<p> Getting common denominators to subtract (or add) two fractions.</p><p>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.</p><p>You probably recognize that and have a common denominator of 12. </p><p>You might also recognize a multiplier of 2 to get into twelfths, .</p><p>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.</p><p>Here’s a visual description of why we must change the numerator and denominator together: 01</p><p>Break this interval up into six equivalent parts.</p><p>The half-way mark is and the full-way mark is .</p><p>01Notice36512 that is close to 1, which is the same as .</p><p>Let’s now break up the interval into twelve equivalent parts. 011210Notice1265 that is close to </p><p>Compare the two diagrams. In the upper diagram, is close to 1, but in the lower diagram, is close to . </p><p>The point is this, just rewriting the denominator as 12 — without changing the numerator — does not create an equivalent fraction.</p><p>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:</p><p>· = </p><p>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.</p><p>And 1 can look like a lot of things: , , , , and so on.</p><p>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 – :</p><p>– </p><p>= · – · </p><p>= – </p><p>=</p><p>This is the same as saying 10 twelfths – 3 twelfths = 7 twelfth.</p><p>Here it is visually: 011012Thea356147 difference difference of between and is </p>