301 views

The difference between simple interest and compound interest on a sum for $2$ years at $8\%$, when the interest is compounded annually is ₹ $16$. If the interest were compounded half-yearly, the difference in $2$ years would be nearly :

1. ₹ $16$
2. ₹ $16.80$
3. ₹ $21.85$
4. ₹ $24.64$

The interest of the first year will be the same in both simple and compound annually, the only difference will be in the second year, which will be the extra interest on the first year’s interest.

Let the simple interest of $1$ year be ₹ $x$,

$8\% x = 16 \implies x = \frac{16}{0.08} = 200$

Half yearly interest would be $\frac{200}{2}=$ ₹ $100$,

Half yearly rate would be $\frac{8}{2} = 4\%$

Extra interest if it were half-yearly

\begin{align} &= 100 \times(1.04^3 + 1.04^2 + 1.04 - 3) \\ &= 100 \times 0.246464 \\ &= 24.6464 \end{align}

Option D is correct.

After we get the interest value, we can find the exact principle value, which is actually not required here,

Let principle value be $p$,

$8\%~p = 200 \implies p = \frac{200}{0.08} = 2500$

Compound interest for 2 years compounded half-yearly $= 2500 \times ( 1.04^4 – 1) = ₹ 424.6464$

Simple interest for 2 years will be simple $2\times 200 = ₹400$

difference $= 424.6464 – 400 = 24.6464$

866 points

1
501 views
1 vote