<5F INTEREST AND ANNUITIES. 
.Let Q — 
n . n 
l . A 
2 . ih 
nA 
; then will 
3000 Q f 2« — 1 .400 
13 
be the 
6Q . ¿a + 3« — 4 + i . «-=^2 . Mu 
rate per cent, required. 
Thus, for example, let n — s, A zz 120, and m s 
56 . 120 
1200; then will Q — 
2 . 210 
14, anil the rate 
- ... 42000 + 6000 - , 
itself zz zz 6,28^, as above.. 
84 x 90 + 75 
The preceding examples explain the different cases 
of annuities in arrear; in the following ones the rules 
for the valuation of annuities are illustrated. 
Examp. l. To find the present value of 100/ annuity, 
to continue seven years, allowing 4 per cent, per annum 
compound interest. 
Here we have given R r 3,04, A zz 100, and n — fa 
imd therefore, by Theorem i, log. v { zz log. A {- 
log. 1 — JL_ — log. 11— 1) — log. 100 + log. 
R B 
1 \= log. ,04 ±z 2,778206; and consequently 
l,0il 7 
c zz 600,2 zz 600/. 4s. which is the value that was to 
l>e found. 
Examp. 2. What annuity, or yearly income, to con 
tinue 20 years, maybe pnrchased for 1000/. at 3| per 
•cent, f 
In this case, R zz 1,035, n zr 20, ' v zz 1000: 
Whence, by Theorem 2, we have log. A ( zz log. v 
+ log. R 
log. 1 - 
R* 
) 
1,847336; and 
consequently A zz 70, 36, or 70/, 7 s. 2</.