RULE.

1. Take any number and perform the same operation with it, as is described to be performed in the question. 2. Then say; as the result of the operation is to the given sum in the question:: so is the supposed number: to the true one required.

The method of proof is by substituting the answer in the question.

EXAMPLES.

1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half as many, one-third and one-fourth as many, I should then have 148; How many scholars had he?

Suppose he had 12

As 57: 148 :: 12: 48 Ans.

2. What number is that which being increased by 1,1, and 4 of itself, the sum will be 125 ?

Ans. 60.

3. Divide 93 dollars between A, B and C, so that B's -share may be half as much as A's, and C's share three times as much as B's.

Ans. A's share $31, B's $151, and C's $461.

4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 33 times as much as A, and C took up as much as A and B both; what share of the gain had each?

Ans. A $40, B $140, and C $180. 5. Delivered to a banker a certain sum of money, to receive interest for the same at 6l. per cent. per annum, simple interest, and at the end of twelve years received 7811. principal and interest together; what was the sui delivered him at first? Ans. £425. 6. A vessel has S cocks, A, B and C; A can fill it in 1 hour, B in 2 hours and Cin 4 hours; in what time will they all fill it together? Ans. 34min. 17 sec.

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

3. Multiply the first position by the last error, and the last 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 products 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; and 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 4 dollars more than A, and C 8 dollars more than B, and D twice as many as C; what is each one's share of the money? 1st. Suppose A 6

B 10

C. 18

D 36

70

2d. Suppose A 8

B 12

10

1st. error 30

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

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

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 0 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 $150, and C $250.

3. A man bequeathed 100l. 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 81. 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 laborer was hired 60 days upon this condition that for every day he wrought he should receive 4s. and for every lay 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 50l. to G one-third, and to H the rest, which was 101. less than the share of G; I demand the sum given, and each man's part?

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

7. Two men, A and B, lay out equal sums of money in trade; A gains 1267. and Blooses 871 and A's money 18 now double to B's; what did each lay out?

Ans. £100.

8. A farmer having driven his cattle to market, eceived for them all 1301. being paid for every ox 71. for every cow 51. and for every calf 1. 10s. there were twice as many cows as oxen, and three times as many caves 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 15 more; C got a 5th part of both their sums added togeth er; how many did each get?

Ans. A got 1274, B 142), C 54.

PERMUTATION OF QUANTITIES, Is the shewing 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 numbers from one up to the given number, continually together, and the last product will be the answer required.

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

6.

cab

Ans. 362880.

3. Seven gentlemen met at an inn, and were so well pleased with their host, and with each other, that they agreed to tarry so long as they, together with their host, could sit every day in a different position at dinner; how long must they have staid at said inn to have fulfilled their agreement? Ans. 110170 years.

ANNUITIES OR PENSIONS,

COMPUTED AT

COMPOUND INTEREST.

CASE I.

365

To find the amount of an annuity, or Pension, in arıcars, at Compound Interest.

RULE.

1. Make 1 the first term of a geometrical progression, and the amount of $1 or £1 for one year, at the given rate per cent, the ratio.

2. Carry on the series up to as many terms as the given number of years, and find its sum.

3. Multiply the sum thus found, by the given annuity, and the product will be the amount sought.

EXAMPLES.

1. If 125 dols. yearly rent, or annuity, be forborne, (or unpaid) 4 years; what will it amount to, at 6 per cent. per annum, compound interest?

1+1,06+1,1256+1,1910164,574616 sum of the series.*Then, 4,374616×125=5346,827 the amount sought.

OR BY TABLE II.

Multiply the Tabular number under the rate and opposite to the time, by the annuity and the product will be the amount sought.

*The sum of the series thus found, is the amount of 11. or 1 dollar annuity, for the given time, which may be found in Table II. rendy calculated.

Hence, either the amount or present worth of annuities may be readily found by Tables for that purpose.