The information in the table shows how much a specific monthly contribution can result in after 3 years of investment with a 4.2% annual interest rate, compounded monthly. On average, the annual cost to attend a 2-year public institution in the 2020-2021 academic year is $3,900. If the goal is to cover the cost for two years at a 2-year public institution after 3 years of investment, what is the minimum amount a student should be saving each month?
To determine the minimum amount a student should be saving each month, we need to find the future value of the monthly contributions after 3 years and ensure it covers at least the cost of attending a 2-year public institution for two years.
Using the compound interest formula:
Future Value = P * (1 + r/n)^(nt)
Where:
P = monthly contribution
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, n is 12 since interest is compounded monthly, and t is 3 years. The annual interest rate is 4.2% or 0.042.
Let's calculate the future value of the monthly contributions after 3 years:
Future Value = P * (1 + 0.042/12)^(12*3)
Future Value = P * (1 + 0.0035)^(36)
Future Value = P * 1.137472
Now, we need this future value to at least cover the cost of attending a 2-year public institution for two years, which is $3,900 per year. Therefore, the minimum amount a student should be saving each month is:
Minimum Monthly Saving = (2 * $3,900) / Future Value
Minimum Monthly Saving = (2 * $3,900) / (P * 1.137472)
Simplifying further:
Minimum Monthly Saving = $7,800 / (P * 1.137472)
So, the student should be saving at least $7,800 divided by (P * 1.137472) each month to cover the cost for two years at a 2-year public institution after 3 years of investment.