CAT 2022Slot 3QAQuestion & Solution
Question
Nitu has an initial capital of ₹20,000. Out of this, she invests ₹8,000 at 5.5% in bank A, ₹5,000 at 5.6% in bank B and the remaining amount at x% in bank C, each rate being simple interest per annum. Her combined annual interest income from these investments is equal to 5% of the initial capital. If she had invested her entire initial capital in bank C alone, then her annual interest income, in rupees, would have been
Options
700
800
900
1000
Solution
1. Concept Used
- Topic: Simple Interest — distributing principal across multiple banks and solving for unknown rate
- Formula: $$ \text{Simple Interest} = \frac{P \times R \times T}{100} $$
2. Calculation
First, identify the amount invested in each bank. Nitu's total capital is ₹20,000. She invests ₹8,000 in Bank A and ₹5,000 in Bank B, so the amount invested in Bank C is $$20000 - 8000 - 5000 = 7000$$
Next, compute the combined annual interest income target. Since it equals 5% of ₹20,000: $$\text{Total Interest} = \frac{5}{100} \times 20000 = ₹1000$$
Now set up the equation by summing the individual simple interests from all three banks and equating to ₹1000: $$\frac{5.5 \times 1 \times 8000}{100} + \frac{5.6 \times 1 \times 5000}{100} + \frac{x \times 1 \times 7000}{100} = 1000$$
Compute the interest from Bank A: $$\frac{5.5 \times 8000}{100} = \frac{44000}{100} = 440$$
Compute the interest from Bank B: $$\frac{5.6 \times 5000}{100} = \frac{28000}{100} = 280$$
Substitute back into the equation: $$440 + 280 + \frac{7000x}{100} = 1000$$
$$720 + 70x = 1000$$
$$70x = 280$$
$$x = 4%$$
Now, if Nitu invests her entire initial capital of ₹20,000 in Bank C alone at 4% per annum: $$\text{Interest} = \frac{20000 \times 4 \times 1}{100} = \frac{80000}{100} = ₹800$$
3. Solution
Answer = Option B ✅
The final calculated annual interest income from Bank C alone is ₹800.