CAT 2025

Slot 1

QA

Question & Solution

1. Concept Used

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

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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 = $$2 \times 2 = 4$$.

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

$$(1.075)^2 = 1.155625$$

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

$$A = 9600 \times 1.33547 \approx 12820.50$$

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

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3. Solution

Answer = Option A (3221) ✅

The total compound interest earned is approximately Rs 3221.