CAT 2025

Slot 1

QA

Question & SolutionQues & Sol

At a certain simple rate of interest, a given sum amounts to Rs 13920 in 3 years, and to Rs 18960 in 6 years and 6 months. If the same given sum had been invested for 2 years at the same rate as before but with interest compounded every 6 months, then the total interest earned, in rupees, would have been nearest to

1. Concept Used

• Topic: Simple Interest (to find Principal & Rate) → Compound Interest (compounded half-yearly)
• Formula: $$SI = \frac{P \times r \times t}{100}, \quad A = P\left(1 + \frac{r/2}{100}\right)^{2t}$$

2. Calculation

Let the principal be $P$ and annual simple interest rate be $r%$.

Setting up equations from SI conditions:

Amount after 3 years: $P + \frac{P \times r \times 3}{100} = 13920 \quad \Rightarrow \quad 13920 - P = \frac{3Pr}{100} \quad \cdots (1)$

Amount after 6.5 years (6 years and 6 months): $P + \frac{P \times r \times 13}{200} = 18960 \quad \Rightarrow \quad 18960 - P = \frac{13Pr}{200} \quad \cdots (2)$

Dividing equation (1) by equation (2):

$\frac{13920 - P}{18960 - P} = \frac{\frac{3Pr}{100}}{\frac{13Pr}{200}} = \frac{3}{100} \times \frac{200}{13} = \frac{6}{13}$

Cross-multiplying: $13(13920 - P) = 6(18960 - P)$

$180960 - 13P = 113760 - 6P$

$180960 - 113760 = 13P - 6P$

$67200 = 7P \quad \Rightarrow \quad P = 9600$

Finding the rate using equation (1):

$13920 - 9600 = \frac{9600 \times r \times 3}{100}$

$4320 = 288r \quad \Rightarrow \quad r = 15%$

Computing Compound Interest (compounded every 6 months) for 2 years:

When compounded half-yearly, the rate per period = $\frac{15}{2} = 7.5%$ and number of periods = $2 \times 2 = 4$.

$A = 9600 \times \left(1 + \frac{7.5}{100}\right)^4 = 9600 \times (1.075)^4$

$(1.075)^2 = 1.155625$

$(1.075)^4 = (1.155625)^2 = 1.33546...$

$A = 9600 \times 1.33547 \approx 12820.50$

$\text{Total Interest} = A - P = 12820.50 - 9600 = 3220.50 \approx 3221$

3. Solution

Answer = Option A (3221) ✅

The total compound interest earned is approximately Rs 3221.