Question 4 of 8 Step 1 of 1 Use the formula for present value of money to calculate the amount you need to invest now in one lump sum in order to have \( \$ 25,000 \) after 10years with an APR of \( 11 \% \) compounded quarterly. Round your answer to the nearest cent, if necessary.

To calculate the present value (PV) of an investment using the formula for present value with compound interest, we can use the following formula: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( PV \) = present value (the amount to invest now) - \( FV \) = future value (the amount you want in the future, which is \$25,000) - \( r \) = annual interest rate (as a decimal, so 11% = 0.11) - \( n \) = number of times the interest is compounded per year (quarterly means \( n = 4 \)) - \( t \) = number of years the money is invested (10 years) Now, we can plug in the values: - \( FV = 25000 \) - \( r = 0.11 \) - \( n = 4 \) - \( t = 10 \) Now, we can calculate the present value: \[ PV = \frac{25000}{(1 + \frac{0.11}{4})^{4 \times 10}} \] First, calculate \( \frac{0.11}{4} \): \[ \frac{0.11}{4} = 0.0275 \] Now, calculate \( 1 + 0.0275 \): \[ 1 + 0.0275 = 1.0275 \] Next, calculate \( 4 \times 10 = 40 \), and then raise \( 1.0275 \) to the power of 40: \[ (1.0275)^{40} \approx 3.030856 \] Now, substitute this back into the present value formula: \[ PV = \frac{25000}{3.030856} \approx 8255.66 \] Thus, the amount you need to invest now in one lump sum to have \$25,000 after 10 years at an APR of 11% compounded quarterly is approximately: \[ \boxed{8255.66} \]