Divide $120 between three persons, so that their shares shall be to each other as 1, 2, and 3, respectively.

Ans. $20, $40, and $60. Three persons make a joint stock. A put in $185,66, B $98,50, and C $76,85; they trade and gain $222; what is each person's share of the gain?

83

36101

Ans. A $104,17710, B $60,57-82437, and C 47,25 38774. Three merchants, A, B, and C, freight a ship with 340 tuns of wine; A loaded 110 tuns, B 97, and C the rest. In a storm the seamen were obliged to throw 85 tuns overboard; how much must each sustain of the loss?

Ans. A 271, B 241, and C 331. A ship worth $860 being entirely lost, of which belonged to A, to B, and the rest to C; what loss will each sustain, supposing $500 of her to be insured?

Ans. A $45, B $90, and C $225.

A bankrupt is indebted to A $277,53, to B $305,17, to C $152, and to D $105. must it be divided?

His estate is worth only $677,50; how

Ans. A $223,812580, B $246,286, C $122,668338, and D $84,739665

6930

A and B, venturing equal sums of money, clear by joint trade $154. By agreement A was to have 8 per cent. because he spent his time in the execution of the project, and B was to have only 5 per cent. ; what was A allowed for his trouble?

Ans. $35,5511.

Three graziers hired a piece of land for $60,50. A put in 5 sheep for 41 months, B put in 8 for 5 months, and C put in 9 for 6 months; how much must each pay of the rent?

Ans. A $11,25, B $20, and C $29,25. Two merchants enter into partnership for 18 months; A put into stock at first $200, and at the end of 8 months he put in $100 more; B put in at first $550, and at the end of 4 months took out $140. Now at the expiration of the time they find they have gained $526; what is each man's just share?

70

Ans. A's $192,95118, B's $333,04114 A, with a capital of $1000, began trade January 1, 1776, and meeting with success in business he took in B a partner, with a capital of $1500 on the first of March following. Three months

after that, they admit C as a third partner, who brought into stock $2800, and after trading together till the first of the next year, they find the gain, since A commenced business, to be $1776,50. How must this be divided among the partners? Ans. A's $457,46364

B's 571,832

C's 747,19346

ALLIGATION.

128. WE shall not omit the rule of alligation, the object of which is to find the mean value of several things of the same kind, of different values; the following examples will sufficiently illustrate it.

A wine merchant bought several kinds of wine, namely; 130 bottles which cost him 10 cents each,

It will be easily perceived,

the cost of a bottle of the mixture. that we have only to find the whole cost of the mixture and the number of bottles it contains, and then to divide the first sum by the second, to obtain the price required.

Now, the 130 bottles at 10 cents cost 1300 cents

5737 divided by 463 gives 12,39 cents, the price of a bottle of the mixture.

The preceding rule is also used for finding a mean of different results, given by experiment or observation, which do not agree with each other. If, for instance, it were required to know the distance between two points considerably removed from each other, and it had been measured; whatever care might have been used in doing this, there would always be a Arith.

15

little uncertainty in the result, on account of the errors inevita bly committed by the manner of placing the measures one after the other.

We will suppose that the operation has been repeated several times, in order to obtain the distance exactly, and that twice it has been found 3794yds. 2ft. that three other measurements have given 3795yds. 1ft. and a third 3793yds. As these numbers are not alike, it is evident that some must be wrong, and perhaps all. To obtain the means of diminishing the error, we reason thus ; if the true distance had been obtained by each measurement, the sum of the results would be equal to six times that distance, and it is plain that this would also be the case, if some of the results obtained were too little, and others too great, so that the increase, produced by the addition of the excesses, should make up for what the less measurements wanted of the true value. We should then, in this last case, obtain the true value by dividing the sum of the results by the number of them.

This case is too peculiar to occur frequently, but it almost always happens, that the errors on one side destroy a part of those on the other, and the remainder, being equally divided among the results, becomes smaller according as the number of results is greater.

According to these considerations we shall proceed as follows;

6 results, giving in all 22768 1.

Dividing 22768yds. 1ft. by 6, we find the mean value of the required distance to be 3794yds. 2ft.

