Question:
Amrita buys a car for which she makes a down payment of ₹ 2,50,000 and the balance is to be paid in 2 years by monthly installments of ₹ 25,448 each. If the financer charges interest at the rate of 20\% p.a., find the actual price of the car.
[Given \( \left(\frac{61}{60}\right)^{-24} = 0.67253 \)]
Amrita buys a car for which she makes a down payment of ₹ 2,50,000 and the balance is to be paid in 2 years by monthly installments of ₹ 25,448 each. If the financer charges interest at the rate of 20\% p.a., find the actual price of the car.
[Given \( \left(\frac{61}{60}\right)^{-24} = 0.67253 \)]
[Given \( \left(\frac{61}{60}\right)^{-24} = 0.67253 \)]
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For loan calculations, always use the present value formula, and remember to convert annual interest rates to monthly rates by dividing by 12. Add the down payment to the present value of the loan to find the total price.
Updated On: Feb 11, 2025
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Solution and Explanation
Step 1: Use the present value formula to calculate the loan amount:
The formula for present value \( PV \) is:
\[
PV = R \cdot \frac{1 - (1 + i)^{-n}}{i},
\]
where \( R \) is the monthly installment, \( i \) is the monthly interest rate, and \( n \) is the number of installments.
Given:
\[
R = 25,448, \quad i = \frac{20}{12 \cdot 100} = 0.01667, \quad n = 24.
\]
Step 2: Substitute the given values into the formula:
\[
PV = 25,448 \cdot \frac{1 - \left(\frac{61}{60}\right)^{-24}}{0.01667}.
\]
Step 3: Simplify the terms:
From the given data, \( \left(\frac{61}{60}\right)^{-24} = 0.67253 \).
Substitute this into the formula:
\[
PV = 25,448 \cdot \frac{1 - 0.67253}{0.01667}.
\]
Step 4: Calculate the present value:
\[
PV = 25,448 \cdot \frac{0.32747}{0.01667}.
\]
\[
PV = 25,448 \cdot 19.638 = 4,99,500.
\]
Step 5: Add the down payment to get the total price of the car:
\[
{Total Price} = {Down Payment} + PV.
\]
\[
{Total Price} = 2,50,000 + 4,99,500 = 7,49,500.
\]
Final Answer: The actual price of the car is ₹ 7,49,500.
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