A fraction is classified by three independent questions. The first is how much it is worth compared with 1, which gives proper and improper fractions, plus the whole-number case that Spanish-language and some other textbooks call an apparent fraction (US books usually just say the fraction equals a whole number). The second is how it is written, which brings in the mixed number. The third is what properties it has: whether it is a unit fraction, reducible or decimal. That is why one fraction can get several names at once, and 12/4 is both improper and apparent without any contradiction.
| Criterion | Question | Types and example |
|---|---|---|
| By value | Is it worth less than 1, exactly 1, or more? | Proper 3/7; improper 9/5; apparent 12/4 = 3 |
| By notation | Is there a whole number in front of the fraction? | Mixed number 1 4/5 |
| By properties | What do its terms say? | Unit 1/9; reducible 10/15; decimal 7/100 |
The table works as an index: the value decides whether a fraction is proper or improper, the notation decides whether it is mixed, and the properties are checked last. To convert from one form to another there are tools that do the calculation, such as improper fraction to mixed number and mixed number to fraction; this guide explains what each type is.
By value: proper, improper and apparent fractions
With a positive numerator and denominator, a fraction is proper if the numerator is less than the denominator. Its value lies between 0 and 1, like 3/7. It is improper if the numerator is greater than or equal to the denominator, and then it is worth 1 or more: 9/5 equals 1 4/5. A fraction is called apparent in some books (a term from Spanish-language textbooks; US books usually just say it equals a whole number) if its numerator is a multiple of the denominator, so that it equals a whole number: 12/4 = 3.
| Fraction | Value | Proper or improper | Apparent? |
|---|---|---|---|
| 3/7 | less than 1 | Proper | No |
| 5/5 | 1 | Improper | Yes |
| 9/5 | 1 4/5 | Improper | No |
| 12/4 | 3 | Improper | Yes |
The borderline case is a numerator equal to the denominator. This guide uses the “greater than or equal to” convention, so 5/5 is improper. Some books call 5/5 only apparent, or reserve “improper” for values greater than 1. Before a test, check the definition in your textbook, because when the numerator equals the denominator the expected answer changes. Every positive apparent fraction is improper, but not the other way around: 9/5 does not equal a whole number.
Worked example: the three questions with one fraction
Worked example: classify 11/4
Classify 11/4 by its value, by its notation and by its properties.
- Compare the terms. 11 is greater than 4, so the fraction is improper and worth more than 1.
- Divide. 11 ÷ 4 is 2 with a remainder of 3, so 11/4 = 2 3/4 and the fraction can be written as a mixed number.
- Look for a common factor. The GCF of 11 and 4 is 1, so it is in lowest terms.
- Check the numerator. It is not 1, so the fraction is not a unit fraction, and 11 is not a multiple of 4, so it is not apparent.
Result: improper, in lowest terms, equivalent to the mixed number 2 3/4.
Check: 2 × 4 + 3 = 11, the starting numerator.
By notation: the mixed number
A mixed number joins a whole number and a proper fraction: 1 4/5 means 1 + 4/5. It is another way of writing an improper fraction, not a different number. To go from improper to mixed, divide; the quotient is the whole number and the remainder is the new numerator. To go back, multiply the whole number by the denominator and add the numerator. Operations with mixed numbers are in the mixed numbers calculator.
By properties: unit, reducible and decimal
These labels describe the terms as written, and they combine with the earlier ones.
| Type | Criterion | Example |
|---|---|---|
| Unit | The numerator is 1 | 1/9, which is proper |
| Reducible | The terms have a common factor greater than 1 | 10/15 = 2/3 |
| In lowest terms | Their only common factor is 1 | 2/3 |
| Decimal | The denominator is 10, 100, 1,000… | 7/100 = 0.07 |
A fraction can be proper and reducible at the same time. So 2/18 is proper and reducible, and it equals 1/9, which is a unit fraction: the label “unit” depends on how the fraction is written. Simplifying a fraction changes its notation, not its value, so it does not turn a proper fraction into an improper one. To simplify, use simplify fractions, and to generate other forms with the same value, equivalent fractions.
Not every fraction is decimal in this classification, even though every fraction has a decimal expansion. A decimal fraction has a denominator of the form given above, like 7/100. The fraction 1/3 does not, and its expansion, 0.333…, never ends.
Practice
Classify each fraction according to this guide and write all of its names.
- 8/8
- 5/9
- 14/6
- 1 2/7
Show the answers
- Improper and apparent: it equals 1.
- Proper, in lowest terms and not a unit fraction.
- Improper, reducible (14/6 = 7/3) and equivalent to the mixed number 2 1/3.
- Mixed number: 1 + 2/7 = 9/7, which is improper.
For more classification problems with answers, there are worksheets on proper and improper fractions and on improper fractions to mixed numbers. The parts of a fraction, and why the denominator is not zero, are explained in numerator and denominator, and the terms on this page are defined in the fraction glossary. To read fractions aloud, see reading and writing fractions, and to place fractions among the numbers, fractions and rational numbers.
Sources
The definitions of proper and improper (and the term apparent, used mainly outside US books) vary slightly between books; the convention used on this page is stated next to the table of values.