f = (a*((1+r)^n-1))/r
f is the future value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods
f is the future value
a is the annuity
r is the interest rate per time period
n is the number of time periods
in your problem:
a = 572.62
r = 3% / 4 = .75% / 100 = .0075 per quarter
n = number of quarters
to graph this formula, let y = f and let x = n
the formula becomes:
y = (a*((1+r)^x-1))/r
when a = 572.62 and r = .0075, the formula becomes:
y = (572.62 * ((1 + .0075) ^ x - 1)) / .0075
that's the formula that you would graph for the annuity formula.
the simple interest formula is f = p * (1 + r * n)
to graph that formula, replace f with y and n with x.
the interest rate is 5% per year / 100 .05 per year / 4 = .0125 per quarter.
the formula becomes f = 6900 * (1 + .0125 * x)
you graph both formulas and find the intersection point of both.
that's when the future value is the same.
the graph is shown below:
that graph tells you that x = 13.428.
this is the number of quarters required for the future value of both formulas to be the same.
that graph tells you that y = 8058.201
that's the future value of both formulas at the intersection point.
the actual number of quarters is 13.42842181 that is not shown because this graph roundes the answer to 3 decimal places.
in order to get the more axact number, i used a financial calculator with the following inputs.
present value = 0
future value = 8058.201
interest rate = 3/4 = .75%
quarterly payments are 572.62.
payments are made at the end of each quarter.
i then had the calculator get me the number of quarters which are what is shown above.
the compounding formula became:
y = (572.62 * ((1 + .0075) ^ 13.42842181 - 1)) / .0075 = 8058.200997 which rounds to 8058.201.
the simple interest formula became:
y = 6900 * (1 + .0125 * 13.42842181) = 8058.201381 which rounds to 8058.201.