When would an ordinary annuity consisting of quarterly payments of $556.19 at 5% compounded quarterly be worth more than a principal of $7400 invested at 4% simple interest?
+130555 That last inequality that Melody had to deal with IS tricky......here's a graph of the solution using Desmos:
https://www.desmos.com/calculator/inmq4qnrny
(I used "x" for "n," but notice that Melody was almost spot on...the "true" answer is about 13.975 quarters, but that's close enough to 14 to make it insignificant.....)
+118725 I'll look at the annuity first. It is just the future value of an ordinary annuity.
$$\boxed{FV=R\times \right[\frac{(1+i)^n-1}{i}}\\\\
i=0.05/4=0.0125\\
i+1=1.0125\\
R=$556.19\\\\
FV=556.19\times \right[\frac{(1.0125)^n-1}{0.0125}$$
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Now I will look at the simple interest scenario.
interest = 1% per quarter
$$\\\boxed{FV=P(1+rt)} \qquad \mbox{In this interest t is number of quarters=n}\\\\
FV=7400(1+0.01n)\\$$
So we need to find the smallest n where
$$\\556.19\times \right[\frac{(1.0125)^n-1}{0.0125}\;>\; 7400(1+0.01n)\\\\
(1.0125)^n-1\;>\; \frac{0.0125\times 7400(1+0.01n)}{556.19}\\\\
(1.0125)^n-1\;>\; 0.16633998(1+0.01n)\\\\
(1.0125)^n-1\;>\; 0.16633998+ 0.0016633998\;n\\\\
(1.0125)^n-0.0016633998\;n\;>\; 1.16633998 \\\\$$
Now this looks pretty horrible so I just solved it by trial and error.
When n=10 LHS=1.11 So n is bigger than 10
When n=20 LHS=1.248 So n can be smaller than 20
When n=15 LHS=1.179 n can be smaller but its getting close
When n=13 LHS=1.154 So n must be bigger than 13
When n=14 LHS = 1.1666 this satisfies the condition.
So the smallest value of n that satisfies this condition is n=14
14/4=3.5years
the annuity will be bigger than the simple interest scenario in 3.5 years time. 
+130555 That last inequality that Melody had to deal with IS tricky......here's a graph of the solution using Desmos:
https://www.desmos.com/calculator/inmq4qnrny
(I used "x" for "n," but notice that Melody was almost spot on...the "true" answer is about 13.975 quarters, but that's close enough to 14 to make it insignificant.....)
+118725 Hi Chris,
The money is compounded quarterly. I have assumed that is is also paid quarterly. That is what my calculations are based on. So the exact answer is 3 and a half years. It is not true until that final quarter payment is made. (It should be a step graph)
Of course if the payments are made daily you are completely correct. 
