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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 gallon, must be mixed with 3 gallons at 4s. per gallon, so that the mixture may be worth 5s. 4d. per gallon? Ans. 3gals. 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 Ss. per lb.?

Ans. 10lb. at 6s. 10lb. at 10s. and 20lb. 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 20 lb. worth 13 cents per 16.?

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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. 5gals. at 10d. 5gals. at 8d. and 20gals. 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 40oz. of 18 carats fine; how much of each sort is necessary?

Ans. {1607, of 15 carats, Soz. of 17 carats, 4oz.

of 20 carats, and 12oz. 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 answer 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 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 12
And C paid

3

A paid 240
B paid 80Ans.
C paid 20

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

2. A person after spending and of his money,

60L. left; how much had he at first?

had

Ans. 144L.

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.

of

5. What is the age of a person who says, that if of the years I have lived be multiplied by 7, and 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, §, and of them, I shall have 333; how many had he?

Ans. 108.

4

7. A certain sum of money is to be divided among persons in such a manner that the first shall have of it, the second 4, the third, and the fourth the remainder, which is 28 dollars; what is the sum?

Ans. 112 dollars.

8. What sum, at 6 per cent. per annum, to 860L. in 12 years?

DOUBLE POSITION.

will amount

Ans. 500L.

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?

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80-50-30 error too little. 80--65-15 error too little.

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2. D, E and F, would divide 100L. among them, so as that E may have 3L. more than D, and F4L. more than E; what is the share of each?

Ans. D's share 30L. E's 33L. F's 37L.

3. A, B and C, owe 1000L. of which B is to pay 100L. more than A, and C is to pay as much as both A and B: how much is each man's share of the debt?

Ans. A's share is 200L. B's 300L. and C's 500L. 4. Bought linen at 4s. per yard, and muslin at 2s. per yard; the number of yards of both was 8, and the whole cost 20s.: how many yards were there of each? Ans. 2 yards of linen, and 6 yards of muslin. 5. The head of a certain fish is 9 inches long; its tail is as long as its head and half of its body; and the length of its body is equal to the length of its head and tail: what is the whole length? Ans. 6 feet.

6. A labourer hired for 40 days upon this condition, that he should receive 20 cents for every day he wrought, and should forfeit 10 cents for every day he was idle; at settlement he received 5 dollars. How many days did he work, and how many days was he idle Ans. Wrought 30 days, idle 10. 7. A father dying, left to his three sons A, B, and C, his estate in money, dividing it as follows, viz. to A he gave half the estate, wanting 44L.; to B he gave a third of it, and 14L. over; and to C he gave the remainder, which was 821.. less than the share of B. What was the whole sum left, and what was each son's share? The sum left was 588L. of which A had 250L. B 210L. and C 128L.

Ans.

8. Two persons, A and B, have both the same income; A saves one fifth of his every year; but B, by spending 150 dollars per annum more than A, at the end of 8 years finds himself 400 dollars in debt: What is their income, and what does each spend per annum?

Ans. Their income is 500 dollars per annum. |
A spends 400dols. and B 550.

ARITHMETICAL PROGRESSION.

Any rank or series of numbers, increasing or decreasing by a common difference, is said to be in arithmetical progression; as 2, 4, 6, 8, 10, and 6, 5, 4, 3, 2, 1.

.

The numbers which form the series are called the The first and last terms are called the ex

terms. tremes.

Note.-In any series of numbers in Arithmetical progression, the sum of the two extremes is equal to the sum of any two terms equally distant from them; as in the latter of the above series 6+1 4+3, and =5+2.

When the number of terms is odd, the double of the middle term is equal to the sum of the two extremes, or any two terms equally distant from the middle term; as in the former of the foregoing series 6x2=2+10, and 4+8.

CASE 1.

The first term, common difference, and number of terms given, to find the last term, and sum of all the

terms.

RULE.

1. Multiply the number of terms, less 1, by the common difference, and to the product add the first term, the sum is the last term.

2. Multiply the sum of the two extremes by the number of terms, and half the product will be the sum of all the terms.

EXAMPLES.

1. The first term of a certain series in arithmetical progression is 2, the common difference is 2, and the

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