Why do you have to multiply or divide the numerator and denominator in a fraction by the same number top and bottom?

We do this so that the ratio of the values will stay the same, even as we manipulate the numbers. For instance:

If we have the problem  · , we would need to do this:

First, we would need to find the same denominator for both fractions. We can do this by multiplying the denominators by 2. However, we need to keep the ratios the same, so we also multiply the numerators by 2.

This is simply done so that the ratio of values is the same. Does this make more sense now?