DOUBLE POSITION, TEACHES to resolve questions by making two supp 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 results in the question. 5. 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 pro 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; 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 4 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 B 10 70 10 80 100 1st. error 2d. error 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 increased or diminished by some given number, which is no known part of the number required. 2d. Suppose A 8 B 12 C 20 D 40 30 The errors being alike, are both too small, there Err. 30 Pos. 6 Proof 10)120(12 A's part. 2. A, B, and C, built a house which cost 500 do of which A paid a certain sum; B paid 10 dollars than A, and C paid as much as A and B both; how did each man pay? Ans. A paid $120, B $150, and C $25 3. A man bequeathed 100l. to three of his friends, a this manner: the first must have a certain portion, second must have twice as much as the first, wanting and the third must have three times as much as the f wanting 151.; I demand how much each man must ha Ans. The first £20 10s. second £33, third £46 4. A laborer was hired 60 days upon this conditi that for every day he wrought he should receive 4s. for every day he was idle should forfeit 2s., at the eration of the time he received 71. 10s.; how many d did he work, and how mary was he idle? Ans. He wrought 45 days, and was idle 15 days 5. What number is that which being increased by its, and 18 more, will be doubled ? Ans. 72 6. A man gave to his three sons all his estate in mon viz.. to F half, wanting 50l. to G one-third, and to H rest, which was 10l. less than the share of G; I dema the sum given, and each man's part? Ans, the sum given was £360, whereof F had £1. G £120, and H £110. 7. Two men, A and B, lay out equal sums of money in trade; A gains 1261. and Blooses 877. 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 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 together; how many did each get? Ans. A got 1271, B 142), C54. 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. EXAMPLES. 1. How many changes can be made of the three first letters of the alphabet ? Proof, bc a cb 2 3 bac 4b ca 6 cab 1x2x3=6 Ans. 2. How many changes may be rung on 9 bells? 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. 110178 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 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 Then, 4,374616×125-8546,827 the amount series. sought. OR BY TABLE II. Multiply the Tabular number under the rate and op posite 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 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 yearly, be forborne 20 years, at 6 per cent. compound in terest; what is the amount ? Under 6 per cent. and opposite 20, in Table II, you will find, Tabular mumber-36,78559 60 Annuity. Ans. $2207,13540-2207, 13cts. 5m.+ 3. Suppose an Annuity of 100l. 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. 3,024d. (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 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 50l. to continue 4 years, at 5l, per cent. compound interest? of 4th the ratio,}=1,215506)50,00000(41,13515+ From 50 41,13513 Divis. 1,05-1-05)8,86487 177,297 £177 5s. 11td. Ans. |