CAT 2022Slot 2QAQuestion & Solution
Question
Mr. Pinto invests one-fifth of his capital at 6%, one-third at 10% and the remaining at 1%, each rate being simple interest per annum. Then, the minimum number of years required for the cumulative interest income from these investments to equal or exceed his initial capital is
Solution
1. Concept Used
- Topic: Simple Interest — Weighted Average Rate & Minimum Years
- Formula: $$\text{Simple Interest} = \frac{P \times R \times T}{100}$$
The total cumulative interest from all three investments must satisfy: $$\text{Total Interest} \geq \text{Initial Capital}$$
2. Calculation
Let the total capital be $$15x$$ (LCM of 5 and 3, chosen to avoid fractions), and let the number of years required be $$T$$.
Breaking down the investments:
- One-fifth of $$15x$$ at $$6%$$ → Principal $$= 3x$$
- One-third of $$15x$$ at $$10%$$ → Principal $$= 5x$$
- Remaining at $$1%$$ → Principal $$= 15x - 3x - 5x = 7x$$
Now, the total interest earned over $$T$$ years is: $$\text{Total Interest} = \frac{3x \times 6 \times T}{100} + \frac{5x \times 10 \times T}{100} + \frac{7x \times 1 \times T}{100}$$
Simplifying each term: $$= \frac{18xT}{100} + \frac{50xT}{100} + \frac{7xT}{100}$$
$$= \frac{75xT}{100}$$
Setting up the inequality (total interest $$\geq$$ initial capital): $$\frac{75xT}{100} \geq 15x$$
Dividing both sides by $$x$$ (since $$x > 0$$): $$\frac{75T}{100} \geq 15$$
$$T \geq \frac{15 \times 100}{75}$$
$$T \geq 20$$
Therefore, the minimum integer value of $$T$$ is 20 years.
3. Solution
Answer = 20 ✅
The minimum number of years required for the cumulative interest income to equal or exceed Mr. Pinto's initial capital is 20 years.