EXAMPLE II.

If a man, walking 7 hours a day, travels 230 leagues in 30 days, how many days would he require to perform a journey of 600 leagues, walking 10 hours a day?

This problem may be reduced to the single rule of three, if we consider, that to travel 30 days, employing 7 hours each day, is the same thing as to travel 30 times 7 hours, or 210 hours. The question, therefore, may be changed in the following manner: If 210 hours are required to travel 230 leagues, how many hours will be requisite to travel 600 leagues? When the number of hours which answer the question have been found, the required number of days may be found by dividing these hours by 10, as the traveller employs 10 hours each day. We must, therefore, find the fourth term of the proportion, the first three of which are as follows;

Leag

Leag.

Hours. Hours.

230: 600 :: 210 x

RULE OF FELLOWSHIP.

As this rule is merely an application of what has been said respecting the rule of three, we shall only give a few examples; to illustrate the use of it.

EXAMPLE I.

A privateer, belonging to three merchants captured a prize worth £80000. what will each partner's share amount to, the first having advanced to purchase and fit out the vessel £2000. the second £6000. and the third £12000?

It is here evident, that each partner must have a share of the prize proportioned to the money he advanced.

We must, therefore, make this proportion: As the sum advanced by each partner, is to the whole money advanced, so is the share of each to the whole prize. Hence we shall have the following proportions, where

the second and fourth terms, that is to say, the sum total of the money advanced and the value of the prize, are common to all of them:

Three persons having entered into partnership, the first advanced £3000. for six months, the second £4000. for five months, and the third £8000. for nine months; at the end of that time they found that their gain amounted to £150000. how much will each partner's share be worth?

As this problem belongs to the compound rule of three, we shall take, as the first terms, the product of the money advanced by each partner, multiplied by the time it was employed; and for the second, the sum of these products, in the following manner:

It may be readily seen, that by means of the rule of three, any sum, such as the amount of a legacy, for example, may be easily divided among several persons, in such a manner, that the shares shall be in the ratio of certain determinate numbers, as 3, 4, 6. In this case, these numbers must be considered as three sums advanced by three partners, and their sum as the total of the money advanced: if we then call the legacy to be divided a, we shall have the following proportion:

RULE OF ALLIGATION.

Alligation is of two kinds.

The first consists in find

The

ing the common price of several things, supposed to be mixed together; as if a goldsmith, for example, should make a composition of gold, silver, &c. and be desirous to know the value of an ounce of this mixture. same rule is employed to determine the mean price of several liquors, or different kinds of merchandise, mixed together.

This rule is exceedingly easy; for nothing is necessary, in solving questions of this kind, but to divide the whole value of the articles by the quantity of each article employed for the mixture, and the quotient will be the answer.

EXAMPLE I.

A wine merchant mixes together 200 bottles of Madeira, at 5 shillings, 500 of Port at 3s. 800 of Malaga at 4s. and 300 of Tokay at 8s.; how much is a bottle of this mixture worth?

Now, if 8100 shillings, or the whole value of the wine, be divided by 1800, the number of the bottles, the quotient will express the value of each bottle of the 8100 81

mixture. Consequently

4s. 6d.

EXAMPLE II.

A gentleman employed 300 workmen, 50 of which were paid at the rate of 8s. a day; 70 at the rate of 6s. and 180 at the rate of 4s. ; how much did each of them, taking one with another, cost him per day?

The sum total, which is 1540 shillings, must be di vided by 300, the number of the workmen, and the quotient will be what each of them, taken one with another, cost him per day.

The object of the second kind of alligation, is to determine in what proportion several things, of different values, ought to be mixed, in order to have an article of a certain mean price.

To obtain this result, the prices of the things to be mixed must be arranged, as seen in the following examples :

A

EXAMPLE I.

4 grocer, who has tea at 3s. 4s. 7s. and 9s. per pound, is desirous of having a mixture which he can sell at 5s per pound. In what proportions must he mix these four kinds of tea, so as to be able to sell the mixture at 5s.?

Arrange the prices of the things to be mixed as seen at A; placing those which are greater than the mean price at the top, and those which are less at the

bottom.

Then compare in succession with the mean prices the prices of all the things to be mixed, and place the differences as in the above example.

Thus, the difference between 3 and 5 is 2, which must be set down opposite to 7, and that between 4 and 5 is 1, which must be placed opposite to 9. Then proceed to the prices greater than that of the mean

price, and compare them with that price in the following manner: The difference between 7 and 5 is 2, which place opposite to 3; and that between 9 and 5 is 4, which place opposite to 4.

The right hand column, the sum of which is 9, shews that to have 9 pounds of tea, at the mean price of 5s. the mixture must consist of 2 pounds at 3s. 4 pounds at 4s. 2 pounds at 7s. and 1 pound at 9s.

It may here be readily seen, that 9 pounds of tea, at the mean price of 5s. will amount exactly to the value of the quantities mixed.

It may sometimes happen that the figures, expressing the different values of the things to be mixed, will not be equal in number both below and above the mean price; on this account, if there are, for example, three figures above it, and only two below, the difference of the third figure at the top must be placed opposite to the second at the bottom, along with the difference of the second at the top; and the difference of the second figure at the bottom must be set down twice; that is to say, it must be placed opposite to the second and the third at the top. See fig. B.

That is to say, to form 11 pounds of tea, of the mean price of 5s. one pound at 2s. two pounds at 3s. two pounds at 4s. three pounds at 6s. and three at 7s. must be mixed together.

EXAMPLE II.

A goldsmith has gold of 23 carats fine, and some of 13 carats, which he is desirous of mixing, so as to form gold of 18 carats, what quantity of each must he take? See Fig. C.