Gopal's Grandsons Compound Interest Calculation

The problem requires us to find the initial amount Gopal deposited for Atul, given a total sum, ages, target age, and an annual compound interest rate. The key condition is that both grandsons receive the same amount when they reach 23 years old.

Determining Time Periods

First, calculate the number of years each grandson's deposit will grow:

• Akshay's age: 13 years. Target age: 23 years. Time period (\(n_{\text{Akshay}}\)): \(23 - 13 = 10\) years.
• Atul's age: 14 years. Target age: 23 years. Time period (\(n_{\text{Atul}}\)): \(23 - 14 = 9\) years.

Compound Interest Formula

The formula for the future amount (A) with compound interest is:

\( A = P \times (1 + r)^n \)

Where:

• \(A\) = Future Amount
• \(P\) = Principal (initial deposit)
• \(r\) = Annual interest rate (10% or 0.10)
• \(n\) = Number of years

Equating Future Values

Let \(P_{\text{Akshay}}\) be Akshay's deposit and \(P_{\text{Atul}}\) be Atul's deposit.

The future value for Akshay is \(A_{\text{Akshay}} = P_{\text{Akshay}} \times (1 + 0.10)^{10} = P_{\text{Akshay}} \times (1.1)^{10}\).

The future value for Atul is \(A_{\text{Atul}} = P_{\text{Atul}} \times (1 + 0.10)^{9} = P_{\text{Atul}} \times (1.1)^{9}\).

Since \(A_{\text{Akshay}} = A_{\text{Atul}}\):

\( P_{\text{Akshay}} \times (1.1)^{10} = P_{\text{Atul}} \times (1.1)^{9} \)

Solving for \(P_{\text{Akshay}}\) in terms of \(P_{\text{Atul}}\):

\( P_{\text{Akshay}} = \frac{P_{\text{Atul}} \times (1.1)^{9}}{(1.1)^{10}} \)

\( P_{\text{Akshay}} = \frac{P_{\text{Atul}}}{1.1} \)

Calculating Atul's Deposit

The total amount distributed is ₹25200:

\( P_{\text{Akshay}} + P_{\text{Atul}} = 25200 \)

Substitute the expression for \(P_{\text{Akshay}}\):

\( \frac{P_{\text{Atul}}}{1.1} + P_{\text{Atul}} = 25200 \)

Factor out \(P_{\text{Atul}}\):

\( P_{\text{Atul}} \left( \frac{1}{1.1} + 1 \right) = 25200 \)

\( P_{\text{Atul}} \left( \frac{1 + 1.1}{1.1} \right) = 25200 \)

\( P_{\text{Atul}} \left( \frac{2.1}{1.1} \right) = 25200 \)

Solve for \(P_{\text{Atul}}\):

\( P_{\text{Atul}} = 25200 \times \frac{1.1}{2.1} \)

\( P_{\text{Atul}} = 25200 \times \frac{11}{21} \)

\( P_{\text{Atul}} = \frac{25200}{21} \times 11 \)

\( P_{\text{Atul}} = 1200 \times 11 \)

\( P_{\text{Atul}} = 13200 \)

Therefore, Gopal deposited ₹13200 in the name of Atul.