Supplement to Alligation.

1. What is Alligation?

QUESTIONS.

2. Of how many kinds is Alligation? 3. What is Alligation MEDIAL?

4. What is the rule for operating?

5. What is Alligation ALTERNATE ?

6. When a number of ingredients of different prices are mixed together, how do we proceed to find the mean price of the compound or mixture?

7. When one of the ingredients is limited to a certain quantity, what is the method of procedure?

8. When the whole compofition is limited to a certain quantity, how do you proceed?

9. How is Alligation proved?

EXERCISES.

1. A Grocer would mix three forts of fugar together; one fort at 10d. per lb. another at 7d. and another at 6d. how much of each fort must he take that the mixture may be fold for 8d. per lb. ?

Ans. 3lb. at 10d. 2 at 7d. and 2 at 6d.

2. A Goldfmith has feveral forts of gold; fome of 24 carats fine, fome of 22, and fome of 18 carats fine, and he would have compounded of these forts the quantity of 60 oz. of 20 carats fine; I demand how much of each fort he must have?

Ans. 12 oz. 24 carats fine, 12 at 22 carats fine, and 36 at 18 carats fine.

§ 10. Position.

POSITION is a rule which, by falfe or fuppofed numbers, taken at pleafure, discovers the true one required. It is of two kinds, Single and Double.

SINGLE POSITION,

Is the working with one fuppofed number, as if it were the true one, to find the true number.

RULE.

1. Take any number and perform the fame operations with it as are defcribed to be performed in the question.

2. Then fay; as the fum of the errors is to the given fum, fo is the fuppofed number to the true one required.

PROOF. Add the feveral parts of the fum together, and if it agree with the fum, it is right.

EXAMPLES.

1. Two men, A and B, having found a bag of money, difputed who fhould have it; A faid the half third, and one fourth of the money made 130 dollars, and if B could tell how much was in it, he should have it all, otherwise he fhould have nothing; I demand how much was in the bag?

DOUBLE POSITION.

DOUBLE POSITION is that which discovers the true number, or number fought, by making ufe of two fuppofed numbers.

RULE.

1. Take only two numbers and proceed with them according to the conditions of the question.

2. Place each error against its refpective pofition or fuppofed number; if the error be too great, mark it with+; if too small with

3. Multiply them cross wife, the first position by the last error, and the last pofition by the first error.

4. If they be alike, that is, both greater or both lefs than the given number, divide the difference of the products by the difference of the errors, and the quotient will be the anfwer; but if the errors be unlike, divide the sum of the products by the fum of the errors, and the quotient will be the answer.

EXAMPLES.

1. A man lying at the point of death, left to his three fons all his estate, viz. to F half wanting 50 dollars; to G one third; and to H the rest, which was 10 dollars lefs than the fhare of G. I demand the fum left, and each

The divifor is the fum of the errors 90+and 10-

2. There is a fifh whofe head is 10 feet long; his tail as long as his head and half the length of his body, and his body as long as his head and tail; what is the whole length of the fifh?

Ans. 80 feet.

Bb

3. A certain man having driven his Swine to market, viz. Hogs, Sows, and Pigs, received for them all 501. being paid for every hog 18s. for every fow 16s. for every pig 2s. ; there were as many hogs as fows, and for every fow there were three pigs; I demand how many there were of each fort? Ans. 25 hogs, 25 fors, and 75 pigs.

1

4. A and B laid out equal fums of money in trade; A gained a fum equal to of his stock, and B loft 225 dollars; then A's money was double that of B's; what did each one lay out? Ans. 600 dollars.

5. A and B have the fame income; A faves of his; but B, by spending 30 dollars per annum more than A, at the end of 8 years finds himself 40 dollars in debt; what is their income, and what does each spend per annum ?

Ans. their income is 200 dolls. per ann. A spends 175 dolls. & B 205 per ann.

§ 11. Discount.

DISCOUNT is an allowance made for the payment of any fum of money before it becomes due, and is the difference between that fum, due fometime hence, and its present worth.

The prefent worth of any fum, or debt due fome time hence, is fuch a fum, as, if put to intereft, would in that time and at the rate per cent. for which the discount is to be made, amount to the fum or debt, then due.

RULE.

As the amount of 100 dollars, for the given time and rate is to 100 dollars, fo is the given fum to its prefent worth, which fubtracted from the given fum, leaves the discount.

EXAMPLES.

1. What is the discount of Dolls. 321,63 due 4 years hence, at 6 per cent?

2. What is the present worth of 426 dollars, payable in 4 years and 12 days, discounting at the rate of 5 per cent.

OPERATION.
Dolls.
Ans. Dolls. 354,515
6 interest of 100 dolls. 1 year.

4 years.

24

100

124 amount.

Then, As 124: 100 321,63

321,63

124)32163,00(259,379

321,63 given fum.
259,379 present worth.

Ans. 62,251 discount.

§ 12. Equation of Payments.

EQUATION of Payments is the finding of a time to pay at once, feveral • debts due at different times fo that neither party fhall fuftain lofs.

RULE.

Multiply each payment by the time at which it is due; then divide the fum of the products by the fum of the payments, and the quotient will be the equated time.