Answer: Linda needs to pay approximately $85.51 into the annuity each quarter.
Explanation:
We can use the formula for the future value of an ordinary annuity:
FV = PMT × (((1 + r/n)^(n×t) - 1) / (r/n))
where:
FV = future value (which we know is $98,000)
PMT = payment made at the end of each period
r = annual interest rate (7.8% in this case)
n = number of compounding periods per year (4, since interest is compounded quarterly)
t = number of years (18 in this case)
Plugging in the numbers, we get:
98000 = PMT × (((1 + 0.078/4)^(4×18) - 1) / (0.078/4))
Simplifying the right-hand side:
98000 = PMT × ((1.0195)^72 - 1) / 0.0195
98000 × 0.0195 = PMT × ((1.0195)^72 - 1)
1908 = PMT × 22.322
PMT = 1908 / 22.322 ≈ $85.51
Therefore, Linda needs to pay approximately $85.51 into the annuity each quarter.
I hope this helps!