You are on vacation in New York City. You didn't bring your car, and it's currently around $-50^\circ C$, so it's probably a good idea to take the NYC metro subway to move around.
You need a metro card to ride the subway, and after paying an initial fee of $\$1.00$ for the card, you can add balance to it.
Reading up on the system, it's kinda dumb. http://web.mta.info/nyct/fare/FaresatAGlance.htm
Here are the important parts:
- Each ride is $\$2.75$.
- Your deposit to the card must be a multiple of $5$ cents (e.g. $\$101.15$ is allowed, but not $\$200.17$).
- If your deposit is $\$5.50$ or greater, you get a $5\%$ bonus to the deposited amount. The bonus is rounded to the nearest integer number of cents.
It's dumb because you're going to waste money. For example, let's say you want to ride the metro $3$ times. When you deposit $\$8.25$, we get a $5\%$ bonus of $\$8.66$. After those three rides, we end up with $\$0.41$.
Now imagine how painful it would be to use all of that! It's nearly impossible to have an empty account ever again with such an ugly number. So, you're wasting money! Unacceptable!
You know you're going to use the metro more than once. And we need to be frugal, so you want to deposit the right amount so that after an integer number of rides, your account balance will be $\$0.00$.
Assuming that you are only going to make a single deposit, how much money should you deposit into your metro card? Perhaps more objectively, what is the least amount of money that you can deposit onto your metro card to satisfy the above conditions?
(I like this problem because it is a very real world application of seemingly useless math!)