Nitu has an initial capital of Rs. $20,000$. Out of this, she invests Rs. $8,000$ at $5.5\%$ in bank $A$, Rs. $5,000$ 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 back $C$ alone, then her annual interest income, in rupees, would have been

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Understanding the Problem:

Nitu has Rs. 20000 total. She splits this into three investments and earns a combined annual interest equal to 5% of her initial capital. We need to find how much she would earn if she invested everything in bank C.

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Find the remaining amount for bank C:

Total capital = Rs. 20000

Amount in bank A = Rs. 8000

Amount in bank B = Rs. 5000

Amount in bank C = Rs. 20000 - Rs. 8000 - Rs. 5000

= Rs. 7000

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Calculate interest from each bank:

Simple Interest Formula: Interest = $\frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}$

Since we're dealing with annual interest, Time = 1 year.

Interest from bank A = $\frac{8000 \times 5.5 \times 1}{100}$

= $\frac{44000}{100}$

= 440

Interest from bank B = $\frac{5000 \times 5.6 \times 1}{100}$

= $\frac{28000}{100}$

= 280

Interest from bank C = $\frac{7000 \times x \times 1}{100}$

= $\frac{7000x}{100}$

= 70x

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Use the given condition:

The problem states: "Her combined annual interest income equals 5% of the initial capital"

5% of initial capital = $\frac{5 \times 20000}{100}$

= 1000

So, Total interest = Rs. 1000

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Set up and solve the equation:

Interest from A + Interest from B + Interest from C = Total interest

$440 + 280 + 70x = 1000$

$720 + 70x = 1000$

$70x = 1000 - 720$

= 280

$x = \frac{280}{70}$

= 4

Therefore, the rate of interest in bank C is 4%.

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Calculate the required answer:

If she invested her entire Rs. 20000 in bank C at 4% per annum:

Interest = $\frac{20000 \times 4 \times 1}{100}$

= $\frac{80000}{100}$

= 800

Answer: Rs. 800

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Key Takeaway: When we solve problems involving multiple investments, we always set up equations based on the given conditions. Here, the total interest condition helped us find the unknown rate, which we then used to calculate the final answer.