11. If 6 bush. of oats serve 4 horses 8 days, how many days will 21 bush. serve 16 horses? Ans. 7 days.

12. If the carriage of 3 cwt. 2 qr. for 14 miles be 3s. 4d., what weight should be conveyed 49 miles for £1, 15s.? Ans. 10 cwt. 2 qr.

13. If 1s. 1 d. be paid for 6 lb. 8 oz. 4 dr. of bread, when wheat is worth 6s. 8d. per bush., what must be paid for the peck loaf, which weighs 17 lb. 6 oz., when wheat sells at 8s. 6d. per bush.? Ans. 3s. 103d.

14. If £100 in 12 months gain £5 of interest, what will £60 gain in 9 months. Ans. £2, 5s. 15. If £100 in 12 months gain £5 of interest, what principal will gain £2, 58. in 9 months? Ans. £60.

16. If the interest of £60 for 9 months be £2,5s., required the rate per cent. per annum? Ans. £5.

17. If 2 horses plough 4 ac. 2 ro. of land in 6 days, when they work 8 hours a-day; how many acres will 16 horses plough in 156 days, when they work 12 hr. 30 min. per day? Ans. 1462 ac. 2 ro.

18. A wall 40 ft. long, 2 thick, and 16 high, was built by 18 men in ten days, how many men will build a wall 960 ft. long, 3 thick, and 24 high, in 72 days? Ans. 135 men.

19. If a chest 10 ft. long, 6 deep, and 6 broad, contain 40 qr. of grain, find the length of another 4 ft. deep and 3 broad, to contain 8 qr.? Ans. 6 ft.

20. If a tailor can make two suits of clothes with 6 yd. of cloth, which is 1 yd. 2 qr. wide, how many suits can he make from a web 48 yd. long and 3 qr. wide? Ans. 8 suits.

DISTRIBUTIVE PROPORTION.

Distributive proportion teaches to find the respective profits or losses of partners in trade, &c. CASE I. When the times are equal.

RULE. As the whole stock is to each partner's stock, so is the sum to be divided to the several parts required.

EXERCISES.

1. Three partners, A, B, and C, form a joint stock: A puts in £15, B £19, and C £30; they gained £19; what is each man's share of the gain? Ans. A's share £4, 9s. 02d., B's = £5, 12s. 9 d., C's £8, 18s. 11⁄2d.

2. Four farmers, A, B, C, and D, hired a shepherd at £31 per annum: A committed to his care 70 sheep, B 110, C 170 and D 290; what part of his wages must each pay? Ans. A's share= £3, 78.9 d., B's =£5, 6s. 63d., C's = £8,48.84d., D's £14, 08. 114d.

3. A bankrupt's money and his effects amount to £112, and he owes A £111, B £217, C £313, and D £383; required each creditor's share of the debtor's stock? Ans. A's share = £12, 2s. 9 d., B's £23, 14s. 8d., C's £34, 4s. 84d., D's = £41, 17s. 9 d.

=

=

4. E, F, G, and H make a joint stock, of which E contributed £31, F £41, G £53, and H £67; in the course of their trading they gain £366: required each man's share of the gain? Ans. E's share= £59, 1s. 101d., F's = £78, 3s. 1 d., G's = £101, 0s. 74d., and H's = £127, 14s. 4 d.

5. Four persons engaged in a speculation, on which there was a loss of £62; what part of the damage must each sustain, A having advanced £23, B £29, C £37, and D £39? Ans. A's loss

= £11, 2s. 9åd., B's = £14, Os. 11 d., C's = £17, 18s. 5 d., D's = £18, 17s. 9 d.

6. An assessment of £528 is granted to the poor of a parish, in which there are four heritors, whose estates are valued at £1351, £2873, £3465, and £3575; what part of the assessment must each pay? Ans. £63, 6s. 6 d., £134, 13s.54d., £162, 8s. 51d., £167, 11s. 62d.

CASE. II. When the times are unequal.

RULE.-Multiply each sum by its respective time, and proceed with the products by Rule I.

