Questions: If 1000 is invested in an account earning 3% compounded monthly, how long will it take the account to grow in value to 1500? Round to the nearest month. years, months

Solution Steps

To solve this problem, we need to use the formula for compound interest to find the time it takes for the investment to grow to a certain amount. The formula for compound interest is:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

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.

We need to solve for \( t \) when \( A = 1500 \), \( P = 1000 \), \( r = 0.03 \), and \( n = 12 \).

1. Use the compound interest formula to set up the equation.
2. Solve for \( t \) by isolating it on one side of the equation.
3. Use logarithms to solve for \( t \).
4. Convert the time in years to months and round to the nearest month.

We start with the compound interest formula:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

where:

• \( A = 1500 \) (the amount we want to reach),
• \( P = 1000 \) (the initial investment),
• \( r = 0.03 \) (the annual interest rate),
• \( n = 12 \) (the number of times interest is compounded per year).

Rearranging the formula to solve for \( t \):

\[ t = \frac{\log\left(\frac{A}{P}\right)}{n \cdot \log\left(1 + \frac{r}{n}\right)} \]

Substituting the known values:

\[ t = \frac{\log\left(\frac{1500}{1000}\right)}{12 \cdot \log\left(1 + \frac{0.03}{12}\right)} \]

After performing the calculations, we find:

\[ t \approx 13.5324 \text{ years} \]

To convert this to months:

\[ t_{\text{months}} = t \times 12 \approx 162.3888 \text{ months} \]

Rounding to the nearest month gives:

\[ t_{\text{months}} = 162 \text{ months} \]

To express \( t_{\text{months}} \) in terms of years and months:

\[ \text{Years} = \left\lfloor \frac{162}{12} \right\rfloor = 13 \] \[ \text{Months} = 162 \mod 12 = 6 \]

Final Answer