Question

2. A couple wants to save money for their child's college education. They deposit their money in the bank at 6% interest. How many thousand dollars should be deposited in the bank annually from the age of 2 to the age of 17 so that a child from 18 to 21 years old can receive $ 501072 worth of money each year? (Interest will continue to be applied to the money other than the amount withdrawn from the account.)

Answer 1

Let say money depoisted from age 2 to 17 is $X every year

I will calculate the present value of money saved every year as well as money withdrawed every year and put the equal.

Present value of money saved when the child is 2 is [X / (1 + 0.06)^⁰]

Present value of money saved when the child is 3 is [X / (1 + 0.06)^¹]

Present value of money saved when the child is 4 is [X / (1 + 0.06)^²]

......

Present value of money saved when the child is 17 is [X / (1 + 0.06)^¹⁵]

Sum of present value of money = [X / (1 + 0.06)^⁰] + [X / (1 + 0.06)^¹] + [X / (1 + 0.06)^²] + ............ + [X / (1 + 0.06)^¹⁵] whose sum can be calculated as [a * (1 - r^ⁿ] / (1 - r)]

where a = [X / (1 + 0.06)^⁰]

r (ratio of two consecutive terms) = 0.9434

Sum of series = [X / (1 + 0.06)^⁰] * (1 - 0.9434^¹⁶) / (1 - 0.9434) = 10.71X

Year	Amount Withdrawed	Present value of amount withdrawed in year 2
18	501,072	197,245.13
19	501,072	186,080.31
20	501,072	175,547.46
21	501,072	165,610.82
		724,483.72

Present value of money deposited and withdrawed must be equal to each other

10.71X = 724,483.72

X = 67,631.34