Question

A sum of money at compound interest amounts to eight times of itself in 3 years. In how many years will it be 512 times of itself?

Answer 1

Compound Interest

Formula. If $A$ be the amount yielded by a sum of $P$ at compound interest $r\%$ yearly for $t$ years, then we have the relation,

$\quad\quad A=P\left(1+\frac{r}{100}\right)^{t}$.

Solution.

Let the sum be $P$ and $r\%$ be the rate of compound interest.

Condition 1.

Given in $t=3$ years, the sum amounts to eight times of itself.

$\Rightarrow 8P=P\left(1+\frac{r}{100}\right)^{3}$

$\Rightarrow \left(1+\frac{r}{100}\right)^{3}=8$

$\Rightarrow 1+\frac{r}{100}=2$

$\Rightarrow \frac{r}{100}=1$

$\Rightarrow \color{red}{r=100}$.

Condition 2.

The sum $P$ will yield itself into $512$ times in $\color{red}{t}$ years (say).

$\Rightarrow 512P=P\left(1+\frac{100}{100}\right)^{t}$

$\Rightarrow (1+1)^{t}=512$

$\Rightarrow 2^{t}=2^{9}$

$\Rightarrow \color{red}{t=9}$

Answer. Therefore in $9$ years, the sum will be $512$ times of itself.

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