gain $1875, then $1 gains

of $1875

=

$1878, or of a dollar; and $2475 (R's capital) will gain 2475 times as much $245, or $825; and since $1 gains $, then $3150 will gain 3150 times as much = $31.50, or $1050.

=

Ans. R, $825; S, $1050. Proof. R's share, $825, added to S's share, $1050, will give $1875.

By noticing the above Analysis we see the reason of the following

RULE.

Make the whole gain or loss the numerator, and the whole stock employed the denominator, of a fraction; then multiply this fraction successively by each man's stock.

Another rule frequently given is this: say,

As the whole stock is to each man's stock, so is the whole gain or loss to each man's share of the gain or loss.

EXAMPLES.

2. A and B jointly rented a pasture for $24; A put in 36 cows and B 24 cows; how much of the rent ought each to pay? Ans. A, $14.40; B, $9.60. 3. Thomas and Smith traded in company. Thomas invested $840 and Smith $480. At the end of two years they had lost $264; how ought this loss to be shared?

Ans. Thomas, $168; Smith, $96. 4. Three men, A, B and C, jointly own a house costing $9000; A furnishing $4500, B, $2500 and C, $2000; the house rents for $810; what share of the rent ought each to have? Ans. A, $405; B, $225; C, $180. 5. P, Q and R hired a carriage for $15.75, each agreeing to pay in proportion to the number of miles he rode. P rode 90 miles, Q, 75, and R, 60 miles; what part of the hire ough Ans. P, $6.30; Q, $5.25; R, $4.20.

each to pay?

6. D, E and F jointly erected a cotton-factory, D furnishing $23450 of the capital, E, $27975, and F, $32425. After a certain time they found their profits were $13975; how ought his gain to be divided?

Ans. D, $39081; E, $46621; F, $5104. ART. 295. In the Examples thus far given the capital of rach partner has been employed for an equal length of time, but in practical business persons frequently enter into partnerships where their capital is employed for unequal lengths of time.

1. H and R engaged in trade; H put in $540 for 6 months, and R, $450 for 8 months. They gained $513; what was each man's share of the gain?

Analysis. $540 for 6 months is the same, as 6 times as many dollars for 1 month - $3240 for 1 month; and $450 for 8 months is the same as 8 times as many dollars for 1 month = $3600 for 1 month. Hence, $3240 (H's money) for 1 month, added to $3600 (R's capital) for 1 month, is equivalent to $6840 for 1 month. Then, since the whole capital of $6840

gains $513, $1 gains of $513 $3240 will gain 3240 times as much

=

513

$, or $; and

$2720, or $243; and

40

$3600 will gain 3600 times as much as $1, which will give $10800, or $270.

40

Ans. H, $243; R, $270.

From the Analysis the reason is obvious for the

RULE.

I. Multiply each man's stock by the time it was continual in trade.

II. Make the sum of these products the denominator of a fraction, and the whole gain or loss the numerator; then mul tiply this fraction successively by the product of each man's stock into its time.

Those who prefer it may employ the following

RULE.

Multiply each man's stock by the time it was continued in trade; then say: As the sum of these products is to each separate product, so is the whole gain or loss to each man's share of the gain or the loss.

Solving the above Example by this latter Rule, we have the following proportions:

$6840: $3240 :: $513: H's share

$243.

$270.

For the mental training of the pupil we recommend the method by Analysis as preferable to any rules.

EXAMPLES.

2. D and E formed a partnership in trade. D put in $750 for 12 months, and E, $600 for 18 months. They gained $594; How should the gain be divided? Ans. D, $270; E, $324.

3. Z and W engaged in speculation. Z employed $945 for 6 months, and W $810 for 8 months. They lost $729; how should this loss be divided? Ans. Z, $340.20; W, $388.80.

4. Three men, A, B and C, rented a pasture for $70.56. A put in 36 cows for 5 months; B, 48 cows for 4 months; and C, 72 cows for 3 months; what part of the rent ought each to pay? Ans. A, $21.60; B, $23.04; C, $25.92.

5. A gentleman employed three mowers to cut hay. The first mowed 9 days, at 10 hours a day; the second worked 12 days, at 9 hours a day; and the third mowed 15 days, at 12 hours a day. Their joint wages amounted to $45.36. What share of the wages ought each to receive?

Ans. First, $10.80; second, $12.96; third, $21.60. 6. C and H engaged in trade. C put in $924 for 10 months,

and H $855 for 12 months. the gain be divided?

They gained $585; how should Ans. C, $277.20; H, $307.80.

ALLIGATION.

ARTICLE 296. ALLIGATION MEDIAL explains how to find the average price of a mixture of different articles when the quantity of each article and its price are given.

Ex. 1. A farmer combines 9 bushels of oats worth 90 cents a bushel, with 9 bushels of rye worth $1.20, and 12 bushels of barley worth $1.25; what is a bushel of the mixture worth?

Hence, if 30 bushels = $33.90, 1 bushel is worth of

$33.90 = $1.13.

Ans. $1.13 per bushel.

The Analysis shows the reason of the following

RULE.

Find the value of each article, then divide the total value of the articles mixed by the number of articles.

NOTE. An ingredient having no commercial value, such as water, may enter into a compound. Although its value is nothing, still its quantity must be added as constituting a part of the number or sum of the articles.

EXAMPLES

2. A grocer mixes 20lb. of clarified sugar worth 15 cents per pound, with 24lb. of A sugar at 10 cents, and 28lb. of brown sugar at 9 cents; what is a pound of the mixture worth? Ans. 11 cents. 3. A whisky-seller mixes 24 gallons of corn whisky worth

64 cents per gallon, with 8 gallons of water, which costs nothing; what is a gallon of the mixture worth?

Ans. 48 cents.

4. A farmer bought 12 pigs; for 3 of them he gave $4 apiece; for 5 of them $8 apiece, and for the remainder $5 each; what was the average price of each? Ans. $6.

5. A grocer mixes 15lb. of tea worth 80 cents a pound, with 20lb. worth 60 cents, and 25lb. worth 90 cents; what is a pound of the mixture worth? Ans. 771 cents.

6. A locomotive ran as follows: for 4 hours at 20 miles an hour, 5 hours at 18 miles per hour, 3 hours at 24 miles, and 1 hour at 31 miles; what was the average hourly speed?

Ans. 21 miles.

7. A thermometer stood for 4 days at 76°, 3 days at 78°, 5 days at 80°, and 6 days at 84°; what was the average temperature? Ans. 8010.

ALLIGATION ALTERNATE.

ART. 297. ALLIGATION ALTERNATE is the method of finding how much of several articles whose values are given must be taken to form a compound of a particular value.

Ex. 1. A grocer has sugar worth 7 cents, 8 cents, 13 cents and 14 cents a pound, which he wishes to mix together, so that he may be able to sell the mixture at 11 cents per pound; how much of each kind must he take?

Analysis. It is plain that if he puts in 1 pound of the 7. cent sugar and sells it for 11 cents, he gains 4 cents; then, to gain 1 cent, he must put in only of a pound. Again, if he puts in 1 pound of the 14-cent sugar, he loses 3 cents; then, to lose 1 cent, he must put in of a pound. Now, the 1 cent gained on the of a pound of the 7-cent sugar exactly balances the 1 cent lost on the of a pound of the 14-cent sugar. Hence, the quantities of each kind of sugar must be to each