REST. 287 3. In like manner, find the present wort and the sum of all these will be the pres sought. o find the annuity, rent, Sc. Or, divide the annuity, &c. by that pow denoted by the number the number of years, and subtract the quoted by the number of this remainder being divided by the ratio the present worth. EXAMPLES. 1.* What ready money will purchase a f the ratio, signified by ue 4 years, at 61. per cent. compound inteuotient, and the product Ratio = Ratio 2 3 = Ratio= 4 First Method. 1.06)60.0000056-603-p == enoted by the number of 1.1236)60·00000(53.399 ed by the time, by the 1-191016) 60.00000 50.377= Ratio 1.26247696)60-00000 (47.525 = = the former. he ratio, denoted by the be the annuity. ill £.207.904 purchase, 3382255776 · Lar-1 26247696 ing continually divided by r till nothing remain, the numbe be equal to t. Let t exprefs the number of half years, or quarters, n the half payment, and the fum of 11. and or year's intereft, then all the will be applicable to half yearly, and quarterly payments,the fame as to The amount of an annuity may also be found for years and parts a of 1. Find the amount for the whole years, as before. --601. 2. Find the intereft of that amount for the given parts of a year. 3. Add this intereft to the former account, and it will give the whole required. The prefent worth of an annuity for years and parts of a year may be found thu 1. Find the prefent worth for the whole years, as before. 2. Find the prefent worth of this prefent worth, difcounting for the given parts of a year, and it will be the whole prefent worth required. Questions in this cafe may also be answered by firft finding the amount of the given annuity by Cafe I. of annuities in arrears, page 280, and then the present worth, or principal, by Case II. of Compound Intereft, page 278. Under 61. per cent and opposite 4 years, you will find •28859-annuity which 11. will purchase in 4 years. Multiply by 2079 259-31 202013 577180 59-997861.60. 2. What salary, to continue 20 years, will 688D. 65c. purchase, at 6 per cent. compound interest? Ans. D.60. CASE III. When the annuity, present worth and ratio, are given, to find the time. RULE. Divide the annuity by the product of the present worth and ratio subtracted from the sum of the present worth and annuity, and the quotient will be that power of the ratio, denoted by the number of years, which, being divided by the ratio, and this quotient by the same, till nothing remain, the number of divisions will show the time: Or, the above quotient being sought in Table 1st. under the given rate, in a line with it, you will see the time. EXAMPLES. 1. For how long may an annuity of 601. per annum be purchased for £.207.906336762, at 61. per cent. compound interest? 47.525619795=divisor. 47-525619795)60.000000000(1.26247696 Divide by 1.06)1-26247696 =1.26247696, which 207-906336762+60-207-906336762×1.06 being sought in Table 1, under the given rate, in a line with it, is 4-4 years. 2. How long may a lease of D.300 yearly fent, be had for D.2132-341 allowing 5 per cent. compound interest, to the purchaser ? Ans. 9 years. ANNUITIES, LEASES, &c. TAKEN IN REVERSION AT COMPOUND INTEREST. CASE I. When the annuity, time and ratio, are given, to find the present worth of the annuity in reversion. RULE.*-1. Divide the annuity by that power of the ratio denoted by the time of its continuance. 2. Subtract this quotient from the annuity: divide 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, time to come, before the annuity commences) and the quotient will be the present worth of the annuity in re version. Or, 1. Multiply the annuity by that power of the ratio denoted by the time of its continuance, minus unity, for a dividend. 2. Multiply that power of the ratio denoted by the time of its continuance, that power of it denoted by the time of reversion, and the ratio less 1, continually together for a divisor, and the quotient arising from the division of these two numbers will be the present worth of the annuity in reversion. EXAMPLES. 1. What is the present worth of 601. payable yearly, for 4 years; but not to commence till two years hence, at 61. per cent. ? First Method. 636 Ratio=1.06 Or, in Table 4th, find the present 1.06 value of 11. at the given rate, both for the time in being and the time in reversion added together, and subtract the present worth of the time in being from the other, multiply the remainder by the annuity, and the produc will be the answer. 1060 2d. power=1.1236 Carried over. 1.1236 Pres. * Letv denote the time in reverfion, and the other letters as before. Then the two cafes under this rule will be expreffed by the following Theorems. Brought over. 1.1236 Pres. worth of the time in being and reversion =4.91732 67416 Present worth of the time } 22472 11236 3.08402 60 11236 £.185.04120 Div. by 4th pow. 1.26247696)60-000000000000(47.525619794281 Subtract the quotient-47.525619794281 Divide by 1.06—1-06)12·474380205719 Divide by 1.06x1·06=1·1236) 207.9063367619(185-035899-1851. the present worth of the annuity in reversion. Os. 84d. 60-47.5256 = 207.906 1.06-1 2. What is the present worth of a reversion of a lease of D.60 per annum, to continue 20 years, but not to commence till the end of 8 years, allowing 6 per cent. to the purchaser ? Ans. D.431.782 (nearly.) An annuity, several times in reversion, and rate being given, to find the sexeral present values. Find the present value of .1 or D.1 by Table 4, at the given rate, and for the several given times, which, being severally multiplied by the annuity, the products will be the several present values of that annuity, for the several times given; subtract the several present values, the one from the other, and the several remainders will answer the question. 3. A 3. A has a term of 6 years in an estate at 601. per annum. B has a term of 14 years in the same estate, in reversion, after the 6 years are expired; and C has a further term of 16 years, after the expiraI demand the present values of the several terms, tion of 20 years. at 6 per cent.? £. s. d. Pres. value of .1 for 36 years=14.61722×60=877 0 74 Ditto of ditto for 20 years Ditto of ditto for 6 years =11.46992×60=688 3 10 = 4.91732x60-295 0 94-A's term. Therefore, 877 0 72-688 3 10.188 16 9 C's term, and 688 3 10-295 0 94.393 3 1 B's term. 4. For a lease of certain profits for 7 years, A offers to pay D.300 gratuity, and D.300 per annum, B offers D.800 gratuity and D.250 per annum, C bids D.1300 gratuity and D.200 per annum, and D bids D.2500 for the whole purchase, without any yearly rent; which is the best offer, computing at 6 per cent.? By Table 4, the present worth of D.300 per annum, for 7 years, at 6 per cent. is } D. 1674-714 To which add 500. Present worth of D.200 per annum for 7 years =1116.476 To which add 1300 Value of C's offer = 2416-476 Hence it appears that D's offer is the best. D's offer 2500• The above questions may be answered by the 4th. and 2d. Tables. 1. Multiply the tabular number in Table 4, corresponding to the rate and the time of continuance, into the annuity, and the product will be the present worth, to commence immediately. 2. Multiply this present worth by the tabular number in Table 2, corresponding to the rate and the time of reversion, and the product will be the present worth of the annuity in reversion. |