Problem Analysis

The question requires calculating the initial amount deposited for Rakesh. A total of ₹24700 was distributed between two grandsons, Baban (12 years) and Rakesh (13 years). The condition is that both will have equal money when they reach 23 years old, with the money earning 8% compound interest annually.

Calculations

Let:

• Total sum distributed = ₹24700
• Baban's current age = 12 years
• Rakesh's current age = 13 years
• Target age for equal amounts = 23 years
• Annual interest rate (r) = 8% or 0.08
• Initial deposit for Baban = B
• Initial deposit for Rakesh = R

Step 1: Determine the time periods for investment.

• Baban's investment duration: \(t_B = 23 - 12 = 11\) years.
• Rakesh's investment duration: \(t_R = 23 - 13 = 10\) years.

Step 2: Formulate the equation based on equal future values.

Using the compound interest formula \(FV = P(1 + r)^t\), where FV is Future Value, P is Principal, r is the annual interest rate, and t is the time in years.

At age 23, their amounts will be equal:

\(B(1 + 0.08)^{11} = R(1 + 0.08)^{10}\)

\(B(1.08)^{11} = R(1.08)^{10}\)

Step 3: Find the relationship (ratio) between B and R.

Rearrange the equation:

\(\frac{B}{R} = \frac{(1.08)^{10}}{(1.08)^{11}}\)

\(\frac{B}{R} = \frac{1}{1.08}\)

Therefore, \(B = \frac{R}{1.08}\).

Step 4: Calculate Rakesh's initial deposit (R) using the total sum.

We know that the total sum is the sum of their initial deposits: \(B + R = 24700\).

Substitute the expression for B:

\(\frac{R}{1.08} + R = 24700\)

Factor out R:

\(R \left( \frac{1}{1.08} + 1 \right) = 24700\)

\(R \left( \frac{1 + 1.08}{1.08} \right) = 24700\)

\(R \left( \frac{2.08}{1.08} \right) = 24700\)

Step 5: Solve for R.

\(R = 24700 \times \frac{1.08}{2.08}\)

\(R = \frac{26676}{2.08}\)

\(R = 12825\)

Final Answer

The amount Atul deposited in the name of Rakesh is ₹12825.