb., 10 T. 15 cwt. 1 qr., Ans. 68 T. 12 cwt. 3 qr. 8 lb.

4. Add together 56 lb, 7 3,5 3, 2 §, 13 gr., 25 tb, 6 3, 7 3, and 5 lb, 103, 33, 19, 19 gr.

Ans. 88 b, 13, 0 3, 19, 12 gr. 5. Add together 13 bu. 2 pk. 7 qt. 1 pt., 150 bu. 1 pk. 5 qt., and Ans. 365 bu. O pk. 2 qt.

200 bu. 3 pk. 5 qt. 1 pt. 6. Add together 3 hhd. 20 gal. 3 qt., 29 hhd. 13 gal. 2 qt., and 200 hhd. 12 gal. 1 qt.

7. Add together 4 m. 5 fur. 20 34 m. 7 fur. 13 p. 1 yd.

8. Add together 15 yd. 3 qr. 100 yd. 1 qr. 2 na. 1 in.

Ans. 232 hhd. 46 gal 2 qt. p., 29 m. 3 fur. 16 p. 4 yd., and Ans. 69 m. O fur. 9 p. 5 yd. 1 na., 75 yd. 3 gr. 3 na. 1 in., and Ans. 192 yd. 0 qr. 2 na. 2 in.

9. Add together 24 A. 3 R. 20 P., 100 Ă. 2 R. 16 P. 4 sq. yd., and 95 A. 1 R. 29 P. 20 sq. yd.

Ans. 220 A. 3 R. 25 P. 24 sq. yd. 10. Add together 200 A. 1 R. 24 P. 20 sq. yd., 50 A. 2 R., and 500 A. 3 R. 19 P. 16 sq. yd.

Ans. 751 A. 3 R. 4P. 52 sq. yd. 11. A farmer raised from one field 150 bu. 3 pk. of wheat, from another 75 bu. 1 pk. 7 qt., and from another 200 bu. 5 qt. What quantity did he raise in all ? Ans. 426 bu. 1 pk. 4 qt.

12. A merchant has in one piece 34 yd. 3 qr. of cloth, in another 21 yd. 2 qr., and in two others each 19 yd. 3 qr. How many yards has he in the four pieces? Ans. 96 yd.

13. An agriculturist sold at one time 3 T. 19 cwt. 2 qr. of hemp, at another 5 T. 13 cwt., and at another 2 T. 16 cwt. 3 qr. 20 lb. What amount did he sell?

Ans. 12 T. 9 cut.

gr. 20 lb.

SUBTRACTION OF POLYNOMIALS.

RULE XXXVIII.

§ 194. For Polynomial or Compound Subtraction.

1. Set the less polynomial under the greater, with similar terms one under the other.

2. Proceeding from right to left, subtract each lower term from the one above it, and underneath set the remainder.

3. If the lower term exceed the upper, add to the upper term that number of its own denomination which makes a unit of the next higher; from the sum subtract the lower term, and add 1 to the next lower term, before subtracting it.

EXAMPLES.

1. To subtract 85£, 13 s. 7 d. from 250£, 9 s. 10 d.

£, S. d.

250 9 10

85 13 7

164 16 3

Having set the less polynomial under the greater, with pounds under pounds, &c., we subtract 7d. from 10d., and set down the remainder 3 d.

The next lower term 13 s. being greater than the upper term 9 s., we add 20 s. to 9 s., since 20 s.=1£. From the sum 29 s. we subtract 13 s., and set down the remainder 16 s.

Then adding 1 to the 5, we say 6 from 10 leaves 4, &c.

2. To subtract 45£, 7s. 3 d. from 150£.

£, S. d.

150 0 0

45 7 3

104 12 9

As we cannot take 3 d. from 0 d., we add 12 d. 1 s., and

say 3 d. from 12 d. leaves 9 d. Then 1 to 7 s. makes 8 s., and since we cannot take 8 s. from 0 s., we add 20, and say 8s. from 20 s. leaves 12 s. Then 1 to 5 makes 6, and 6 from 10

leaves 4, &c.

The difference between two polynomials is not changed by adding, to any upper term, a quantity equal to the unit added to the next lower term.

EXERCISES.

1. From 60£, 17 s. subtract 35£, 13 s. 6 d.

Ans. 25£, 3 s. 6 d.

2. From 200 lb. 9 oz. 1 dwt. subtract 180 lb. 10 oz.

Ans. 19 lb. 11 oz. 1 dwt.

3. From 150 T. 13 cwt. subtract 75 T. 3 cwt. 1 qr.

Ans. 75 T. 9 cwt. 3 qr.

4. From 433 b, 3 3, 2 3, subtract 93 b, 10 3.

Ans. 339 b, 5 3,2 3.

5. From 100 bu. 2 pk. subtract 21 bu. 1 pk. 1 qt.

Ans. 79 bu. 0 pk. 7 qt.

6. From 21 T. 2 hhd. 3 gal. subtract 3 T. 13 gal.

Ans. 18 T. 1 hhd. 53 gal.

7. From 275 L. 2 m. subtract 75 L. 1 m. 5 fur.

Ans. 200 L. 0 m. 3 fur.

8. From 150 yd. 3 qr. 2 na. subtract 2 qr. 3 na.

Ans. 150 yd. 0 qr. 3 na.

9. From 123 A. 2 R. subtract 30 A. 3 R. 13 P.

Ans. 92 A. 2 R. 27 P.

10. From 200 A. 3 R. 20 P. subtract 30 A. 50 P.

Ans. 170 A. 2 R 10 P.

11. A jeweler purchased 34 lb. 9 oz. 13 dwt. of silver ware, of which he has sold 19 lb. 4 oz. 18 gr. What quantity has he remaining? Ans. 15 lb. 5 oz. 12 dut. 6 gr.

12. An agriculturist raised 30 T. 13 cwt. 1 gr. of hemp, of which he has sent to market 21 T. 15 cwt. 21 lb. What quantity of hemp has he still on hand?

Ans. 8 T. 18 cwt. 0 qr. 7 lb. 13. A farmer raised 500 bu. 3 pk. 7 qt. of wheat. Having sold 300 bu. 2 pk. 5 qt. of this crop, what quantity of wheat has he still unsold? Ans. 200 bu. 1 pk. 2 qt.

14. A vintner purchased 3 T. 1 hhd. 40 gal. 2 qt. of wine, of which he has sold 1 T. 3 hhd. 47 gal. 1 pt. What quantity of wine has he yet unsold? Ans. 1 T. 1 hhd. 56 gal. 1 qt. 1 pt.