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.