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.
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Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
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In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
