J. (1) A person bought 49 yds. 2 qrs. of cloth for £19 3s. 71d., at what rate must he sell 15 yds. 3 qrs. to gain £1 175. 14d. by the whole?

(2) Bought 150 yds. of calico for £3 1s. 1 d., but yards being spoiled, what should be charged for 12 yds. I qr. of the remainder so as neither to gain nor lose?

(3) A farmer bought 40 sheep at £2 7s. 41d. each, but 4 of them having died, how many of the remainder should he sell for £21 9s. 8d. to gain at the rate of £4 3s. 6d. upon the whole?

(4) Bought 5 hhds. 35 gallons of wine for £166 5s., and paid duty and other charges £23 175. Id., at what rate should 28 gallons be sold to gain £30 upon the whole?

(5) How much water must be mixed with 144 gallons of brandy worth 34s. per gallon to reduce the price to 32s. per gallon?

(6) A person mixes 48 lbs. of tea at 3s. 6d. per lb. with 60 lbs. at 2s. 8d. per lb., how many lbs. of the mixture must he sell for £2 13s. 9d. to gain £2 195. upon the whole?

(7) A person bought a piece of cloth for £21 10s., but 4 yards of it being spoiled he sold the remainder at the rate of 9 yards for £2 18s. 6d., thus gaining £3 4s. upon the whole; how many yards did the piece contain?

(8) After paying an income-tax of 7d. in the pound a person has £347 Is. 51d. left; what is his income? (9) A chemist buys 1 lb. 8 oz. (avoir.) of drugs for 7s. 3d., what should he charge per ounce (apoth.) to gain double his outlay?

(10) A draper buys linen at 7s. 6d. per English ell and retails it at 7s. 6d. per yard; how many yards must he sell to gain 19s. 6d. ?

163

CHAPTER IV.

PRACTICE.

22. When the first of the three given terms of a proportion is I or a unit, the fourth term or answer may generally be found by an abbreviated process called Practice.

23. This rule is so called because of its frequent use by practical men in commercial calculations. It is of two kinds, simple and compound; simple, when only one of the middle terms of the proportion is a compound quantity; compound, when both of the middle terms are compound quantities.

24. An aliquot part of a quantity is such a part as when taken a certain number of times will exactly make up that quantity. Thus 6d., 4d., 3d., 2d., are aliquot parts of 1s.; 14 lbs., 7 lbs. 4 lbs. of 1 qr., and 9, 6, 3, 2 of 18, &c.

25. The same general principle-viz., the finding of the cost, &c., of any aliquot part of a quantity by taking the same aliquot part of the cost, &c. of the quantity-is applicable to all cases of Prac tice.

26. The pupil's progress in this rule will depend upon the readiness with which he is able to resolve any compound expression into aliquot parts of a given unit. Tables of aliquot parts are given in some books of arithmetic, but the pupil had better construct them for himself, by taking any unit, and, dividing it by the numbers 2, 3, 4, &c., up to 12, collect the aliquot parts, and commit them to memory. Thus, taking £1 for the unit, and dividing by 2 we get 10s. is of £1, dividing by 3 we get 6s. 8d. is of £1; but since £1 divided by 7

gives 2s. 10ld. with a remainder, therefore 2s. 101d. is not an aliquot part of a pound. Again, taking the first aliquot part ros., and dividing by 2, we get 5s. is of 10s. ; similarly we find that 3s. 4d. is of 10s. ; 2s. 6d. is of 10s., and so on for the other aliquot parts of a pound.

Examples.

Resolve into aliquot parts of £1 the following quantities :

(1) 155.; (2) 145.; (3) 17s. 6d.; (4) 13s. 6d.; (5) 9s. 7 d.; (6) 19s. 11ld.

Resolve into aliquot parts of 1 cwt. the following quantities :

(1) I qr. 21 lbs. (2) 3 qrs. 18 lbs. (3) 2 qrs. 15 lbs. (4) 3 qrs. 23 lbs.

Resolve into aliquot parts of Is. the following quantities:

A. 9d.; 7d.; 5d.; 41d.; 31d.; 71d.; 81d.; rold.; 91d.; 10ąd.; 5ąd.; 71d.; 111d.

Resolve into aliquot parts of £1 the following quantities :

B. 12S.; 12s. 6d. ; 7s. 6d.; 16s. 8d.; 15s. 6d.; 14s. 8d.; 13s. 6d.; 11s. 9d.; 15s. 73d.; 18s. 9d.; 14s. 5d.; 9s. 72d.; 3s. 11d.; 175. 102d. ; 19s. 63d.; 18s. 11 d.

Resolve into aliquot parts of 1 yard the following quantities :—

C. I qr. 2 nls. ; 3 qrs. 2 nls. ; I qr. 3 nls. ; 2 qrs. 3 nls. ; 3 nls. ; 3 qrs. 3 nls.; 1 ft. 8 in.; I ft. 11 in.; 2 ft. 6 in.; 2 ft. 10 in.

Resolve into aliquot parts of 1 cwt. the following quantities:

D. 3 qrs. 14 lbs. ; 1 qr. 21 lbs. ; 3 qrs. 5 lbs. ; 1 qr. 16 lbs. ; 2 qrs. 18 lbs. ; 3 qrs. 24 lbs. ; 2 qrs. 23 lbs. ; 2 qrs. 19 lbs.

Resolve into aliquot parts of 1 ton the following quantities :

E. 5 cwt. 2 qrs.; 13 cwt. 3 qrs.; 17 cwt. I qr. ; 15 cwt. 3 qrs.; II cwt. I qr. 14 lbs.; 16 cwt. 3 qrs. 18 lbs.; 14 cwt. 2 qrs. 24 lbs.

CHAPTER V.

SIMPLE PRACTICE.

27. There are two cases of Simple Practice. (1) When the third term of the proportion is an abstract number; (2) When the third term is a compound quantity.

Case I.

Examples.-(1) If 36 articles cost is., how many articles may be bought for Is. 8d.?

IS. : Is. 8d. :: 36 articles : No. required.

(2) How many for 15s. 6d., when 120 cost £1?

Omitting the top line in the addition, because the given sum is less than £1, we get the number for 15s. 6d. = 93 articles.

The operation may be conveniently arranged as follows :