The errors being alike, are both too small, therefore, A 12 X 8 20 Proof 100 120 240 10)120(12 A's part. 2. A, B, and C, built a house which cost 500 dollars, of vhich A paid a certain sum ; B paid 10 dollars more than %. and C 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 1001. to three of his friends, after :1 is manner; the first must have a certain portion, the sea corid must have twice as much as the first, wanting 81. 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 £16 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 forieit 2s.; at the expiration of the time he received 71. 10s.; how many days did he vork, 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 , and 18 more, will be doubled ? Ans. 72. 6. A man gave to his three sons all his estate in money riz. to F half, wanting 501., to G one-third, and to II the rest, which was 101. less than the share of G; I demand the sum given, and each man's part ? Ans. the sum gwen was £360, whereof F kad £130 G £120, and H £110 7. Two men, A and B, lay out equal sunis of money il trade; A gains 1261. and B loses 871. and A's money is now double to B's; what did cach lay out? Ans. £300. 8. A farmer having driven his cattle to market, received for them all 1:301. being paid for every ox il. for every com 31. aud for every calf ll." 10s. there were twice as many cows as oxen, and three times its 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 rertain number; B as many as A and 15 niore; 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, EXAMPLES. 1. How many changes can be a b c made of the first three letters of 2 ясь the alphabet ? 3 b a Proof, 4 | bca 5 cba 1x2x3=6 Ans. 6 cab 2. Ilow many changes may be rung on 3 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. 11047 years. COMPUTED AT CASE I. 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 anpaid) 4 years; what will it amount to at 6 per cent. per annum, compound interest ? 1+1,06+1,1236+1,19101654,374616, sum of the series.* --Then, 4,374616 x 125=$546,827, 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 ll. or 1 dollar anquity, 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 fourrd by tables for that p!upose. 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. 5 m. + 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. 3,024d. (by Table II.) 4. What will a pension of 1201. per annum, payablo yearly, amount to in 3 years, at 51. 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 5l. per cent. compound interest ? Atheoretic, of}=1,215506)50,00000(41,13513+ Subtract 41,13513 177,297: €177 5s. 111d. Ans. 50 BY TABLE III. We have 3,54595=present worth of 11. for 4 years. 50=Annuity. Ans. £177,29750=present worth of the annuity. . 2. What is the present worth of an annuity of 60 dols per annum, to continue 20 years, at 6 per cent. compound interest? Ans. $688, 19; cts. + 3. What is 301. per annum, to continue 7 years, worth in ready money, at 6 per cent. compound interest ? Ans. £167 9s. 5d. + (II. To find the present worth of Annuities, Leases, &c. ta ken 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 denotid 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 501. 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 2d power of 1,05=1,1025)177,297(180,8136=£160 16s. 3d. I qr. present worth of the annuity in reversion. OR BY TABLE III. Find the present value of ll. 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 11. for the time in reversion, and multiply the remainder by the annuity; the product will be the answer |