The errors being alike, are both too small, therefore, 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 ach man pay? Ans. A paid $120, B $130, and C $250. 3. A man bequeathed 100%. to three of his friends, after lis manner; the first must have a certain portion, the se cond must have twice as much as the first, wanting 81. and the third must have three times as much 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; hat for every day he wrought he should receive 4s. and for very 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? Ans. He wrought 45 days, and was idle 15 days. 5. What number is that which being increased by its !, ts, 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 II 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 £360, whereof F had £130 G £120, and H £110 7. Two men, A and B, lay out equal sums of money trade; A gains 1267. and B loses 877. and A's money in now double to B's; what did each lay out? Ans. £300. 8. A farmer having driven his cattle to market, received for them all 1307. being paid for every ox 77. for every com 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 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, 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, 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. 110 years. ANNUITIES OR PENSIONS, 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 Der 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 se-Then, 4,374616 × 125=$546,827, the amount ries.*. Bought. 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 an nuity, for the given time, which may be found in Table II. ready calculabed. Hence, either the amount or present worth of annuities may be readily found by tables for that purpose. R 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, 13 cts. 5m.+ 3. Suppose an annuity of 1007. 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 II.) 4. What will a pension of 1207. 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 In terest. RULE. Divide the annuity, &c. by that power of the ratio signified 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. EXAMPLES. 1. What ready money will purchase an annuity of 501. to continue 4 years, at 51. per cent. compound interest? 4th power of }=1,215506)50,00000(41,18513+ the ratio, From 50 41,13513 D-1-05)8,86487 177,297 £177 5s. 111d. Ans. BY TABLE III. Under 5 per cent, and even with 4 years, Multiply by 50=Annuity. years. Ans. £177,29750=present worth of the annuity. interest? 2. What is the present worth of an annuity of 60 dols per annum, to continue 20 years, at 6 per cent. compound Ans. $688, 191⁄2 cts.+ 3. What is 30%. per annum, to continue 7 years, worth in ready money, at 6 per cent. compound interest? Ans. £167 9s. 5d.+ III. To find the present worth of Annuities, Leases, &c. taken in REVERSION at Compound Interest. 1. Divide the annuity by that power of the ratio denoted by the time of its continuance. 2. Subtract the quotient from the annuity: Divide the remainder by the ratio less 1, and the quotient will be the present worth to commence immediately. 3. Divide this quotient by that power of the ratio denoted by the time of Reversion, (or the time to come before the annuity commences) and the quotient will be the present worth of the annuity in Reversion. EXAMPLES. 1. What ready money will purchase an annuity of 50%. payable yearly, for 4 years; but not to commence till two years, at 5 per cent.? 4th power of 1,05=1,215506)50,00000(41,13513 Subtract the quotient=41,13513 Divide by 1,05-1-,05)8,86487 2d power of 1,05=1,1025)177,297(160,8136=£160 16s. 3d. 1 qr. present worth of the annuity in reversion. OR BY TABLE III. Find the present value of 17. at the given rate for the sum of the time of continuance, and time in reversion added together; from which value subtract the present worth of 14. for the time in reversion, and multiply the remainder by the annuity; the product will be the answer |