Write the rates of the simples under each other, and link each rate which is less than the mean rate, with one or more that is greater; place the difference between each rate and the mean price, opposite to the rates with which it is linked; then if only one difference stand against either rate, it will be the quantity required at that rate; but if there be several, their sum will be the quantity.

Note 1.-If all the given rates be greater, or less, than the mean rate, they must be linked to a cipher.

2. Different modes of linking will produce different

answers.

EXAMPLES.

1. How much rye at 4s. per bushel, barley at 33. per bushel, and oats at 2s. per bushel, will make a mixture worth 2s. 6d. per bushel?

2. A vintner has three kinds of wine, viz. one kind at 160 cents per gallon, another at 180 cents, and another at 240 cents; how much of each kind must he take to make a mixture, worth 190 cents per gallon?

Ans.

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50 gals. at 160 cts. 50 gals. at 180 cts. and 40 gals. at 240 cts.

3. How much sugar at 4d. at 6d. and at 11d. per lb. must be mixed together, to make a composition worth 7d. per lb. ? Ans. An equal quantity of each kind.

4 It is required to mix several sorts of wine, viz. at 9s. 15s. and 12s. per gallon, with water, that the mixture may be worth 12s. per gallon; how much of each sort must be taken?

Ans. 3 gals. at 9s. 3 gals. at 15s. and 12 gals. at 12s. with 9 gals. of water. 5. A grocer has several sorts of sugar, viz. one sort at 12 cents per lb. another at 11 cents, a third at 9 cents, and a fourth at 8 cents per lb.; how much of each sort must he take to make a mixture worth 10 cents per lb. ?

When the price of all the simples, the quantity of one of them, and the mean price of the whole mixture are given, to find the several quantities of the rest.

RULE.

Link the several prices, and place their differences as in case 2; then

As the difference opposite to the price of the given quantity,

Is to the differences respectively;

So is the given quantity,

To the several quantities required.

EXAMPLES.

1. A grocer would mix 30 lb. of sugar, at 14 cents per lb. with some at 9 cents, 10 cents, and 13 cents per lb. ; how much of each sort must he mix with the thirty lb. that the mixture may sell at 12 cents per lb. ?

2. How much barley at 30 cents per bushel, rye at 36 cents, and wheat at 48 cents, must be mixed with 12 bushels of oats, at 18 cents, to make a mixture worth 22 cents per bushel ? Ans. 1 bushel of each sort.

3. How much wine at 5s. at 5s. 6d. and at 6s. per galfon, must be mixed with 3 gallons at 4s. per gallon, so that the mixture may be worth 5s. 4d. per gallon?

Ans. 3 gals. at 5s. 6 at 5s. 6d. and 6 at 6s. 4. How much tea at 12s. 10s. and at 6s. per lb. must be mixed with 20 pounds at 4s. per lb. to make a mixture worth 8s. per lb. ?

Ans. 10 lb. at 6s. 10 lb. at 10s. and 20 lb. at 12s.

CASE 4.

When the prices of the several simples, the quantity to be compounded, and the mean price are given, to find the quantity of each simple.

RULE.

Link the several prices, and place their differences as before; then,

As the sum of the differences,

Is to the difference opposite to each price;

So is the quantity to be compounded,

To the quantity required.

EXAMPLES.

1. How much sugar at 10 cents, 12 cents, and 15 cents, per lb. will be required to make a mixture of 29 lb. worth 13 cents per lb. ?

2. A brewer has three sorts of beer, viz. at 10d. 8d. and 6d. per gallon; how much of each sort must he take to make a mixture of 30 gallons, worth 7d. per gallon?

Ans. 5 gals. at 10d. 5 gals. at 8d. and 20 gals. at 6d. 3. A goldsmith has gold of 15, 17, 20, and 22 carats fine, and would melt together of each of these so much, as to make a mass of 40 oz. of 18 carats fine; how much of each sort is necessary ?

Ans.

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16 oz. of 15 carats, 8 oz. of 17 carats, 4 oz. of 20 carats, and 12 oz. of 22 carats fine. 4. How many gallons of water must be mixed with wine, at 4s. per gallon, so as to fill a vessel of 80 gallons, that may be afforded at 2s. 9d. per gallon?

Ans. 25 gallons of water, with 55 of wine.

POSITION.

Position is a rule for finding an unknown number, by one or more supposed numbers. It is divided into two parts, single and double.

SINGLE POSITION.

Single Position teaches to resolve such questions as require only one supposition.

RULE.

Suppose any number to be the true one, and proceed with it agreeably to the tenor of the question; then, As the result of the operation,

Is to the number given;
So is the supposed number,
To the number sought.

PROOF.

Work with the auswer according to the tenor of the question, and the result must equal the given number.

EXAMPLES.

1. A, B, and C, bought a quantity of wine for 340 dolfars, of which sum A paid three times more than B, and B four times more than C; how much did each pay?

As 51:340: 36: 240 sum paid by A.

340 Proof.

2. A person after spending and of his money, had 60L. left; how much had he at first? Ans. 144L. 3 What number of dollars is that, of which the 1, 1, and, make 74? Ans. 120. 4. A person having about him a certain number of crowns, said, if a third, a fourth, and a sixth of them were added together, tne sum would be 45; how many crowns had he? Ans. 60.

5. What is the age of a person who says, that if of the years I have lived, be multiplied by 7, and of them be added to the product, the sum will be 292 ? Ans. 60 years

6. A schoolmaster being asked how many scholars he had, answered, if to double the number I add,, and of them, I shall have 333; how many had he? Ans. 108,

7. A certain sum of money is to be divided among 4 persons, in such a manner that the first shall have of it, the second, the third, and the fourth the remainder, which is 28 dollars; what is the sum? Ans. 112 dols. 8. What sum, at 6 per cent. per annum, will amount to 350L. in 12 years? Ans. 500L.

DOUBLE POSITION.

Double Position teaches to find the true number, by making use of two supposed numbers.

RULE.

Suppose two numbers, and work with each agreeably to the tenor of the question, noting the errors of the results: multiply the errors of each operation into the sup posed number of the other; then,

If the errors be alike, i. e. both too much, or both ton little, take their difference for a divisor, and the difference of the products for a dividend: but if the errors be unlike, take their sum for a divisor, and the sum of the products for a dividend.

PROOF.

As in Single Position.

EXAMPLES.

1. A, B, and C, would divide 80 dollars among them in such a manner, that B may have 5 dollars more than A, and C 10 dollars more than B, required the share of each? Suppose A's share $10 Suppose A's share $15

B's

C's

15

25

50

B's

C's

20

30

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