Suppose the parents of a child begin making quarterly payments of $1,000 into an account paying 7% compounded quarterly. Payments begin on the 10th birthday. How much will be available on the child's 15th birthday?

To solve this problem, we need to calculate the future value of an annuity where regular payments are made into an account with compound interest. Here are the steps:

### 1. Identify the parameters:
- **Quarterly payment** ($PMT$) = $1,000
- **Interest rate** ($r$) = 7% per year compounded quarterly
- **Number of payments per year** = 4 (quarterly)
- **Number of years** = 5 years (from the child's 10th to the 15th birthday)

### 2. Calculate the quarterly interest rate:
The annual interest rate is 7%, so the quarterly interest rate is:
\[
i = \frac{7\%}{4} = \frac{0.07}{4} = 0.0175
\]

### 3. Determine the number of periods:
Since payments are made quarterly over 5 years, the total number of periods ($n$) is:
\[
n = 4 \times 5 = 20 \text{ quarters}
\]

### 4. Use the future value of an annuity formula:
The future value $FV$ of an annuity can be calculated using the formula:
\[
FV = PMT \times \frac{(1 + i)^n - 1}{i}
\]
Substituting the values we have:
\[
FV = 1000 \times \frac{(1 + 0.0175)^{20} - 1}{0.0175}
\]

### 5. Calculate $FV$:
First, calculate the compound factor:
\[
(1 + 0.0175)^{20} \approx 1.40255
\]
Now, subtract 1 and divide by the interest rate:
\[
\frac{1.40255 - 1}{0.0175} \approx \frac{0.40255}{0.0175} \approx 23.003
\]
Finally, multiply by the payment amount:
\[
FV = 1000 \times 23.003 \approx 23003
\]

So, the amount available on the child's 15th birthday is approximately **$23,003**.

Would you like further details on any part of this calculation or have any questions? 

Here are five related questions you might find interesting:
1. How would the future value change if the payments were made monthly instead of quarterly?
2. What would be the future value if the interest rate were 8% instead of 7%?
3. How would the calculation differ if the compounding was done annually instead of quarterly?
4. What is the present value of these future payments at the start of the investment?
5. How much interest is earned during the 5-year period?

**Tip:** Always double-check the number of compounding periods when dealing with compound interest problems to ensure accuracy.

Learn how to calculate the future value of quarterly payments with compound interest. This problem involves determining how much will be available on a child's 15th birthday when parents start making $1,000 quarterly payments at a 7% interest rate compounded quarterly from the child's 10th birthday. Get a detailed step-by-step solution and explore related questions.

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