Answer
See the sample answer below.
Work Step by Step
Monthly compounded interest means that an account balance is re-fed with interest earned 12 times a year, and that increased balance will now earn interest next month, and so on. So, we have a fraction of the $ 5\%$ interest added to the balance, increase it, and add $ 5/12\%$ interest next month.
Continuous compounding takes this to the next level. Not only do we increase the balance by a fraction 12 times a year, we do it every moment in time, every second, every tenth of a second, every 100th, etc. It turns out that this is an infinite process. The number $e $ is the value that a sequence $(1+\displaystyle \frac{1}{n})^{n}$ approaches, when n gets very large, which justifies the use of $e$ in continuous compounding.
During one month, if compounding monthly, the balance remains the same until the end of the month, when monthly compounding increases it by $(5/12)\%$.
If compounding is continuous, the balance constantly changes, increasing by a small fraction day to day, hour to hour, second to second, causing the balance at the end of the month to be slightly higher than after the monthly compounding.
