202 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 re'sults in the question. 3. Multiply the first position by the last crror, 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 3. 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 A 8 B 10 C 18 D 36 B 12 100 --- 2d. eiror 20 *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 inereased or diminished by some given number, which is no known part of the number required. The errors being alike, are both too small, therefore, Pos. 6 Err 30 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 paid as much as A and B both; how much did each man pay? Ans. A paid $120, B $130, and C $250. 3. A man bequeathed 100%. 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 87. and the third must have three times as ...uch 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 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 many was he idle è 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 ? An . 72. 6. A man gave to his three sons all his estate in money, viz. to F half, wanting 50%. to G one-third, and to H the rest, which was 107. less than the share of G; I demand the sum given, and each man's part? Ans. the sum given was £300, whereof F had £130 € £120, and H £110. 204 PERMUTATION OF QUANTITIES. 7. Two men, A and B, lay out equal sums of money in trade; A gains 1267. 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, receiv ed for them all 1307. being paid for every ox 71. for every cow 51. and for every calf 17. 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 togeth er; how many did each get? Ans. A got 127, B 1421, C 54. PERMUTATION OF QUANTITIES, Is the snowing how many different ways any given num ber 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. 1X2X36 Ans. cab 2. How many changes may be rung on nine bells? Ans. 362830 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. 110 years. ANNUITIES OR PENSION” COMPUTED AT COMPOUND INTEREST. CASE I. 36 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 $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+1,191016=4,374616 sum of the series. Then, 4,374616×125-$546,82, 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 I. rady 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 yearly, 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. $2207,13540=$2207, 13cts. 5m.+ 3. Suppose an Annuity, of 100%. be 12 years in arrears, it is required to find what is now due, compound interest being allowed at 57. per cent. per annum ? Ans. £159. 14s. 3,024d. (by Table II.) 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 Interest. RULE. 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 ratio less 1, the quotient will be the present value of the Annuity sought. EXAMPLES. 1. What ready money will purchase an Annuity of 50. to continue years, at 57. per cent. compound interest? 4th power of the ratic, {=1,21550€ 50,00000(41,13513+ 50 |