If `$300` is invested at `3%`, compounded annually, how log (to the nearest year) will it take for the money to increase by `50%` ?

Options

A. 11
B. 12
C. 13
D. 14

Explanation

To solve the problem of how long it will take for an investment of $300 at an interest rate of 3% compounded annually to increase by 50%, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Principal (P) = $300
- Rate (R) = 3% = 0.03 (as a decimal)
- Amount (A) after a 50% increase = P + 50% of P = P * 1.5 = $300 * 1.5 = $450

2. **Use the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P \left(1 + r\right)^t \]
where:
- A = final amount
- P = principal amount
- r = annual interest rate (in decimal)
- t = time in years

3. **Substitute the Known Values into the Formula**:
We substitute A, P, and r into the formula:
\[ 450 = 300 \left(1 + 0.03\right)^t \]
Simplifying this gives:
\[ 450 = 300 \left(1.03\right)^t \]

4. **Divide Both Sides by 300**:
\[ \frac{450}{300} = \left(1.03\right)^t \]
This simplifies to:
\[ 1.5 = \left(1.03\right)^t \]

5. **Take the Logarithm of Both Sides**:
To solve for t, we take the logarithm of both sides:
\[ \log(1.5) = \log\left((1.03)^t\right) \]
Using the property of logarithms, we can rewrite the right side:
\[ \log(1.5) = t \cdot \log(1.03) \]

6. **Solve for t**:
Rearranging gives:
\[ t = \frac{\log(1.5)}{\log(1.03)} \]

7. **Calculate the Value of t**:
Using a calculator:
- \(\log(1.5) \approx 0.1761\)
- \(\log(1.03) \approx 0.0128\)

Thus:
\[ t \approx \frac{0.1761}{0.0128} \approx 13.78 \]

Rounding to the nearest year:
\[ t \approx 14 \]

### Final Answer:
It will take approximately **14 years** for the investment to increase by 50%.