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DOUBLE POSITION, TEACHES to resolve questions by making two suppositions of false numbers.*
RULE. 1. Take any two convenient numbers, and proceed with each according to the conditious 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 smält.
EXAMPLES. 1. A purse of 100 dollars is to be divided among 4 men, A, B, C and D, so that B may lave 4 dollars more than A, and C 8 dollars more than B, and D twice as many as l'; what is each one's share of the money ? 1st. Suppose A6
2d. Suppose A 8 B 10
C 20 D56
* Those yaestions, in which the results are not propor. tional to their positions, belong to this rule ; such as those in thich the number sought is increased or diminished by son given number, which is no known part of the number required.
The errors being alike, are both too small, therefore,
$ CA 12 B 16 с
D 48 8 20
Proof 100 240 120 120
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 much as A and B both ; how much did each man pay?
Ans. A paid $120, B 8130, and C 8250. - 3. A man bequeathed 1001. 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 sl. and the third must have three times as much as the first, wanting 15l.; 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 day he was idle should forfeit 28.; at the expiration of the time he received 71. 10s.; how many days aid 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 4, 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 H the rest, which was 101. less than the skare of G; I demand the sum given, and each man's part? Liis. the sum giren vastsed, whereof P had' £130,
G 120, and H "110.
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, receiv. ed for them all 1301. being paid for every ox 71. for every cow 5l. and for every calf 1l. 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 15 more; C got a 5th part of both their sums added together; 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.
1. How many changes can be
a b c made of the three first letters of
2 the alphabet ?
3 bac Proof,
5 cb a 1x2x3=6 Ans.
6 2. How many changes may be rung on 9 bells ?
3. Seven gentlemen met at an inng 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. 110479 years.
ANNUITIES OR PENSIONS,
CASE I. To find the amount of an annuity, or Pension, in arrears,
at Compound Interest.
RULE. 1. Make 1 the first term of a geometrical progression, and the amount of 81 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 tne 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 +1,1236+1,191016=4,374616 sum of the series.*-Then, 4,374616X125=8546,827 the amount sought.
OR BY TABLE IL. 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 1l. or 1 dollar annuity, for the given time, which may be found in Table II. ready calculated.
Hence, either the amount or present worth of annuities may be readily found by Tables for that purpose.
2. If a salary of 60 dollars per annum to be paid year y, be forborne 20 years, at 6 per cent. compound interest; wliat is the amount?
Under 6 per cent. and opposite 20, in Table II, you will find, Tabular number=36,78559
Ans. $2207,13540=82207, 13cts. 5m.+ 3. Suppose an Annuity of 1001. be 12 years in arrears, it is required to find what is now due, compound interest being allowed at 5l. per cent. per annum?
Ans. £1591 14s. 3,024d. (by Table III.) 4. What will a pension of 1201. per annum, payable yearly, amount to in 3 years, at 5l. per cent. compound interest ?
Ans. £378 6s. II. To find the present worth of Annuities at Compound
Divide the annuity, &c. by that power of the ratio sig. nified by the number of years, and subtract the quotient from the annuity : This remaiuder being divided by the ratio less 1, the quotient will be the present value of the Annuity sought.
1. What ready money will purchase an Annuity of 501. to continue 4 years, at gl. per cent. compound interest 4th power
the ratio of} =1,215506)50,00000(41,18513+
Divis. 1,05-1505) 8,86487
177,297=£177 5$. 111d. Ans.