4.7 Introduction and Key Insights

Now that you are comfortable reducing fractions into their lowest terms, we want to build some familiarity with rewriting fractions. Specifically, we want to rewrite two fractions so that they have the same denominator.

Consider the fractions $3/12$ and $4/12$. These two expressions have a common denominator (12). Now practice your fraction-reducing skills from the previous section to note that

$\frac{3}{12} = \frac{1}{4} \text{ and } \frac{4}{12} = \frac{1}{3}$

The fractions $1/3$ and $1/4$ can each be re-expressed so that they have a common denominator of 12. Something similar is also true for the fractions $20/60$ and $24/60$; they have a common denominator of $60$, and each of these fractions reduces to $1/3$ and $2/5$.

The process of finding a single common denominator for two fractions is in some sense the reverse of what we have just discussed: start with two different-denominator fractions and then rewrite them so they have the same denominator.

This video discusses how to find "the least common denominator". For our two previous examples, the least common denominator of $1/4$ and $1/3$ is 12, but the least common denominator of $1/3$ and $2/5$ is 15 (not 60, the larger denominator we used above). Indeed, instead of writing

$\frac{1}{3} = \frac{20}{60} \text{ and } \frac{2}{5} = \frac{24}{60}$.

We could have written:

$\frac{1}{3} = \frac{5}{15} \text{ and } \frac{2}{5} = \frac{6}{15}$.

To find the least common denominator, you will use the least common multiple of the denominators.