Questions: In order to accumulate enough money for a down payment on a house, a couple deposits 444 per month into an account paying 3% compounded monthly. If payments are made at the end of each period, how much money will be in the account in 4 years? What is the amount in the account after 4 years? (Round to the nearest cent as needed.)

In order to accumulate enough money for a down payment on a house, a couple deposits 444 per month into an account paying 3% compounded monthly. If payments are made at the end of each period, how much money will be in the account in 4 years?

What is the amount in the account after 4 years?

(Round to the nearest cent as needed.)

Transcript text: In order to accumulate enough money for a down payment on a house, a couple deposits $\$ 444$ per month into an account paying $3 \%$ compounded monthly. If payments are made at the end of each period, how much money will be in the account in 4 years? What is the amount in the account after 4 years? \$ (Round to the nearest cent as needed.)

Solution

Solution Steps

To solve this problem, we need to calculate the future value of an annuity. The couple makes regular monthly deposits into an account with interest compounded monthly. We will use the future value of an annuity formula, which is:

\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]

where:

$ P $ is the monthly deposit amount ($444),
$ r $ is the monthly interest rate (annual rate divided by 12),
$ n $ is the total number of deposits (number of years times 12).

Solution Approach

Convert the annual interest rate to a monthly interest rate.
Calculate the total number of deposits over 4 years.
Use the future value of an annuity formula to find the total amount in the account after 4 years.

Step 1: Calculate Monthly Interest Rate

The annual interest rate is given as $ 3\% $, which can be expressed as a decimal: \[ \text{annual interest rate} = 0.03 \] To find the monthly interest rate, we divide by $ 12 $: \[ r = \frac{0.03}{12} = 0.0025 \]

Step 2: Calculate Total Number of Deposits

The couple makes monthly deposits for $ 4 $ years. The total number of deposits is: \[ n = 4 \times 12 = 48 \]

Step 3: Calculate Future Value of the Annuity

Using the future value of an annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Substituting the known values: \[ FV = 444 \times \frac{(1 + 0.0025)^{48} - 1}{0.0025} \] Calculating this gives: \[ FV \approx 22613.4565 \] Rounding to the nearest cent: \[ FV \approx 22613.46 \]