Reducing Fractions to the Least Common Denominator

Reducing two fractions to a common denominator $$ \frac{a}{b} \ , \ \frac{c}{d} $$ means rewriting them as two equivalent fractions that share the same denominator. $$ \frac{a \cdot k}{b \cdot k} \ , \ \frac{c \cdot j}{d \cdot j} \ \ where \ \ b \cdot k = d \cdot j $$

To reduce two fractions to a common denominator, I apply the invariant property of fractions to each fraction until they have the same denominator.

There are infinitely many possible ways to do this.

Example. These two fractions have different denominators $$ \frac{1}{2} $$ $$ \frac{1}{3} $$ I apply the invariant property to both fractions to obtain a common denominator. I multiply the numerator and denominator of the first fraction by 3. Then I multiply the numerator and denominator of the second fraction by 2. This gives two fractions equivalent to the originals with the same denominator (6). $$ \frac{1}{2} \sim \frac{1 \cdot \color{red}3}{2 \cdot \color{red}3} = \frac{3}{6} $$ $$ \frac{1}{3} \sim \frac{1 \cdot \color{red}2}{3 \cdot \color{red}2} = \frac{2}{6} $$ Of course, there are infinitely many other equivalent fractions with the same denominator. For example $$ \frac{1}{2} \sim \frac{3}{6} \sim \frac{6}{12} \sim \frac{12}{24} \ ... $$ $$ \frac{1}{3} \sim \frac{2}{6} \sim \frac{4}{12} \sim \frac{8}{24} \ ... $$

In practice, we usually choose the representation with the smallest possible common denominator.

In this case, we speak of reducing fractions to the least common denominator.

Example. The least common denominator of the two fractions above leads to $$ \frac{1}{2} \sim \frac{3}{6} $$ $$ \frac{1}{3} \sim \frac{2}{6} $$

To reduce two fractions to the least common denominator, it is enough to compute the least common multiple (LCM) of the denominators and then apply the invariant property.

A practical example

Consider the following two fractions with different denominators.

$$ \frac{1}{12} $$

$$ \frac{2}{10} $$

I compute the least common multiple (LCM) of the denominators.

$$ mcm(12,10) = 60 $$

Note. I factor both numbers into prime factors $$ 12 = 2^2 \cdot 3 $$ $$ 10 = 2 \cdot 5 $$ Then I multiply each prime factor once, taking the highest exponent $$ mcm(12,10) = 2^2 \cdot 3 \cdot 5 = 60 $$ This result is the least common multiple of the two numbers

I apply the invariant property to the first fraction to obtain an equivalent fraction with denominator equal to 60.

In this case, I multiply both the numerator and the denominator by 5.

$$ \frac{1}{12} \sim \frac{1 \cdot 5}{12 \cdot 5} = \frac{5}{60} $$

Note. To find the multiplication factor, divide the target denominator (60) by the current denominator (12) $$ 60:12 = 5 $$

I then apply the invariant property to the second fraction to obtain an equivalent fraction with denominator equal to 60.

In this case, I multiply both the numerator and the denominator by 6.

$$ \frac{2}{10} \sim \frac{2 \cdot 6}{10 \cdot 6} = \frac{12}{60} $$

Note. Again, to find the multiplication factor, divide the target denominator (60) by the current denominator (10) $$ 60:10 = 6 $$

Now the two fractions have the same denominator.

$$ \frac{1}{12} \sim \frac{5}{60} $$

$$ \frac{2}{10} \sim \frac{12}{60} $$

The two fractions are now expressed with the least common denominator.

Note. Sometimes, to speed up the process, the two denominators are multiplied together without computing the least common multiple. This is a common shortcut, but generally not advisable because, except in rare cases, it produces equivalent fractions with much larger numbers, making subsequent calculations more cumbersome. For example, consider again the fractions $$ \frac{1}{12} \ , \ \frac{2}{10} $$ If I multiply the denominators 12·10 without computing the LCM, I obtain a common denominator of 120 instead of 60 $$ \frac{1 \cdot 10}{12 \cdot 10} \ , \ \frac{2 \cdot 12}{10 \cdot 12} $$ $$ \frac{10}{120} \ , \ \frac{24}{120} $$ These fractions are equivalent to the originals, but involve larger numbers than those obtained using the least common multiple. Working with larger numbers makes calculations more difficult and increases the risk of errors. For this reason, it is good practice to compute the least common multiple whenever reducing fractions to a common denominator.

And so on.