An investor evaluates two compounding options offered by a financial institution at an interest rate of 20% per annum. The compound interest accrued on a certain capital investment when compounded annually over a 2-year period is Rs. 9,200 more than the compound interest earned on the same capital when compounded semi-annually over a 1-year period. What is the total capital amount invested (in Rs)?

The correct answer is 40000.

Let the total capital amount invested be $P$ (in Rs.).

The interest rate offered is $R=20\%$ per annum.

Step 1: Calculate the compound interest accrued when compounded annually over 2 years ($C{I}_1$)

For annual compounding over a 2-year period:

Rate per compounding period = $20\%$

Number of compounding periods ($n$) = $2$

The total compound amount $A_1$ is calculated as:

$A_1=P{\left(1+\frac{20}{100}\right)}^2$

$A_1=P{\left(1.2\right)}^2=1.44P$

The compound interest earned ($C{I}_1$) is:

$C{I}_1=A_1-P=1.44P-P=0.44P$

Step 2: Calculate the compound interest accrued when compounded semi-annually over 1 year ($C{I}_2$)

When interest is compounded semi-annually (every 6 months):

Interest rate per half-year ($r$) = $\frac{20\%}{2}=10\%$

Number of compounding periods in 1 year ($n$) = $2$

The total compound amount $A_2$ is calculated as:

$A_2=P{\left(1+\frac{10}{100}\right)}^2$

$A_2=P{\left(1.1\right)}^2=1.21P$

The compound interest earned ($C{I}_2$) is:

Step 3: Determine the total capital investment ($P$)

According to the problem statement, $C{I}_1$ is Rs. 9,200 more than $C{I}_2$:

$C{I}_1-C{I}_2=9200$

$0.44P-0.21P=9200$

$0.23P=9200$

$P=\frac{9200}{0.23}$

$P=40000$

Thus, the total capital amount invested is Rs. 40,000.