A man has Rs 10,000 for investment. He invests Rs 4,000@5% per annum and Rs 3,500@4% per annum. At what rate of interest the remaining money should be interested so that his total income become Rs 500 at the end of the year?

Options

A. `6.4%`
B. `8.6%`
C. `8.5%`
D. `6.7%`

Answer

To solve the problem step by step, we will calculate the total interest earned from the investments and then determine the rate of interest required for the remaining amount to achieve a total income of Rs 500.

### Step 1: Calculate the interest from the first investment
The first investment is Rs 4,000 at an interest rate of 5% per annum.

\[ \text{Interest} = \frac{P \times R \times T}{100} \]
Where:
- \( P = 4000 \)
- \( R = 5 \)
- \( T = 1 \)

\[ \text{Interest} = \frac{4000 \times 5 \times 1}{100} = \frac{20000}{100} = 200 \]

### Step 2: Calculate the interest from the second investment
The second investment is Rs 3,500 at an interest rate of 4% per annum.

\[ \text{Interest} = \frac{P \times R \times T}{100} \]
Where:
- \( P = 3500 \)
- \( R = 4 \)
- \( T = 1 \)

\[ \text{Interest} = \frac{3500 \times 4 \times 1}{100} = \frac{14000}{100} = 140 \]

### Step 3: Calculate the total interest from the first two investments
Now, we add the interest from both investments:

\[ \text{Total Interest from first two investments} = 200 + 140 = 340 \]

### Step 4: Determine the total interest required
The total income desired is Rs 500. Therefore, the interest required from the remaining amount is:

\[ \text{Interest from remaining amount} = 500 - 340 = 160 \]

### Step 5: Calculate the remaining amount to be invested
The total amount available for investment is Rs 10,000. The amount already invested is:

\[ \text{Total invested} = 4000 + 3500 = 7500 \]

Thus, the remaining amount is:

\[ \text{Remaining amount} = 10000 - 7500 = 2500 \]

### Step 6: Set up the equation for the remaining amount
Let \( x \) be the rate of interest for the remaining Rs 2,500. The interest from this amount can be expressed as:

\[ \text{Interest} = \frac{2500 \times x \times 1}{100} = \frac{2500x}{100} = 25x \]

### Step 7: Solve for \( x \)
We know that this interest must equal Rs 160:

\[ 25x = 160 \]

Dividing both sides by 25:

\[ x = \frac{160}{25} = 6.4 \]

### Conclusion
The rate of interest at which the remaining money should be invested is **6.4%**.

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