How do I compare fractions when their denominators are different?
Quick answer
Rename both fractions as equivalent fractions with the same denominator. Then compare their numerators: the fraction with the larger numerator is larger. For example, rewrite 3/4 and 5/6 as 9/12 and 10/12. Since 9 is less than 10, 3/4 is less than 5/6. Check that each rewrite multiplied the numerator and denominator by the same number.
The question, the details, the next step.
What should I do first when the denominators are different? I want to compare 3/4 and 5/6, but 4 and 6 do not tell me which fraction is bigger.
Give both fractions a shared denominator, so you are comparing the same-sized parts of a whole. For 3/4 and 5/6, 12 works: it is divisible by both 4 and 6. Rename each fraction in twelfths:
- 3/4 = 9/12, because 4 × 3 = 12, so also multiply 3 × 3 = 9.
- 5/6 = 10/12, because 6 × 2 = 12, so also multiply 5 × 2 = 10.
Fractions with different denominators can be represented as equal-sized subdivisions of a whole; equivalent fractions sharing a denominator make their numerators comparable. [S1]
Once I have 9/12 and 10/12, how do I decide which original fraction is larger?
The denominators now match, so compare the numerators: 9 is less than 10. That means 9/12 is less than 10/12, and therefore 3/4 is less than 5/6. In the original fractions, the numbers 3 and 5 alone were not enough to compare them because the denominators described different-sized parts. After renaming, both fractions count twelfths, so comparing the counts answers the question. [S1]
How can I check that changing both fractions into twelfths did not change their values?
Check each conversion separately: did you multiply the numerator and denominator by the same nonzero number? For 3/4, both were multiplied by 3; for 5/6, both were multiplied by 2. Multiplying top and bottom by the same number creates an equivalent fraction, rather than changing the amount represented. You can also reverse the steps: 9/12 reduces to 3/4 by dividing both numbers by 3, and 10/12 reduces to 5/6 by dividing both by 2. [S1]
A quick written check is:
- 3/4 × 3/3 = 9/12
- 5/6 × 2/2 = 10/12
- 9 < 10, so 3/4 < 5/6
Do I always need to find the smallest shared denominator, and what can I do if I cannot see one right away?
No. Any denominator that is a multiple of both original denominators will work; the smallest shared one usually keeps the arithmetic simpler. If you cannot spot it, list a few multiples of each denominator until one appears in both lists. For 4 and 6, the lists begin 4, 8, 12 and 6, 12, so 12 is a convenient choice. You can then use the same-value check before comparing. This method shows the relationship between the fractions; it does not require guessing from the original numerators. [S1]
A practical checklist
Write down the two fractions and identify their denominators.
Choose a number divisible by both denominators.
For each fraction, multiply its numerator and denominator by the same factor to reach that denominator.
Check that each renamed fraction reduces back to the original.
Compare the numerators and state the comparison using the original fractions.
A few more useful answers
What if one denominator is already a multiple of the other?
Use the larger denominator as the shared denominator. For example, for 2/3 and 5/6, rewrite 2/3 as 4/6, then compare 4/6 with 5/6. [S1]
Can I compare fractions without making a shared denominator?
Sometimes a number line or a clear relationship makes the order easy to see. A shared denominator is a dependable method when that reasoning is not obvious; it lets you compare equal-sized parts. [S1]
What is the most common mistake in this method?
Changing the denominator without changing the numerator by the same factor. That does not make an equivalent fraction, so check both parts of every rewrite. [S1]
Sources
References used for this guide.
- [S1] 4.1 Visualize Fractions — Prealgebra 2e openstax.org · Checked