ANNUITIES. 



thereto is 901. Us.0$d. for this isthe sum, 

 which, put out at the same rate of interest, 

 will, at the end of two years, amount to 

 1001. In like manner, if a person has 1001. 

 due to him at the end of three years, and 

 lie wishes to have the same advanced im- 

 mediately, the sum which ought to be 

 given as an equivalent thereto is 86/. 7s. 

 8d. for this is the sum which, at the same 

 rate of interest, will at the end of three 

 years amount to 1001. And if these three 

 ralues are added tog-ether, they will make 

 272/. 6s. 6d. being the sum which ought 

 to be paid down for an annuity of 1001. for 

 three years ; as this sum improved at the 

 given rate of interest is just sufficient to 

 make the three yearly payments. 



As the amount or present worth of 17. 

 for any given term is usually adopted as 

 the foundation of calculations relating to 

 annuities, let r represent the amount of 

 17. in one year; that is, one pound in- 

 creased by a year's interest; then r, or r 

 raised to the power whose exponent is 

 any given number of years, will be the 

 amount of II. in those years ; its increase 

 in the same time is r" 1 : now the 

 interest for a single year, or the annui- 

 ty corresponding with the increase, is r 

 1 ; therefore as i -- 1 is to r" 1, so is u 

 (any given annuity) to a its amount: 

 hence we have 



EXAMPLE. To what sum will an an- 

 nuity of 50/. amount in 6 years, at 5 per 

 cent, per annum, compound interest ? 



50 x"TO-S"> 6 " 1 = 340J. 19s. Id. 



"~705~~ 



In this manner the amount of an annuity 

 for any number of years, at any given rate 

 of interest, may be found. But when the 

 term of years is considerable, it will be 

 more convenient to work by logarithms, 

 by which the labour of all calculations re- 

 lating to compound interest is greatly 

 abridged. There is, however, little occa- 

 sion in general to calculate the amount, or 

 present worth of annuities, except for 

 particular rates of interest, as the follow- 

 ing tables, and others of a similar nature, 

 for different rates of interest, which are 

 given in most books on compound inter- 

 est, save much time and labour in com- 

 mon practice, and are therefore in gene- 

 ral use. 



TABLE I. 



Shewing the amount of an annuity of 17. 

 in any number of years not exceeding 

 100, at 5 per cent, per annum, com- 

 pound interest. 



EXAMPLE 1. To what sum will an an- 

 nuity of 1051. amount, in 19 years, at 5 

 per cent, compound interest ? 



The number in the table opposite to 19 

 years is 30,5390, which multiplied by 105 

 gives the answer, 3206/. 11s. lOd 



EXAMPLE 2. In what time will an an- 

 nuity of 251. amount to 3575Z. at 5 per 

 cent, compound interest? 



Divide 35751. by 25/. the quotient is 143; 

 the number nearest to this in the table is 

 142,9933, and the number of years cor- 

 responding, or 43 years, is the answer. 



The present worth of an annuity, or the 

 sum to be given in one present payment 

 as an equivalent for an annuity for any 

 given number of years, is found on simi- 

 lar principles ; for as 11. is the present 

 value of r" (its amount in n years,) and as 

 the present value of any other amount, 

 and consequently 



it X f n 1 



of : must bear the same propor- 

 tion to that amount, we have 



