Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $25,000
 over the next 3
 years. What amount must be deposited quarterly into an account that earns 5.1%
 interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary.



To solve this problem, we need to determine the amount to deposit quarterly to accumulate $25,000 in 3 years at a 5.1% annual interest rate, compounded quarterly.

We can use the **future value of an ordinary annuity formula**, which is used for periodic deposits into an interest-bearing account:

\[
A = P \left( \frac{(1 + r)^n - 1}{r} \right)
\]

Where:
- $A$ is the future value (the amount you want to save, which is $25,000),
- $P$ is the periodic deposit amount (which we want to find),
- $r$ is the interest rate per period (quarterly),
- $n$ is the total number of periods.

### Given:
- The annual interest rate is 5.1%, so the quarterly interest rate is $ \frac{5.1\%}{4} = \frac{0.051}{4} = 0.01275 $.
- The number of periods over 3 years with quarterly deposits is $ 3 \times 4 = 12 $.
- The future value $A = 25,000$.

Now we can solve for $P$.

\[
25,000 = P \left( \frac{(1 + 0.01275)^{12} - 1}{0.01275} \right)
\]

Let me calculate the value of $P$.The amount that must be deposited quarterly is approximately **$1,941.25** in order to accumulate $25,000 over 3 years at a 5.1% interest rate, compounded quarterly.

Would you like more details on the steps? Here are 5 related questions to expand on this:

1. How would the deposit change if the interest rate were higher or lower?
2. What is the total interest earned after 3 years?
3. How does the compounding frequency affect the final amount?
4. Can this formula be applied for different time periods, such as monthly or yearly deposits?
5. How would the calculation differ if the future value goal was different?

**Tip:** Regular deposits and compound interest significantly accelerate savings, so starting early is beneficial!

Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $25,000 over the next 3 years. What amount must be deposited quarterly into an account that earns 5.1% interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary.

Learn how to calculate the quarterly deposit required to save $25,000 in 3 years with a 5.1% interest rate compounded quarterly. This problem uses the future value of an ordinary annuity formula and covers key concepts like compound interest and periodic deposits. Suitable for high school students in Grades 10-12.

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