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?
\[ \textbf{Step 1: Divide the sum.} \] Let the total sum be \( 7x \). \[ \text{First part} = 2x, \quad \text{Second part} = 5x \] --- \[ \textbf{Step 2: Compound Interest (CI) on first part.} \] \[ A = P(1+r)^t \] Here, \( P = 2x, \ r = 20\% = 0.2, \ t=2 \). \[ A = 2x (1.2)^2 = 2x \times 1.44 = 2.88x \] \[ CI = A - P = 2.88x - 2x = 0.88x \] --- \[ \textbf{Step 3: Simple Interest (SI) on second part.} \] \[ SI = \frac{P \times R \times T}{100} \] Here, \( P = 5x, \ T = 2 \). \[ SI = \frac{5x \times R \times 2}{100} = \frac{10xR}{100} = \frac{xR}{10} \] --- \[ \textbf{Step 4: Equating interests.} \] \[ \frac{xR}{10} = 0.88x \] \[ R = 0.88 \times 10 = 8.8 \] --- \[ \boxed{8.8\% \ \text{per annum}} \]
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