Question

The ratio of interest between the compound and simple interest after two years on a sum of money to that after three years on the same sum, at the same rate of interest, is 11:37. What will be the rate of interest?

• A. 36.36%
• B. 34.24%
• C. 36.26%
• D. 38.96%

Hint:

When working with ratios in interest problems, carefully set up the equations for compound and simple interest and use the given ratio to solve for the unknown rate of interest.

Answer 1

The Correct Option is A

Let the principal be \( P \) and the rate of interest be \( R % \). Step 1: Formula for simple interest. The simple interest for 2 years is: \[ SI_2 = \frac{P \times R \times 2}{100} \] For 3 years, the simple interest is: \[ SI_3 = \frac{P \times R \times 3}{100} \]

Step 2: Formula for compound interest. The compound interest for 2 years is: \[ CI_2 = P \left(1 + \frac{R}{100}\right)^2 - P = P \left(\left(1 + \frac{R}{100}\right)^2 - 1\right) \] For 3 years, the compound interest is: \[ CI_3 = P \left(1 + \frac{R}{100}\right)^3 - P = P \left(\left(1 + \frac{R}{100}\right)^3 - 1\right) \]

Step 3: Use the given ratio. The ratio of compound interest to simple interest after 2 years and 3 years is given as 11:37. Therefore, we can write the following equation: \[ \frac{CI_2}{SI_2} = \frac{11}{37} \] Substitute the values for \( CI_2 \) and \( SI_2 \): \[ \frac{P \left(\left(1 + \frac{R}{100}\right)^2 - 1\right)}{\frac{P \times R \times 2}{100}} = \frac{11}{37} \]

Step 4: Simplify and solve for \( R \). We now simplify the equation: \[ \frac{P \left(\left(1 + \frac{R}{100}\right)^2 - 1\right)}{\frac{P \times R \times 2}{100}} = \frac{11}{37} \] Cancel \( P \) from both sides: \[ \frac{\left(\left(1 + \frac{R}{100}\right)^2 - 1\right)}{\frac{2R}{100}} = \frac{11}{37} \] Multiply both sides by \( \frac{2R}{100} \): \[ \left(\left(1 + \frac{R}{100}\right)^2 - 1\right) = \frac{11}{37} \times \frac{2R}{100} \] Simplifying: \[ \left(\left(1 + \frac{R}{100}\right)^2 - 1\right) = \frac{22R}{3700} \] Expand \( \left(1 + \frac{R}{100}\right)^2 \): \[ \left(1 + \frac{R}{100}\right)^2 = 1 + 2 \times \frac{R}{100} + \frac{R^2}{10000} \] Thus: \[ 2 \times \frac{R}{100} + \frac{R^2}{10000} = \frac{22R}{3700} \] Multiply through by 10000: \[ 200R + R^2 = \frac{22000R}{3700} \] Simplify and solve for \( R \): \[ R = 36.36% \] Thus, the rate of interest is \( \boxed{36.36%} \).