Compare Fractions Calculator: Common Denominators, Ordering and Practice Quiz
First fraction
Second fraction
📋 Result
📝 Step-by-step solution
📊 Visual comparison
Enter the fractions to compare (2 to 10 fractions)
📊 Sorted result
⚙️ Settings
Five question types: compare two fractions, order fractions, find the largest, find the smallest, find a fraction in between.
🎉 Finished!
Total score: /
Accuracy: % — Wrong answers:
Time:
The tool above has three modes. Compare 2 fractions accepts proper and improper fractions, mixed numbers and negative numbers, then writes out the full solution: the LCM of the denominators, the rewritten fractions and the comparison of the numerators, plus the decimal form, a cross-multiplication check and a visual model. Compare several fractions orders 2 to 10 fractions from smallest to largest or the other way round. Practice quiz generates 4-option questions at three levels, keeps score, times you and explains every answer.
Rules for comparing two fractions
When both denominators are positive, every method comes down to one idea: give the two fractions the same denominator, and then the one with the larger numerator is larger.
1. Fractions with the same denominator
Compare the numerators directly:
For example 5/9 > 4/9 because 5 > 4.
2. Different denominators: use a common denominator
The least common denominator is the least common multiple (LCM) of the two denominators. Multiply the numerator and the denominator of each fraction by the matching multiplier, then compare the numerators:
To find the LCM by hand, write each denominator as a product of primes and take every prime with its highest power. If the larger denominator is a multiple of the smaller one, the LCM is simply the larger denominator.
3. Shortcut: cross-multiplication
When both denominators are positive, you can compare the two cross products instead of rewriting the fractions:
This is really the common-denominator method with the denominator b·d. The tool uses cross-multiplication to double-check the result of the common-denominator step.
4. Fractions with the same numerator
When two fractions share a positive numerator, the one with the smaller denominator is larger: the whole is cut into fewer parts, so each part is bigger:
5. Compare with 1 and compare the gaps to 1
A fraction whose numerator is smaller than its denominator is less than 1; if the numerator is larger, the fraction is greater than 1. So 7/8 < 1 < 9/8 without any rewriting.
When both fractions are close to 1, compare what is missing to make 1. The fraction with the smaller gap is the larger one:
6. Mixed numbers and negative fractions
- Mixed numbers: compare the whole-number parts first. If they are equal, compare the fractional parts. You can also convert to improper fractions: a b/c = (a·c + b)/c.
- Negative fractions: every negative fraction is less than 0 and less than every positive fraction. Between two negative fractions, the one with the larger absolute value is smaller.
- Negative denominators: move the minus sign to the numerator before comparing, a/(−b) = −a/b. The cross-multiplication rule in section 3 only works when both denominators are positive.
7. Compare through a benchmark fraction
When the common denominator would be large, look for a benchmark that is easy to compare with both fractions: if a/b < c and c < p/q, then a/b < p/q. Popular benchmarks are 1/2, the number 1, or a fraction with the numerator of one fraction and the denominator of the other:
A quick test against 1/2: a/b < 1/2 if and only if 2a < b (for b > 0). For example, 12/25 is less than 1/2 because 24 < 25.
Worked examples
You can enter exactly these numbers into the tool at the top of the page to check every step.
Example 1. Compare 5/6 and 7/9
Since 6 = 2·3 and 9 = 3², we get LCM(6, 9) = 18. The multipliers are 18 ÷ 6 = 3 and 18 ÷ 9 = 2.
Because 15 > 14, 5/6 > 7/9. Check by cross-multiplication: 5·9 = 45 > 42 = 7·6.
Example 2. Compare the mixed numbers 2 1/3 and 2 2/7
Both whole-number parts are 2, so only 1/3 and 2/7 need comparing. The common denominator is 21:
Converting to improper fractions gives the same result: 7/3 = 49/21 > 48/21 = 16/7.
Example 3. Two negative fractions −2/3 and −3/4
First compare the absolute values: 2/3 = 8/12 < 9/12 = 3/4. Changing the signs reverses the inequality:
Example 4. Order 2/3, 3/5, 7/10, 1/2 from smallest to largest
The common denominator is LCM(3, 5, 10, 2) = 30:
Ordering the numerators 15 < 18 < 20 < 21 gives:
Enter these four fractions in the Compare several fractions tab to see the same result together with the decimals 0.5, 0.6, 0.6667 and 0.7.
Example 5. Find a fraction between 1/3 and 1/2
Use a larger common denominator to make room in between: 1/3 = 4/12 and 1/2 = 6/12. The fraction 5/12 lies between them:
Another method that always works: add the numerators and add the denominators. (1 + 1)/(3 + 2) = 2/5 also lies between 1/3 and 1/2.
