How long will it take to double your investment if you place $7500 in an account at 4.25% compounded semi-annually? Round to the nearest half year.



Answer :

To determine how long it will take to double your initial investment of [tex]$7500 at an annual interest rate of 4.25%, compounded semi-annually, we need to use the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after \( t \) years, including interest. - \( P \) is the principal amount (the initial sum of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times the interest is compounded per year. - \( t \) is the time the money is invested for in years. Here are the known values: - \( P = 7500 \) - \( r = 4.25 / 100 = 0.0425 \) - \( n = 2 \) (since the interest is compounded semi-annually) - \( A = 2P = 2 \times 7500 = 15000 \) (since we want to double the investment) We need to solve for \( t \). Rearranging the compound interest formula to solve for \( t \): \[ 2P = P \left(1 + \frac{r}{n}\right)^{nt} \] Dividing both sides by \( P \): \[ 2 = \left(1 + \frac{r}{n}\right)^{nt} \] Taking the natural logarithm (ln) of both sides: \[ \ln(2) = \ln\left[\left(1 + \frac{r}{n}\right)^{nt}\right] \] Using the logarithm power rule: \[ \ln(2) = nt \cdot \ln\left(1 + \frac{r}{n}\right) \] Solving for \( t \): \[ t = \frac{\ln(2)}{n \cdot \ln\left(1 + \frac{r}{n}\right)} \] Substitute the known values: \[ t = \frac{\ln(2)}{2 \cdot \ln\left(1 + \frac{0.0425}{2}\right)} \] Evaluating this expression yields: \[ t \approx 16.482024930376596 \] To round this to the nearest half year: \[ t \approx 16.5 \] Thus, it will take approximately 16.5 years to double your initial investment of $[/tex]7500 at an annual interest rate of 4.25%, compounded semi-annually.