Method of Finding Common Denominator

Before we move on, we need to recognize that you can simply multiply the numbers to easily find a common denominator. However, this method does not always produce the least common denominator.

Consider the two fractions $3/8$ and $5/6$. You can compute the least common denominator by calculating the least common multiple LCM(8, 6), but if you only need a single common denominator, then you can simply multiply $8\times 6 = 48$. Multiply the fraction with the eight denominator by sixes (one on top, one on bottom), and multiply the fraction with the six denominator by eights (one on top, one on bottom) as follows:

$\frac{3}{8} = \frac{6\times 3}{6\times 8} = \frac{18}{48} \text{ and } \frac{5}{6} = \frac{8\times 5}{8\times 6} = \frac{40}{48}$

Now, we have common denominators! You can do this for any fractions $a/b$ and $c/d$$:

$\frac{a}{b} = \frac{d\times a}{d\times b} \text{ and } \frac{c}{d} = \frac{b\times c}{b\times d}$