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This problem involves calculating the regular payment required to reach a future value for an ordinary annuity.

The formula for the future value of an ordinary annuity is:

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

Where:
- $ FV $ is the future value of the annuity (in this case, $12,000).
- $ P $ is the regular payment per period (which we need to find).
- $ r $ is the interest rate per period.
- $ n $ is the total number of periods.

### Given dataThis problem involves calculating the regular payment required to reach a future value for an ordinary annuity.

The formula for the future value of an ordinary annuity is:

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

Where:
- $ FV $ is the future value of the annuity (in this case, $12,000).
- $ P $ is the regular payment per period (which we need to find).
- $ r $ is the interest rate per period.
- $ n $ is the total number of periods.

### Given 
- Future Value $ FV = 12,000 $
- Annual interest rate = 5.4% = 0.054
- Compounded quarterly, so the quarterly interest rate $ r = \frac{0.054}{4} = 0.0135 $
- Time = 10 years, and since the payments are made quarterly, the number of periods $ n = 10 \times 4 = 40 $

Now, let’s solve for $ P $ using the formula.

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

Substituting the values:

\[
P = \frac{12,000 \times 0.0135}{(1 + 0.0135)^{40} - 1}
\]

Let me calculate this.The Morris family needs to pay approximately **$228.23** into the annuity each quarter in order for it to grow to a total value of $12,000 after 10 years.

Would you like further details or have any questions?

Here are five related questions to explore:

1. How would the required payment change if the interest rate were compounded monthly instead of quarterly?
2. What would happen if the interest rate were higher, say 6%, but the same payment period applied?
3. How would the final value change if the Morris family wanted to invest for 15 years instead of 10?
4. How much interest would the Morris family have earned by the end of 10 years?
5. How would the formula change for a continuous compounding interest scenario?

**Tip:** Always double-check whether payments are at the beginning or end of the period, as this determines whether to use an ordinary annuity or an annuity due formula.

The Morris family wants to save money to travel the world. They plan to invest in an ordinary annuity that earns 5.4% interest, compounded quarterly. Payments will be made at the end of each quarter. How much money do they need to pay into the annuity each quarter for the annuity to have a total value of $12,000 after 10 years? Do not round intermediate computations, and round your final answer to the nearest cent.

Learn how to calculate the required quarterly payment for an ordinary annuity to reach a future value of $12,000 in 10 years with a 5.4% interest rate compounded quarterly. This detailed step-by-step solution includes essential financial formulas and compound interest principles.

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