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pepper, and 3 lb. white pepper are worth 10 lb. Calcutta ginger, and 4 lb. ginger cost 279?

Ans. $2.80.

4. If 14 lb. Rio coffee costs as much as 11 lb. Java, and 6 lb. Java as much as 7 lb. Laguayra, and 17 lb. Laguayra as 18 lb. Jamaica, and 16 lb. Jamaica as 17 lb. Manila, what cost 20 lb. Rio if 15 lb. Manila cost $2.40? Ans. $3.30.

5. If 11 shares of United Co's of N. J. are worth 16 shares of Norristown, and 4 shares of Norristown are worth 10 of Pennsylvania, and 3 of Pennsylvania are worth 9 of Reading; how many shares of Reading are worth 22 of United Co's?

Ans. 240.

6. What will be the cost of 3 kegs of 2 d. sheathing nails if 2 kegs of 2 d. nails are worth 3 kegs of 4 d. nails, and 5 kegs of 4 d. nails are worth 6 kegs of 8 d. nails, and kegs of 8 d. nails are worth 10 kegs of 10 d. nails, and 2 kegs of 10 d. nails are worth $6.20? Ans. $18.60.

7. What will be the cost of 6 boxes of old layer raisins if 7 boxes are worth 120 lb. new Valencia raisins, and 4 lb. new Valencia raisins are worth 7 lb. currants, and 11 lb. currants are worth 2 lb. shelled Languedoc almonds, and 6 lb. shelled Languedoc almonds are worth $1.98? Ans. $10.80.

8. If 12 shillings in Boston equaled 15 shillings in Philadelphia, and 45 shillings in Philadelphia equaled 28 shillings in Charleston, S. C., and 14 shillings in Charleston equaled 24 shillings in New York, how many shillings in New York were equal to 72 shillings in Boston? Ans. 96.

9. If James earns as much in 6 months as John does in 8 months, and John earns as much in 4 months as Jonathan in 7 months, and Jonathan earns as much in 5 months as Josiah in 6 months, how long will it take Josiah to earn as much as James in 18 months? Ans. 50 months.

10. What is the cost of 200 lb. of buckwheat flour if 900 lb. cost as much as 11 barrels of rye flour, and 75 barrels of rye flour as 44 barrels Minnesota Extra, and 26 barrels Minnesota Extra as 371⁄2 barrels Extra Round Hoop Ohio; and 10 barrels Extra Round Hoop Ohio cost $52?

Ans. $10.75§.

MEDIAL PROPORTION.

579. Medial Proportion is the process of combining two or more quantities of different values.

580. The Mean Value is the average value of the combination.

NOTE--The subject has been called Alligation, from alligo, I bind, a Lame suggested by the method of linking the figures with a line in solving the problems.

CASE I.

581. Given, the quantity and value of each, to find the mean value.

NOTE.-This case was formerly called Alligation Medial.

1. A merchant mixed 24 lb. of sugar at 10 cents a pound, 30 lb. at 14 cents, and 26 lb. at 20 cents; what is the aver age price of the mixture?

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Rule. Find the sum of the values of the ingredients and divide it by the sum of the ingredients.

WRITTEN EXERCISES.

2. A person mixed 25 lb. of tea at 50 cents a pound, 34 lb. at 80 cents, and 41 lb. at $1.10; what is the mean price or quality of the mixture ? Ans. $.844.

3. A person mixed 18 gal. of wine at $.50, 26 gal. at $.80, 20 gal. at $1.20, with 6 gal. of water; what was the value of a gallon of the mixture? Ans. $.76.

4. A goldsmith combined 8 oz. of gold 21 carats fine, 12 oz. 22 carats fine, 18 oz. 20 carats fine, with 28 oz. of alloy; required the fineness of the composition. Ans. 12 carats.

5. A person mixed 12 gal. of alcohol 90% strong, 7 gal

80% strong, 10 gal. 75% strong, and 11 gal. 70% strong; what per cent. of alcohol in the mixture? Ans. 79%.

6. A drover bought 30 cows at $20 a head, 40 at $25 a head, 30 at $28 a head; he sells them at a gain of 25%; what is the average price per head received? Ans. $30.50.

CASE II.

582. Given, the mean value and the value of each ingredient, to find the proportional quantity of each. NOTE. This and the following cases were formerly called Alligation Alternate.

1. A grocer wishes to mix sugars worth 5, 7, 12, and 14 cents a pound, forming a mixture worth 9 cents a pound; in what proportion must the sugars be mixed?

SOLUTION.-If we take 1 lb. at 5 cents for the mixture worth 99, we gain on it 4, and to gain 1 cent we would take of a pound. If we take 1 9 lb. at 14, we will lose 5, and to lose

1 cent, what we have just gained, we

OPERATION.

