How much would you need to deposit in an account now in order to have \( \$ 4000 \) in the account in 10 years? Assume the account earns \( 8 \% \) interest compounded monthly.

To find out how much you need to deposit now in order to have \( \$4000 \) in 10 years with an interest rate of \( 8\% \) compounded monthly, we can use the formula for the present value of a future sum: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( PV \) is the present value (the amount to deposit now), - \( FV \) is the future value (the amount desired in the future, which is \( \$4000 \)), - \( r \) is the annual interest rate (as a decimal, so \( 0.08 \)), - \( n \) is the number of times that interest is compounded per year (monthly, so \( n = 12 \)), - \( t \) is the number of years the money is invested (which is \( 10 \) years). Now, we can plug in the values: - \( FV = 4000 \) - \( r = 0.08 \) - \( n = 12 \) - \( t = 10 \) First, calculate \( r/n \): \[ r/n = \frac{0.08}{12} = \frac{0.08}{12} \approx 0.00666667 \] Next, calculate \( nt \): \[ nt = 12 \times 10 = 120 \] Now we can substitute these values into the present value formula: \[ PV = \frac{4000}{(1 + 0.00666667)^{120}} \] Calculating \( (1 + 0.00666667)^{120} \): \[ (1 + 0.00666667)^{120} \approx (1.00666667)^{120} \approx 2.21964 \] Now substitute this back into the present value formula: \[ PV = \frac{4000}{2.21964} \approx 1801.44 \] Thus, the amount you need to deposit now is approximately: \[ \boxed{1801.44} \]