Number
Grade 2-4Simplifying & Ordering Fractions
Fractions questions are rarely just about the arithmetic, they are about presenting an answer properly. This lesson covers simplifying a fraction fully, converting between mixed numbers and improper fractions, and ordering fractions that have different denominators, all skills that quietly sit underneath almost every other fractions topic.
Written by Asad, Co-Founder of Teachably
Simplifying fractions
A fraction is fully simplified when the numerator and denominator share no common factor other than 1. To simplify, divide both numbers by their highest common factor.
It is fine to simplify in more than one step, dividing by a smaller common factor first and checking again, as long as you keep going until no common factor remains. 8/12 can be simplified by dividing by 4 to reach 2/3 directly, or by dividing by 2 twice to reach the same answer.
Equivalent fractions
Equivalent fractions represent the same value, found by multiplying or dividing both the numerator and denominator by the same number. 2/5 is equivalent to 8/20, because both parts were multiplied by 4.
This is the same idea used in reverse when simplifying, and it is essential for ordering fractions and adding or subtracting them later.
Mixed numbers and improper fractions
A mixed number combines a whole number and a fraction, like 3 1/4. An improper fraction has a numerator bigger than its denominator, like 13/4. They represent the same value in two different forms.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator. For 3 1/4: 3 × 4 = 12, then 12 + 1 = 13, giving 13/4.
Ordering fractions
Fractions can only be compared directly once they share the same denominator. Find a common denominator, usually the lowest common multiple of the denominators, convert every fraction to it, then compare the numerators.
The fractions are then put back in their original form for the final answer, the common denominator is just a tool for comparing them, not part of the answer itself.
Worked Examples
Three exam-style questions, fully solved.
Simplify 8/12 fully.
Easy- 1.Find the highest common factor of 8 and 12, which is 4
- 2.Divide both the numerator and denominator by 4: 8 ÷ 4 = 2, 12 ÷ 4 = 3
- 3.Check there is no common factor left between 2 and 3
Answer: 2/3
Write 3 1/4 as an improper fraction.
Medium- 1.Multiply the whole number by the denominator: 3 × 4 = 12
- 2.Add the numerator: 12 + 1 = 13
- 3.Keep the same denominator
Answer: 13/4
Order these fractions from largest to smallest: 5/6, 3/4, 7/9, 11/12
Hard- 1.Find a common denominator for 6, 4, 9 and 12: the lowest common multiple is 36
- 2.Convert each fraction: 5/6 = 30/36, 3/4 = 27/36, 7/9 = 28/36, 11/12 = 33/36
- 3.Order the numerators from largest to smallest: 33, 30, 28, 27
- 4.Convert back to the original fractions in that order
Answer: 11/12, 5/6, 7/9, 3/4
Avoid These
The most common mistakes students make.
Simplifying a fraction partway and stopping, for example simplifying 8/12 to 4/6 and not noticing it can still be simplified further to 2/3.
Converting a mixed number to an improper fraction by multiplying the whole number and denominator but forgetting to add the numerator.
Comparing fractions by looking at the numerators or denominators alone, without converting to a common denominator first.
Choosing a common denominator that is not actually a common multiple of every denominator involved, which produces fractions that will not convert cleanly.
Creating an equivalent fraction by multiplying only the numerator or only the denominator, instead of multiplying both by the same number.
FAQ
Questions parents and students ask.
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