Charles put $1,000 in a saving account that ears 2% compound interest each year. The account would earn approximately $126 in how many years ?

Options

A. 5
B. 6
C. 7
D. 8

Explanation

To solve the problem, we need to determine how many years it will take for Charles' investment of $1,000 to grow to approximately $1,126 with a compound interest rate of 2% per year.

### Step-by-Step Solution:

1. **Understand the Problem**:
- Charles invests $1,000.
- The interest rate is 2% compounded annually.
- He wants to find out how long it will take for his investment to grow to $1,126.

2. **Set Up the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P(1 + r)^t \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for in years.

In this case:
- \( A = 1126 \)
- \( P = 1000 \)
- \( r = 0.02 \)

3. **Plug in the Values**:
\[ 1126 = 1000(1 + 0.02)^t \]
Simplifying this gives:
\[ 1126 = 1000(1.02)^t \]

4. **Divide Both Sides by 1000**:
\[ \frac{1126}{1000} = (1.02)^t \]
This simplifies to:
\[ 1.126 = (1.02)^t \]

5. **Take the Logarithm of Both Sides**:
To solve for \( t \), take the logarithm of both sides:
\[ \log(1.126) = \log((1.02)^t) \]

6. **Use the Power Rule of Logarithms**:
By using the power rule of logarithms, we can bring down the exponent:
\[ \log(1.126) = t \cdot \log(1.02) \]

7. **Solve for \( t \)**:
Rearranging gives:
\[ t = \frac{\log(1.126)}{\log(1.02)} \]

8. **Calculate the Values**:
Using a calculator:
- \( \log(1.126) \approx 0.0513 \)
- \( \log(1.02) \approx 0.0086 \)

Thus:
\[ t \approx \frac{0.0513}{0.0086} \approx 5.97 \]

9. **Round the Result**:
Rounding 5.97 gives approximately 6 years.

### Conclusion:
Charles' investment will take approximately **6 years** to grow to $1,126 at a 2% compound interest rate.