What Are Fractions?

A fraction is a way of describing a part of a whole, or a part of a group. Whenever something is split into equal pieces, a fraction tells you how many of those pieces you are talking about. A fraction is written as two numbers separated by a bar, like \(\frac{3}{4}\).

The bottom number, called the denominator, tells you how many equal parts the whole has been split into. The top number, called the numerator, tells you how many of those parts you actually have. So \(\frac{3}{4}\) means the whole was cut into 4 equal parts, and you are looking at 3 of them.

Fractions can also stand for division. The fraction \(\frac{3}{4}\) is another way to write \(3 \div 4\). This is a useful way to think about fractions when they appear alongside decimals: every fraction can be turned into a decimal by dividing the numerator by the denominator.

Once you understand numerator and denominator, it helps to recognize a few common categories of fractions.

An improper fraction and a mixed number are just two ways of writing the same amount. To rewrite \(\frac{7}{4}\) as a mixed number, divide 7 by 4: it goes in 1 whole time with 3 left over out of 4, so \(\frac{7}{4} = 1\frac{3}{4}\). Working comfortably between these two forms matters when you start multiplying improper fractions and mixed numbers or adding and subtracting mixed numbers.

Equivalent fractions look different but describe the same size of piece. Multiplying or dividing both the numerator and the denominator by the same number never changes the value of a fraction, because you are only cutting the pieces into smaller pieces (or joining them back up) — the total amount stays the same. For example, \(\frac{1}{2}\), \(\frac{2}{4}\), and \(\frac{3}{6}\) are all equivalent.

Benchmark fractions are simple, familiar fractions such as \(0\), \(\frac{1}{4}\), \(\frac{1}{2}\), \(\frac{3}{4}\), and \(1\). They act as reference points on a number line, so you can quickly estimate where a less familiar fraction, like \(\frac{5}{9}\), belongs, even before doing any exact calculation.

Fractions show up whenever a quantity is shared or divided unevenly, from cooking measurements to reading a clock to understanding probability. Once you are confident naming and comparing fractions, you can move on to combining them: subtracting fractions that already share a denominator (see subtracting fractions with like denominators) is often the first operation students practice, before tackling multiplication or dividing fractions and mixed numbers.

Is \(\frac{9}{5}\) proper, improper, or a mixed number? Since the numerator (9) is greater than the denominator (5), it is an improper fraction. Dividing 9 by 5 gives 1 whole with 4 left over out of 5, so as a mixed number it is \(1\frac{4}{5}\).

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