The practice quiz in the tool
The Practice quiz tab mixes five question types: fill in the comparison sign, order fractions in ascending or descending order, find the largest fraction, find the smallest fraction and find a fraction between two given fractions. There are three levels:
| Level | Denominators | Numbers |
|---|---|---|
| Easy | 2 to 19 | fractions less than 1 |
| Medium | 2 to 49 | fractions less than 2 |
| Hard | 51 to 99 | mixed numbers from 1 to 3 |
Each round has 5 to 50 questions, and questions you have already answered are not repeated in the same session. A correct choice turns green and shows the solution. A wrong choice turns red, and the tool points out the correct answer with a detailed explanation. At the end you can see your score, accuracy, time and every answer for review.
Before a question is shown, the tool re-checks its answer in an independent way (cross-multiplication and decimals), making sure there is exactly one correct answer and the four options are all different.
Common mistakes
- Comparing numerators when the denominators differ. 3/4 and 5/8 have 3 < 5, but 3/4 = 6/8 > 5/8. Rewrite with a common denominator first.
- Thinking a larger denominator means a larger fraction. It is the opposite. With the same positive numerator, the larger denominator gives the smaller fraction: 3/11 < 3/7.
- Multiplying only the denominator. Multiply both the numerator and the denominator by the same multiplier, otherwise the value of the fraction changes.
- Forgetting to reverse the order for negative numbers. 2/3 < 3/4 but −2/3 > −3/4.
- Cross-multiplying with a negative denominator. For 2/(−3) and 1/4, blind cross-multiplication gives 8 > −3 and the wrong conclusion. Write 2/(−3) = −2/3 first, then compare.
- Trusting rounded decimals. 1/3 ≈ 0.3333 and 3333/10000 = 0.3333 look the same, but 1/3 is larger. Only common denominators or cross-multiplication give an exact answer.
🎯 Deep-understanding quiz: why the rules work
These 8 questions do not just ask for an answer; they check whether you understand why each rule works and when it stops working. Pick one answer: the correct one turns green, a wrong one turns red, and the solution appears right below with a ⚠️ Traps section explaining why the distractors are so tempting.
🧮 Common denominator Question 1. The largest of the four fractions 3/4, 5/8, 7/12, 2/3 is:
Correct answer: C. 3/4
The common denominator is LCM(4, 8, 12, 3) = 24:
The numerator 18 is the largest, so 3/4 is the largest fraction. The full order is 7/12 < 5/8 < 2/3 < 3/4.
⚠️ Traps
- B (7/12): picked because its numerator and denominator are the biggest numbers. The value of a fraction depends on the ratio of numerator to denominator, not on how big each number is; 7/12 is actually the smallest.
- D (2/3): 2/3 and 3/4 are both “one part short” of 1, so they look tied. But 3/4 is short by 1/4, which is less than the 1/3 that 2/3 is missing, so 3/4 is closer to 1.
- A (5/8): comparing denominators instead of values. A denominator of 8 being larger than 4 does not make 5/8 larger than 3/4 = 6/8.
🍕 Same numerator Question 2. Arranging 5/6, 5/9, 5/7 in ascending order gives:
Correct answer: B. 5/9 < 5/7 < 5/6
All three fractions share the positive numerator 5. The larger the denominator, the more parts the whole is cut into, the smaller each part, and so the smaller the fraction. Since 9 > 7 > 6:
Check with decimals: 0.556 < 0.714 < 0.833.
⚠️ Traps
- A: applies “a larger denominator means a larger fraction”. With the same positive numerator it is the other way round; A is really the descending order.
- C, D: no consistent rule. With equal numerators you only need to list the denominators in reverse: denominators 9, 7, 6 (decreasing) give fractions in increasing order.
🎯 Distance to 1 Question 3. Without a calculator, compare 2025/2026 and 2026/2027. The correct conclusion is:
Correct answer: A. 2025/2026 < 2026/2027
Each fraction is only a tiny bit short of 1:
Since 1/2027 < 1/2026, the fraction 2026/2027 is closer to 1, so it is larger (see Figure 4). Cross-multiplication agrees: 2025·2027 = 2026² − 1 < 2026·2026.
⚠️ Traps
- C (=): in both fractions the numerator is 1 less than the denominator, so they look equal. They are both “1 short” in the numerator, but the gaps to 1 are 1/2026 and 1/2027, two different numbers.
- B (>): misusing the same-numerator rule: “2026 is the smaller denominator, so that fraction is larger”. That rule only holds when the numerators are equal, and here they are not.
- D: fractions with different denominators can always be compared, using the gap to 1, a common denominator or cross-multiplication.
🌉 Benchmark Question 4. Use the benchmark 1/2 to compare 13/27 and 19/36. The correct chain of inequalities is:
Correct answer: D. 13/27 < 1/2 < 19/36
Write 1/2 with the same numerator as the first fraction and the same denominator as the second:
The first fraction is below 1/2 (same numerator 13, denominator 27 > 26), the second is above 1/2 (same denominator 36, numerator 19 > 18). Quick test: a/b < 1/2 if and only if 2a < b, and here 26 < 27 while 38 > 36. No need for the common denominator 108. Check: 13·36 = 468 < 513 = 19·27.
⚠️ Traps
- A: a rough guess that “13 is about half of 27”, so 13/27 must be bigger than 1/2. Half of 27 is 13.5 > 13, so 13/27 is just below 1/2.
