When you are comparing two fractions why does the denominator have to be the same?

Answer 1

Strictly speaking, to compare (meaning, in your mind's eye to size up next to each other) two fractions, they aren't "required" to have (don't say: "be") the same denominator. (That's the bottom number of a fraction.)

It simply makes things a whole lot easier for the human mind to grasp, unless, of course, you're very blessed to be a mathematical genius.

For instance, most all of us can visualize a few simple fractions such as 1/2 , 1/3, 1/5, 3/4 etc. as parts left of a whole pie without any trouble and imagine their comparative sizes side by side, as we were shown to do as school kids.

I don't think many adults have difficulty seeing in their mind's eye that 1/2 is more (pie, glassful or whatever) than 1/3 or, similarly, that 3/4 is more (of whatever stuff) than 1/2.

But it gets considerably trickier to visualize and mentally grasp the difference in size or amount between, say, 2/3 and 3/5.

In such a case, converting the way we write two fractions so that their denominators are the same, or are 'common,' immediately makes visualizing their relative sizes easy.

To refresh the memory about how to do this:

1.) Multiply both fractions' denominators together. In our case: 5 x 3 = 15.

2.) Multiply each numerator (top number) by the other fraction's denominator.

So, the 3 of 3/5 becomes 3 x 3 = 9.

In the same way, the 2 of 2/3 becomes 2 x 5 =10.

3.) Now, underline each of the new numerators and write the new common denominator below, in our case 15. ('common' as in 'to have in common', not as in 'ordinary' or 'plain.')

4.) Our two new fractions are 9/15 and 10/15.

These two fractions are exactly the same size, amounts or numbers as they were before; they're simply written in a different way, but...

Suddenly when we look at 9/15 and 10/15 side by side, it's obvious that ten 15ths of anything is one 15th more (of whatever) than nine 15ths of it.

5.) Now choose your slice of pie. Want the 3/5 piece or the 2/3 piece?