Put two to eight fractions in order, with all three methods shown on your own numbers.
Put them in order
Three methods, ordered exactly.
Two to eight of them. Commas work as well as new lines; decimals and mixed numbers are read as fractions.
3 values
2/3 < 3/4 < 5/6
Largest first: 5/6 > 3/4 > 2/3
Largest
5/6
by exact comparison
Smallest
2/3
by exact comparison
Common denominator
12
multiplying them all would give 72
Closest pair differ by
1/12
0.08(3)
Method 2 — over one denominator
Each value rewritten over the common denominator, with its decimal and rank
Value
Over 12
Decimal
Place
3/4
9/12
0.75
2
2/3
8/12
0.(6)
3
5/6
10/12
0.8(3)
1
Once every value is over the same denominator the numerators alone decide it — and those numerators are also what you would add, which is why this is the method that carries on being useful.
Method 1 — cross multiplication, pair by pair
2/3 < 3/4 because 2 × 4 = 8 < 3 × 3 = 9
3/4 < 5/6 because 3 × 6 = 18 < 5 × 4 = 20
Fast for two, and it never needs a common denominator. The catch is that it only works while both denominators are positive — which is why a negative denominator is moved to the numerator before any of this runs.
Method 3 — by decimal. The two closest values here separate at the 2nd decimal place. Round any shorter than that and they look identical.
The order is decided on the exact fractions, not on their decimals — so 3 values that agree to six places still sort correctly.
The shared denominator is 12, not 72: the denominators have factors in common, so the least common multiple is 6× smaller than multiplying them all together.
The order above is decided on exact fractions. The decimals are shown for reading, never for deciding.
What this tool shows
The order is decided on exact fractions, never on their decimals. The page also works out how many decimal places the two closest values need before they stop looking identical — which is the whole argument against comparing by decimal.
Ordering up to eight fractions at once
Cross multiplication, pair by pair, with the products
Everything rewritten over one shared denominator
The decimal column, and where it would mislead
Equal values recognised and given the same place
Negative denominators normalised before comparing
Three methods, same numbers Ordering decided exactly The sign trap made explicit How close is too close for decimals
Exact ordering; decimals shown but never used to decide.
Updated 7 September 2026 · Works in any browser, no installation
To compare two fractions, cross multiply: a/b is smaller than c/d exactly when ad is smaller than cb — provided both denominators are positive. For more than two, rewrite them all over one common denominator and compare the numerators, which is the only method that scales. Comparing the decimals works until two values agree to however many places you kept, which is why 1/3 and 0.33 look equal and are not.
At a glance
Formula shown
For two: a/b < c/d exactly when ad < cb, provided b and d are both positive. For any number of them: rewrite each over L = lcm(all denominators) and compare the numerators.
Scenario support
Deciding which of two offers is the better fraction off; putting test scores written as fractions in order; checking a student\u2019s ordering of fractions with unlike denominators.
Educational estimate
Planning support from the values you enter — not professional advice.
Cross multiplication, and its trap
For exactly two fractions this is the fastest method and it needs no common denominator: to compare a/b with c/d, compare ad with cb. 3/4 against 2/3 becomes 9 against 8, so 3/4 is larger.
The rule is almost always stated without its condition, which is that b and d must both be positive. Multiplying an inequality by a negative reverses it, and the denominators are what you are multiplying by. Compare 1/2 with 1/−3 by cross multiplying as written and you get 1 × −3 = −3 against 1 × 2 = 2, concluding that 1/2 is larger — which is true, but only by luck of which side you put where.
This page moves any negative sign onto the numerator before comparing anything, and says so when it does. After that the rule is safe, which is why every other step on the site assumes a positive denominator.
One denominator for all of them
Cross multiplication handles two. For three or more it becomes a tangle of pairwise comparisons, and the common denominator is the method that actually scales: rewrite everything over the least common multiple, and the numerators alone decide the order.
3/4, 2/3 and 5/6 over twelfths are 9/12, 8/12 and 10/12. Reading the order off 8, 9, 10 takes no thought at all, which is the point.
Using the LEAST common multiple rather than multiplying the denominators keeps the numbers small — twelve rather than seventy-two here. It also leaves you with fractions you can go on to add, which cross multiplication does not.
