Final answer:
Katie will make approximately 4566 full payments of $440 each to save up $8100 for her tuition, and she will have a final smaller payment of -$2,004,340.
Step-by-step explanation:
To determine the number of full payments and the size of the final smaller payment, we can use the present value formula and the concept of a geometric series.
The present value (PV) is $8100 and the interest rate (j₁₂) is 8%. The quarterly payment is $440.
Using the formula for the present value of an annuity, we can set up the equation:
PV = Payment * [1 - (1 + j₁₂)⁻ᴺ]/j₁₂
Substituting the given values, we have:
$8100 = $440 * [1 - (1 + 0.08)⁻⁴ᴺ]/0.02
Simplifying the equation, we get:
[1 - (1.02)⁻⁴ᴺ] = 8100/440
[1 - (1.02)⁻⁴ᴺ] = 18.409
Now, we solve for N using logarithms:
N = log((1.02)⁻⁴⁵⁶⁶) / log(1.02)
N ≈ 4566.25
Therefore, Katie will make approximately 4566 full payments over the course of her tuition saving plan. To find the final smaller payment, we subtract the total amount paid in the full payments from the total tuition:
Final Payment = $8100 - ($440 * 4566)
Final Payment ≈ $8100 - $2,005,440
Final Payment ≈ -$2,004,340
Since the final smaller payment is negative, we can submit it as -$2,004,340.