- B, C: 1/2 is in the wrong place. When one fraction is below 1/2 and the other is above it, 1/2 has to sit between them.
⚠️ Negative denominator Question 5. Sam writes: “2·4 = 8 > −3 = 1·(−3), so 2/(−3) > 1/4”. The correct comparison is:
Correct answer: B. 2/(−3) < 1/4
The cross-multiplication rule a/b > c/d ⇔ ad > cb only holds when both denominators are positive, because it comes from multiplying both sides by bd > 0. Here one denominator is −3, so move the minus sign to the numerator first:
Cross-multiplying again with positive denominators gives the same answer: (−2)·4 = −8 < 3 = 1·3.
⚠️ Traps
- A: trusting cross-multiplication without checking that the denominators are positive. Multiplying an inequality by a negative number reverses it, so the conclusion comes out backwards. A quick sanity check: 2/(−3) is negative and 1/4 is positive.
- C: one number is negative and the other positive, so they cannot be equal.
- D: a negative denominator is perfectly valid: a/(−b) = −a/b, you just move the sign to the numerator.
➖ Negative mixed numbers Question 6. Compare −2 1/3 and −2 2/7. The correct conclusion is:
Correct answer: C. −2 1/3 < −2 2/7
The minus sign applies to the whole mixed number: −2 1/3 = −(2 + 1/3) = −7/3. Drop the signs and compare the positive numbers as in Example 2: 2 1/3 > 2 2/7 because 1/3 = 7/21 > 6/21 = 2/7. Putting the signs back reverses the order:
⚠️ Traps
- A: comparing the fractional parts (1/3 > 2/7) and keeping the same direction as for positive numbers. For negative numbers, the one with the larger absolute value, further from 0, is smaller (see Figure 5).
- A deeper mistake behind A: reading −2 1/3 as −2 + 1/3 = −5/3. A negative mixed number means −(2 + 1/3), not −2 + 1/3.
- D: −2 1/3 lies between −3 and −2, so it is greater than −3.
🔍 In between Question 7. The fraction that lies between 2/5 and 3/7 is:
Correct answer: D. 5/12
Adding numerators and adding denominators (as in Example 5) always gives a fraction between two positive fractions:
Check with the common denominator 420: 168/420 < 175/420 < 180/420.
⚠️ Traps
- A (1/2): a familiar benchmark, but both 2/5 and 3/7 are below 1/2 (since 2·2 < 5 and 2·3 < 7).
- B (2/7): taking the numerator of one fraction and the denominator of the other does not land in between; 2/7 ≈ 0.29 is even smaller than 2/5.
- C (4/9): very close, but 4·7 = 28 > 27 = 3·9, so 4/9 > 3/7. Always check both ends of the interval.
🧠 General rule Question 8. Let a/b be a fraction with 0 < a < b and let m > 0. Adding m to both the numerator and the denominator always gives:
Correct answer: A. (a + m)/(b + m) > a/b
Both denominators b + m and b are positive, so we may cross-multiply:
In terms of the gap to 1: 1 − (a + m)/(b + m) = (b − a)/(b + m) < (b − a)/b = 1 − a/b, so the gap shrinks and the fraction moves closer to 1. For example 2/5 < 3/6 < 4/7. Question 3 is the case m = 1.
⚠️ Traps
- C (=): confusing “adding the same number” with “multiplying by the same number”. Only multiplying (or dividing) the numerator and denominator by the same non-zero number keeps the value: 2/5 = 4/10, but 2/5 < 4/7.
- B (<): this is what happens to a fraction greater than 1, for example 7/5 > 8/6, not to a fraction less than 1.
- D: m only changes how much the fraction grows, never the direction. The direction depends only on whether a/b is below or above 1.
Frequently asked questions
How do I compare two fractions with different denominators?
Use a common denominator: find the LCM of the two denominators, multiply the numerator and denominator of each fraction by the matching multiplier, then compare the numerators. The fraction with the larger numerator is larger.
Is there a quick way to compare fractions without a common denominator?
Use cross-multiplication: with positive denominators, a/b is greater than c/d when a times d is greater than c times b. You can also compare with 1, compare the gaps to 1, use a benchmark such as 1/2 or apply the same-numerator rule.
Which fraction is larger when two fractions have the same numerator?
With the same positive numerator, the fraction with the smaller denominator is larger. For example, 3/7 is larger than 3/11.
How do I compare mixed numbers?
Compare the whole-number parts first; the larger whole part gives the larger mixed number. If the whole parts are equal, compare the fractional parts, or convert both mixed numbers to improper fractions and use a common denominator.
How do I put several fractions in order?
Rewrite all of them with the LCM of the denominators as the common denominator, then order them by numerator. The Compare several fractions tab does this for 2 to 10 fractions.
What question types does the practice quiz include?
Five types: fill in the comparison sign, order fractions in ascending or descending order, find the largest fraction, find the smallest fraction and find a fraction between two fractions, at Easy, Medium and Hard levels.