Question

How can someone compare two denominators if they are not the same

Answer 1

As you might know, fractions don't have an unique way to be expressed. For example, you can write 1/2 as

$\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{10}{20} = \dfrac{50}{100} = \ldots$

How do you generate an equivalent fraction? You multiply numerator and denominator by the same number! For example, 1/2 and 2/4 are the same fraction, because you multiplied both numerator and denominator by 2. Or, 1/2=50/100 because you multiplied them by 50.

So now, suppose you have two generic fractions, a/b and c/d. You want to compare them, for example you want to know which one is bigger, or you want to sum them. You can manipulate both fractions so that they have the same denominator:

$\dfrac{a}{b} = \dfrac{a\cdot d}{b\cdot d} = \dfrac{ad}{bd}$

$\dfrac{c}{d} = \dfrac{c\cdot b}{d\cdot b} = \dfrac{bc}{bd}$

So now the two fractions have the same denominator.

Note that this is an approach that will always work, but may not be optimal. Take for example 5/6 and 7/9. You may multiply the first fraction by 9/9 and the second by 6/6:

$\dfrac{5}{6} = \dfrac{45}{54},\quad\dfrac{7}{9} = \dfrac{42}{54}$

But it is enought to multiply the first by 3/3 and the second by 2/2:

$\dfrac{5}{6} = \dfrac{15}{18},\quad\dfrac{7}{9} = \dfrac{14}{18}$

In other words, you have to look for the least common multiple of the two denominators, and multiply both fractions accordingly.