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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 ques tion.

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 37 : 148 : : 12 : 48 Ans. as many 12

48
as many
6

24
as many =
4

16
as many
3

12
Result, 37

Proof, 143 2. What number is that which being increased by }, j, and ! of itself, the sum will be 125 ?

Ans. 50. 3. Divide 93 dollars between A, B and C, so thai B's chare

may be half as much as A's, and C's share three ti, nes As much as B's.

Ans. A's share $31, B's $15), and C's $46?. 4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 3 times as much as A, and C took up as niuch 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 61. per cent, per annum, simple interest, and at the end of twelve years received 7311, principal and interest together; what was the sun delivered to him at first?

Ans. £425. 6. A vessel has 3 cocks, A, B and C; A can fill it in 1 hour, B in 2 hours, and C in 4 hours: in what time will they all fill it together?

Ans, 31 min, 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 mueb 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 four 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

2d. Suppose A8 B 10

B 12 C 18

C 20 D 36

D 40

70 100

80 100

Ist error, 30

2d error, 20

* 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 increased or diminished by some given number, which is no lonown part of the number required.

30

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

Pos. Err.
6

A 12
B 16
С 24
D 48

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X

8

20

Proof 100

120

240
120

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 iniuch as A and B both ; how much did each inan pay

?

Ans. A paid $120, B $131, and C $250. 3. A man bequeathed 1001. to three of his friends, after this manner; the first minst have a certain portion, the second must have twice as much as the first, wanting 81. and the ihird must have three times as much as the first, wanting 15l.; I demand how much each man inust have ?

Ans. The first £20 10s. second £:33, third £16 10s. 4. A labourer was hired for 60 days upon this condition ; that for every daw he wrongla: he should receive 4s. and for every day he was idle should forfest 2s.; at the expiration of the time he received 71. 10s.; how many days did he work, and how many was he idle ?

Ins. 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 501., to G one-third, and to the rest, which was 101. less than the share of G; I demand the sum give!, and each man's port?

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

every cow

7. Two men, A and B, lay out equal sums of money in trade; A gains 1261. and B loses 871. and A's money is now double to B's; what did each lay out?

Ans. £300. 8. A farmer having driven his cattle to market, received for them all 1301. being paid for every ox 71. for 51. and for every calf il. 10s. there were twice as mauy 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 rertain pumber; B as many as A and 15 more; C got a 5th part of both their sums added togethes; how many did each get?

Ans. A got 127, B 142, C 51.

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 numbers from one up to the given number, continually together, and the last product will be the answer required.

EXAMPLES.

1. How many changes can be

a b c made of the first three letters of

a cb he alphabet ?

3 bac Proof,

4 bca

5 cba 1x2336 Ans.

6 cab 2. How many changes may be rung on 3 bulls ?

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. 11037 years.
ANNUITIES OR PENSIONS,

COMPUTED.AT
COMPOUND INTEREST.

CASE I.
To find the amount of an Annuity, or Pension, in arrears,

at Compound Interest.

RULE. 1. Make I 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 +1,1236+1,1910164,374616, sum of the series. * -Then, 4,374616 X 125=9516,827, the amount fought.

OR BY TABLE II. Multiply the Tabular nuniber under the rate, and opposite to the time, by the annuity, and the product will be the amount sought.

* The sum of the scries thus found, is the amount of ll. or 1 dollar annuity, for the giveu time, which may be found in Table Il. ready calcula. ted.

Hence, cither the amount or present worth of annuities may be readily round by tables for that purposc,

R

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