A man lends 10,000 in four parts. If he gets 8% on `2000,7(1)/(2)%` on 4000 and `8(1)/(2)%` on 1400, what percent must he get for the remainder, if his average annual interest is 8.13% ?

Options

A. `7%`
B. `9%`
C. `9(1)/(4)%`
D. `10(1)/(2)%`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the total interest he should receive
The man lends a total of ₹10,000 and wants an average annual interest of 8.13%.

\[ \text{Total Interest} = \text{Principal} \times \text{Rate} = 10000 \times \frac{8.13}{100} = 813 \]

### Step 2: Calculate the interest from the first three parts
1. **Interest from ₹2000 at 8%:**
\[ \text{Interest} = 2000 \times \frac{8}{100} = 160 \]

2. **Interest from ₹4000 at 7.5% (7(1)/(2)%):**
\[ \text{Interest} = 4000 \times \frac{7.5}{100} = 300 \]

3. **Interest from ₹1400 at 8.5% (8(1)/(2)%):**
\[ \text{Interest} = 1400 \times \frac{8.5}{100} = 119 \]

### Step 3: Calculate the total interest from the first three parts
Now, we will sum the interests calculated above:

\[ \text{Total Interest from first three parts} = 160 + 300 + 119 = 579 \]

### Step 4: Calculate the remaining interest needed
To find out how much interest he still needs to earn to reach the total interest of ₹813:

\[ \text{Remaining Interest} = 813 - 579 = 234 \]

### Step 5: Calculate the remaining principal
The total principal lent is ₹10,000. The sum of the first three parts is:

\[ \text{Sum of first three parts} = 2000 + 4000 + 1400 = 7400 \]

Thus, the remaining principal is:

\[ \text{Remaining Principal} = 10000 - 7400 = 2600 \]

### Step 6: Calculate the required interest rate for the remaining principal
We need to find the interest rate \( r \) such that the interest from the remaining principal of ₹2600 gives us the remaining interest of ₹234.

Using the formula for interest:

\[ \text{Interest} = \text{Principal} \times \frac{Rate}{100} \]

We can rearrange it to find the rate:

\[ 234 = 2600 \times \frac{r}{100} \]

Solving for \( r \):

\[ r = \frac{234 \times 100}{2600} = \frac{23400}{2600} = 9 \]

### Final Answer
The percentage he must get for the remainder is **9%**.

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