3. A merchant would mix 4 sorts of wine, of several prices, viz. at 75c. $1 25c. $1 50c. and $1 621c. per gallon; of these he would have a mixture of 72 gallons, worth $1 374c. per gallon; what quantity of each sort must he have?

Ans. 8 at 75c. 16 at $1 25c. 40 at $1 50c. and 8 at $1 62c. Or, 16 at 75c. 8 at $1 25c. 8 at $1 50c. and 40 at $1 621c.

4. How many gallons of water of no value, 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?

When more than one of the simples are limited.

RULE.

Find, by Alligation Medial, what will be the rate of a mixture made of the given quantities of the limited simples only; then, consider this as the rate of a limited simple, whose quantity is the sum of the first given limited simples, from which, and the rates of the unlimited simples, by Case II. calculate the quantity.

EXAMPLES.

1. How much wine, at 80c. and at 871c. per gallon, must be mixed with 8 gallons at 75c. and 12 gallons at 90c. per gallon, that the mixture may be worth 821c. per gallon?

Limited simples {

S8 gallons, at 75c.=$ 6
12 gallons, at 90 = 10 80c.

20

Gal. $ c.

16 80

Gal. c.

As 20: 16 80 :: 1: 84 per gallon.

Now, having found the rate of the limited simples, the question may stand thus: How much wine, at 80c. and 874c. per gallon, must be mixed with 20 gallons at 84c. per gallon, that the mixture may be worth 821c. per gallon?

*The three last Cases need no demonstration, as the 2d and 3d evidently result from the first, and the last from Alligation Medial, and the second Case in Alternate.

W w

75 90

2. How much gold, of 14 and 16 carats fine, must be mixed with 6oz. of 19, and 12 of 22 carats fine, that the composition may be 20 carats fine? Ans. 1oz. of each sort.

POSITION.

POSITION is a rule, which, by false or supposed numbers, taken at pleasure, discovers the true ones required. It is divided into two parts; single and double.

SINGLE POSITION.

Single Position teaches to resolve those questions, whose results are proportional to their suppositions: such are those which require the multiplication or division of the number sought by any proposed number; or when it is to be increased or diminished by itself a certain proposed number of times.

RULE.*

1. Take any number, and perform the same operations with it as are described to be performed in the question.

2. Then say, as the sum of the errours is to the given sum, so is the supposed number, to the true one required.

Proof. Add the several parts of the sum together, and if it agrees with the sum, it is right.

EXAMPLES.

1. A school master, being asked how many scholars he had, said, If I had as many more as I now have, three quarters as many, half as many, one fourth and one eighth as many, I should then have 435: Of what number did his school consist?

* The operations contained in the question being performed upon the answer or number to be found, will give the result contained in the question. The same operations, performed on any other number, will give a certain result. When the results are proportional to their supposed numbers, it is manifest that one result must be to the result in the question, as the supposed number is to the true one or answer. In any cases, when the results are not proportional to their sup positions, the answer cannot be found by this rule.

2. A person lent his friend a sum of money unknown, to receive interest for the same at 6 per cent. per annum, simple interest, and at the end of 12 years, received for principal and interest $860: What was the sum lent? Ans. $500.

3. A, B and C joined their stocks, and gained $353 121c. of which A took up a certain sum, B took up four times so much as A, and C, five times so much as B: What share of the gain had each? ($14 12 c. A's share.

4. A, B, C and D spent 35s. at a reckoning, and, being a little dipped, they agreed that A should pay, B, C, and D : What did each pay in the above proportion?

s. d.

A, 13 4

5. A certain sum of money is to be divided between 5 men, in such a manner as that A shall have, B, C, D, and E the remainder, which is £40: What is the sum? Suppose £200, then +++=120.

200-120 80. As 80 40 :: 200: 100 Ans. 6. A person, after spending and of his money, had £263 left: 18 What had he at first?

Ans. £160.

7. A and B, talking of their ages, B said his age was once and an half the age of A; C said his was twice and one tenth the age of both, and that the sum of their ages was 93: What was the age of each? Ans. A's 12, B's 18, and C's 63 years.

