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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 question.

EXAMPLES.

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1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half ag many, one-third and one-fourth as many, I should then have 148: How many scholars had he? Suppose he had 19 As 57 : 143 : : 12 : 48 And as many 12

48
as many
6

24.
} as many
4

16
# as many

19 Result, S7

Proof, 148 2. What number is that which being increased by 1, 1, and $ of itself, the sam 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.

Hits as share 51, 's 15), and C’s 461 dolls. 4. A, B and Cojoined their stock and gaineti 360 dols. of which Ä took up a certain sum, I took 3} times as much as A, and C took up as much as A and B both; what share of the gain liad each ?

Ans. 4 $40, D $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 7311. principal and interest together : What was the sum 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, S4min. 171 sec.

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DOUBLE POSITION, TEACHBS to resolve questions in making two suprositions of faise imbers. *

RULE. 1. Take any two conveniet hirs, and proceed with each according to the core if the question.

2. Find low much the posti di viskrent from the Results in the question.

S. Multiply the first position by the last error, and the - Tast position by the first error.

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

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

Note. The errors are said to be like when they are both too great, or both too small: and unlike, when one

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 1), so that may have 4 dollars more than A, and C & civilars piore dan David D twice as many as C: what is each one's share of the money ? Ist. Suppose 6

21. Suppose A8 B 10

B 12 C13

C20 Dsü

D 40

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* 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 giren number, which is no known part of the number tegrired

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

Pos. Err.
6 SO

A 12
B 16
C24

D 48 8 20

Proof, 100 240 120 120

X

10) 120(12 A's part.

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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 150, and C 250 dols. S. A man bequeathed 100l. to threc 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 15l. : 1 demand how much each man must have ?

Ans. The first £ 20 10s. second £33, third £,46 10s.

4. A laborer 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 71. 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 s, its , and 18 inore, will be doubled ? Ans. 72.

6. A man gave to his three sons all his estate in money, viz. to F hall, 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 giten was £360, whereof F had £150,

G f100. and H 6110

7. Two men, A and B, lay out equal sums of money In trade; A gains 126l. and B looses 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, recived for them all 1301. being paid for every ox 7l. for every cow 51. and for every call ll. 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 oxell, 10 cous, 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 fifth part of both their sums added together: how many did cach get ?

918. A 127), B 1421, C 54.

PERMUTATION OF QUANTITLES, Is the showing how many diferent ways any given nuinber 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's and the last product will be the answer required.

EXAMPLES. 1. Ilow many changes can be made of the three first letters of the alphabet ?

31 bac Proof,

41bca 5

съа ix2x3=6us.

6

cab 2. How many changes may be ruing on 9 bells

28. 362800.

a

a cb

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 diferent position at dinner ; how long must they have staid at sail inn to have fulfilled their agreement ?

Ans. i10179 years.

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ANNUITIES OR PENSIONS,

COMPUTED AT
COMPOUND IVTEREST.

CASE I. To find the amount of an annuity, or Pension, in arrcars, at Compound Interest.

Ihre No RULE. 1. Make 1 the firet 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 29 the given number of years, and find its sur

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

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-+-1,1936-1,19 1016=24,574616 sum of the series.* ---Then, 4,374613X125=6546,827 the amount sought.

OR BY TABLE I Multiply the Tabular number under the rate and op. posite to the time, by the annuity, and the product will be the amount sought.

EXIBITLES.

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

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

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