Question

A sum of money was divided into 2 parts in the ratio 2:5. 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. \(8.8\%\)
• B. Cannot be determined
• C. \(10.2\%\)
• D. \(9.6\%\)

Answer 1

The Correct Option is A

Calculating the Required Simple Interest Rate 

- Let the total sum of money be S. Then:

• The first part is \( \frac{2}{7}S \)
• The second part is \( \frac{5}{7}S \)

 

- The interest earned on the first part (compounded annually) is given by:

\[ A_1 = P \left( 1 + \frac{r}{100} \right)^t \]

Where:

• \( P = \frac{2}{7}S \)
• \( r = 20\% \)
• \( t = 2 \text{ years} \)

 

Substituting the values:

\[ A_1 = \frac{2}{7}S \left( 1 + \frac{20}{100} \right)^2 = \frac{2}{7}S \times 1.44 = \frac{2.88}{7}S \]

- The interest earned on the second part (simple interest) is given by:

\[ A_2 = \frac{5}{7}S \times \frac{r}{100} \times 2 \]

- For the interest to be the same in both cases, we equate the two expressions:

\[ \frac{2.88}{7}S = \frac{5}{7}S \times \frac{r}{100} \times 2 \]

- Cancelling common terms and simplifying:

\[ 2.88 = 10r \div 100 \] \[ r = \frac{2.88 \times 100}{10} = 8.8 \]

Thus, the rate of simple interest must be 8.8% for the second part to yield the same interest as the first part.

Conclusion: The correct answer is 8.8%.