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: Calculate using the Simple Interest (SI) formula for 2 years. \[SI = \frac{P \times R \times T}{100}\] Given \( SI = 14400, \, T = 2 \): \[14400 = \frac{P \times R \times 2}{100}\] \[14400 = \frac{2PR}{100} \(\Rightarrow\) 14400 = \frac{PR}{50}\] \[PR = 14400 \times 50 = 720000\] Step 2: Calculate using the Compound Interest (CI) formula for 2 years. \[CI = P \left( \left(1+\frac{R}{100}\right)^2 - 1 \right)\] \[16560 = P \left( \frac{R}{100} + \frac{R^2}{10000} \right) \times 2\] Alternative formula: \[CI - SI = \frac{P \times (R/100)^2 \times 2}{2}\] Simplified for 2 years: \[CI - SI = \frac{P \times R^2}{100^2}\] Step 3: Determine the difference CI - SI. \[CI - SI = 16560 - 14400 = 2160\] \[2160 = \frac{P \times R^2}{100^2}\] \[P \times R^2 = 2160 \times 10000 = 21600000\] Step 4: Divide the equations from Steps 1 and 3. From Step 1: \( PR = 720000 \). From Step 3: \( PR^2 = 21600000 \). \[\frac{PR^2}{PR} = \frac{21600000}{720000}\] \[R = 30%\] Step 5: Calculate the principal (P). \[PR = 720000 \(\Rightarrow\) P \times 30 = 720000\] \[P = \frac{720000}{30} = 24000\] \boxed{\text{Principal = ₹ 24,000, Rate = 30% per annum}}