129. Questions sometimes occur, which are to be solved by a method, the reverse of that above given. It may be required, for example, to find what quantity of two different ingredients it would take to make a mixture of a certain value. It is evident, that if the value of the mixture required exceeds that of one of the ingredients just as much as it falls short of that of the other, we should take equal quantities of each to make the compound,

1

So also, if the value of the mixture exceeded that of one twice as much as it fell short of that of the other, the proportion of the ingredients would be as one half to one. To mix wine at $2 per gallon with wine at $3, so that the compound shall be worth $2,50, we should take equal quantities, or quantities in the proportion of 1 to 1. If the mixture were required to be worth $2,663, the quantities would be in the proportion of 1 to 1, or of 1 to and generally, the nearer the mixture rate is to 663/ 33/31 that of one of the ingredients, the greater must be the quantity of this ingredient with respect to the other, and the reverse; hence, To find the proportion of two ingredients of a given value, necessary to constitute a compound of a required value, make the difference between the value of each ingredient and that of the compound the denominator of a fraction, whose numerator is one, and these fractions will express the proportion required; and being reduced to a common denominator, the numerators will express the same proportion, or show what quantity of each ingredient is to be taken to make the required compound.

When the compound is limited to a certain quantity, the proportion of the ingredients, corresponding to it, may be found by saying; as the whole quantity, found as above, is to the quantity required, so is each part, as obtained by the rule, to the required quantity of each.

Let it be required, for example, to mix wine at 5s. per gallon and 8s. per gallon, in such quantities that there may be 60 gallons worth 6s. per gallon. The difference between 6s. and 5s. is 1, and between 6s. and 8s. is 2, giving for the required quantities the ratio of to, or 2 to 1; thus, taking x equal to the quantity at 5s. and y equal to the quantity at 8s. we have these proportions; 3: 60 :: 2 : x, and 3:60::1:y, giving, for the answer, 40 gallons at 5s. and 20 gallons at 8s. per gallon.

Also, when one of the ingredients is limited, we may say; as the quantity of the ingredient found as above, is to the required quantity of the same, so is the quantity of the other ingredient to the proportional part required.

For example, I would know how many gallons of water at es. per gallon, I must mix with thirty gallons of wine at 6s. per

1

gallon, so that the compound may be worth 5s. per gallon. First, the difference between Os. and 5s. is 5; and the difference between 6s. and 5s. is 1; the quantity of water therefore will be to that of the wine, as to, or as 1 to 5. Then, from this ratio, we institute the proportion, 5: 30:: 1:x, which gives 6, for the number of gallons required.

As we have found the proportion of two ingredients necessary to form a compound of a required value, so also we may consider either of these in connexion with a third, with a fourth, and so on, thus making a compound of any required value, consisting of any number whatever of simple ingredients. The two ingredients used, however, must always be, one of a greater and the other of a less value, than that of the compound required.

A grocer would mix teas at 12s. and 10s. with 40lbs. at 4s. per pound, in such proportions that the composition shall be worth 8s. per lb. If he mix only two kinds, the one at 4s, and the other at 10s. their quantities will be in the ratio of 1 to 1, or 1:2; and if he mix the tea at 4s. also with that at 12s. their ratio will be that of to, or of 1 to 1. Adding together the proportions of the ingredient, which is taken with each of the others, we find the several quantities, at 4s. 10s. and 12s. to be as 2, 2, and 1. And taking x for the number of lbs. at 10s. and y for the quantity at 12s. we have the following proportions;

2:40::2: x; and 2:40::1:y;

giving, for the answer, 40lb. at 10s. and 20lb. at 12s. per pound. The problems of the two last articles are generally distinguished by the names of alligation medial, and alligation alternate. A full explanation of the latter belongs properly to algebra.

Examples.

A composition being made of 5lb. of tea at 7s. per pound, 91b. at 8s. 6d. per pound, and 141⁄2lb. at 5s. 10d. per pound; what is a pound of it worth? Ans. 6s. 101d.

How much gold, of 15, of 17, and of 22 carats† fine, must be mixed with 5oz. of 18 carats fine, so that the composition may be 20 carats fine? Ans. 5oz, of 15 carats fine, 5 of 17, and 25 of 22.

† A carat is a twenty fourth part; 22 carats fine means 22 of pure metal. A carat is also divided into four parts, called grains of a carat.