Question

how much difference is there between investing $5,000 at 4% simple interest for 5 years and investing that same amount at 4% compounded quarterly

Answer 1

To find the difference between the tow investments we are going to use tow formulas. Simple interest formula for our simple interest investment, and compound interest formula for our compound interest investment.
- Simple interest formula: $A=P(1+rt)$
  where
  $A$ is the final investment value
  $P$ is the initial investment 
  $r$ is the interest rate in decimal form 
  $t$ is the time in years
For our problem we know that $P=5000$, $r= \frac{4}{100} =0.04$, and $t=5$, so lets replace those values in our simple interest formula to find $A$:
$A=5000(1+(0.04)(5))$
$A=5000(1.2)$
$A=6000$

Now that we know the final investment value of our simple interest investment, lets use the compound interest formula to find the final investment value of the other one:
- Compound interest formula: $A=P(1+ \frac{r}{n})^{nt}$
  where
  $A$ is the final investment value
  $P$ is the initial investment 
  $r$ is the interest rate in decimal form
  $n$ is the number of times the interest is compounded per year
  $t$ is the time in years 
For our problem we know that $P=5000$, $r= \frac{4}{100} =0.04$, and $t=5$. Since we know that the interest is compounded quarterly (each 4 months), it mean that is compounded $\frac{12months}{4months} =3$ times per year, so $n=3$. Now that we have all the vales, lets replace them in our formula:
$A=5000(1+ \frac{0.04}{3} )^{(3)(5)}$
$A=5000(1+ \frac{0.04}{3} )^{15}$
$A=6098.95$

now that we know the final amounts of our investments, lets find how much difference is between them: 
$6098.95-6000=98.95$

We can conclude that the difference between invest $5000 in a compound interest investment vs a simple interest investment is $98.95.