to the same denominations; thus the above proportion will be at first & × : 1 :: xx: 1; then, making the denominators disappear, we shall have 3×5: 8 45 ells ells 13

:: xX2: 3, whence X

2X5×3

8×2

16

which is the number of ells demanded.

These examples are sufficient to show the spirit of the rule, and to convince us, that every question which gives rise to the Rule of Three indirect or inverse, may always be solved by the direct compound rule of three.

PROOF OF THE RULE OF THREE.

To be certain that the rule of three has been well performed, it is necessary to put, instead of x, its value in the proportion established, before any reduction be made, and then make the reductions by as different ways as possible, in order to avoid the faults into which we might have fallen; and, if we find the produce of the extremes, equal to that of the means, may conclude that the value of x is right, and that the sum has been well performed.

we

RULE OF FELLOWSHIP.

The intent of this rule, is principally to divide the gain or loss in trade, proportionably to the respective stocks of the partners.

This rule may be either simple or compound. It is simple, when all the stocks of the partners are in

trade during the same length of time; it is compound, and called Fellowship with time, when the stock of the partners are not in trade during the same length of time; but to solve the question in this case, we must always reduce it to one of simple fellowship. The method to be pursued in order to solve a question in fellowship, is founded on this evident proportion, As the whole stock is to the whole gain, so is the particular stock to its particular gain. We are going to make an application of this in the following questions.

sired to know each man's share of the profit, in proportion to the sum he put in..

Making use of this proportion; as the whole stock is to the whole gain, so is the particular stock to its gain, it is very easy to solve the question; for we shall have these three proportions, which will give the share of each one.

whole flock. whole gain. ftock of ift.

15392£ : 4000£ :: 2000£: to the gain of 1ft=519£.15s.0d. 60 15392 4000 :: 5844: to the gain of 2d=1518-14-2 310 15392: 4000 :: 7548: to the gain of 3d 1961 - 10-9 11

481

461

Proof, 4000.0-0.

The proof of this rule is made, by adding together all the particular gains, the sum of which must be equal to the whole gain.

REMARK. It is necessary to observe, that the first ratio of each of these proportions being the same, it is very advantageous, in order to abridge the calculation, to reduce this ratio to its simplest expression, and to effect that, we may make use of the method taught, for reduction, by way of trial.

Here this reduction, being made, we have; first proportion; 481: 125 :: 200 : to the gain of the 1st,&c.

COMPOUND FELLOWSHIP;

OR FELLOWSHIP WITH TIME.

First Question. Three merchants have put in partnership; the 1st 2000 £. which remained six months in trade; the 2nd 3454£. which were in 8 months; the 3d 5482£. which remained in one year; the profit resulting from these three stocks, omounted to 4000£. what was the share of each one in proportion to his stock, and to the time that each stock remained in trade?

To solve this question, we must reduce it to one of Simple Fellowship, which can always be done, by bringing each stock to a same unit of time in trade. Then we shall take a month for the unit of time, and we shall reduce all the stocks to it, by pursuing this reasoning.

The 2000 £. the first rnan's stock, which remained six months in trade, must produce the same profit, that 6 times 2000. or 12000. remaining in for one month, would produce. In like manner,

"

The 3454. the second man's stock which remained 8 months in trade, must produce the same profit, that 8 times 3454. or 27632£. remaining in for one month, would produce :-In fine,

The 5482. the 3d man's stock, which remained 12 months in trade, must produce the same profit that 12 times 5482. or 65784. remaining in one month, would produce; thus the three stocks are reduced to a unit of time, and now the question may be expressed in this manner;

Three merchants formed a company for one month, and they gained 4000£.

The first put in

The second

12000£.

27632

65784

105416

what share of the profit has each man ?

This now becomes a question of simple fellowship, which is solved by the three following proportions.

£. s. d.

4963

£. £. £. 105416: 4000::12000: to the gain of the 1st-455-6- 9,1015 105416:4000::27632: to the gain of the 2d=1048-9-10-697-17 105416:4000::65784: to the gain of the 3d-2496-3-477

Proof, 4000-0-0--0

NOTE. The Ratio of the whole stock, to the whole gain, may be reduced to that of 13177: 500; which simplifies the calculation in each proportion.

REMARK. Though the sum of the particular gains be equal to the whole gain, yet the division of

the gains may be false; because some error of calculation may have been made in reducing the stocks to a unit of time; thus, then, it is very necessary, before we establish the proportions, to be certain, that the calculation of the reduction of the stocks, to a unit of time, is right.

Second Question. Three merchants formed a company, and gained in two years, the sum of 1755£. the 1st put in 8000. which remained all the time in trade; the 2nd put in at first 7000£. but 9 months after, he put in 3000. more: the 3d put in at first 10000₤. but a year after he withdrew the half of it; it is requi red to tell each man's share of the whole gain, proportionably to his stock, and to the time it remained in trade?

We must first reduce all the stocks to a unit of time, and here again we shall take a month for the unit. Though this question is more complicated than the preceding, on account of the changes, which the two last merchants made in their stocks, it is nevertheless easily solved by the same method, by only considering the different stocks of each of the merchants, one after another, and from one period of time to another. Let us enter into details.

The first merchant, who put in 8000. which remained in trade all the time, has the same right to the whole profit, as if he had put in 24 times 8000£ or 192000. for one month.

The second merchant, having at first put in 7000. which remained in trade 9 months, and at that period of time, having put in 3000, more, making then a sum of 10000. which are found to have remained 15 months in trade, has the same right to the whole gain, as if he had put in for one month, on one