Question

Suppose a certain sum doubles in $2$ year at $r\%$rate of simple interest per annum and $R\%$ rate of interest per annum compounded annually.

Then,
(a) $r<R$
(b) $R<r$
(c) $R=r$
(d) Cannot be determined

Answer 1

Let the principal be $P.$

When the rate is ${}^{\prime }{r}^{\prime }$ as simple interest.

Time, $T=2$ years

When the sum is doubled.

i.e, Amount $=2P$

Amount = Principal + Interest

$2P=P+\frac{P\times r\times T}{100}$

$\Rightarrow 2P-P=\frac{P\times r\times 2}{100}$

$\Rightarrow 50P=Pr$

$r=50\%$

When the rate is ${}^{\prime }{R}^{\prime }$ as compound Interest

Time $=2$ years

When the Sum is doubled. Amount will be $2P.$

Amount is given by the formula in case of compound interest

$A=P(1+\frac{R}{100}{)}^2$

$\Rightarrow 2P=P(1+\frac{R}{100}{)}^2$

$\Rightarrow 2=(1+\frac{R}{100}{)}^2$

$\Rightarrow (1+\frac{R}{100})=\sqrt{2}$

$\Rightarrow (1+\frac{R}{100})=1.414$

$\Rightarrow \frac{R}{100}=0.414$

$\therefore R=41.4\%$

So, $R<r$