How to solve it
- If the denominators are equal, the fraction with the larger numerator is larger.
- If the numerators are equal, the fraction with the smaller denominator is larger.
- Otherwise cross-multiply: the numerator of one by the denominator of the other.
- The larger product marks the larger fraction.
3 × 3 = 9; 2 × 5 = 10
9 < 10 → 3/5 < 2/3
Result: 3/5 < 2/3
Common mistake:Wrong: 5/9 > 2/3.Right: 5/9 < 2/3.With different denominators the numerator alone does not decide: 5 is greater than 2, but 5/9 is less than 2/3.
To compare two fractions, enter one in each field of the calculator. The sign that appears between them, <, = or >, is the answer, and below it you see the two cross products that justify it. The example below shows the calculation by hand.
Worked example: which is greater, 3/7 or 4/9?
Two hikes cover 3/7 and 4/9 of the same trail. Which one is longer?
- Cross-multiply, first product. The numerator of the first fraction times the denominator of the second: 3 × 9 = 27.
- Cross-multiply, second product. The numerator of the second fraction times the denominator of the first: 4 × 7 = 28.
- Compare the two products. 27 is less than 28, so the first fraction is the lesser.
Result: 3/7 < 4/9, and the 4/9 hike is the longer one.
Check: with common denominator 63, 3/7 = 27/63 and 4/9 = 28/63. These are the same 27 and 28 as the cross products.
Which method fits which fractions
Before cross-multiplying, see whether the fractions allow a shortcut. The first three cases in the table can usually be solved without multiplying. Cross products always work, and the common denominator (27/63 < 28/63 in the example) is the same calculation written another way.
| If the fractions have | What you do | Example |
|---|---|---|
| The same denominator | Compare the numerators | 3/8 < 5/8 |
| The same numerator | The smaller denominator wins | 3/5 > 3/8 |
| One on each side of 1/2 | Compare each with 1/2; a fraction is less than 1/2 if twice its numerator is less than its denominator | 2/5 < 1/2 < 4/7 |
| Any other case | Cross products | 3 × 9 = 27 < 28 = 4 × 7 |
The middle row works because 2 × 2 = 4 is less than 5, so 2/5 falls below 1/2, and 2 × 4 = 8 is greater than 7, so 4/7 falls above it. If both fall on the same side, the benchmark does not decide and you move on to cross products. The benchmark fractions guide adds other useful reference points, such as 0 and 1.
Cross-multiplication and the butterfly method
Cross-multiplication, which some people call the butterfly method for the two arrows drawn between the fractions, is exactly the cross-product rule. Each arrow joins a numerator with the denominator opposite it, and each product is the numerator of its fraction over the common denominator b × d.
That explains why the method does not change the order. To compare a/b with c/d, write both with denominator b × d: a/b = (a × d)/(b × d) and c/d = (c × b)/(b × d). The denominators are equal and positive, so the greater numerator decides, which is the greater of the two products. Here the common denominator is 63, and the numerators 27 and 28 are the products from the example. If you prefer a different common denominator, for example the smallest possible one, the result is the same, as you can see in least common denominator.
Negatives and negative denominators
With signs, the order flips compared with the unsigned values. For −5/8 and −3/5, the products are (−5) × 5 = −25 and (−3) × 8 = −24. Since −25 < −24, we get −5/8 < −3/5. The fraction −5/8 is farther from zero and therefore lesser, even though without the signs the relation would be the opposite.
A negative denominator has to be fixed first. The input 3/−5 is normalized to −3/5, and the calculator does that for you. If you cross-multiplied without fixing it, you would get (−5) × (−5) = 25 against 3 × 8 = 24, and −5/8 would seem to be greater, which is the opposite of the truth. Comparisons with zero are simple, because 0/7 equals 0, as does 0/−7, and any positive fraction is greater than any negative one.
Two fraction traps
The second trap shows up with two very close fractions, such as 5/6 and 6/7. At a glance they look equal, but the products are 5 × 7 = 35 and 6 × 6 = 36, so 5/6 < 6/7. You can reason it out without multiplying. 5/6 is 1/6 short of 1, and 6/7 is 1/7 short of 1. The second is shorter by less because 1/7 is less than 1/6, so the second comes closer to 1. The difference between them is 1/6 − 1/7 = 1/42, the same one you get from subtracting 6/7 − 5/6.
A rounded decimal can also hide that difference. That is why the calculator always compares with integers, without going through decimals.
What this calculator compares and what it does not
It compares two values with integer numerator and denominator, up to 100 digits each, and rejects any denominator equal to zero. It also accepts decimals (0.35), mixed numbers (1 2/5) and whole numbers (3) as they are, without converting them first. For other cases there is a better page.
- For three or more values, order fractions sorts them from least to greatest or the reverse.
- To find out how far apart they are, subtract: 6/7 − 5/6 = 1/42 comes out with subtract fractions.
- If you are looking for a fraction between two others, fraction between fractions gives an in-between value.
- To see the position, the fraction number line places them to scale.
- If the products come out equal, as 6 × 15 and 9 × 10 do for 6/10 and 9/15, the fractions are equivalent, two ways of writing the same value.
Common fractions in order
The table lists nine widely used fractions from least to greatest, with their decimals. In repeating decimals, the repeating block is in parentheses. It helps you place a new fraction from memory by comparing it with the nearest row.
| Fraction | Decimal |
|---|---|
| 1/8 | 0.125 |
| 1/4 | 0.25 |
| 1/3 | 0.(3) |
| 2/5 | 0.4 |
| 1/2 | 0.5 |
| 5/8 | 0.625 |
| 2/3 | 0.(6) |
| 3/4 | 0.75 |
| 7/8 | 0.875 |
Practice and further reading
Practice with three comparisons
- Compare 5/9 and 4/7.
- Compare −1/3 and −2/7.
- Compare 6/10 and 9/15.
See the answers
- 5/9 < 4/7, because 5 × 7 = 35 is less than 4 × 9 = 36.
- −1/3 < −2/7, because (−1) × 7 = −7 is less than (−2) × 3 = −6.
- 6/10 = 9/15, because 6 × 15 = 90 = 9 × 10.
There are more exercises with solutions in the compare and order worksheets, and fraction practice checks them instantly. If you are comparing measurements of different objects, first make sure the whole is the same, with the same whole guide. OpenStax explains the order of fractions on a number line.
Frequently asked questions
How do I compare a decimal with a fraction?
Enter the decimal as it is (0.35) and the calculator compares it with 3/8. By hand, 0.35 = 7/20 and 7 × 8 = 56 is less than 3 × 20 = 60, so 0.35 < 3/8. To see that conversion step by step, use decimal to fraction.
How do I compare a mixed number with a fraction?
Enter the mixed number as it is, with a space between the whole number and the fraction, and the calculator compares it with 4/3. By hand, 1 2/5 = 7/5 and the products are 7 × 3 = 21 and 4 × 5 = 20, so 1 2/5 > 4/3. If you want to practice the step before that, there is mixed number to fraction.
How do I compare a whole number with a fraction?
Enter 3 and 11/4 in the fields. By hand, 3 = 3/1, and the products are 3 × 4 = 12 and 11 × 1 = 11, so 3 > 11/4. That makes sense, because 11/4 is 2 wholes and 3/4, less than three units.
How do I compare three or more fractions?
This page compares only two values at a time. For a list of 2 to 50 values, with fractions, decimals and mixed numbers mixed together, use order fractions: it sorts them from least to greatest or the reverse and keeps together the ones that have the same value.