Getting common denominators to subtract (or add) two fractions.

We cannot subtract (or add) the fractions and until they both have the same (common) denominator. However, getting a common denominator of two fractions is more than just making the denominators the same; we need to make the denominators the same without changing the value of each fraction.

You probably recognize that and have a common denominator of 12.

You might also recognize a multiplier of 2 to get into twelfths, .

However, in creating the common denominators, we can’t just write the new fractions with a denominator of 12; in other words, we can’t just rewrite the fractions as and . (Notice that the denominators have changed but the numerators did not.) No, we must also change the numerators.

Here’s a visual description of why we must change the numerator and denominator together: 01

Break this interval up into six equivalent parts.

The half-way mark is and the full-way mark is .

01Notice36512 that is close to 1, which is the same as .

Let’s now break up the interval into twelve equivalent parts. 011210Notice1265 that is close to

Compare the two diagrams. In the upper diagram, is close to 1, but in the lower diagram, is close to .

The point is this, just rewriting the denominator as 12 — without changing the numerator — does not create an equivalent fraction.

Instead, you might see that and are both close to 1 and appear to occupy the same position on the interval between 0 and 1. The truth is, and are equivalent fractions and we can rewrite into by multiplying both the numerator and denominator by 2:

· =

The point is this, when getting common denominators, we can build up the fractions to have the same denominator as long as we multiply each fraction by 1.

And 1 can look like a lot of things: , , , , and so on.

Where is all of this leading? Understanding that we can build up fractions (to have common denominators) allows us to subtract (or add) fractions like – :

–

= · – ·

= –

=

This is the same as saying 10 twelfths – 3 twelfths = 7 twelfth.

Here it is visually: 011012Thea356147 difference difference of between and is