When decimals lie
Converting to decimals and comparing feels like the modern answer and is the one method here that can be wrong.
1/3 and 33/100 are 0.333… and 0.33. To two places both are 0.33. To three they are 0.333 and 0.330, and only then does the order appear. Any comparison that rounded to two places would have called them equal.
The tool computes the exact gap between the closest pair in your list and tells you the first decimal place at which they separate. If that number is larger than the precision you were planning to keep, the decimal method is not safe on your numbers — and it will not warn you itself.
Bigger bottom, smaller piece
With the same numerator, a larger denominator means a smaller number: 1/8 is less than 1/7 is less than 1/6. This is the comparison that most often goes wrong, because every other kind of number gets bigger when the digits do.
The reason is what the denominator means. It is not a count, it is a size: how many equal pieces the whole was cut into. Cutting a cake into eight gives smaller slices than cutting it into six.
It is worth checking any answer against this. If a list of unit fractions comes out with 1/8 above 1/6, something has been read as a count that was really a size.
Equal is an answer
1/2, 2/4 and 3/6 are the same number written three ways. A comparison tool that ranks them 1st, 2nd and 3rd has said something false.
This page gives equal values the same place and states that they are equal in the answer line, using = rather than <. Two fractions are equal exactly when their reduced pairs are identical, which is a comparison of two whole numbers and needs no arithmetic at all.
That is one of the practical reasons for reducing before you do anything else: it turns “are these the same?” into a question you can answer by looking.
Which method to use when
Two fractions, quick answer: cross multiply. Two multiplications and a comparison, no common denominator needed — just check the denominators are positive first.
Three or more, or you will add them next: common denominator. It scales, it leaves you with something usable, and the numerators make the order obvious.
You need a rough sense, fast: decimals are fine, as long as you know how close the values are. Use the separation figure above to decide whether they are safe here.
Benchmarking: for a sanity check, compare each against 1/2. 5/8 is above, 3/7 is below, and that alone settles many pairs without any arithmetic.
Sources and methodology
Comparison of rationals is settled; these are where the conventions and the caveats are written down.
Method. Every value is parsed to an exact fraction with the sign normalised onto the numerator, and the ordering is computed by comparing cross products of BigInts — no decimal is involved in the decision. The shared denominator is the least common multiple, found by Euclid rather than by multiplying. The gap between the closest pair is computed exactly and compared against successive powers of ten to find the first decimal place at which they differ. That engine is verified on every change against 58 hand-written assertions, including that the reported ranking is a genuine ordering on every one of three hundred generated sets, checked by cross multiplication independently of how the engine sorted them. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
FractionAdd, subtract, multiply and divide fractions and mixed numbers exactly, with the least common denominator chosen, the cross-cancelling done first, and the true repeating decimal beside the answer.
Simplify FractionsReduce a fraction to lowest terms with both routes to the divisor shown side by side — Euclid line by line and the shared primes — and a stated proof when nothing can be cancelled.
Fraction to DecimalConvert a fraction to its exact decimal, with the length of the repeat worked out from the denominator before dividing and the long division shown remainder by remainder.
Decimal to FractionTurn any decimal into its exact fraction, repeating ones included, with the four algebra lines that cancel the infinite tail instead of rounding it away.
Mixed NumberAdd, subtract, multiply and divide mixed numbers with the borrow shown where it happens, plus a second independent check worked through improper fractions.
LCMThe least common multiple of two to six numbers, with the prime table that produces it, the multiples themselves, and how far counting would have had to go.
If you need to combine these values rather than order them, the Fraction Calculator does the four operations. If two of them turn out to be the same number written differently, the Simplify Fractions Calculator shows why — both reduce to the same pair.
Educational use disclaimer
This is an educational tool. The ordering is exact; the decimal column is shown for reading and is deliberately never used to decide the order, because that is the failure this page exists to demonstrate.
Published the comparing fractions page: cross multiplication, a shared denominator and the decimal column shown together on the same inputs.
The ordering is decided on exact rationals and never on the decimals, and the page reports the first decimal place at which the closest pair separate — the failure mode of comparing by decimal, made concrete.
Cross multiplication is stated with the condition the schoolbook rule omits: it only holds while both denominators are positive, so a negative denominator is normalised first.
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