The total amount Edgar paid for a slice of pizza and a tip of exactly $24\%$ was between $\$2.50$ and $\$3.00$. What was the price of the pizza slice? 
The total amount Edgar paid for a slice of pizza and a tip of exactly 24% was between $\$2.50$ and $\$3.00$. What was the price of the pizza slice?   

Well, I did the trial and error method and managed to get $\$2.25$, but I feel like there'd be a more mathematical way to solve this. For example, if the price was $x$, then I could try solving it as $2.5 < 1.24 x < 3$, but that doesn't necessarily make the problem easier, I think. And above all, it may be easier if I could use a calculator for this question, but I'm not allowed to, so I'd like to find a way to solve this without a calculator...
Uh, just to avoid confusion, there wasn't any limit, but how I understand the problem is that 1.24x (the x here is the same variable I used above)should terminate right at the hundredth digit...?
 A: Well, note that $2.5=\frac{5}{2}$, and that $1.24=\frac{31}{25}$. 
Dividing the two gives us that $x>\frac{125}{62}$, and in the same method we earn $x<\frac{75}{31}$. 
The decimal expression for each is $2.01612\dots$, and $2.419354 \dots $. 
If you could only pay the original money with cents, the values would be $2.02$,$2.03$, $\dots$ $2.41$ dollars. 
If you could pay with dimes, the values would be $2.1$, $2.2$, $2.3$, $2.4$ dollars. 
If you could pay with quarters, the only value would be $2.25$ dollars. 
But in order for the tip $0.24x=\frac{6x}{25}$ to be an integer, then $x$ needs to be divisible by $25$, so you have to pay with quarters.  
A: 
The total amount Edgar paid for a slice of pizza and a tip of exactly 24% was between $\$2.50$ and $\$3.00$. What was the price of the pizza slice?

This models as
\begin{align}
2.5 \le & \,\, p + 0.24 \, p \le 3 \iff \\
2.5 \le & \,\, 1.24 \,p \le 3 \iff \\
2.01 \le \frac{2.5}{1.24} \le & \,\, p \le \frac{3}{1.24} \le 2.42
\end{align}
where $p$ is the price of the slice of pizza. Used was that one can divide both sides of an inequality with a positive number, which does not change the ordering, the less than operator stays.

And above all, it may be easier if I could use a calculator for this
  question, but I'm not allowed to, so I'd like to find a way to solve
  this without a calculator...

The only calculations are the divisions.
We can write
$$
\frac{2.5}{1.24} = \frac{250}{124} = \frac{125}{62}
$$
to reduce the fractional numbers to a plain fraction of integers.
Then the usual division algorithms for manual calculation apply, e.g after Adam Riese.
Similar
$$
\frac{3}{1.24} = \frac{300}{124} = \frac{150}{62} 
= \frac{75}{31}
$$