EXERCISES.

1. Three butchers pay £84, 6s. 8d. for a season of a park: A put in 200 sheep for 4 months, B 400 for 6 months, and C 800 for 7 months; what part of the rent must each pay? Ans. A's share = £7, 13s. 4d., B's = £23, C's = £53, 13s. 4d.

2. Three merchants enter into company: A puts £135 in for 3 months, B £365 for 7 months, C £406 for 8 months, they clear £97; find each person's share? Ans. A's share = £6, 6s. 63d., B's = £39, 18s. 54d., C's = £50, 15s.

3. Two merchants enter into partnership for 19 months: A put in at first £31, at the end of 3 months he put in £26 more, and 5 months after he took out £38; B put in at first £21, at the end of 7 months he put in £52 more, and 9 months after he took out £46; they gain by trading £115; required each man's share? Ans. A's share= £45, 17s. 2 d., B's £69, 2s. 9 d.

4. E commenced business with a capital of £1000; in 1 month D joined him in partnership; 3 months afterward they admitted C; at the end of 2 months B, and A in 2 months more; in 11 months from the commencement of E the profits

amounted to £333; required the respective shares, supposing each partner to have advanced the same sum as E? Ans. E's share= £101, 15s., D's = £92, 10s., C's = £64, 15s., B's = £46, 5s., A's = £27, 15s.

EQUATION OF PAYMENTS.

Equation of payments teaches to find the time at which two or more sums due at different dates may be paid at once.

RULE.-Multiply each sum by the time it has to run, then divide the sum of the products by the amount of the debt, the quotient is the time required.

EXERCISES.

1. A owes B £150 at 2 months, £100 at 3 months, £225 at 4 months, and £200 at 6 months; when must the whole be paid at once? months.

Ans. 4

2. I have a bill of £74 due in 42 days, one of £37 in 47 days, one of £148 in 59 days, and one of £222 in 71 days; what is the equated time for receiving one bill for the whole? Ans. 61 days.

3. A debt of £403 is to be paid as follows, viz. £26 at present, £169 in 42 days, £91 in 54 days, and £117 in 104 days; when should the whole be paid? Ans. 60 days.

4. A debt of £52 is due to me at present, £338 at 2 months, £234 at 8 months, £182 at 17 months; when should one payment be made for the whole ? Ans. 7 months.

EQUATION OF RATES.

Equation of rates teaches to find the rate at which the sum of several quantities would produce the same amount as the sum of their separate values.

D

RULE.-Multiply each quantity by its rate, then divide the sum of the products by the sum of the quantities, and the quotient will be the rate of the compound.

EXERCISES.

1. What is the average rate of 51 qr. of barley at 31s., 85 qr. at 32s., and 153 qr.at 38s.? Ans. 358. 2. A grocer mixes 7 lb. of tea at 7s. 3d. per lb., 14 lb. at 8s. 5d., 35 lb. at 9s. 7d., and 21 lb. at 7s. 2d.; what is the value of a lb. of the mixture? Ans. 8s. 6d.

3. A vintner mixes 3 galls. of wine at 4s. 10d. per gall. with 9 galls. at 4s. 8d., 6 galls. at 5s., and 15 galls. at 5s. 5d.; how much is a gall. of the mixture worth? Ans. 5s. 1d.

4. What is the average rate of 27 qr. of wheat at £2, 10s. per qr., 36 qr. at £3, 21 qr. at £3, 6s., and 12 qr. at £4? Ans. £3, 1s.

VULGAR FRACTIONS.

DEFINITIONS AND PRINCIPLES.

A fraction is a part or several parts of unity, and is expressed by two numbers, placed one above the other with a line between them, thus, the 3 (numerator). 7(denominator). The denominator shows into how many parts the whole is divided, and the numerator expresses how many of these are taken.

fraction three-sevenths is written

as

A proper fraction is less than unity, as .
An improper fraction is not less than unity, as J.
A compound fraction is a fraction of a fraction,
of 1.

A mixed number is composed of a whole number and a fraction, as 8.