Lesson 4 of 10 · Grades 3 to 5
Compare fractions
Comparing fractions means asking which one is larger. Sometimes the answer is quick: 5/8 is more than 3/8. Other times you have to work a little, like comparing 2/3 with 3/4. This lesson gives you three safe ways to decide.
Which one is bigger?
Comparing fractions means deciding which of two fractions is larger. The answer is a bit like comparing prices: sometimes you can tell at a glance, and sometimes you need to convert both to the same units first. This lesson gives you three methods. Pick the one that fits the numbers in front of you.
Method 1: same denominator, compare numerators
When two fractions have the same denominator, their parts are the same size. Comparing them is then just counting: 5/8 is bigger than 3/8, because five equal eighths is more than three equal eighths. The fraction with the larger numerator is the larger fraction. Nothing else changes.
Method 2: same numerator, compare denominators
When the numerators are the same, the denominators tell a different story. A bigger denominator means the whole was cut into more pieces, and more pieces means each piece is smaller. So 1/3 is bigger than 1/6, and 2/5 is bigger than 2/7. When the numerators match, the fraction with the smaller denominator is the larger fraction.
Method 3: make a common denominator
When both numbers are different, rewrite one or both fractions so they share a denominator. You already know how to do this: multiply the top and the bottom of a fraction by the same number. A safe choice is to multiply the two denominators together, though a smaller common denominator is neater when you can spot one.
Compare 2/3 and 3/4. The denominators are 3 and 4, so 12 works as a common denominator. Multiply 2/3 by 4 on top and bottom to get 8/12. Multiply 3/4 by 3 on top and bottom to get 9/12. Now the denominators match: 9/12 is bigger than 8/12, so 3/4 is bigger than 2/3.
A quick shortcut: compare to 1/2
Benchmark fractions are useful when the numbers are close to familiar values. A fraction is equal to 1/2 when its numerator is half of its denominator, for example 4/8 and 6/12. A fraction is bigger than 1/2 when the numerator is more than half the denominator, and smaller than 1/2 when it is less. So 3/5 is bigger than 1/2 because 3 is more than half of 5, while 2/5 is smaller. Compare each fraction with 1/2 first, and if they fall on opposite sides, you have your answer. If they fall on the same side, convert.
Worked example
Compare 4/5 and 7/9. The denominators are 5 and 9, and they do not share a smaller common multiple, so use 5 x 9 = 45. Convert 4/5: multiply top and bottom by 9 to get 36/45. Convert 7/9: multiply top and bottom by 5 to get 35/45. Now compare numerators: 36 is bigger than 35, so 4/5 is bigger than 7/9. The two fractions are close, less than one ninth apart, but the method still gives a clear answer.
Watch out for this
Do not compare numerators to denominators. A fraction is not a pair of separate numbers competing with each other; it is one value. And when the denominators differ but the numerators match, the bigger denominator loses, not wins. Finally, check that your common-denominator rewrite keeps the value the same: multiply the top and the bottom by the same number every time.
Remember
- Same denominator: the bigger numerator is the bigger fraction.
- Same numerator: the smaller denominator is the bigger fraction.
- Different numbers: rewrite both with a common denominator, then compare.
- 1/2, 1 and other benchmarks can give you an answer quickly.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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Which is bigger: 2/7 or 5/7?
Show answer
5/7. The denominators are the same, and 5 is bigger than 2.
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Which is bigger: 1/4 or 1/5?
Show answer
1/4. The numerators are the same, and quarters are bigger pieces than fifths.
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Compare 3/4 and 2/3.
Show answer
3/4. Write both in twelfths: 3/4 = 9/12 and 2/3 = 8/12.
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Which is smaller: 5/8 or 1/2?
Show answer
1/2. Write 1/2 as 4/8, and 4/8 is smaller than 5/8.
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Put < or > in the gap: 4/6 __ 2/3
Show answer
= (they are equal). 4/6 simplifies to 2/3.
Puzzle · Mystery fraction
Puzzle time
I am a fraction. My denominator is 8. I am more than 1/2. I am less than 3/4. Which fraction am I?
Show a hint
Write 1/2 and 3/4 as eighths first. Then look for an eighth that sits between them.
Show the answer
Answer: 5/8. 1/2 is the same as 4/8, and 3/4 is the same as 6/8. The only eighth between 4/8 and 6/8 is 5/8. 6/8 is not less than 3/4. It is equal to it.