CASE 2.

When the prices of several simples are given, to find how much of each, at their respective rates, must be taken to make a compound or mixture at any proposed price.

RULE.

Set the prices of the simples one under another, and link every price which is not greater than the mean rate, to one or more that are greater than that rate; place the difference between each price and the mean rate opposite to the price or prices with which it is linked then, if only one difference stand opposite to either particular price, it will be the quantity required at that price; but if there be several differences, their sum will be the quantity.

Note.-Different modes of linking will produce dif

ferent answers.

EXAMPLES.

1. How much rye at 4 s. per bushel, barley at 3 s. per bushel, and oats at 2 s. per bushel, will make a mixture worth 2 s. 6 d. per bushel?

bu.

S.

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. {50 gals. at 160 cts., 50 gals. at 180 cts.,

and 40 gals. at 240 cts.

per

lb.

3. How much sugar at 4 d. at 6 d. and at 11 d. mus': be mixed together to make a composition worth 7 d. per lb.? Ans. an equal quantity of each kind. 4. It is required to mix several sorts of wine, viz. at 9 s. 15 s. and 21 s. per gallon, with water, that the mixture may be worth 12 s. per gallon; how much of each soit must be taken?

Ans.

{

3 gals. 9 s., 3 gals. 15 s., and 12 gals. at 21 s. 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 Ib. with some at 9 cents, 10 cents, and 13 cents per Ib.; now 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 5 s., at 5 s. 6 d., and at 6 s. per gallon, must be mixed with 3 gallons at 4 s. per gallon, so that the mixture may be worth 5 s. 4 d. per gallon? Ans. 3 gals. at 5 s., 6 at 5 s. 6 d., and 6 at 6 s. 4. How much tea at 12 s., 10 s., and at 6 s. per lb. must be mixed with 20 pounds at 4 s. per lb. to make a mixture worth 8 s. per lb.?

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

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 20 lb. worth 13 cents per lb.?

2 As 8:2::20: 5lb. at 10 cts.

2 8:4:20: 10 lb. at 15 cts. Ans.

3+1=4

8:2::20: 5lb. at 12 cts.

8 Sum of differences.

2. A brewer has three sorts of beer, viz. at 10 d., 8 d., and 6 d. per gallon; how much of each sort must he take to make a mixture of 30 gallons, worth 7 d. per gallon? Ans. 5 gals. at 10d., 5 gals. at 8d., and 20 gals. at 6 d. 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.

S16 oz. of 15 carats, 8 oz. of 17 carats, 4 oz.

4. How many gallons of water must be mixed with wine, at 4 s. per gallon, so as to fill a vessel of 80 gallons, that may be afforded at 2 s. 9 d. per gallon?

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

POSITION.

Position is a rule for nading an unknown number, by one or more supposed numbers.

two parts, single and double.

It is divided into

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 answer according to the tenor of he question, and the result must equal the given number.

EXAMPLES.

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

$

Suppose A paid 36
Then B paid

$

A paid 240
B paid 80 Ans.

12

And C paid

3

C

para 0

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

2. A person after spending and of his money, had 60 L. left; how much had he at first? Ans. 144 L. 3. What number of dollars is that, of which the 4, , 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, the 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 1⁄2, 1, 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 dollars. per annum, will amount Ans. 500 L.

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 supposed number of the other; then,

If the errors be alike, i. e. both too much, or both too 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?