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A elementary school student brought cupcakes to their class to celebrate their birthday! However, there's a problem: There are 25 students in the classroom, and only 24 cupcakes!

The students have recently been studying fractions. One of the students quickly puts forth a solution: "Everybody can have 24/25 of a cupcake! We can cut 1/25 off of every cupcake, combine those into one cupcake, and then we'll have 25." However, nobody seems willing to accept a "cupcake" made out of 24 cupcake crumbs.

It is up to you to figure out how to divide the cupcakes in a way that is both fair, and doesn't reduce the treats to pastry dust. What strategy can you use to ensure that the smallest slice you have to make is as large as possible?

This puzzle is purely mathematical: Every student must get exactly 24/25 of a cupcake, every cupcake is exactly the same size, the students will wait patiently while you measure and cut the cakes, etc..

This puzzle was inspired by a page from Math Curse, where the main character faces a similar situation. They resolve it by simply not eating a cupcake, but here you have no such option...

This is my first post on PuzzlingSE, so please tell me if I'm doing anything wrong!

My smallest piece is

$frac625$

You split the cakes as follows:

$3$ cakes as $frac625+frac625+frac625+frac725$
$13$ cakes as $frac725+frac925+frac925$
$8$ cakes as $frac825+frac825+frac925$

This gives $9$, $16$, $16$, and $34$ pieces of sizes $frac625$ to $frac925$.

Then put them together as

$9$ cakes of $frac625+frac925+frac925$
$16$ cakes of $frac725+frac825+frac925$

How about

- Cut 19 cupcakes into five fifths each. You have 95 fifths.

- Take one fifth out of each of the other 5 cupcakes. You have 100 fifths and 5 leftovers.

- Give each student 4 fifths.

- Cut the 5 leftovers each into 5 equal pieces. Give each student 1.

Each student got the same number of the same sized bits, so it must be fair.

The smallest piece is

The fifth of the leftover, which would be 4/25 (16%) of a cupcake.

Let's call a whole cupcake

100 unit and what we are aiming is 96 dividing these 100s into pieces

so the obvious answer from the question was

$4$

so I take

9 of $100$ unit into $5$ of 20 unit pieces which makes $45$ pieces of 20 units.

and secondly

15 of $100$ unit into $2$ of 32 unit pieces and $1$ of 36 unit pieces which makes $30$ of 32 unit pieces and $15$ of $36 unit pieces.

as a table

So from now on, I form our cakes as below:

We do this procedure

$4$ times more and we will not have any cake leftover.

so the answer becomes

$frac525$