F 
j 
OF INTEEESt AND ANNUITIES. 
Examp. 3. For how long time may one, with 6oo7. 
purchase an annuity of 100/. at 4 per cent. 9 
In this example, we have R — 1,04, A =: loo, and 
v — 600; and therefore, by Theorem 3, n [ — 
log. A — log. A + v —rR . _ i e 
— J2— ) — 7 the number of 
log. R 
years required. 
Examp. 4. To determine at what rate of interest an 
annuity of 50/. to continue 10 years, may be purchased, 
for 400/. 
Here A — 50/. n — 10, and v> — 400: whence, by 
Theorem 4, R*+i — — + lx R n l- A being — o, we 
V v° 
have R* 1 — l,125R ,c> + ,125 = 0; which equation 
resolved, gives the required value of R = 1,042775; 
and consequently the rate of interest, 4,2775/. per 
annum~ 
The solution of this last case being somewhat tedious, 
the following approximation (which will be found to 
answer very near the truth, w r hen the number of years 
is not very large) may be of use. 
Assume Q = 11 n 1 - ^ ; so shall 
2«A — 2tf 
3000Q — 2«+ 1 X 400 
6Q . 5Q — 3n — 4 + { . » + 2 . 1 1« + 13 
rate per cent, very nearly. 
express the 
Thus, for example, let A (as above) be = 50, 
10 x ll x 50 
a — 10, and u — 400; then Q being — 
82500 — 8400 
= 27.5, we have 
165 x 103,5 + 246' 
for the rate, per cent, the same as before. 
1000 — 800 
or 4,277-5*