Here,

that is in 5 years 6 months, from what principal would the same sum of interest arise in 8 years 9 months. as the time in which the given sum of annual interest is allowed to accumulate, is greater than that first mentioned, the principal producing such a sum of interest must of course be smaller than that in the middle term, which is therefore multiplied by the least and the product divided by the greatest number of months; and the quotient £ 1142 .. 18. 11. 2 is the principal sum which, at the rate expressed in the question, would in 8 years and 9 months produce £ 500 of in

terest.

In performing the same example by one operation, the given sum of principal 682.. 12 (reduced to 13652 shillings) is the middle term; and the number 394170, obtained by multiplying the given interest of this sum, by the time connected with the interest of the sum required by the question, becomes the first term: the third term 660000 being composed of the given interest of the unknown principal multiplied into the time connected with the interest of the given principal. We have now a fresh series of proportionals, as 394170 to 13652 shillings, so 660000 to a fourth number of shillings 22858, which together with the value of the remainder, is equal to £1142.. 18. 11. 2.

OF FELLOWSHIP.

This rule, also called Distributive Proportion, serves to divide amongst a number of partners the profits or loss arising from a common stock, in proportion to the share which each partner has contributed. From the nature of proportionals it follows that of any series, the sum of all the antecedents is to the sum of all the consequents, as each antecedent is to its consequent: that is, that the sum of all the shares is to the sum of all the sums of profit or loss, as each individual share of the stock is to the profit or loss attached to such individual share.

Suppose

223

Suppose, for example's sake, it were required to divide 120 into 3 parts, in the proportion to one another of 3, 4, and 5; the first share would be to the second as 3 to 4; the second would be to the third as 4 to 5; and the first would be to the third as 3 to 5. But the sum of all the antecedents being to the sum of all the consequents in the same proportion as each antecedent is to its consequent, we will have the sum of the given antecedents 3, 4, and 5, that is 12, to the sum of the consequents or 120, as each of these antecedents, 3 for instance, to its consequent 30, 4 to its consequent 40, and 5 to its consequent 50; and as the numbers 3, 4, and 5, added together make up 12, the amount of the proportional parts, so the several shares 30, 40, and 50, make up 120, the total number that was to be divided. Again: Suppose three persons to form a joint stock for the purpose of trade, to which A contributed the value of £ 390, B £520, and C £650. After some time, on settling their accounts, they found the total gain amounted to £312. How much of this sum would fall to each partner, in proportion to his share of the joint stock?

£

A 390

B 520

C 650 Gain A's Stock

£

Stock £1560; 312 :: 390

The question here is to discover what each partner's share of the profits of the adventure should be, in proportion to his share of the stock; add therefore the three shares £ 390, £520, and 650, together, and then state the proportion, as the amount of all the shares, £1560, is to the amount of the whole gain, £312, so is a partner's stock, A's, for instance, £390, to his proportion of the gain; which by the operation turns out to be £78. Working in the same way, we find B's gain to be £104; and C's £130: and as a proof that the operation has been rightly performed, by adding these several gains together we have £312, agreeing with the total gain given in the question.

Example 2d.

Example 2d. Three partners made a joint stock: D put in £ 556, EL 368, and F 256: but at the end of three months D withdrew his share, and E at the end of five months; whilst F carried on the adventure for eight months, when the profits, amounting to £285.. 15, were to be diyided amongst the partners. How much ought each to receive, in proportion to his share of the common stock, and the time during which his money. remained employed?

In this example there are two series of proportions, the one of the several shares contributed to form the joint stock, and the other of the several periods of time for which each share was employed. Hence as D employed his stock £556 for 3 months, his share of the profit would be the same as if he had contributed 3 times that sum for 1 month: E's stock of 368, for 5 months would be of the same value as 5 times that sum for 1 month: and F's stock of £256 for 8 months, would be of the same value as 8 times that stock for 1 month. Hence it becomes necessary to multiply each stock, as above, by the number of months it was employed in the adventure; by which we obtain the compound quantity 1668 for D, 1840 for E, and 2048 for F; which being added together give for the 1st term of the proportions 5556; for the 2d term we have £285.. 15; the given profit on the whole; and for the 3d these several compound quantities. The first operation of proportion gives £ 85 .. 15 8 3. for D's share of the gain, with a remainder of 1380: The second operation (here omitted, but which the student will easily perform, as well as the third) gives for E's share £ 94 .. 12 .. 7 .. 3, with a remainder of 2868; and the third gives £105.06..7.. 1, and 1308 for a remainder. These sums added together, as above, produce £ 285 .. 14 .. 11 ..3, wanting one farthing to make up the total sum of gain,