Finding a Common Denominator
There are two ways to find a common denominator. One is simpler, while the other gives a better result.
Suppose the two fractions we want to compare are a/b and x/y, where b and y are different. One way to get a common denominator is to simply multiply both the top and bottom of the first fraction by y, and multiply both the top and bottom of the second fraction by b. This works because y/y = 1, and multiplying by 1 does not change the number.
This gives a/b * y/y = ay/by, and x/y * b/b = bx/by. We now have a common denominator! Going back to our pizza example, this would be like cutting each half of the first pizza into eight more slices, and cutting each slice of the second pizza in half; in each case, the pizza ends up with sixteen slices.
Of course, you generally don't want to have your pizza cut into 16 slices; they'll be awfully small! Generally, we will actually want to find the lowest common denominator. We can do this by using a factor tree to find the prime factors of each number, then multiply together all of the unique prime factors. In the example above, 2 = 2 and 8 = 2 * 2 * 2; the only prime factor is 2 and the most that is appears is three times, so the lowest common denominator is 23, or 8. Multiplying the first number by 4/4 and the second by 1/1 puts both numbers in terms of eights, so we can compare them.