x+y× 10x+y2, we have r+yx 4=10x+y, or 4x+4y=10 x+y, and by transposition 3y=6x, therefore, y=2x. This being fubftituted for y in the former equation, we have 10x+2x)2=144×x+2x or 144x2=144× 3*, and dividing by 144x we have x=3 and y=2x=6; therefore, the fide of the cube is 36. CHAP. X. OF THE VALUE OF LIVES; OR, DOCTRINE OF ANNUITIES. SECT. I. THE VALUE OF AN ANNUITY FOR A SINGLE LIFE. AN Annuity is a fum of money payable yearly, halfyearly, or quarterly; to continue either for life, for a certain number of years, or for ever. When an annuity remains unpaid after it is due, it is faid to be in arrear. When the purchaser of an annuity does not immediately enter upon poffeffion, the annuity is faid to be in reverfion, The But com The intereft upon annuities in arrear may be computed either in the way of fimple or compound interest. pound interest being found most equitable, both for buyer and feller, is in most general ufe. Annuities may be divided into certain and uncertain. A certain annuity is that which continues for a certain time, or for ever. An uncertain annuity depends upon one or more lives. Before I proceed to give the doctrine of contingent annuities, it will be neceffary to deliver the rules for calculating of annuities certain. To find the Amount of an Annuity for a given Term of Years, at a given Rate of Interest. EXAMPLE. What will an annuity of 50l. amount to, at the end of 8 years, at the rate of 5 per cent. per annum, simple intereft? In this example, the interest being at 5 per cent. multiply the rate of intereft of 11. for 1 year, or .05 by 50 the annuity, and the product by 8, the number of years, and the product hence arifing is 20: the half whereof (10) multiplied by the number of years, made lefs by one, (7,) produces 70, the fimple interest; which added to the product of 50, and 8, (400,) give 470, the amount required. PROBLEM II. To find the Amount of an Annuity, at compound Intereft. RULE. Multiply the amount of 1l. for 1 year, as often into itself as there are years, except one; or, which is the same, raise it to the power whofe index is equal to the number of years, and from the refult fubtract 1; then divide the remainder by the intereft of 1l. for 1 year, and multiply the quotient by the annuity, and the product will be the amount required. EXAMPLE. What is the amount of an annuity of 50l. for 3 years, at 5 per cent. per annum, compound intereft? Here the amount of 11. for 1 year is 1.05, which multiplied twice into itself, produces 1,157625, and 1 fubtracted from this, the remainder is .157625, which divided by .05, the quotient is 1525; this multiplied by 50, produces 157.625, or 157. 12s. 6d. the anfwer required. NOTE. If the payments are half-yearly or quarterly, the amount, and intereft of 1. must be taken for a half, or a quarter of a year. And then the double or quadruple of the pime must be taken. And the amount of 1l. for half a year at compound intereft is equal to the fquare root of the amount for a year; and the amount for a quarter of a year is equal to the fquare root of that for half a year. PROBLEM III. To find the prefent Value of an Annuity, having the Time and Rate. RULE. Multiply the amount of one year as often into itfelf as there are years, lefs 1; or involve it to the power denoted by the time: by this result, divide 1, and subtract the quotient from 1, divide the remainder by the interest of 17. for a year; then multiply this last quotient by the annuity, and the product will be the present value. EXAMPLE. What is the prefent value of an annuity of 401. for 5 years, discounting 5 per cent. per annum, compound intereft? Here 1.05 involved to the fifth power is 1.27628. By which dividing 1, the quotient is .78353, which subtracted from 1, leaves.21647; this divided by .05 gives 4.3294, which multiplied by 40 is 173.176, or 1731. 35. 6d. the prefent worth. PROBLEM PROBLEM IV. Having the prefent Worth, Rate, and Time, to find the Annuity. RULE. Find the prefent value of 17. annuity at the given rate and time; and then by the rule of three, fay, as the present worth, thus found, is to 17. annuity, fo is the present worth given to its annuity; that is, divide the given present worth by that of 1. annuity. EXAMPLE. What annuity will 1731. 35. 7d. purchase to continue 5 years, allowing compound intereft at 5 per cent.per annum? 05:1::1:20%. 1.05 x 1.05 x 1.05 x 1.05 x 1.05 1.2762815625 20 15.6705 4.3295 prefent worth of 11. annuity. 4.329)173.179(40%. annuity, Answer. Annuities for ever, or Freehold Efiates. In calculating the value of an annuity for ever, commonly called an Annuity in fee fimple, three things are to be confidered: 1. The annuity, or yearly rent. 2. The price, or prefent worth. 3. The rate of interest. Having the Rent and Rate of Intereft, to find the Price or Value. RULE. As the interest of 17. is to 1. fo is the rent to the price or value. EXAMPLE. EXAMPLE. What is the present worth of an annuity of 401. per annum in fee fimple, compound intereft 3 per cent. per ann.? As .035 the intereft of 11. for a year is to il. fo is 40%. the rent of the annuity to 1142.857142 or 11421. 175. 14. PROBLEM II. Having the Price and Rate of Intereft, to find the Annuity. RULE. AS 11. is to its intereft, fo is the price to the annuity. EXAMPLE. What annuity will 4000l. purchase, at 41 per cent. per ann. compound interest? As 1. is to .045, fo is 4000l. to 180l. the annuity. PROBLEM III. Having the Price and Rent of the Annuity, to find the Rate of Intereft. RULE. As the price is to the rent, fo is 17. to the rate of intereft. EXAMPLE. If an annuity of 180l. coft 4000l. what is the rate of intereft compound per ann. ? As 4000:180::1: .045 or 4 per cent. rate of interest. Having the Rate of Intereft, to find how many Years Purchase an Eftate is worth. RULE. Divide 1 by the rate of intereft, and the quotient is the answer. EXAMPLE. How many years purchase is an annuity, when the purchaser has 2 per cent. for his money? .025)1.coo( 40 years purchase. PROBLEM |