What Is a Fraction? Equal Parts of a Whole

Cutting a whole into equal parts

Take a pizza. Cut it into $4$ equal slices. Take $3$ of them.

You have three quarters:

$$\frac{3}{4}$$

Number	Name	What it says
$4$ (bottom)	denominator	how many equal parts the whole was cut into
$3$ (top)	numerator	how many of those parts you have

The bottom number comes first in your thinking. It tells you how big one part is. Then the top counts them.

The parts must be equal

This is the rule that makes a fraction mean anything.

Cut a pizza into $4$ pieces where one is huge and three are tiny. Taking $3$ of them is not three quarters. It is three pieces, and nobody knows how much.

Equal parts is what lets one number describe every piece. Without it, the bottom number counts pieces but says nothing about size.

Unit fractions are the building blocks

A unit fraction has $1$ on top:

$$\frac{1}{2}, \quad \frac{1}{3}, \quad \frac{1}{4}, \quad \frac{1}{8}$$

Each one is a single part. And every other fraction is made of copies of one:

$$\frac{3}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4}$$

So $\tfrac{3}{4}$ is three copies of $\tfrac{1}{4}$. That is worth saying aloud, because it turns a fraction into something you can count.

$$\frac{5}{8} = 5 \text{ copies of } \frac{1}{8}$$

Counting in unit fractions

Because a fraction is copies of one part, you can count in them the way you count in ones:

$$\frac{1}{4}, \quad \frac{2}{4}, \quad \frac{3}{4}, \quad \frac{4}{4}, \quad \frac{5}{4}$$

Each step adds one more quarter. Nothing about the parts changes. There are more of them.

This is also multiplication:

$$3 \times \frac{1}{4} = \frac{3}{4}$$

Three copies of a quarter is three quarters, in the same way three copies of $5$ is $15$.

When a fraction is a whole number

Count on to $\tfrac{4}{4}$. That is all four of the four parts — the whole thing.

$$\frac{4}{4} = 1$$

Top and bottom the same means one whole. That works for any bottom number:

$$\frac{2}{2} = 1 \qquad \frac{5}{5} = 1 \qquad \frac{100}{100} = 1$$

Keep counting past it and you go over one:

$$\frac{5}{4} = 1 \text{ whole and } \frac{1}{4} \text{ more}$$

Whole numbers can be written as fractions too:

$$3 = \frac{3}{1} \qquad 2 = \frac{8}{4}$$

$\tfrac{8}{4}$ is eight quarters, and every four of them make one whole. Eight quarters is two wholes.

Why the bottom number grows as the parts shrink

A bigger bottom number means smaller parts, which feels backwards.

$$\frac{1}{8} \; \text{is smaller than} \; \frac{1}{4}$$

Cutting the same pizza into $8$ slices instead of $4$ gives you more slices, so each one has to be smaller. The whole did not change — only how finely it was cut.

So with unit fractions, the bigger the bottom number, the smaller the fraction.

Worked examples

Common mistakes

Practice problems