438. When any three of the quantities A, P, n, r are given, the fourth may be found. Ex. Required the rate when $500 in 4 years at simple interest amounts to $610. 439. Since P will in n years amount to A, it is evident that P at the present time may be considered equivalent in value to A due at the end of n years; so that P may be regarded as the present worth of a given future sum A. Ex. Find the present worth of $600, due in 2 years, the rate of interest being 6 per cent. When compound interest is payable semi-annually, The amount of P dollars in Ex. Find the present worth of $500, due in 4 years, at 5 per cent compound interest. 441. If the sum set apart at the end of each year put at compound interest be represented by 8, then, The sum at the end of the nth year = to be = S + SR + SR2 +····· + SR”-1. That is, the amount AS+SR+SR2 + ··· +SR”-1. ARSRSR2 + SR3++SR”. (1) If $10,000 be set apart annually, and put at 6 per cent compound interest for 10 years, what will be the amount? By logarithms the amount is found to be $131,740 (nearly). (2) A county owes $60,000. What sum must be set apart annually, as a sinking fund, to cancel the debt in 10 years, provided money is worth 6 per cent? NOTE. The amount of tax required yearly is $3600 for the interest and $4555 for the sinking fund; that is, $8155. ANNUITIES. 442. A sum of money that is payable yearly, or in parts at fixed periods in the year, is called an annuity. I. To find the amount of an unpaid annuity when the interest, time, and rate per cent are given. The sum due at the end of the Ex. An annuity of $1200 was unpaid for 6 years. What was the amount due if interest be reckoned at 6 per cent? II. To find the present worth of an annuity when the time it is to continue and the rate per cent are given. Let P denote the present worth. Then the amount of P for n years will be equal to A, the amount of the annuity for n years. But the amount of P for n years and .. PR R-1 8 441. (1) Find the present worth of an annual pension of $105, for 5 years, at 4 per cent interest. (2) Find the present worth of a perpetual scholarship that pays $300 annually, at 6 per cent interest. III. To find the present worth of an annuity that begins in a given number of years, when the time it is to continue and the rate per cent are given. Let p denote the number of years before the annuity begins, and q the number of years the annuity is to continue. Then the present worth of the annuity to the time it terminates is S Rp+q - 1 and the present worth of the annuity to the time it begins is |