A certain amount grows at an an nual interest rate of `12%`, compounded monthly. Which of the following equations can be solved to find the number of years, y. that it would take for the investment to increase by a factor of 64 ?

Options

A. `64 = (1.01)^(12y)`
B. `1/(64)=(1.04)12y`
C. `64=(1.04)12y`
D. `8=(1.01)6y`

Answer

To solve the problem, we need to find the equation that represents the growth of an investment at an annual interest rate of 12%, compounded monthly, and determine how long it will take for the investment to increase by a factor of 64.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the principal amount (initial investment) be \( P \).
- The annual interest rate is \( 12\% \), which can be expressed as a decimal: \( r = 0.12 \).
- Since the interest is compounded monthly, the monthly interest rate is \( \frac{12\%}{12} = 1\% = 0.01 \).
- The number of compounding periods per year is \( n = 12 \).
- We want the investment to grow by a factor of 64, so the final amount \( A \) will be \( 64P \).

2. **Use the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
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).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for in years.

3. **Substitute the Known Values**:
We substitute \( A = 64P \), \( r = 0.12 \), and \( n = 12 \) into the formula:
\[ 64P = P \left(1 + \frac{0.12}{12}\right)^{12y} \]
Simplifying this gives:
\[ 64 = \left(1 + 0.01\right)^{12y} \]
or
\[ 64 = (1.01)^{12y} \]

4. **Final Equation**:
The equation we can solve to find the number of years \( y \) is:
\[ 64 = (1.01)^{12y} \]

### Conclusion:
The equation that can be solved to find the number of years \( y \) that it would take for the investment to increase by a factor of 64 is:
\[ 64 = (1.01)^{12y} \]