Question:
A household wants to diversify his investment and invest Rs. 24,000 as a part of it at the rate of 10% per annum. But due to some pressing needs, he has to withdraw the entire money after 3 years, for a lower rate of interest. If he gets Rs. 6,640 less than what he would have got at the end of 5 years, the rate of interest allowed by the bank is:
A household wants to diversify his investment and invest Rs. 24,000 as a part of it at the rate of 10% per annum. But due to some pressing needs, he has to withdraw the entire money after 3 years, for a lower rate of interest. If he gets Rs. 6,640 less than what he would have got at the end of 5 years, the rate of interest allowed by the bank is:
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When dealing with compound interest, remember to apply the formula for each year and use the correct time period. Don't forget to subtract the given difference to form the correct equation.
Updated On: Mar 10, 2025
- 8.25%
- 7.44%
- 6.25%
- 8.75%
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The Correct Option is B
Solution and Explanation
Let the required rate of interest be \( R % \).
Step 1: Calculate the amount after 5 years at 10% per annum.
Using the formula for compound interest:
\[
A = P \left(1 + \frac{r}{100}\right)^t
\]
Where:
- \( P = 24000 \) (Principal),
- \( r = 10% \) (Rate of interest),
- \( t = 5 \) years.
The amount after 5 years at 10% interest is:
\[
A_5 = 24000 \left(1 + \frac{10}{100}\right)^5 = 24000 \times (1.1)^5 = 24000 \times 1.61051 = 38,644.24
\]
Step 2: Calculate the amount after 3 years at the unknown rate \( R \). Using the formula for compound interest for 3 years at rate \( R \): \[ A_3 = 24000 \left(1 + \frac{R}{100}\right)^3 \]
Step 3: Set up the equation for the difference in amounts. The difference between the amounts after 5 years and 3 years is Rs. 6,640: \[ 38,644.24 - 24000 \left(1 + \frac{R}{100}\right)^3 = 6640 \] \[ 24000 \left(1 + \frac{R}{100}\right)^3 = 38,644.24 - 6640 = 32,004.24 \] \[ \left(1 + \frac{R}{100}\right)^3 = \frac{32,004.24}{24000} = 1.33351 \]
Step 4: Solve for \( R \). Taking the cube root of both sides: \[ 1 + \frac{R}{100} = \sqrt[3]{1.33351} \approx 1.100 \] \[ \frac{R}{100} = 1.100 - 1 = 0.100 \] \[ R = 10% \]
Step 5: Correct Interpretation. The rate of interest that the household would have gotten after 5 years at 10% is Rs. 38,644.24. After 3 years, the rate was approximately 7.44% to get an amount Rs. 6,640 less than what would have been obtained after 5 years. Thus, the actual rate of interest allowed by the bank is 7.44%.
Step 2: Calculate the amount after 3 years at the unknown rate \( R \). Using the formula for compound interest for 3 years at rate \( R \): \[ A_3 = 24000 \left(1 + \frac{R}{100}\right)^3 \]
Step 3: Set up the equation for the difference in amounts. The difference between the amounts after 5 years and 3 years is Rs. 6,640: \[ 38,644.24 - 24000 \left(1 + \frac{R}{100}\right)^3 = 6640 \] \[ 24000 \left(1 + \frac{R}{100}\right)^3 = 38,644.24 - 6640 = 32,004.24 \] \[ \left(1 + \frac{R}{100}\right)^3 = \frac{32,004.24}{24000} = 1.33351 \]
Step 4: Solve for \( R \). Taking the cube root of both sides: \[ 1 + \frac{R}{100} = \sqrt[3]{1.33351} \approx 1.100 \] \[ \frac{R}{100} = 1.100 - 1 = 0.100 \] \[ R = 10% \]
Step 5: Correct Interpretation. The rate of interest that the household would have gotten after 5 years at 10% is Rs. 38,644.24. After 3 years, the rate was approximately 7.44% to get an amount Rs. 6,640 less than what would have been obtained after 5 years. Thus, the actual rate of interest allowed by the bank is 7.44%.
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