A person donated $160,000 to a university to establish a scholarship fund. What is unusual is that the entire amount came from what the person was able to save each month over
76 years while working at a low-wage job
(a) How much would the person have to put into a savings account at the end of every 3 months to accumulate $160,000 over 76 years? Assume an interest rate of 4.75%
compounded quarterly
(b) Answer part (a) using a 3% and a 8% interest rate.
(a) The necessary deposit is $4.75%
(Do not round until the final answer. Then round to the nearest cent as needed)
To calculate the necessary deposit at an interest rate of 4.75% compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = total amount accumulated ($160,000)
P = initial deposit
r = interest rate per period (4.75% or 0.0475)
n = number of compounding periods per year (4 for quarterly)
t = total number of years (76)
We are solving for P, so rearrange the formula to solve for P:
P = A / (1 + r/n)^(nt)
Substitute the values into the formula:
P = $160,000 / (1 + 0.0475/4)^(4*76)
P = $160,000 / (1 + 0.011875)^304
P = $160,000 / (1.011875)^304
P = $160,000 / 18.767348
P ≈ $8,522.40
So, the person would have to put $8,522.40 into a savings account at the end of every 3 months to accumulate $160,000 over 76 years with an interest rate of 4.75% compounded quarterly.