Ordering Fractions Calculator
Arrange 2 to 15 fractions, mixed numbers, whole numbers, and decimals from least to greatest or greatest to least, with the LCD method shown step by step.
How to use this calculator
Type a list of 2 to 15 values, separated by commas — any mix of whole numbers, fractions, mixed numbers, and decimals, like 3/8, 1/2, 5/6, 1 1/4, 0.7. Choose a direction with the toggle (least → greatest, or greatest → least), and the calculator sorts every value and shows the full working below.
The 5-step method
- Convert to fractions. Every whole number and mixed number becomes a fraction:
2 = 2/1,3 5/8 = 29/8. Values already written as fractions or decimals are converted too. - Find the LCD. Take the denominator of every value (as entered) and find their least common multiple — that's the least common denominator (LCD) for the whole list.
- Rewrite as equivalents. Multiply each fraction's numerator and denominator so every value shares the LCD as its denominator.
- Sort by numerator. Once every value has the same denominator, ordering the numerators orders the values.
- Map back to the original notation. The final answer shows each value exactly as you typed it — a fraction stays a fraction, a decimal stays a decimal — just rearranged into sorted order.
Worked example
Order least to greatest: 2, 3/4, 9/12, 3 5/8, -12/16
- Convert:
2 = 2/1,3/4stays,9/12stays (kept as entered, not simplified),3 5/8 = 29/8,-12/16stays. - Denominators:
1, 4, 12, 8, 16. Their LCM is48, so LCD = 48. - Rewrite over 48:
96/48, 36/48, 36/48, 174/48, -36/48. - Sort the numerators:
-36 < 36 = 36 < 96 < 174. - Map back:
-12/16 < 3/4 = 9/12 < 2 < 3 5/8.
Notice 3/4 and 9/12 land on the same value (9/12 reduces to 3/4), so they're joined with "=" and keep the order they were typed in.
Handling negative values and mixed numbers
Negative fractions and mixed numbers compare the same way any signed number does — the more negative a value, the smaller it is. In the example above, -12/16 (a negative three-quarters) is the smallest value in the whole list, ahead of every positive fraction. Mixed numbers like 3 5/8 are converted to an improper fraction (29/8) internally before comparing, then displayed back in mixed-number form in the final order.
An alternative: ordering by decimal
Instead of finding a common denominator, you can convert every value to a decimal and compare those directly — 3/8 = 0.375, 1/2 = 0.5, 0.7 stays 0.7, 5/6 = 0.8333…, 1 1/4 = 1.25 — giving the identical order 3/8 < 1/2 < 0.7 < 5/6 < 1 1/4. Both methods always agree; the LCD method is exact for every input, while the decimal method is easier to eyeball but gets truncated (with a trailing "…") for a repeating decimal like 5/6.
More worked examples
| Values | Direction | Result |
|---|---|---|
| 3/8, 1/2, 5/6, 1 1/4, 0.7 | Least → greatest | 3/8 < 1/2 < 0.7 < 5/6 < 1 1/4 |
| 3/8, 1/2, 5/6, 1 1/4, 0.7 | Greatest → least | 1 1/4 > 5/6 > 0.7 > 1/2 > 3/8 |
| 1/2, 2/4, 1/3 | Least → greatest | 1/3 < 1/2 = 2/4 |
| -1/2, 1/4, -2 | Least → greatest | -2 < -1/2 < 1/4 |
Precision note
Every value is handled with arbitrary-precision integers (BigInt) — the LCD, the rewritten equivalents, and the numerator comparison that decides the final order are always exact, never floating-point approximations. Only the optional decimal display has a natural limit: a repeating decimal is shown to 6 digits with a trailing "…" rather than continuing forever, but it never changes the order the exact LCD method already found.
Frequently asked questions
- How do I order a list of fractions from least to greatest?
- Convert every whole number and mixed number to a fraction, find the least common denominator (LCD) of all the denominators, rewrite each value over that LCD, then sort by numerator. The numerator order matches the value order once every fraction shares the same denominator.
- Can this calculator order mixed numbers, whole numbers, and decimals together?
- Yes. Type any mix of whole numbers ("2"), fractions ("3/8"), mixed numbers ("1 1/4"), and decimals ("0.7") separated by commas — each is converted to a fraction before ordering, so any combination of shapes sorts correctly against any other.
- What happens when two values in the list are equal?
- Equal values are joined with "=" instead of the less-than or greater-than sign, and they keep their original left-to-right order from the list you typed. For example, "1/2, 2/4, 1/3" orders as 1/3 < 1/2 = 2/4, with 1/2 staying ahead of 2/4 exactly as entered.
- How does switching from least-to-greatest to greatest-to-least work?
- The direction toggle re-sorts the whole list in the opposite order and swaps the comparison sign from "<" to ">". Tied values still keep their original input order in both directions — only the overall ranking flips.
- How does this calculator handle negative fractions and mixed numbers?
- Negative values compare exactly like positive ones on the number line — more negative is smaller. Type a leading minus sign directly on a fraction ("-1/2") or a mixed number ("-1 1/4"); the calculator finds the LCD and orders every value, positive or negative, in the same pass.
- How many values can I order at once?
- Between 2 and 15 values in a single list, separated by commas. Below 2 there is nothing to compare; above 15 the input is rejected so the LCD stays a manageable size to display.
- Is there a shortcut that skips finding the LCD?
- Yes — converting every value to a decimal and comparing those directly gives the same order, since dividing each fraction's numerator by its denominator preserves size. The LCD method and the decimal method always agree; use whichever is faster for the numbers in front of you.