8 months; Y, $430 for 9 months; and Z, $400 for 7 months: they gained $900; how much was each man's share of the profits?

734. Three men built a house: A furnished 7 workmen for 7 days; B, 9 workmen for 11 days; and C, 13 workmen for 4 days. They received $790 for the job; how much should each of the contractors receive?

735. Three men hired a pasture for $90: A pastured 8 cows 90 days; B, 13 cows 20 days; and C, 115 cows 11 days; how much should each pay?

736. Two merchants enter into partnership for two years, each investing $1000. At the end of 9 months A put in $500 more, and B took out $500; at the end of 18 months A doubled his capital, and B put in twice as much as he already had in; at the expiration of the two years they had gained $8713.50: what was each man's share of the gain?

737. Two persons engaged in business with a capital of $5000. A's 'share of the profits amounted to $473.13, and B's share of the profits amounted to $274.12; what amount of the capital did each furnish?

738. It took two persons 21 days to build a wall, for which they received $25; A's share of the money amounted to $6.50, and B received the remainder; how many days did each work?

739. Two men engaged in business with a capital of $5000. A's capital was in the business 5 months, and he received, as his share of the profits, $473.12; B's capital was in the business 9 months, and he received, as his share of the profits, $325.38; how much of the capital did each' furnish?

QUESTIONS.-Give the rule for finding the amount of a bill of exchange that can be purchased for a given sum of U. S. currency.

(289., a.) Give the analysis of finding the cost of a bill of exchange on France. (290., a.) Give the analysis for finding the amount of a bill on France that can be purchased for a given sum of U. S. currency. (291., a.)

SECTION XII.

LESSON I.

ALLIGATION.

307. Alligation Medial is the process of finding the mean price or quality of a mixture, when the quantity of each ingredient and its price or quality are known.

A wine merchant mixes 12 gallons of wine worth $1.50 per gallon, and 9 gallons of brandy worth $2. per gallon, with 5 gallons of water; what is the value of the mixture?

ANALYSIS.-1. Since 12 gallons of wine at $1.50 per gal. are worth $18., and 9 gallons of brandy at $2. are worth $18., and 5 gallons of water are worth nothing, the whole 26 gallons will be worth the sum of $18. and $18., which is $36.00.

2. If 26 gallons of the mixture are worth $36.00, one gallon is worth of $36., which is $1.38.

RULE.-Divide the entire cost of the given simples by the entire quantity, and the result will be the mean price.

1. A grocer mixed 13 gallons of water with 40 gallons of brandy worth $1.25 per gallon; will he gain, or lose, by selling the mixture at 5 cts. per gill?

2. A grocer mixed together 20 gallons of molasses worth 175 cts., 13 gallons worth 50 cts., 30 gallons worth $1., and 50 gallons worth 30 cts.; what are 15 gallons of the mixture worth?

3. A farmer mixed together 5 bushels of oats worth 40 cts., 9 bushels of rye worth 50 cts., and 30 bushels of corn worth 75 cts.; what is one bushel of the mixture worth?

QUESTIONS. When is a bill of exchange said to be at a premium? (282.) When at a discount? (283.) What is the difference between present worth and discount? (259.) (260.) What is a bank? (261.) When is a note said to be at maturity? (266.) What are the proceeds of a note? (267.) What is per cent.? (183.) What is an agent? (188.) What is a charter? (192.) What is the market value of stocks? (200.) What are profit and loss? (206.)

LESSON II.

308. Alligation Alternate is the process of finding what quantity of simples, whose prices or qualities are given, must be taken to make a mixture of any given price or quality.

309. To find what quantity of each simple must be taken to form a mixture of a given value.

I have different qualities of sugar worth respectively 10, 11, 12, 14, and 15 cts. per lb.; what proportion of each must I take to make a mixture worth 13 cents?

*NOTE.-The difference between the total loss and the total 'gain is one cent, which added equally to the losses makes 14 cents for each.

18 lbs.

"+lb. lb.

9 lbs.

ANALYSIS.-1. For convenience arrange 'the prices of the simples in a column, in the order of their values, with the given mean at the left.

2. Since the total gain on the quantity of simples used must equal the total loss, first take that portion of each simple that will equal the gain of 1 cent upon selling at the mean price.

3. If, by using one pound of sugar at 10 cts. in a mixture worth 13 cts., the gain is 3 cts.; to gain 1 ct., use of a pound.

4. If, by using one pound of sugar at 11 cts. in a mixture worth 13 cts., the gain is 2 cts.; to gain 1 ct., use of a pound.

5. If, by using one pound of sugar at 12 cts. in a mixture worth 13 cts., the gain is 1 ct.; to gain 1 ct., use 1 pound.

6. If, by using one pound of sugar at 14 cts. in a mixture worth 13 cts., the loss is 1 ct.; to lose 1 ct., use 1 pound.

7. If, by using one pound of sugar at 15 cts. in a mixture worth 13 cts., the loss is 2 cts. ; to lose 1 ct., use of a pound.

8. Since the total gain is 3 cts., and the total loss is 2 cts., add the difference, 1 ct., equally to the losses, making 1 cent for each.

9. If 1 pound of sugar at 14 cts. must be used, to lose 1 cent; to lose cent,pound must be used, which added to 1 pound makes 1 lbs., the proportional quantity.

142

10. If lb. of sugar at 15 cts. must be used, to lose 1 cent; to lose cent, lb. must be used, which added to lb. makes lb., the proportional quantity.

Therefore, according to the conditions of the questions, the proportional quantities are § lb. at 10 cts., lb. at 11 cts., 1 lb. at 12 cts., 11⁄2 lb. at 14 cts., and 2 lb. at 15 cts. Multiplying by 12*, the least common multiple of the denominators, we have 4 lbs. at 10 cts., 6 lbs. at 11 cts., 12 lbs. at 12 cts., 18 lbs. at 14 cts., and 9 lbs. at 15 cts.

RULE.-I. Arrange the prices or qualities in a column,

*NOTE. If proportional quantities be multiplied by any number, they will still be proportional. The most convenient method of changing two or more fractional numbers to integers without changing their ratios is to multiply them by the least common multiple of their denominators.

in the order of their values, with the given mean price at the left.

II. Compare the price of each simple with the mean price, and take that portion of each which will gain one of the mean price.

III. Find the difference between the total loss, and the total gain; if it be a gain, add it equally to the losses; if a loss, add it equally to the gains, adding that portion of each quantity required by the added gain or loss to the portion already found, and the result will be the proportional quantity required.

IV. If the proportional quantities are fractional, multiply them by the least common multiple of the denominators.

EXAMPLES FOR PRACTICE.

4. A farmer wishes to mix corn at 75 cts. per bushel with rye at 55 cts., oats at 50 cts., and wheat at 95 cts.; what quantity of each must he take to make a mixture worth 70 cts. per bushel?

5. I have different kinds of salt worth respectively 25, 30, 35, and 50 cts. per bushel; how much of each kind must be taken in order that the mixture may be sold without loss at 42 cts. per bushel?

6. How much tea at 40 cts., 50 cts., 60 cts., and 70 cts. per pound, must be taken, that the mixture may be worth 45 cts. per pound? 55 cents per lb.? 68 cents per lb.?

LESSON III.

310. To find what quantity of the other simples is re quired to form a mixture at a given price, when the quan tity of one of the simples is limited..

A merchant has tea at 40 cts., 50 cts., 75 cts., 95 cts,