now, £3500 + £5000 = £8500 is the whole cause,

and £3500 and £5000 are the partial causes: also £1000 is the whole effect and the partial effects are the required shares: whence, we have

and these sums make up the £1000 gained.

Here, the ratio of the shares depending solely upon the amounts subscribed, the example is termed an instance of Single Fellowship.

Ex. 2. A field of grass is rented by two persons for £27: the former keeps in it 15 oxen for 10 days and the latter 21 oxen for 7 days; find the rent paid by each.

Here, the portions of the rent must evidently be as the numbers of oxen and the numbers of days jointly: also, the partial causes are

15 × 10 150 and 21 × 7 = 147 :

=

and therefore the whole cause is 150+147 or 297; whence, £. £. S. d. f.

10

297: 150 :: 27: 13. 12. 8. 11, the 1st portion: 296: 147:: 27: 13. 7.34.1, the 2nd portion : and the sum of both portions is £27 as it ought to be.

This is an instance of Double Fellowship, the portions of rent depending upon two particulars, the number of oxen put in and the number of days they are kept there.

159. The principles of these examples being independent of the number of interests concerned enable us to lay down the following Rule.

RULE. Find the values of the partial causes and also their sum: then, as this sum is to each part of it, so is the whole effect to its corresponding part.

In this rule it is understood that every agent is employed under exactly the same circumstances: as, for instance, in the last example each of the oxen is supposed

to consume the same quantity of grass, the pasturage being uniform throughout: but whenever their relative qualities are assigned, it will easily be seen that similar methods must be pursued.

Ex. If £100 be distributed among 6 men, 9 women and 12 children; what will be received by them, when the shares of a man, a woman and a child are as the numbers 3, 2, 1?

Here,

6 × 3 = 18

VII. THE RULE OF ALLIGATION.

160. DEF. Alligation sometimes called Alligation Medial is the rule by means of which the rate or quality of a composition or mixture is found from the rates or qualities of the ingredients of which it is made up.

Ex. If 12 bushels of wheat at 6s. a bushel and 15 bushels at 78. a bushel, be mixed together, what will be the value of a bushel of the mixture?

Here, from the most obvious principles, we have

72}

15 × 7 = 105

the values of the ingredients:

therefore 72 + 105 = 177s. is the value of the mixture which contains 12 + 15 = 27 bushels: whence,

bush. bush. £. S. d. f.

27 : 1 :: 177 : 6. 64. 3, the price of a bushel.

The usual form of the operation is as follows:

and the number of ingredients being any whatever, we have the following Rule.

RULE. Divide the sum of the products of the ingredients and their respective rates by the sum of the ingredients, and the quotient will be the rate of the mixture.

Examples for Practice.

(1) If £75 be due in 4 months, £125 in 5 months and £150 in 7 months: what is the equated time?

Answer: 5 months.

(2) What will be the equated time of payment of £200 due at 3 months, £300 at 8 months and £500 at 12 months?

Answer: 9 months.

(3) Find the equated time of payment, when of a sum of money is due at 3 months, at 8 months and the remainder at 15 months.

Answer: 7 months.

(4) A owed B £750 to be paid in 15 months, but at 12 months he paid him £250: at what time was the remainder due?

Answer: 16 months.

(5) Divide £1000 among three persons, so that their shares shall be as the numbers 2, 5, 9.

Answer: £125, £312. 10s. and £562. 10s.

(6) Of £2180, A's share is to B's share as 2 to 3, B's is to C's as 4 to 7 and C's is to D's as 5 to 11: find the share of each.

Answer: A's is £200, B's is £300, C's is £525, and D's is £1155.

(7) Three partners put into business the sums of £300, £400, and £500, and at the end of a certain time they gained £600: find the share of each. Answer: £150, £200, and £250.

(8) Three persons forming a joint stock of £45000, gain by trading £15000; and of this their shares are

£7500, £5000, and £2500: find the portion of stock contributed by each.

Answer: £22500, £15000, and £7500.

(9) A person bequeathed by will the following legacies: £1500 to A. £875 to B, £525 to C and £350 to D: but when his property was sold it produced only £2437. 10s.: how much did he really leave to each ? Answer: £1125 to A, £656. 5s. to B, £393. 15s. to C and £262. 10s. to D.

(10) If A advance £1500 for 9 months and B £1200 for 6 months: what share of a gain of £1150 belongs to each ?

Answer: £750 to A and £400 to B.

(11) If A contribute £6000 for 5 months, B £5000 for 6 months, C £4000 for 7 months and D £2500 for 12 months, in the formation of a joint stock: divide a profit of £4760 equitably among them.

Answer: The share of each is £1190.

(12) Three merchants A, B, C engage in commerce; A with £1000 for 12 months, B with £1800 for 7 months and C with £2500 for 4 months, and they gain £350: what share of the gain belongs to each?

Answer: £121. 78. 83d. 1. to A, to B and £101. 3s. 1d. 14

66

£127. 98. 1d. ff. f. to C.

(13) Three persons with a joint stock gain £3650: the first advances of the capital for of the time, the second of the capital for of the time and the third the remainder of the capital for the whole time: find their shares.

Answer: £486. 138. 4d., £730, and £2433. 68. 8d.

(14) A prize of £3825 is to be divided among 3 officers, 12 assistants and 100 men, in proportion to their pay and time of service jointly: the officers have £5 a month and have served 9 months: the assistants who have £2. 10s. a month have served 6 months and the men have served 3 months at £1. 10s. a month. What is the share of each?

Answer: £225 of each officer, £75 of each assistant and £22. 10s. of each of the men.

(15) A wine merchant mixes 20 gallons of wine at 12s. a gallon, 25 gallons at 148. and 36 gallons at 16s.: what will be the price of a gallon of the mixture?

(16) A mixture is made of 10 bushels of flour at 38. 8d., 21 bushels at 3s. 10d. and 35 bushels at 4s.: what is the price of a bushel of it?

Answer: 3s. 10 d. 35 f.

VIII. THE DOCTRINE OF EXCHANGES.

161. DEF. 1. Exchange is the rule by means of which it is ascertained what sum of money of one country is equivalent to a given sum of another, according to some settled rate of commutation: and the operations necessary to calculate this must, from the nature of the case, be applications of the Rule of Proportion.

The Course of Exchange is used to express the sum of money of any place given in exchange for a fixed sum of that of another: and the Par of Exchange denotes the sum of money of any place, which is of the same intrinsic value as that fixed sum.

Ex. How many pounds Flemish can I receive for £1050 sterling, the course of exchange being 35 shillings Flemish for £1 sterling?

Here, from the nature of the question, we have

2, 0) 3 67 5, 0 shillings Flemish :

£1837 10 the sum Flemish required.

In questions of this kind, all that is necessary to be known is the course of exchange and the subdivisions of the monies to be commuted.

162. DEF. 2. The Arbitration or Comparison of Exchanges is the determining what rate of exchange,