Ans.

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would take lb.; hence we take lb. at 5% as often aslb. at 14%, or in whole numbers, 20 times, which is 5 of the first, as often as 20 times }, which is 4 of the fourth. In a similar manner we find that we must take 3 lb. at 79, as often as 2 lb. at 12; hence the quantities may be mixed in the proportion of 5, 3, 2, and 4.

Rule.-I. Write the several prices or qualities in a column, and the mean price or quality of the mixture at the left.

II. Select two quantities, the one less and the other greater than the average, write the reciprocal of the difference between each quantity and the average opposite the quantity, and reduce these to integers by multiplying by the least common denominator, and proceed in the same manner until all the prices have been used.

III. Add two or more proportional numbers if they stand opposite a given quantity; the results will be the proportional numbers required.

NOTES.-1. When there are three quantities, compare the one which is greater or less than the average with both the others, and take the sum of the two numbers opposite this one.

2. A common factor may be inserted in any couplet or omitted from it without changing the proportional parts; it is thus seen that there may be any number of answers in the same proportion.

WRITTEN EXERCISES.

2. A grocer has teas worth 7, 10, 16, and 18 dimes a pound; what relative quantities of each must be taken to form a mixture worth 12 dimes a pound? Ans. 6; 4; 2; 5.

3. A merchant has 4 pieces of muslin, worth 10, 14, 20, and 22 a yard, respectively; how many yards must he sell of each that the price may average 187? Ans. 1; 1; 2; 2.

4. How shall I combine gold 16 carats, 18 carats, and 22 carats, to make a mixture of 20 carats fine, if I wish to mix equal quantities of 1st and 2d? Ans. 1st, 1; 2d, 1; 3d, 3. 5. What relative quantities of rice worth 121, 183, and 20 cents a pound, must be taken to form a mixture worth 16 cents a pound? Ans. 27; 15; 15.

6. A farmer bought pigs at $4 each, sheep at $5 each, and calves at $63 each; how many must he sell of each so that the average price may be $5 each? Ans. 9; 2; 3.

7. A man has a quantity of 3, 5, 25, and 50 cent pieces, which he wishes to exchange for 10 cent pieces; what is the relative number of pieces exchanged? Ans. 40; 15; 5; 7.

CASE III.

583. Given, the mean value, the value of each ingredient, and the quantity of one or more, to find the other quantities.

1. A farmer bought 20 hens at 10 dimes each; how many must he buy at 4 and 5 dimes each, so that the average price may be 8 dimes each?

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235 at 10. But he bought 20, or 4 times 5, at 10 dimes, hence he must buy 4 times 1 or 4 at 4 dimes, and 4 times 2 or 8 at 5 dimes.

Rule.-I. Find the proportional quantities by Case 11.

II. Divide the given quantity by the proportional quantity iimited, and multiply each of the other proportional quantities by the quotient.

WRITTEN EXERCISES.

2. A merchant bought some hats for $5 each, some vests for $7 each, and 48 coats for $16 each; the average price was $12; how many vests and hats did he buy? Ans. 16 of each. 3. A publisher sold Mentals @ 15, Primaries @ 10, Grammars @30, and 360 Spellers @ 40; required the number of each, if the average price was 35¢. Ans. 36.

4. A merchant wishes to mix 40 lb. of sugar at 69 and 40 at 89, with some at 14 and 15, so that the mixture may be worth 10g; how much of the latter kinds must he take? Ans. 20 lb. at 149; 32 lb. at 159.

5. A grocer wished to mix 15 lb. of tea at $1 a pound, 21 lb. at $1, with that worth 70 cents and 50 cents, so that the mixture may be worth 80 cents; how much must he take of the 3d and 4th? Ans. 42 lb. at 709; 35 lb. at 50%.

6. A man has some 3 ct. pieces, some 5 ct. pieces, some 10 ct. pieces, and 290 fifty-cent pieces, which he exchanges for 25 ct. pieces; how many must he exchange of each kind i Ans. 250 three-cent; 50 five-cent; 50 ten-cent.

CASE IV.

584. Given, the mean value, value of each ingredient, and entire quantity, to find the quantity of each ingredient.

1. A person has a sum of money in ten-cent pieces, which he wishes to exchange for 3, 5, 25, and 50 cent pieces, having 255 pieces in all; how many of each will he obtain?

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times 51, hence we must take 5 times as many of each, which gives respectively 200, 15, 5, and 35.

Rule.-I. Find the proportional quantities by Case II. II. Divide the required quantity by the sum of the pro

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