8. A vessel has 3 cocks, A, B and C; A can fill it in an hour, Bin of an hour, and C in of an hour: In what time will they all fill it together? Ans. hour.

9. A person having about him a certain number of dollars, said that,,, and of them would make 57: Pray, how many had he?

Ans. 60.

10. A Gentleman bought a chaise, horse and harness, for $500, the horse cost more than the harness, and the chaise more than the horse What was the price of each?

:

11. A and B, having found a purse of money, disputed, who should have it: A said that, and of it amounted to £35, and, if B could tell him how much was in it, he should have the whole, otherwise he should have nothing: How much did the purse contain? Ans. £100.

12. A gentleman divided bis fortune among his sons; to A he gave $9, as often as to B $5, and to C but $3 as often as to B $7, yet C's portion came to $1059: What was the whole estate? Ans. $7979 80c. 13. Seven eighths of a certain number exceeds four fifths by 6: What is that number? Ans. 80. 14. What number is that, which, being increased by, and of itself, the sum will be 2343?

DOUBLE POSITION.

Ans. 90.

Double Position teaches to resolve questions by making two suppositions of false numbers.

Those questions, in which the results are not proportional to their positions, belong to this rule: such are those, in which the number sought is increased or diminished by some given number, which is no known part of the number required.

RULE.*

1. Take any two convenient numbers, and proceed with each according to the conditions of the question.

2. Place the result or errours against their positions or suppos

ed numbers, thus,

Pos.
30.

Err.
12

X

201

6

and if the errour be too great, mark it

with; and if too small with

3. Multiply them crosswise; that is, the first position by the last errour, and the last position by the first errour.

The rule is founded on this supposition, that the first errour is to the second, as the difference between the true and first supposed number is to the difference between the true and second supposed number: When that is not the the exact answer to the question cannot be found by this rule.

case,

That the rule is true according to the supposition may be thus demonstrated. Let A and B be any two numbers produced from a and b by similar operations, it is required to find the number from which N is produced by a like operation.

Put x number required, and let N-A-r, and N-Bs. Then according to the supposition on which the rule is founded, r; s :: x-a: x-b, whence, by multiplying means and extremes, rx-rb=sx-sa; and by transposition, rxrb-sa

scrb-sa; and by division, r == number sought; and if r and s be both negative, the Theorem is the same, and if r or s be negative, x will be rb+sa

4. If the errours be alike, that is, both too small or both too great, divide the difference of the products by the difference of the errours, and the quotient will be the answer.

5. If the errours be unlike; that is, one too small, and the other too great, divide the sum of the products by the sum of the errours, and the quotient will be the answer.

Note. When the errours are the same in quantity, and unlike in quality, half the sum of the suppositions is the number sought.

EXAMPLES.

1. A lady bought damask for a gown, at 8s. per yard, and lining for it at 3s. per yard; the gown and lining contained 15 yards, and the price of the whole was £3 10s.: How many yards were there of each ?

Suppose 6 yards damask, value 48s.
Then she must have 9 yards lining, value 27s.

Sum of their values

75s.

So that the first errour is 5 too much, or + 5
Again, suppose she had 4 yards of damask, value 32s.
Then she must have 11 yards of lining, value 33s.

Sum of their values = 65s.

So that the second errour is 5 too little or

5s.

£ s. d.

5 yards at 8s. 5- 10 yards at 3s.

= 200

= 1 10 0

20

30

3 10 0 proof.

20

Sum of errours = 5+5=10)50

Ans. 5yds. damask, and 15—5—10yds, lining.

Or, 6+4÷2=5 as before.

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

Suppose

80. 120+ Ans. Their income is £200 per ann.

160

X

40+ Also, A spends £175 and B £205 per annum. Then, 80-10-70 A's expense per annum, and 70+30 =100, B's expense per annum. Then 100 × 8-80×8=160, which should have been 40; therefore, 160-40-120 more than it should be, for the first errour. In like manner proceed for the second

errour.

3. A and B laid out equal sums of money, in trade: A gained a sum equal to of his stock, and B lost $225, then A's money was double that of B: What did each lay out? Ans. $600.