Question

Why can you multiply both quantities in a ratio by the same number to find an equivalent ratio

Answer 1

Let's call the numbers x and y.
The ratio is x:y

Now, if we multiply them by the same number (let's call it z), we have :

zx:zy

And since they are both multiplied by z, we can simplify it and get x:y (basically reversing the step) and the ratio will stay the same

So no matter what number you multiply, as long as both of them are multiplied to the same number, you will be able to get an equivalent ratio. (Apologies if it doesn't make any sense..)