The compound interest on a principal at 20% p.a. for 1.5 years, compounded annually, is ₹3,400. Find the principal.

The correct option is ₹10,625.

Given Data:

Rate of interest per annum ($R$) = 20%

Time period ($T$) = 1.5 years = $1\frac{1}{2}$ years

Compound Interest ($CI$) = ₹3,400

Formula:

When interest is compounded annually, the total amount ($A$) for $n\frac{1}{2}$ years (where the full years are $n$ and fraction is $\frac{1}{2}$) is given by:

$A=P\times \left(1+\frac{R}{100}\right)\times \left(1+\frac{\frac{1}{2}\times R}{100}\right)$

Step-by-Step Calculation:

Here, full years $n=1$ and fractional year is $\frac{1}{2}$.

Substitute $R=20$ into the amount formula:

$A=P\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{10}{100}\right)$

$A=P\times \left(\frac{120}{100}\right)\times \left(\frac{110}{100}\right)$

$A=P\times \left(\frac{6}{5}\right)\times \left(\frac{11}{10}\right)$

$A=P\times \frac{66}{50}=1.32P$

We know that Compound Interest is the difference between the total amount and the principal:

$CI=A-P$

Substitute the value of $CI$ and $A$:

$3400=1.32P-P$

$3400=0.32P$

$P=\frac{3400}{0.32}$

$P=\frac{340000}{32}$

$P=10625$

Thus, the principal amount is ₹10,625.