4.1 Homework

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Question 10
How much would you need to deposit in an account now in order to have $\$ 6000$ in the account in 10 years? Assume the account earns $2 \%$ interest compounded monthly.
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Step 1 ：The problem is asking for the present value of a future sum of money. This is a common problem in finance and can be solved using the formula for the present value of a future sum, which is: \(PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}\) where: PV is the present value, FV is the future value, r is the annual interest rate (in decimal form), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.

Step 2 ：In this case, we know that FV = $6000, r = 2% = 0.02, n = 12 (since interest is compounded monthly), and t = 10. We can plug these values into the formula to find the present value.

Step 3 ：Substituting the given values into the formula, we get: \(PV = \frac{6000}{(1 + \frac{0.02}{12})^{12*10}}\)

Step 4 ：Solving the above expression, we find that the present value is approximately $4913.20.

Step 5 ：Final Answer: You would need to deposit approximately \(\boxed{4913.20}\) in the account now in order to have $6000 in the account in 10 years, assuming the account earns 2% interest compounded monthly.