Question

On a certain principal, CI and SI at a certain rate of interest for 2 years is ₹ 16560 and ₹ 14400 respectively. Find the principal and rate of interest per annum.

• A. 21,600, 13%
• B. 20,000, 30%
• C. 24,000, 30%
• D. 24,000, 35%

Hint:

For 2 years, use the relation \( CI - SI = \frac{P \times R^2}{100^2} \). This shortcut saves time compared to expanding the full CI formula.

Answer 1

The Correct Option is C

Step 1: Use formula for SI for 2 years. \[ SI = \frac{P \times R \times T}{100} \] Here, \( SI = 14400, \, T = 2 \). \[ 14400 = \frac{P \times R \times 2}{100} \] \[ 14400 = \frac{2PR}{100} \(\Rightarrow\) 14400 = \frac{PR}{50} \] \[ PR = 14400 \times 50 = 720000 \] Step 2: Use formula for CI for 2 years. \[ CI = P \left( \left(1+\frac{R}{100}\right)^2 - 1 \right) \] \[ 16560 = P \left( \frac{R}{100} + \frac{R^2}{10000} \right) \times 2 \] But more directly: \[ CI - SI = \frac{P \times (R/100)^2 \times 2}{2} \] Actually formula: For 2 years, \[ CI - SI = \frac{P \times R^2}{100^2} \] Step 3: Calculate CI - SI. \[ CI - SI = 16560 - 14400 = 2160 \] So, \[ 2160 = \frac{P \times R^2}{100^2} \] \[ P \times R^2 = 2160 \times 10000 = 21600000 \] Step 4: Divide two relations. From Step 1: \( PR = 720000 \). From Step 3: \( PR^2 = 21600000 \). \[ \frac{PR^2}{PR} = \frac{21600000}{720000} \] \[ R = 30% \] Step 5: Find principal. \[ PR = 720000 \(\Rightarrow\) P \times 30 = 720000 \] \[ P = \frac{720000}{30} = 24000 \] \boxed{\text{Principal = ₹ 24,000, Rate = 30% per annum}}