George invests $1,000 into an account that he hopes will earh 12% interest annually. How many years (rounded to the nearest year) will it take his investment to double in value?

Options

A. 4
B. 6
C. 7
D. 8

Explanation

To determine how many years it will take for George's investment of $1,000 to double at an annual interest rate of 12%, we can use the formula for compound interest.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Principal (P) = $1,000
- Final Amount (A) = $2,000 (since we want to double the investment)
- Rate (r) = 12% = 0.12 (as a decimal)
- Time (t) = ? (in years)

2. **Use the Compound Interest Formula**:
The formula for compound interest is:
\[ A = P(1 + r)^t \]
Plugging in the values we have:
\[ 2000 = 1000(1 + 0.12)^t \]

3. **Simplify the Equation**:
Divide both sides by 1000:
\[ 2 = (1.12)^t \]

4. **Take the Logarithm of Both Sides**:
To solve for \( t \), we take the logarithm of both sides:
\[ \log(2) = \log((1.12)^t) \]
Using the property of logarithms \( \log(a^b) = b \cdot \log(a) \):
\[ \log(2) = t \cdot \log(1.12) \]

5. **Solve for t**:
Rearranging gives:
\[ t = \frac{\log(2)}{\log(1.12)} \]

6. **Calculate the Logarithms**:
Using a calculator:
- \( \log(2) \approx 0.3010 \)
- \( \log(1.12) \approx 0.0492 \)

Now substitute these values into the equation:
\[ t = \frac{0.3010}{0.0492} \approx 6.12 \]

7. **Round to the Nearest Year**:
Rounding 6.12 to the nearest whole number gives us:
\[ t \approx 6 \text{ years} \]

### Final Answer:
It will take approximately **6 years** for George's investment to double at an annual interest rate of 12%.

---