Question

Sanju puts equal amount of money, one at $10\%$ per annum compound interest payable half-yearly and the second at a certain rate per annum compound interest payable yearly. If he gets equal amounts after $3$ years what is the value of the second rate percent?

• A. $10\frac{1}{4}\%$
• B. $10\%$
• C. $9\frac{1}{2}\%$
• D. $8\frac{1}{4}\%$

Answer 1

Answer: (A)

$10\frac{1}{4}\%$

Given that, Sanju puts the same amount of money, one at $10\%$ per annum compound interest, compounded half-yearly, and other at some interest per annum compound interest, compounded annually.

He receives the same amount after $3$ years.

To find out: The rate of interest for the second investment.

Let the principal amount be $Rs.x$. And the amount received after $3$ years be $A$

Also, let the rate of interest in the second case be $y\%$ per annum.

In the first case, the payment is made half-yearly at $10\%$.

So the number of time in $3years=3÷\frac{1}{2}=6$ and

the rate $=10\%÷2=5\%$

In CI, we know that the amount $\left(A\right)=P{(1+\frac{R}{100})}^T$

Here, $P=x,R=5\%andT=6$

$\therefore A=x{(1+\frac{5}{100})}^6$.

Now, in the second case, $P=x,R=y\%andT=3$

$\therefore A=x{(1+\frac{y}{100})}^3$.

Given that, the amount received in both cases is the same.

Hence, $x{(1+\frac{y}{100})}^3=x{(1+\frac{5}{100})}^6$

$\Rightarrow {(1+\frac{y}{100})}^3={\left\{{(1+\frac{5}{100})}^2\right\}}^3$

$\Rightarrow (1+\frac{y}{100})={(1+\frac{5}{100})}^2=\frac{441}{400}$

$\Rightarrow \frac{y}{100}=\frac{441}{400}-1$

$\Rightarrow \frac{y}{100}=\frac{41}{400}$

$\therefore y=\frac{41}{4}=10\frac{1}{4}\%$

Hence, the required rate of interest for the second case is $10\frac{1}{4}\%$.