Question

The interest is supposed compound unless the contrary is state.
A man has a capital of $\$20000$ for which he receives interest at $5$ per cent; if he spends $\$1800$ every year, show that he will be ruined before the end of the ${17}^{th}$ year; having given
$\log 2=.3010300,\log 3=.4771213,\log 7=.8450980$.

Answer 1

Let the present value of an annuity of $\$1800$ be $V$

$V=A\frac{1-r^{-n}}{r-1}$

$V=1800\frac{1-r^{-n}}{r-1}$

$r=1-\frac{R}{100}$

$1-r=1-1+\frac{5}{100}$

$1-r=\frac{5}{100}$

A man will be runied if $V$ is greater than $\$20000$

$1800\frac{1-r^{-n}}{r-1}>20000$

$1-r^{-n}>\frac{5}{9}$

$r^{-n}<\frac{4}{9}$

$r^n>\frac{9}{4}$

For $n=17$

$\log r^{17}=17\log \frac{21}{20}$

$\log \frac{9}{4}=17\log \frac{21}{20}$

$\log \frac{9}{4}=2\log 3-2log2$

$\log \frac{9}{4}=0.35218$