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.
Similar Questions
- In what time will ₹ 4400 become ₹ 4576 at 8% per annum interest compounded half – yearly?
Important Questions from Compound Interest
At what rate percent per annum will Rs. 7200 amount to Rs. 7938 in one year, if interest is compounded half yearly?
What is the compound interest (in Rs.) on a sum of Rs. 8192 for \(1 \frac{1}{4}\) years at 15% per annum, if interest is compounded 5-monthly ?
What is the difference (in Rs.) between the interests on Rs. 50,000 for one year at 8% per annum compounded half yearly and yearly?
A sum of money becomes Rs. 11,880 after 4 years and Rs. 17,820 after 6 years on compound interest, if the interest is compounded annually. What is the half of the sum (in Rs.)?
A sum invested at compound interest amounts to Rs. 7,800 in 3 years and Rs. 11,232 in 5 years. What is the rate per cent?
