Comparing Fractions with Unlike Denominators
Learn how to tell which fraction is bigger or smaller when the bottom numbers are different.
Reading is good — doing is better. Practice Comparing Fractions with Unlike Denominators as an interactive lesson.
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When two fractions have different denominators (bottom numbers), you cannot compare them directly just by looking. You first need to rewrite them so they share the same denominator, then compare the numerators (top numbers) to decide which fraction is greater, lesser, or equal.
Remember the rule
Make the bottoms the same, then compare the tops. Bigger top = bigger fraction (when denominators match).
Key words
- Fraction
- A number that shows part of a whole, written with a top number and a bottom number, like 3/4.
- Numerator
- The top number in a fraction — it tells how many parts you have.
- Denominator
- The bottom number in a fraction — it tells how many equal parts the whole is split into.
- Unlike Denominators
- When two fractions have different bottom numbers, like 1/2 and 1/3.
- Common Denominator
- A number that both denominators can divide into evenly, so the fractions can be compared fairly.
- Equivalent Fraction
- A fraction that looks different but has the same value, like 1/2 and 2/4.
- Benchmark Fraction
- A familiar fraction like 1/2 that you can use as a reference point to help compare other fractions.
- Least Common Denominator (LCD)
- The smallest number that both denominators go into evenly — the easiest common denominator to use.
Worked examples
Compare 1/2 and 1/3. Which is greater?
→ 1/2 is greater than 1/3. Rewrite with a common denominator of 6: 1/2 = 3/6 and 1/3 = 2/6. Since 3 > 2, we get 3/6 > 2/6, so 1/2 > 1/3. · Even though both fractions have a 1 on top, the bigger denominator means each piece is smaller.
Compare 3/4 and 2/3. Which is greater?
→ 3/4 is greater than 2/3. The LCD of 4 and 3 is 12. Rewrite: 3/4 = 9/12 and 2/3 = 8/12. Since 9 > 8, we get 3/4 > 2/3. · The difference is small — only one twelfth — so finding the common denominator is the only reliable way to see it.
Compare 2/5 and 3/10. Which is greater?
→ 2/5 is greater than 3/10. The LCD of 5 and 10 is 10. Rewrite: 2/5 = 4/10. Now compare 4/10 and 3/10. Since 4 > 3, we get 2/5 > 3/10. · When one denominator is already a multiple of the other, you only need to convert one fraction.
Compare 5/6 and 7/8. Which is greater?
→ 7/8 is greater than 5/6. The LCD of 6 and 8 is 24. Rewrite: 5/6 = 20/24 and 7/8 = 21/24. Since 21 > 20, we get 7/8 > 5/6.
Compare 3/5 and 1/2 using a benchmark.
→ 3/5 is greater than 1/2. Think: 1/2 of 5 is 2.5, so a fraction equal to 1/2 would be 2.5/5. Since 3 > 2.5, we know 3/5 is more than 1/2. So 3/5 > 1/2. · Using 1/2 as a benchmark is a quick mental math strategy when you cannot easily find the LCD.
Are 4/6 and 2/3 equal, or is one greater?
→ They are equal. Simplify 4/6: divide top and bottom by 2 to get 2/3. So 4/6 = 2/3. They are equivalent fractions. · Always check if one fraction simplifies to the other before assuming they are different.
Common mistakes
- Comparing numerators without making denominators the same first — for example, thinking 3/8 > 3/5 just because 8 > 5, when actually 3/5 is larger.
- Forgetting to multiply both the top AND bottom by the same number when making equivalent fractions — changing only the denominator gives a wrong fraction.
- Assuming the fraction with the bigger denominator is always bigger — bigger denominator actually means each piece is smaller.
- Picking any common denominator but making arithmetic errors when converting; always double-check that numerator × same factor = new numerator.
- Stopping after finding a common denominator but forgetting to compare the new numerators to write the final answer with > , <, or =.
FAQs
Why do we need the same denominator to compare fractions?
The denominator tells you the size of each piece. Comparing pieces of different sizes is like comparing apples and oranges — you need to cut everything into pieces of the same size first so the comparison is fair.
Does it have to be the LEAST common denominator, or can I use any common denominator?
Any common denominator works to get the right answer. The least common denominator just keeps the numbers smaller and easier to work with. For example, for 1/2 and 1/3 you could use 6 or 12 or 18 — all give the correct comparison.
What is the fastest way to find a common denominator?
Multiply the two denominators together — that always gives you a common denominator. For 3/4 and 2/5, use 4 × 5 = 20 as your common denominator. Then convert each fraction: 3/4 = 15/20 and 2/5 = 8/20.
Can I compare fractions without finding a common denominator?
Yes! Two shortcuts: (1) Use a benchmark like 1/2 — decide if each fraction is above or below 1/2, and if one is above and one is below, you are done. (2) Use cross-multiplication — multiply 3 × 5 and 2 × 4 in 3/4 vs 2/5 (you get 15 and 8), and the bigger product goes with the bigger fraction. But finding a common denominator is the most reliable method.
What if the fractions have the same numerator but different denominators?
When numerators are the same, the fraction with the SMALLER denominator is greater. Example: 3/4 > 3/7 because fourths are bigger pieces than sevenths.
How do I know my answer is correct?
Convert both fractions to decimals on a calculator or by dividing numerator ÷ denominator, and see which decimal is larger. This is a great way to check your work.
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