Rational and Irrational Numbers: Background You’ll Need 3

  • Determine which of two fractions is larger

Imagine you’re at a music festival with two stages. One stage is promising to play [latex]3[/latex] out of [latex]4[/latex] of your favorite songs, while the other stage promises to play [latex]5[/latex] out of [latex]6[/latex]. Which stage do you go to for the best experience? The answer lies in understanding and comparing fractions. Just like choosing the best stage can enhance your festival experience, knowing how to determine which fraction is larger can be a game-changer in both academics and everyday life.

Common Denominator

When comparing fractions, having a common denominator makes everything easier. A common denominator is a special number that both denominators in the fractions can divide into evenly, without leaving a remainder. Think of it as the universal translator in the world of fractions; it allows two fractions to be expressed in the same “language,” making them directly comparable.

common denominator

A common denominator is a number that all denominators in a set of fractions can divide into evenly.

To find a common denominator, you can either multiply the two denominators together or find the least common denominator (LCD). The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. 

least common denominator

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

To find the LCM of two numbers, you can use the following methods:

  • Listing Multiples: List the multiples of each number until you find the smallest multiple that appears in both lists.

    • For example, to find the LCM of [latex]4[/latex] and [latex]5[/latex], you’d list the multiples of [latex]4[/latex] [latex](4, 8, 12, 16, 20...)[/latex] and [latex]5[/latex] [latex](5, 10, 15, 20...)[/latex] and identify the smallest multiple they share, which is [latex]20[/latex].

  • Prime Factorization: Break each number down into its prime factors and multiply each factor the greatest number of times it occurs in either number.

    • For instance, the prime factors of [latex]12[/latex] are [latex]2 \times 2 \times 3[/latex] and of [latex]15[/latex] are [latex]3\times5[/latex]. The LCM is [latex]2 \times 2 \times 3 \times 5 = 60[/latex].


After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. To do this, you’ll need to multiply the numerator and denominator of each fraction by a specific factor so that the denominator matches the LCD. 

If you’re comparing [latex]\frac{3}{4}[/latex] and [latex]\frac{5}{6}[/latex] and the LCD is [latex]12[/latex], you’d multiply the numerator and denominator of [latex]\frac{3}{4}[/latex]by [latex]3[/latex] to get [latex]\frac{9}{12}[/latex],and [latex]\frac{5}{6}[/latex]by [latex]2[/latex] to get [latex]\frac{10}{12}[/latex]. Now, both fractions have the same denominator, making them directly comparable.

Once the denominators are the same, focus on the numerators. The fraction with the larger numerator will be the larger fraction.

In our example above, [latex]\frac{10}{12}[/latex] is larger than [latex]\frac{9}{12}[/latex] because [latex]10[/latex] is greater than [latex]9[/latex].