Imagine you had a cake you were going to be sharing at a party with a small group of - let's say - 5 friends. You want to give each friend an equal share, so you slice the cake up into 5 equal slices. In math, when we have one whole and we split the whole into a number of equal parts, we call each of those parts a fraction of the whole.
These fractions have names, too, and we call those names denominators. The denominator depends on the number of parts we split the whole up into. Here, we split the whole cake up into 5 equal parts, so we call those parts 5ths and put a 5 in the denominator (the bottom half) of the fraction. The top half, which we call the numerator, tells us how many of those parts we have.
Back to the cake. We start out with all 5 equal pieces, or [tex] \frac{5}{5} [/tex], of the cake. Each friend gets an equal piece - everyone's happy!
At, that's what you're thinking when your phone starts ringing. It's one of your friends, and she tells you she's going to be bringing four more friends! Okay, you think, this is a little much, but I can make it work! Your phone rings again. It's another friend, and now he wants to bring four more friends, too! What's worse, over the next hour, you get calls from your three other friends saying the exact same thing.
Holy crud, you think, I've gotta feed five times as many people now! You're a little flustered, sure, but you're always up for a good challenge. You pull out some paper and get to work.
You sketch out that cake of yours, sliced into 5 beautifully even 5ths. Somehow, this cake is going to have to feed 5 times as many mouths as it was before, which means that, in order to share the cake fairly, you'll need split it up into 5 x 5, or 25 equal pieces!
You want to be smart about this, so you think, alright, how do I use the slices I already have to divide this all up? Oh! What if I split each piece into 5 equal pieces?
Originally, you'd sliced your cake up so that you had 5 equal slices, or [tex] \frac{5}{5} [/tex] of the whole cake. Slicing each slice up into 5 equal, smaller slices gives 5 times as many slices, or 5 x 5 = 25 total, [tex] \frac{25}{25} [/tex] of the whole cake.
Ring ring! Ring ring! It's your friends! Or, two of them, at least. Something came up, and sadly, they won't be able to make it, and neither will their extra four friends! Out of respect, you decide to save their two big slices to give to them later.
Now you're left with 5 - 2 = 3 big slices, or [tex]\frac{3}{5} [/tex] of the whole cake. You've still split each of those 3 big ones up into 5 smaller slices, though, so how many of those smaller slices are you gonna be serving? Well, 3 groups of 5 slices is 3 x 5 = 15. Those 15 slices make up [tex] \frac{15}{25} [/tex] of the whole cake, since we had 25 pieces altogether.
Notice how we never changed the amount of cake we had; we only changed the number of pieces it was cut into. Mathematically, when we took those 5 big slices and cut them into 5 equal smaller slices, we did:
[tex] \frac{5\times5}{5\times5}= \frac{25}{25} [/tex]
and when we took away two of the bigger slices, leaving us with 3 slices, we just changed that original fraction to:
[tex]\frac{3\times5}{5\times5}= \frac{15}{25} [/tex]
I put a picture of that process of slicing up slices at the end, if you need some help visualizing what it looks like!
The important takeaway from this is that multiplying the denominator (the size of the parts) and the numerator (the number of parts) by the same number, we aren't changing the number itself! The bigger the denominator, the smaller the pieces; if you make your pieces two times smaller, but double the number of pieces, you end up with the same amount of stuff you started with!
