
Juan plans to set aside money for his young daughter’s college tuition. He will deposit money in an ordinary annuity that earns 3.6% interest, compounded quarterly. Deposits will be made at the end of each quarter.

How much money does he need to deposit into the annuity each quarter for the annuity to have a total value of $71,000 after 16 years?

To determine how much Juan needs to deposit into the annuity each quarter, we can use the **future value of an ordinary annuity formula**:

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

Where:
- $FV$ is the future value of the annuity ($71,000 in this case),
- $P$ is the deposit made at the end of each quarter (what we're solving for),
- $r$ is the interest rate per period (quarter),
- $n$ is the total number of periods (quarters).

### Step-by-Step Breakdown:

1. **Annual Interest Rate** ($i$) = 3.6% per year.
   - Since the interest is compounded quarterly, we divide this rate by 4 (the number of quarters in a year):
   \[
   r = \frac{3.6\%}{4} = \frac{0.036}{4} = 0.009
   \]
   So, the interest rate per period is 0.009.

2. **Number of Periods** ($n$) = 16 years × 4 quarters per year = 64 quarters.

Now, we plug the values into the formula for the future value of an annuity:

\[
71,000 = P \times \frac{(1 + 0.009)^{64} - 1}{0.009}
\]

We'll solve for $P$, the quarterly deposit amount.

### Let's calculate it:Juan needs to deposit approximately **$825.23** into the annuity at the end of each quarter to have a total of $71,000 after 16 years.

Do you need further clarification or any additional details on this calculation?

Here are some related questions:
1. How does the frequency of compounding affect the required deposit amount?
2. What would be the required deposit if the interest rate were 5% instead of 3.6%?
3. How much total interest will Juan have earned by the end of 16 years?
4. How would the deposit amount change if Juan started depositing 5 years later?
5. What is the difference between an ordinary annuity and an annuity due?

**Tip:** Increasing the compounding frequency (e.g., from yearly to quarterly) results in more interest being accumulated over time, which can reduce the required deposit amounts.

Juan plans to set aside money for his young daughter’s college tuition. He will deposit money in an ordinary annuity that earns 3.6% interest, compounded quarterly. Deposits will be made at the end of each quarter. How much money does he need to deposit into the annuity each quarter for the annuity to have a total value of $71,000 after 16 years?

Learn how to calculate the quarterly deposits Juan needs to make into an annuity to reach $71,000 in 16 years. This problem involves compound interest, annuities, and algebraic concepts suitable for Grades 10-12. Includes detailed formula breakdown and step-by-step solution.

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