A unit fraction is one with 1 in the numerator, like 1/2, 1/7 or 1/10. Of two unit fractions, the one with the smaller denominator is greater, because the whole is split into fewer pieces. That is why 1/5 is greater than 1/8. With this rule you can order a list without switching to decimals. If you want to write a fraction as a sum of different unit fractions, the Egyptian fraction calculator does that; here we look at what they are and how to compare them.
What makes a fraction a unit fraction
In a unit fraction the numerator is 1 and the denominator is a positive whole number. It counts a single part of a whole divided into equal parts. So 1/4 and 1/100 are unit fractions, while 2/5 and 3/4 are not, because their numerator is not 1. A denominator of 0 does not define a fraction, so 1/0 is ruled out.
The word describes how the fraction is written, not how much it is worth. 1/2 and 1/100 are both unit fractions, and one is fifty times the other. If you need to review the parts of a fraction, the guide on numerator and denominator separates them step by step, and to draw a unit fraction you can use the visual fractions tool. Grade 3 math starts here: the Common Core standards describe 1/b as the quantity formed by one part when a whole is partitioned into b equal parts.
Why a larger denominator gives a smaller part
Split one bar into 5 equal parts and an identical bar into 8. To fit eight pieces they have to be cut thinner, so each eighth is smaller than each fifth. The argument needs no decimals: n copies of 1/n always fill the whole, and if more copies are needed, each one is smaller.
Another way to see it is division, since 1 ÷ 5 = 0.2 and 1 ÷ 8 = 0.125, and dividing 1 by a larger number gives a smaller quotient. The comparison only holds when the whole is the same. Half of a small pie is not equal to half of a large one, a point the guide on fractions of the same whole develops.
Ordering a list of unit fractions
To order unit fractions, put the denominators in order from smallest to largest and write the fractions in that same order. The list then runs from greatest to least without any calculation.
Worked example: ordering four unit fractions
Order from greatest to least the unit fractions with denominators 6, 3, 10 and 4, and find how far apart the two largest are.
- Write down the denominators. They are 6, 3, 10 and 4.
- Put them in order from smallest to largest. That gives 3, 4, 6 and 10.
- Write the unit fractions in that order. 1/3, 1/4, 1/6 and 1/10 already run from greatest to least.
- Subtract the first two. With common denominator 12, 1/3 = 4/12 and 1/4 = 3/12, so the difference is 1/12.
Result: 1/3 > 1/4 > 1/6 > 1/10, and the two largest are 1/12 apart.
Check: as decimals, 0.333… > 0.25 > 0.166… > 0.1, and 1/4 + 1/12 = 1/3, because 3/12 + 1/12 = 4/12.
The denominator rule works when the numerators are equal, and fails as soon as they change.
| Case | Example | What to do |
|---|---|---|
| Numerator 1 in all | 1/7 and 1/9 | The smaller denominator wins: 1/7 |
| Same numerator, other than 1 | 3/7 and 3/8 | The smaller denominator wins here too: 3/7 |
| Different numerators | 3/4 and 2/3 | You have to calculate: 3 × 3 = 9 is more than 2 × 4 = 8, so 3/4 wins |
In the last row the denominator 4 is larger than 3 and yet 3/4 is the greater fraction, so only the first two rows can be settled at a glance. For the rest, the compare fractions calculator uses cross-multiplication, and the order fractions calculator returns the full list. On the number line unit fractions move toward 0 as the denominator grows, and the guide on benchmark fractions shows how to estimate with 0, 1/2 and 1.
The distance between neighboring unit fractions
Two consecutive unit fractions, 1/n and 1/(n + 1), are exactly 1/(n × (n + 1)) apart. It comes from subtracting with a common denominator: 1/n − 1/(n + 1) = (n + 1)/(n × (n + 1)) − n/(n × (n + 1)) = 1/(n × (n + 1)).
| Subtraction | n × (n + 1) | Result |
|---|---|---|
| 1/11 − 1/12 | 132 | 1/132 |
| 1/16 − 1/17 | 272 | 1/272 |
| 1/19 − 1/20 | 380 | 1/380 |
| 1/24 − 1/25 | 600 | 1/600 |
The gap shrinks very quickly, so unit fractions with large denominators crowd together near 0 on the line. In a drawing, those two fractions almost merge, but they are 1/380 apart, which is not zero. For any subtraction, the subtract fractions calculator shows the common denominator.
Adding unit fractions
To add 1/a + 1/b you look for a common denominator, for example the least common multiple (LCM) of a and b. With 1/14 + 1/21 the LCM is 42, and 3/42 + 2/42 = 5/42. The denominators are not added. Doing so would give 2/35, which is less than the addend 1/14, impossible when adding positive amounts, and the guide on why you don’t add denominators explains it with drawings.
In general, 1/a + 1/b = (a + b)/(a × b), and the result is a unit fraction only if the numerator comes out to 1 after simplifying. That happens with 1/9 + 1/18 = 3/18 = 1/6, and not with 1/6 + 1/7 = 13/42. So simplify and look at the final numerator instead of guessing from the addends. The add fractions calculator does the work and simplifies, and the least common denominator calculator shows the shared denominator.
Any fraction from unit fractions
Every fraction a/b is the sum of a copies of 1/b. So 3/11 = 1/11 + 1/11 + 1/11, which is why fractions with the same denominator are added by counting numerators: you put together pieces of the same size.
There is another way to break a fraction down, using unit fractions with different denominators, which is the territory of Egyptian fractions. For example, 7/10 = 1/2 + 1/5, and 5/6 = 1/2 + 1/3 because 3/6 + 2/6 = 5/6. Every positive fraction has a decomposition like this, and it is not unique. The Egyptian fraction calculator finds one, the guide on other ways to represent fractions compares this notation with continued fractions, and the guide on types of fractions places the unit fraction among proper, improper and whole-number fractions.
Practice
Solve by hand first, then check. For the grade-level standards behind unit fractions, see the Common Core State Standards for Mathematics, Grade 3: Number and Operations—Fractions (accessed October 17, 2026).
Practice
- Order from greatest to least 1/15, 1/25, 1/18 and 1/22.
- Find 1/8 − 1/9.
- Find 1/10 + 1/15.
- Write 3/14 as a sum of equal unit fractions.
- Which is greater, 1/60 or 1/61?
Show the answers
- 1/15 > 1/18 > 1/22 > 1/25, because the denominators from smallest to largest are 15, 18, 22 and 25.
- 1/72, because 8 × 9 = 72 and 9/72 − 8/72 = 1/72.
- 1/6, because 3/30 + 2/30 = 5/30 = 1/6.
- 1/14 + 1/14 + 1/14.
- 1/60, because 60 < 61.