John and Rosamond want to retire in 15 years and can save $140 every three months. They plan to deposit the money at the end of each quarter into an account paying 7.72% compounded quarterly. How much will they have at the end of 15 years? (Round your answer to the nearest cent.)

This is a compound interest problem involving regular quarterly deposits. Since they are making regular deposits at the end of each quarter, we can use the **future value of an ordinary annuity formula**:

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

where:
- $ P $ is the deposit amount per period,
- $ i $ is the interest rate per period,
- $ n $ is the total number of periods,
- $ FV $ is the future value of the annuity.

### Step 1: Identify the Variables
- **Deposit per quarter, $ P $**: $140
- **Annual interest rate**: 7.72%, compounded quarterly
- **Time, $ t $**: 15 years

### Step 2: Convert Annual Interest Rate to Quarterly Interest Rate
Since interest is compounded quarterly, divide the annual rate by 4 to find the quarterly interest rate.

\[
i = \frac{7.72\%}{4} = 0.0193
\]

### Step 3: Calculate the Total Number of Periods
Since deposits are made quarterly for 15 years, the total number of periods is:

\[
n = 15 \times 4 = 60
\]

### Step 4: Substitute Values into the Formula
Now we can substitute $ P = 140 $, $ i = 0.0193 $, and $ n = 60 $ into the formula.

\[
FV = 140 \cdot \frac{(1 + 0.0193)^{60} - 1}{0.0193}
\]

### Step 5: Calculate the Future Value
1. Calculate $ (1 + 0.0193)^{60} $.
2. Subtract 1 from the result.
3. Divide by 0.0193.
4. Multiply by 140.

After performing these calculations:

\[
FV \approx 14,947.99
\]

### Final Answer
At the end of 15 years, John and Rosamond will have approximately **$14,947.99**.

Would you like more details or have any questions about the calculations?

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### Further Related Questions:
1. How would the future value change if the interest rate was higher or lower?
2. What would happen if they saved monthly instead of quarterly?
3. How much would they have if they planned to retire in 20 years instead of 15?
4. What if they made deposits at the beginning of each quarter instead?
5. How much interest will they have earned by the end of 15 years?

### Tip:
For annuity problems, always check if payments are made at the beginning or end of each period, as it changes the formula used.

Calculate the future value of quarterly savings for retirement over 15 years with 7.72% interest compounded quarterly. This problem explores key finance concepts, including compound interest and the future value of an annuity. Suitable for high school students (Grades 10-12).

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