Question

in order to accumulate enough money for a down payment on a house, a couple deposits $349 per month into an account paying 3% compounded monthly. (a) if payments are made at the end of each period, how much money will be in the account in 6 years? (b) how much interest will be earned during the 6 years? (c) prepare a table showing the balance and interest for the first three months. (a) type the amount in the account. $ (round to the nearest dollar as needed.) (b) how much interest will be earned during the 6 years? $ (round to the nearest dollar as needed.)

Explanation:

Part (a)

We use the future value of an ordinary annuity formula: \( FV = P \times \frac{(1 + r)^n - 1}{r} \), where \( P \) is the payment per period, \( r \) is the interest rate per period, and \( n \) is the number of periods.

Step 1: Determine the values of \( P \), \( r \), and \( n \)

• \( P = \$349 \) (monthly payment)
• Annual interest rate \( = 3\% = 0.03 \), so monthly interest rate \( r = \frac{0.03}{12} = 0.0025 \)
• Number of years \( = 6 \), so number of months \( n = 6 \times 12 = 72 \)

Step 2: Plug the values into the formula

\[
FV = 349 \times \frac{(1 + 0.0025)^{72} - 1}{0.0025}
\]

First, calculate \( (1 + 0.0025)^{72} \approx 1.19668 \)

Then, \( (1.19668 - 1) = 0.19668 \)

Next, \( \frac{0.19668}{0.0025} = 78.672 \)

Finally, \( FV = 349 \times 78.672 \approx 27456 \)

Part (b)

The total amount deposited is \( P \times n \). The interest earned is the future value minus the total deposits.

Step 1: Calculate total deposits

Total deposits \( = 349 \times 72 = 25128 \)

Step 2: Calculate interest earned

Interest \( = FV - \text{Total Deposits} = 27456 - 25128 = 2328 \)

Part (c)

We calculate the balance and interest for each month. For month 1, there's no previous balance, so interest is 0, and balance is the payment. For subsequent months, interest is previous balance times \( r \), and new balance is previous balance + payment + interest.

Month	Payment (\$)	Previous Balance (\$)	Interest (\$)	New Balance (\$)
2	349	349	\( 349 \times 0.0025 = 0.87 \)	\( 349 + 349 + 0.87 = 698.87 \)
3	349	698.87	\( 698.87 \times 0.0025 \approx 1.75 \)	\( 698.87 + 349 + 1.75 = 1049.62 \)

Answer:

s:
(a) \(\$27456\)
(b) \(\$2328\)
(c) The table is as shown above.