Alexandra is saving money and plans on making quarterly contributions into an account earning an annual interest rate of 5.4% compounded quarterly. If Alexandra would like to end up with $75,000 after 14 years, how much does she need to contribute to the account every quarter, to the nearest dollar?

To determine how much Alexandra needs to contribute every quarter to reach her goal of $75,000 after 14 years with an annual interest rate of 5.4% compounded quarterly, we use the formula for the future value of an annuity due to periodic contributions:

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

where:
- $ FV $ is the future value of the annuity (in this case, $75,000),
- $ P $ is the amount of each periodic contribution,
- $ r $ is the interest rate per period,
- $ n $ is the total number of contributions.

First, we need to find $ r $ and $ n $:

1. **Quarterly interest rate $ r $:**

Since the annual interest rate is 5.4%, compounded quarterly, the quarterly interest rate is:
\[ r = \frac{5.4\%}{4} = \frac{0.054}{4} = 0.0135 \]

2. **Total number of contributions $ n $:**

Since contributions are made quarterly for 14 years:
\[ n = 14 \times 4 = 56 \]

Now we can plug these values into the annuity formula:

\[ 75000 = P \times \frac{(1 + 0.0135)^{56} - 1}{0.0135} \]

First, calculate $ (1 + 0.0135)^{56} $:

\[ (1 + 0.0135)^{56} \approx 2.102 \]

Then, calculate the numerator:

\[ 2.102 - 1 = 1.102 \]

So, the formula becomes:

\[ 75000 = P \times \frac{1.102}{0.0135} \]

Now, calculate the fraction:

\[ \frac{1.102}{0.0135} \approx 81.63 \]

Thus, we have:

\[ 75000 = P \times 81.63 \]

Finally, solve for $ P $:

\[ P = \frac{75000}{81.63} \approx 918.75 \]

Rounding to the nearest dollar, Alexandra needs to contribute approximately **$919** every quarter.

Would you like any details or have any questions about this solution?

Here are 5 questions you might be interested in:
1. How do you calculate the future value of an annuity with monthly contributions?
2. What is the difference between compounded quarterly and compounded annually?
3. Can you explain the formula for compound interest?
4. How does changing the interest rate affect the future value of an investment?
5. What are the advantages of making regular contributions to a savings account?

**Tip:** When dealing with compound interest, always ensure you convert the interest rate to match the compounding period for accurate calculations.

Learn how to calculate quarterly contributions needed to achieve a $75,000 savings goal in 14 years with a 5.4% annual interest rate compounded quarterly. Step-by-step solution using the future value of an annuity formula.

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