n order to accumulate enough money for a down payment on a​ house, a couple deposits 

$ 715$715 

per month into an account paying 

3 %3% 

compounded monthly. If payments are made at the end of each​ period, how much money will be in the account in 

77 

​years?

Question content area bottom

Part 1

Type the amount in the​ account: 

​$enter your response here

​(Round to the nearest​ dollar.)

To solve this problem, we use the formula for the future value of an ordinary annuity, which applies when equal payments are made at the end of each period, and interest is compounded at regular intervals:

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

Where:
- $ FV $ is the future value of the account (the amount of money in the account after 7 years),
- $ P $ is the monthly payment (\$715),
- $ r $ is the annual interest rate (3% or 0.03),
- $ n $ is the number of times the interest is compounded per year (monthly compounding, so $ n = 12 $),
- $ t $ is the number of years the money is deposited (7 years).

### Step-by-Step Calculation:

1. **Monthly interest rate**:
   \[
   \frac{r}{n} = \frac{0.03}{12} = 0.0025
   \]

2. **Total number of periods**:
   \[
   nt = 12 \times 7 = 84 \, \text{months}
   \]

3. **Future Value (FV) formula**:
   \[
   FV = 715 \times \frac{(1 + 0.0025)^{84} - 1}{0.0025}
   \]

Now, let's compute:

\[
FV = 715 \times \frac{(1.0025)^{84} - 1}{0.0025}
\]
\[
FV = 715 \times \frac{1.221391662 - 1}{0.0025}
\]
\[
FV = 715 \times \frac{0.221391662}{0.0025}
\]
\[
FV = 715 \times 88.5566648
\]
\[
FV \approx 63,297
\]

### Final answer:
The amount of money in the account after 7 years is approximately **\$63,297**.

Would you like more details or have any questions?

Here are 5 questions to extend this topic:
1. How does increasing the monthly deposit affect the future value?
2. How would a higher interest rate (e.g., 4%) impact the total amount after 7 years?
3. What would happen if the couple deposits money for 10 years instead of 7?
4. How much would they have if the interest were compounded quarterly instead of monthly?
5. What is the impact of making payments at the beginning of each period instead of the end?

**Tip:** The more frequently interest is compounded, the more interest is earned, leading to a higher future value.

In order to accumulate enough money for a down payment on a house, a couple deposits $715 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 7 years?

Discover how to calculate the future value of monthly deposits of $715 at 3% compounded monthly over 7 years. This solution involves using the future value of an ordinary annuity formula and covers concepts such as compound interest and financial planning. Ideal for high school students in Grades 10-12.

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