Question

A sum of money was divided into 2 parts in the ratio 2:5. The first part was invested for 2 years at the annual interest rate of 20% compounded annually. At what rate of simple interest per annum the second part must be invested for 2 years so that the interest earned in both cases is the same?

• A. 9.6%
• B. 8.8%
• C. Cannot be determined
• D. 10.2%

Hint:

When comparing compound and simple interest, use the appropriate formulas for each type and set the interests equal to solve for the unknown rate.

Answer 1

The Correct Option is B

Step 1: Let the total sum of money be \( S \). The two parts are in the ratio of 2:5. Therefore, the first part is \( \frac{2}{7}S \) and the second part is \( \frac{5}{7}S \). 

Step 2: The interest earned on the first part using compound interest formula is: \[ A = P \left(1 + \frac{r}{100}\right)^t \] Where \( P = \frac{2}{7}S \), \( r = 20 % \), and \( t = 2 \) years. The interest \( I_1 \) for the first part is: \[ I_1 = A - P = \frac{2}{7}S \left( \left( 1 + \frac{20}{100} \right)^2 - 1 \right) \] \[ I_1 = \frac{2}{7}S \left( 1.2^2 - 1 \right) = \frac{2}{7}S \left( 1.44 - 1 \right) = \frac{2}{7}S \times 0.44 \] \[ I_1 = \frac{0.88}{7}S \] 

Step 3: Let the rate of interest for the second part be \( r \)%. The interest \( I_2 \) on the second part using the simple interest formula is: \[ I_2 = \frac{P \times r \times t}{100} = \frac{5}{7}S \times \frac{r \times 2}{100} \] We are given that the interest from both parts is the same, so \( I_1 = I_2 \). Thus: \[ \frac{0.88}{7}S = \frac{5}{7}S \times \frac{r \times 2}{100} \] 

Simplifying: \[ 0.88 = \frac{10r}{100} \] \[ 0.88 = \frac{r}{10} \] \[ r = 8.8 \] 

Thus, the required rate of interest is 8.8%.