A man invests a part of 10,000 at 5% and the remainder at 6%. The 5% Investment yields annu ally 76.50 more than the 6% investment. The amount invested at 6% is

Options

A. 3600
B. 3550
C. 3850
D. 4000

Answer

To solve the problem step by step, we will first define the variables and then set up the equations based on the information given in the question.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( x \) be the amount invested at 5%. Therefore, the amount invested at 6% will be \( 10,000 - x \).

**Hint**: Start by defining what you know and what you need to find.

2. **Calculate Interest from Investments**:
The interest earned from the investment at 5% can be calculated as:
\[ \text{Interest at 5%} = \frac{5}{100} \times x = 0.05x \]
The interest earned from the investment at 6% is:
\[ \text{Interest at 6%} = \frac{6}{100} \times (10,000 - x) = 0.06(10,000 - x) \]

**Hint**: Remember that interest is calculated as the principal amount multiplied by the rate of interest.

3. **Set Up the Equation**:
According to the problem, the interest from the 5% investment is $76.50 more than the interest from the 6% investment:
\[ 0.05x = 0.06(10,000 - x) + 76.50 \]

**Hint**: Translate the word problem into a mathematical equation.

4. **Expand and Rearrange the Equation**:
Expanding the right side gives:
\[ 0.05x = 600 - 0.06x + 76.50 \]
Combine like terms:
\[ 0.05x + 0.06x = 676.50 \]
This simplifies to:
\[ 0.11x = 676.50 \]

**Hint**: Make sure to combine like terms carefully.

5. **Solve for \( x \)**:
Now, divide both sides by 0.11 to find \( x \):
\[ x = \frac{676.50}{0.11} = 6150 \]

**Hint**: When solving for a variable, isolate it on one side of the equation.

6. **Calculate the Amount Invested at 6%**:
Now that we have \( x \), we can find the amount invested at 6%:
\[ \text{Amount at 6%} = 10,000 - x = 10,000 - 6150 = 3850 \]

**Hint**: Use the total amount to find the remainder after calculating one part.

### Final Answer:
The amount invested at 6% is **$3850**.