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


1. Take any two convenient numbers, and proceed with each according to the conditions of lie 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 prer 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 uulike, when one is too great, and the other too small.

EXAMPLES. 1. A purse of 100 dollars is to be divided


4 A, B, C and D, so that B may have four dollars more than A, und 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

1st error, 30

2d error,


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

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

Pos. Err

A 12
B 16
С 24





Proof 100


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 .", and C paid as much as A and B both ; how much did tich man pay?

ins. I paid $120, B $130, and C $250. 3. A mın bequeathed 100l. to three of his friends, aster this manner; the first must h :ve a certain portion, the second must have twice as niuch as the first, wanting &l. and the third must have three times as much as the first, wanting i5l. ; I demand how much each man must have ? Ane. The first £20 10s. second L33, 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 inany was he idle?

Ans. He wrought 49 days, and was idle 15 days. 5. What nurober 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, V12. to F half, wanting 501. to G one-third, and to Hl the rest, which was 101. less than the share of G; I demand me sum given, and cach man's


? Ans. the sum given was £360, whereof F had £130,

G £120, and # £110.

7. Two men, A and B, lay out equal sums of money trade; A gains 1261. and B loses 871. and A's money 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 1301. being paid for every ox 71. for every cos 51. and for every calf 11. 10s. there were twice as many cows as oxen, and three times us many calves as cows how many were there of each sort?

Ans. 5 oxen, 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 rertain number; B as many as A and 15 more;

C got a 5th part of both their sums added together; how many did each get?

Ans. A got 127, B 142}, C 54.

PERMUTATION OF QUANTITIES, IS the showing how many different ways any given num. ber of things may be changed.

To find the number of Permutations, or changes, thu 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.


1. How many changes can be

1 a b c made of the first three letters of

2 ясь she alphabet ?

3 b a G Proof,

4 | bca

5 cba 1x2x36 Ans.

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

Ans. 862880

3. Seven gentlemen met at an inn, and were so well pleased with their host, and with each other; that they agreed :0 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 in to liave fulfilled their agreement ?

Ans. 11037 years.


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

at Compound Interest.

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

2. Carry on the ries up to as many terms as the given d'imber of years, and find its sum.

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



1. JC 125 dols. yearly rent, or annuity, be forborne (or ampaid) 4 years; what will it amount to at 6 per cent. per annun, compound interest ? 1+1,06 +1,1236 +1,19101654,374616, sum of the se.

--Then, 4,374616 x 125=$546,827, the amount sought.

OR BY TABLE II. Multiply the Tal ular number under the rate, and opposite lo the time, ly the amuity, and the product will be the amount sought.

* The sun or the series thus found, is the amount of il. or 1 dollar anguity, for the given time, which may be found in Table 11. ready calcula. red.

llence, cither the ar runt or present work of annui!ics may be readily found by tables for that p'ırpose.

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

Under 6 per cent. and opposite 20, in Table II., you will find, Tabular number=36,78559

60 Annuity. Ans. $2207,135-10=$2207, 13 cts. 5m. + 3. Suppose an annuity of 1001. be 12 years in arrears, it is required to find what is now due, compound interest being allowed at 51. per cent. per annum ?

Ans. £1591 14s. 3,02-1d. (by Table II.) 4. What will a pension of 1201. per annum, payable yearly, aniount to in 3 years, at 5l. per cent. compound interest?

Ans, £378 Os.

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



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.


1. What ready money will purchase an annuity of 501 to continue 4 years, at 51. per cent. compound interest ? 4th power of

} =(

the ratio,

Subtrict 41,13513

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