Three types of fractions
| Type | Rule | Value | Examples |
|---|---|---|---|
| Unit fraction | Numerator is 1 | Less than 1 | 1/2, 1/5, 1/12 |
| Proper fraction | Numerator < denominator | Less than 1 | 3/4, 2/7, 9/10 |
| Improper fraction | Numerator ≥ denominator | 1 or more | 5/4, 7/3, 6/6 |
Every unit fraction (except 1/1) is also a proper fraction.
Unit fractions: the building blocks
A unit fraction is one single part. All other fractions are built by counting unit fractions:
- 3/4 = 1/4 + 1/4 + 1/4
- 5/4 = 1/4 + 1/4 + 1/4 + 1/4 + 1/4
Proper fractions: less than one whole
In a proper fraction, the numerator is smaller than the denominator, so you have fewer parts than make a whole. On a number line, all proper fractions sit between 0 and 1.
Improper fractions: one whole or more
In an improper fraction, the numerator is the same as or bigger than the denominator.
- 4/4 = 1 whole
- 7/4 = 1 whole and 3/4 more
Despite the name, there’s nothing wrong with improper fractions. Mathematicians use them all the time because they’re easier to calculate with.
Improper fractions and mixed numbers
An improper fraction can be written as a mixed number (a whole number plus a proper fraction):
| Improper | Think | Mixed |
|---|---|---|
| 7/4 | 4 quarters make 1, 3 left | 1 3/4 |
| 11/3 | 9 thirds make 3, 2 left | 3 2/3 |
| 10/5 | 10 fifths make 2, none left | 2 |
To convert, divide: 11 ÷ 3 = 3 R2 → 3 2/3.
Picturing them
Shade bars for 5/4: one full bar (4/4) and one more quarter of a second bar. This shows clearly that 5/4 is more than 1.
Worked examples
Example 1: Classify 8/9.
8 < 9: proper fraction.
Example 2: Classify 9/8.
9 > 8: improper fraction (equal to 1 1/8).
Example 3: Write 13/5 as a mixed number.
13 ÷ 5 = 2 R3 → 2 3/5.
Common mistakes
- Thinking "improper" means wrong.
- Forgetting that 5/5 is improper (it equals 1).
- Calling 1/1 a proper fraction. It equals 1, so it is improper.
Try it yourself
Classify each as unit, proper or improper.
- 1/9
- 6/7
- 12/5
- 3/3
- Write 17/4 as a mixed number.
Answers: 1) unit (and proper) 2) proper 3) improper 4) improper 5) 4 1/4
Quick recap
- Unit fractions have numerator 1.
- Proper fractions are less than 1; improper fractions are 1 or more.
- Improper fractions can be written as mixed numbers.