Understanding the Problem

The question asks us to find the rate of interest (R) when the difference between the compound interest (CI) and simple interest (SI) earned on a principal amount (P) of ₹5,000 over a period (T) of 2 years is ₹50.

Formulas for Simple Interest and Compound Interest

We need the formulas for Simple Interest (SI) and Compound Interest (CI) for a 2-year period. Let P be the principal amount, R be the rate of interest per annum, and T be the time period in years.

• Simple Interest (SI) is calculated as: $SI = \frac{P \times R \times T}{100}$
• Compound Interest (CI) for 2 years is calculated as: $CI = P \times \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right)$

Calculating the Difference between CI and SI

The difference between CI and SI for 2 years can be simplified using the formula:

Difference = CI - SI

Difference = $ \left[ P \times \left( \left(1 + \frac{R}{100}\right)^2 - 1 \right) \right] - \left[ \frac{P \times R \times T}{100} \right]$

Substituting T = 2:

Difference = $ P \times \left(1 + \frac{R}{100}\right)^2 - P - \frac{P \times R \times 2}{100}$

Difference = $ P \times \left(1 + \frac{2R}{100} + \frac{R^2}{10000}\right) - P - \frac{2PR}{100}$

Difference = $ P + \frac{2PR}{100} + \frac{PR^2}{10000} - P - \frac{2PR}{100}$

Difference = $ \frac{PR^2}{10000}$

This can also be written as:

Difference = $ P \times \left(\frac{R}{100}\right)^2$

Applying the Given Values

We are given:

• Principal (P) = ₹5,000
• Time (T) = 2 years
• Difference (CI - SI) = ₹50

Now, let's plug these values into the difference formula:

$50 = 5000 \times \left(\frac{R}{100}\right)^2$

Solving for the Rate of Interest (R)

To find R, we rearrange the equation:

1. Divide both sides by 5000: $\frac{50}{5000} = \left(\frac{R}{100}\right)^2$
2. Simplify the fraction: $\frac{1}{100} = \left(\frac{R}{100}\right)^2$
3. Take the square root of both sides: $\sqrt{\frac{1}{100}} = \sqrt{\left(\frac{R}{100}\right)^2}$
4. Calculate the square root: $\frac{1}{10} = \frac{R}{100}$
5. Solve for R by multiplying both sides by 100: $R = \frac{1}{10} \times 100$
6. Calculate the final value: $R = 10$

Therefore, the rate of interest is 10%.

Conclusion

The rate of interest required for the difference between compound interest and simple interest to be ₹50 on ₹5,000 in 2 years is 10%.