Rohan borrowed a sum of Rs.1,20,000 from a financial institution at an annual compound interest rate of 8% for a tenure of 2 years. He invested $\frac{3}{4}$th of this borrowed amount into a mutual fund project that yields 10% compound interest per annum compounded annually for 2 years. If he retains the remaining portion of the money with himself, calculate his total net gain or loss after 2 years.

The correct answer is Rs.1068 loss.

Step 1: Calculate the total amount Rohan owes to the financial institution after 2 years.

Rohan borrowed a principal amount of Rs. 1,20,000 at a compound interest rate of 8% per annum for 2 years.

The formula for total amount under compound interest is:

$A=P{(1+\frac{r}{100})}^n$

Substituting the given values:

$A_{\text{borrowed}}=\mathrm{1,20,000}\times {(1+\frac{8}{100})}^2$

$A_{\text{borrowed}}=\mathrm{1,20,000}\times {\left(1.08\right)}^2$

$A_{\text{borrowed}}=\mathrm{1,20,000}\times 1.1664=\text{Rs. 1,39,968}$

So, Rohan must pay back Rs. 1,39,968 to the financial institution after 2 years.

Step 2: Calculate the amounts invested and retained by Rohan.

Rohan invested $\frac{3}{4}$th of the borrowed amount into a mutual fund project:

$\text{Invested Amount}=\frac{3}{4}\times \mathrm{1,20,000}=\text{Rs. 90,000}$

The remaining portion of the money retained with himself is:

$\text{Retained Amount}=\mathrm{1,20,000}-\mathrm{90,000}=\text{Rs. 30,000}$

Step 3: Calculate the maturity returns from the mutual fund after 2 years.

The mutual fund yields 10% compound interest per annum for 2 years on the invested principal of Rs. 90,000:

$A_{\text{mutual fund}}=\mathrm{90,000}\times {(1+\frac{10}{100})}^2$

$A_{\text{mutual fund}}=\mathrm{90,000}\times {\left(1.1\right)}^2$

$A_{\text{mutual fund}}=\mathrm{90,000}\times 1.21=\text{Rs. 1,08,900}$

Step 4: Calculate Rohan's total money available after 2 years and his net outcome.

Total money Rohan has after 2 years (mutual fund returns + retained cash):

$\text{Total Money Available}=\mathrm{1,08,900}+\mathrm{30,000}=\text{Rs. 1,38,900}$

Now, calculate Rohan's net gain or loss by comparing his total money available with the amount he owes to the financial institution:

$\text{Net Outcome}=\text{Total Money Available}-\text{Repayment Amount}$

$\text{Net Outcome}=\mathrm{1,38,900}-\mathrm{1,39,968}=-\mathrm{1,068}$

Since the total money available (Rs. 1,38,900) is less than the amount to be repaid (Rs. 1,39,968), Rohan incurs a total net loss of Rs. 1068.