Calculating Mandar's Initial Deposit
The problem requires us to find the initial amount deposited for Mandar so that both grandsons receive equal amounts at age 23, given a total deposit and an annual compound interest rate.
Compound Interest Setup
Let \(P_R\) be the principal amount deposited for Rupesh and \(P_M\) be the principal amount deposited for Mandar. The total deposit is ₹23400.
So, \(P_R + P_M = 23400\).
The interest rate is 8% per annum, compounded annually. \(r = 0.08\).
Determining Time Periods
Rupesh is 14 years old and will receive the money at 23. The time period for Rupesh's investment is \(t_R = 23 - 14 = 9\) years.
Mandar is 15 years old and will receive the money at 23. The time period for Mandar's investment is \(t_M = 23 - 15 = 8\) years.
Equal Amount Condition
The future value (Amount, \(A\)) is calculated using the formula \(A = P(1 + r)^t\). Both grandsons should receive the same amount when they turn 23.
Amount for Rupesh: \(A_R = P_R(1 + 0.08)^9 = P_R(1.08)^9\).
Amount for Mandar: \(A_M = P_M(1 + 0.08)^8 = P_M(1.08)^8\).
Since \(A_R = A_M\), we have:
\(P_R(1.08)^9 = P_M(1.08)^8\)Relating Initial Deposits
We can simplify the equation to find the relationship between \(P_R\) and \(P_M\):
\(P_R = P_M \frac{(1.08)^8}{(1.08)^9}\) \(P_R = P_M \frac{1}{1.08}\)Calculating Mandar's Deposit (\(P_M\))
Now substitute the expression for \(P_R\) into the total deposit equation (\(P_R + P_M = 23400\)):
\(P_M \left(\frac{1}{1.08}\right) + P_M = 23400\)Factor out \(P_M\):
\(P_M \left(\frac{1}{1.08} + 1\right) = 23400\) \(P_M \left(\frac{1 + 1.08}{1.08}\right) = 23400\) \(P_M \left(\frac{2.08}{1.08}\right) = 23400\)Solve for \(P_M\):
\(P_M = 23400 \times \frac{1.08}{2.08}\) \(P_M = 23400 \times \frac{108}{208}\)Simplify the fraction \(\frac{108}{208} = \frac{27}{52}\).
\(P_M = 23400 \times \frac{27}{52}\)Perform the calculation:
\(P_M = \frac{23400}{52} \times 27\) \(P_M = 450 \times 27\) \(P_M = 12150\)Therefore, Shivam deposited ₹12150 in the name of Mandar.
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?
