Question

Simple interest on a certain sum of mine at $9\%$ is rupees $450$ in $2$ years. Find the compound interest on the same sum at the same rate for $1$ year if the interest is reckoned half-yearly.

Answer 1

Determine the compound interest:

Simple interest for $1$ year $=\frac{450}{2}$

$=225$

Let $\mathrm{x}$ be the principal $\frac{9\mathrm{x}}{100}=225\mathrm{x}=2500$

Thus, the principal amount $\left(P\right)$ is $\mathrm{Rs}.2500$.

Now let's calculate the compound interest for the available values:

Principal, $P=\mathrm{Rs}2500$

Since the interest is compounded half-yearly, there are $2$ half years in one year.

$\mathrm{Time}=1\mathrm{year}=2\mathrm{half}\mathrm{years}$

$\mathrm{Rate}=\frac{9}{2}\%\mathrm{half}\mathrm{yearly}$

The amount is computed as,

$$\begin{gathered} A=P{\left(1+\frac{\mathrm{rate}}{100}\right)}^{\mathrm{time}} \\ A=2500\times {\left(1+\frac{9}{200}\right)}^2 \\ A=2500\times \frac{209}{200}\times \frac{209}{200} \\ A=\mathrm{Rs}.2730.0625 \end{gathered}$$

Since the amount is $\mathrm{Rs}2730.0625$ and the principal is $\mathrm{Rs}2500$, therefore, compound interest is calculated as:

Compound Interest = Amount - Principal

$$\begin{gathered} \therefore \mathrm{C}.\mathrm{I}.=2730.0625-2500 \\ =\mathrm{Rs}230.0625 \end{gathered}$$

Therefore, the compound interest is $\mathbf{Rs}\mathbf{}\mathbf{230}\mathbf{.}\mathbf{0625}$.