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?

Question

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...?

Answer 1

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.

Answer 2

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 $$ \frac{3}{1.24} = \frac{300}{124} = \frac{150}{62} = \frac{75}{31} $$