Final answer:
To calculate 10% of the monthly pay that was invested quarterly over 8 years to accumulate $7500 at a quarterly compound interest rate of 2.5%, we reverse-engineer the compound interest formula to find the total amount invested and then divide by the number of contributions. The result is approximately $192.26.
Step-by-step explanation:
The student is asking about the application of the compound interest formula to determine the initial investment required to accumulate a certain amount in a mutual fund over a period of time. Given that the final amount is $7,500, compounded quarterly at a rate of 2.5% over 8 years, we need to first calculate how much was invested each quarter.
The formula for compound interest is A = P(1 + r/n)^(nt), where:
- A is the future value of the investment/loan, including interest.
- P is the principal investment amount (the initial deposit or loan amount).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested or borrowed for, in years.
To find the principal P:
P = A / (1 + r/n)^(nt)
By substituting the values:
P = $7,500 / (1 + 0.025/4)^(4*8)
P = $7,500 / (1 + 0.00625)^(32)
P = $7,500 / (1.00625)^32
P = $7,500 / 1.219006942
P ≈ $6,152.20
Since the investments are made quarterly, this total principal is spread out over 8 years with four contributions per year, meaning there are a total of 32 contributions. To find the amount of each contribution (which is 10% of the monthly pay), we divide the total principal by the number of contributions.
Monthly pay contribution = $6,152.20 / 32
Monthly pay contribution ≈ $192.26
Therefore, 10% of the monthly pay is approximately $192.26.