Question

Mr. Dileep Rao has set up a sinking fund so that he can accumulate \(₹ 10,00,000\) in \(10 \)years for his children's higher education. How much amount should Dileep Rao deposit at the beginning of each year to accumulate this amount at the end of\( 10\) years. If the interest rate is  compounded annually? Given that \((1.12)^{11}=3.477\) (Rounded off to the nearest paise)

• A. \(₹ 50000\)
• B. \(₹ 50900\)
• C. \(₹ 51211.10\)
• D. \(₹50912.18\)

Answer 1

The Correct Option is D

To determine the amount Mr. Dileep Rao should deposit annually, we need to use the formula for the future value of a sinking fund when the compound interest is applied. The formula for the future value \(FV\) of a sinking fund with annual payments \(P\), interest rate \(r\), and time \(n\) is given by:

\(FV = P \times \frac{(1+r)^n - 1}{r}\)

In this scenario, Mr. Rao wants to accumulate ₹10,00,000 in 10 years at an annual interest rate of 12% (or 0.12 when expressed as a decimal). Therefore, we set up the equation as follows:

\(10,00,000 = P \times \frac{(1+0.12)^{10} - 1}{0.12}\)

Given \((1.12)^{11} = 3.477\), we first need to calculate \((1.12)^{10}\).

Since \((1.12)^{11} = (1.12) \times (1.12)^{10}\), it follows that:

\( (1.12)^{10} = \frac{3.477}{1.12} \approx 3.104\)

Substituting \((1.12)^{10} - 1\) in the sinking fund formula:

\( 10,00,000 = P \times \frac{3.104 - 1}{0.12} \)

\( 10,00,000 = P \times \frac{2.104}{0.12} \)

Solving for \(P\):

\( P = \frac{10,00,000 \times 0.12}{2.104} \)

\( P = \frac{1,20,000}{2.104} \approx 50912.18 \)

Hence, the amount Mr. Dileep Rao should deposit at the beginning of each year is \(₹50912.18\).