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DOUBLE POSITION,

TEACHES to resolve questions by making two suppo sitions of false numbers.*

RULE.

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

2. Find how much the results are different from .he re sults in the question.

3. Multiply the first position by the last error, and the las position by the first error.

4. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer.

5. If the errors are unlike, divide the sum of the pro ducts by the sum of the errors, and the quotient will be the answer.

NOTE. The errors are said to be alike when they are both too great, or both too small; aud unlike, when one is too great, and the other too small.

EXAMPLES.

1. A purse of 100 dollars is to be divided among 4 men A, B, C and D, so that B may have four dollars more thar A, and C 8 dollars more than B, and D twice as many a C; what is each one's share of the money ? 1st. Suppose A 6

2d. Suppose A 8

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* Those questions in which the results are not proportional to their posi tions, belong to this rule; such as those in which the number sought is in ereased or diminished by some given number, which is no known part of the sumber required.

The errors being alike, are both too small, therefore,

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10)120/12 A's part.

2. A, B, and C, built a house which cost 500 dollars, of which A paid a certain sum; B paid 10 dollars more than A, and C paid as much as A and B both; how much did each man pay?

Ans. A paid $120, B $130, and C $250. 3. A man bequeathed 1007. to three of his friends, after this manner; the first must have a certain portion, the second must have twice as much as the first, wanting 87. and the third must have three times as much as the first, wanting 157.; I demand how much each man must have?

Ans. The first £20 10s. second £33, third, £46 10s. 4. A labourer was hired for 60 days upon this condition; that for every day he wrought he should receive 4s. and for every day he was idle should forfeit 2s.; at the expiration of the time he received 77. 10s.; how many days did he work, and how many was he idle?

Ans. He wrought 45 days, and was idle 15 days. 5. What number is that which being increased by its, its, and 18 more, will be doubled ? Ans. 72.

6. A man gave to his three sons all his estate in money, viz. to F half, wanting 50%. to G one-third, and to H the test, which was 101. less than the share of G; I demand one sum given, and each man's part?

Ans. the sum given was £360, whereof F had £130,
G £120, and H £110.

7. Two men, A and B, lay out equal sums of money trade; A gains 1267. and B loses 877. and A's money i now double to B's; what did each lay out?

Ans. £300.

8. A farmer having driven his cattle to market, receive for them all 1307. being paid for every ox 71. for every co 51. and for every calf 17. 10s. there were twice as many cows as oxen, and three times as many calves as cows how many were there of each sort?

Ans. 5 oxen, 10 cows, and 30 calves. 9. A, B, and C, playing at cards, staked 324 crowns; but disputing about tricks, each man took as many as he could; A got a certain number; B as many as A and lá more; C got a 5th part of both their sums added together; how many did each get?

Ans. A got 1271⁄2, B 1421⁄2, C 54.

PERMUTATION OF QUANTITIES,

IS the showing how many different ways any given number of things may be changed.

To find the number of Permutations, or changes, that can be made of any given number of things all different from each other.

RULE.--Multiply all the terms of the natural series of number from one up to the given number, continually together, and the last product will be the answer required.

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2. How many changes may be rung on 9 bells?

Ans. 362880.

1x2×36 Ans.

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2. If a salary of 60 dollars per annum to be paid yearly be forborne twenty years, at 6 per cent. compound interest what is the amount?

Under 6 per cent. and opposite 20, in Table II., yoɩ will find,

Tabular number=36,78559

60 Annuity.

Ans. $2207,13540-$2207, 13 cts. 5m.+

3. Suppose an annuity of 1007. be 12 years in arrears, it it required to find what is now due, compound interest being allowed at 51. per cent. per annum?

Ans. £1591 14s. 3,024d. (by Table II.)

4. What will a pension of 1207. per annum, payable yearly, amount to in 3 years, at 57. per cent. compound interest? Ans. £378 6s.

II. To find the present worth of annuities at Compound In

terest.

RULE.

Divide the annuity, &c. by that power of the ratio sig nified by the number of years, and subtract the quotient from the annuity: This remainder being divided by the ra tio less 1, the quotient will be the present value of the an nuity sought.

EXAMPLES.

1. What ready money will purchase an annuity of 501 to continue 4 years, at 51. per cent. compound interest? 4th power th Praver of }=1,215506)50,00009(41,13513+

From
Subtrict

50
41,13513

*$*86487

77,291 £177 5s. 1d. Ans.

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