The interest received on a sum of money when invested in scheme A is equal to the interest received on the same sum of money when invested for 2 years in scheme B. Scheme A offers simple interest (p.c.p.a.) and scheme B offers compound interest (compounded annually). Both the schemes offer the same rate of interest. If the numerical value of the number of years for which the sum is invested in scheme A is same as the numerical value of the rate of interest offered by the same scheme, what is the rate of interest (p.c.p.a) offered by scheme A?

Let sum invested in both schemes = $$Rs. P$$

Let rate of interest in both schemes = $$R \%$$

Time period in scheme A = $$R$$ years

=> Simple interest under Scheme A = $$\frac{P \times R \times R}{100}$$

= $$Rs. \frac{P R^2}{100}$$

Also, interest received from both schemes is also same, and time period under scheme B = 2 years

=> Compound interest under scheme B = $$P [(1 + \frac{R}{100})^T - 1]$$

=> $$\frac{P R^2}{100} = P [(1 + \frac{R}{100})^2 - 1]$$

=> $$\frac{R^2}{100} = [1 + (\frac{R}{100})^2 + \frac{2 R}{100}] - 1$$

=> $$\frac{R}{100} = \frac{R}{10000} + \frac{1}{50}$$

=> $$\frac{R}{100} (1 - \frac{1}{100}) = \frac{1}{50}$$

=> $$\frac{R}{100} \times \frac{99}{100} = \frac{1}{50}$$

=> $$R = \frac{100 \times 100}{99 \times 50}$$

=> $$R = \frac{200}{99} = 2 \frac{2}{99} \%$$