Maths
What are improper fractions and mixed numbers?
Quick answer
An improper fraction has a numerator bigger than its denominator, like 7/4. A mixed number shows the same amount as a whole number plus a fraction, like 1 3/4.
In one sentence
An improper fraction is a fraction that is one whole or more, written with a bigger top number, and a mixed number shows the same amount as whole numbers plus a fraction.
Explained
In a proper fraction like 3/4, the top number (numerator) is smaller than the bottom (denominator). It is less than one whole. But sometimes you have more than a whole. If you eat 7 quarter-slices of pizza, you have eaten 7/4. That is an improper fraction, because the numerator is bigger than the denominator. (The name is a bit unfair; it is a perfectly fine number.)
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You can also write 7/4 as one whole pizza (4/4) plus 3 more quarters. That is the mixed number 1 3/4, a whole number and a fraction side by side. Both mean exactly the same.
Improper to mixed: divide the numerator by the denominator. The answer is the whole number, and the remainder is the new numerator over the same denominator.
7 ÷ 4 = 1 remainder 3, so 7/4 = 1 3/4.
Mixed to improper: multiply the whole number by the denominator, add the numerator, and put the result over the same denominator.
2 1/3: 2 × 3 = 6, then 6 + 1 = 7, so 2 1/3 = 7/3.
Mixed numbers are easy to picture and good for measuring (2 1/2 cups of flour). Improper fractions are usually easier to use when you multiply or divide fractions. Both come from the idea that a fraction is a division: 7/4 means 7 ÷ 4. For the basics, see fractions.
Examples
| Improper | Mixed |
|---|---|
| 5/2 | 2 1/2 |
| 9/5 | 1 4/5 |
| 11/3 | 3 2/3 |
| 13/4 | 3 1/4 |
| 17/6 | 2 5/6 |
Common mistakes
- Using the wrong denominator. The bottom number stays the same in both forms.
- Forgetting the remainder. In 11 ÷ 3, the answer is 3 remainder 2, so 3 2/3, not just 3.
- Adding when you should multiply. To convert 2 1/3, first do 2 × 3, then add 1.
Quick quiz
Improper fractions and mixed numbers
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Related topics
Fractions
A fraction is part of a whole that has been split into equal parts. In ¾, the bottom number says how many equal parts and the top says how many you have.
Simplifying fractions
To simplify a fraction, divide the top and bottom by the same number until you cannot go any further. For example, 12/18 simplifies to 2/3 when you divide both by 6.
Division
Division is sharing into equal groups, or finding how many groups fit. The sign is ÷, so 12 ÷ 3 = 4. The answer is called the quotient.
Adding and subtracting fractions
To add or subtract fractions, first give them the same denominator. Then add or subtract only the numerators and keep the denominator. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Comparing fractions
To compare fractions, check whether they share a denominator or a numerator, or rewrite them with a common denominator. The bigger fraction is the one with more of the whole.
Equivalent fractions
Equivalent fractions look different but are the same size, like 1/2, 2/4 and 4/8. You make them by multiplying or dividing the top and bottom by the same number.