Question

If the compound interest on a certain sum of money for $2\mathrm{years}$ is $Rs2,700$ and the simple interest on the same sum for the same period is $Rs2500$, what will be the rate of interest?

Answer 1

Step 1: Use standard formula of compound interest and simple interest

Let the principle amount $\left(P\right)$$=x$

Time period $\left(t\right)$$=2\mathrm{years}$

Compound interest $\left(C.I\right)$$=Rs2700$

Simple interest $\left(S.I\right)$$=Rs2500$

Rate of interest $\left(R\right)$$=r$

Amount$=A$

For Compound Interest

$$\begin{gathered} A=P{\left(1+\frac{R}{100}\right)}^t \\ \Rightarrow x+2700=x{\left(1+\frac{r}{100}\right)}^2\left[\because A=P+C.I\right].....\left(1\right) \end{gathered}$$

For Simple Interest

$$\begin{gathered} S.I=\frac{P\times R\times t}{100} \\ \Rightarrow 2500=\frac{x\times r\times 2}{100} \\ \Rightarrow \frac{125000}{r}=x...........\left(2\right) \end{gathered}$$

Step 2: Find the rate of interest

Now by substituting $x$ in equation $\left(1\right)$ we get

$\begin{array}{rcl}\left(\frac{125000+2700r}{r}\right)& =& \frac{125000}{r}{\left(1+\frac{r}{100}\right)}^2\\ & \Rightarrow & \left(125000+2700r\right)=\frac{25}{2}{\left(100+r\right)}^2\\ & \Rightarrow & 250000+5400r=25\left(10000+200r+r^2\right)\\ & \Rightarrow & 250000+5400r=250000+5000r+25r^2\\ & \Rightarrow & 400r-25r^2=0\\ & \Rightarrow & 25r\left(16-r\right)=0\\ \therefore r& =& 16\%\end{array}$

Hence, the rate of interest will be $16\%$