Fractions

Table of Contents

Fractions appear whenever we want to talk about “parts of a whole” or “sharing” in a precise way. In this chapter we focus on the general idea of fractions, what they look like, how to read them, and how to picture them. Later subchapters will specialize this (proper/improper fractions, mixed numbers, operations).

What a Fraction Looks Like

A (simple) fraction has two integer parts separated by a horizontal bar:
$$
\frac{a}{b}
$$
where:

$a$ is called the numerator
$b$ is called the denominator, and $b \neq 0$

So:

In $\dfrac{3}{5}$, $3$ is the numerator and $5$ is the denominator.
In $\dfrac{7}{2}$, $7$ is the numerator and $2$ is the denominator.

The bar “$/$” in typing (as in 3/5) is just another way of writing the horizontal fraction bar.

Meaning of Numerator and Denominator

The denominator tells you what kind of part you are using: into how many equal parts the whole is divided.

The numerator tells you how many of those parts you have.

For example:

$\dfrac{3}{4}$ means: the whole is divided into $4$ equal parts, and you have $3$ of those parts.
$\dfrac{1}{10}$ means: the whole is divided into $10$ equal parts, and you have $1$ of those parts.

If the denominator is $b$, each single part is “one $b$‑th” of the whole. We often say:

$\dfrac{1}{4}$: “one quarter” or “one fourth”
$\dfrac{1}{3}$: “one third”
$\dfrac{1}{5}$: “one fifth”

Then:

$\dfrac{2}{4}$: “two quarters”
$\dfrac{7}{3}$: “seven thirds”, and so on.

Fractions as Division

Every fraction can be seen as a division:
$$
\frac{a}{b} = a \div b \quad (b \neq 0).
$$

For example:

$\dfrac{6}{2}$ means $6 \div 2$
$\dfrac{3}{4}$ means $3 \div 4$
$\dfrac{1}{8}$ means $1 \div 8$

Often, the result of division is not a whole number. Fractions are a way to write that result exactly:

$3 \div 4 = \dfrac{3}{4}$ (instead of $0.75$—decimals will be treated in another chapter)
$2 \div 5 = \dfrac{2}{5}$
$7 \div 2 = \dfrac{7}{2}$

So a fraction is another way of writing “$a$ divided by $b$,” using integers $a$ and $b$.

Fractions as Numbers

Fractions are numbers, not just “two numbers written on top of each other.” They have a place on the number line.

To picture $\dfrac{3}{4}$ on the number line:

Mark $0$ and $1$.
Divide the segment from $0$ to $1$ into $4$ equal parts (because the denominator is $4$).
Starting from $0$, move $3$ of those parts to the right.

The point you reach is $\dfrac{3}{4}$.

This works for any positive fraction $\dfrac{a}{b}$:

Break the segment from $0$ to $1$ into $b$ equal pieces.
Each piece is $\dfrac{1}{b}$.
$\dfrac{a}{b}$ is $a$ of those pieces from $0$.

Fractions can also be:

Equal to $1$, such as $\dfrac{5}{5}$, $\dfrac{8}{8}$.
Greater than $1$, such as $\dfrac{5}{3}$, $\dfrac{9}{4}$ (these will be handled more in the “Proper and improper fractions” and “Mixed numbers” sections).
Negative, like $-\dfrac{2}{5}$, $-\dfrac{7}{3}$, by placing a minus sign in front of the fraction.

Visual Models of Fractions

Visual models help you understand what a fraction represents. Here are two common ways to picture them.

Area Model

Imagine a shape representing one whole (often a circle or rectangle):

Divide the whole into $b$ equal parts (denominator).
Shade $a$ of those parts (numerator).

Example: $\dfrac{3}{4}$

Draw a rectangle.
Divide it into $4$ equal sections.
Shade $3$ of them.

The shaded region represents $\dfrac{3}{4}$ of the whole.

Number Line Model

The number line is a straight line with numbers in order:

Mark $0$ and $1$.
Divide the interval from $0$ to $1$ into $b$ equal parts.
Count $a$ parts from $0$; mark that point.

This point is $\dfrac{a}{b}$.

Example: $\dfrac{2}{5}$

Divide the segment from $0$ to $1$ into $5$ equal parts.
Move $2$ of those parts to the right of $0$.
That position is $\dfrac{2}{5}$.

The area model focuses on “parts of a shape.” The number line model focuses on “a location among other numbers.”

Equivalent Fractions (Same Amount, Different Look)

Sometimes two different-looking fractions represent the same quantity. These are called equivalent fractions.

For example, $\dfrac{1}{2}$ and $\dfrac{2}{4}$ represent the same amount.

One way to see this is by multiplying (or dividing) both numerator and denominator by the same nonzero number:

Multiply numerator and denominator by $2$:
$$
\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}.
$$
Multiply numerator and denominator by $3$:
$$
\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}.
$$

We have:
$$
\frac{1}{2} = \frac{2}{4} = \frac{3}{6},
$$
even though the numerators and denominators differ.

In general, if $k$ is a nonzero integer, then:
$$
\frac{a}{b} = \frac{a \times k}{b \times k}.
$$

This is the core idea behind simplifying fractions and recognizing when two fractions name the same number. The detailed process of simplifying and comparing fractions will be developed further in later sections of this chapter series.

Proper Form and Simplified Form (Preview Idea)

When working with fractions, two ideas often come up:

Proper vs. improper: whether the numerator is less than, equal to, or greater than the denominator.
Simplified (or reduced) form: a fraction where the numerator and denominator have no common factor other than $1$.

These ideas will be handled in detail in the upcoming subchapters, but they rely on the basic understanding from this chapter:

what numerators and denominators are,
how fractions represent division,
and how different fractions can represent the same number (equivalent fractions).

Common Ways of Saying Fractions

Fractions are often spoken in particular ways:

$\dfrac{1}{2}$: “one half”
$\dfrac{1}{3}$: “one third”
$\dfrac{1}{4}$: “one fourth” or “one quarter”
$\dfrac{2}{3}$: “two thirds”
$\dfrac{5}{8}$: “five eighths”
$\dfrac{7}{10}$: “seven tenths”

The general pattern in English:

For $\dfrac{1}{b}$, we say “one b‑th” (with special words for $2$, $3$, and $4$: half, third, quarter/fourth).
For $\dfrac{a}{b}$ with $a \neq 1$, we use the plural: “a b‑ths” (e.g. “three fifths”, “nine tenths”).

Summary of the Fraction Idea

A fraction $\dfrac{a}{b}$ (with $b \neq 0$) is a number that represents $a$ parts out of $b$ equal parts of a whole.
The numerator $a$ counts how many parts; the denominator $b$ tells how the whole is subdivided.
Fractions are another way to write division: $\dfrac{a}{b} = a \div b$.
Fractions live on the number line and can be pictured by area models or number-line models.
Different-looking fractions can represent the same amount if one is obtained from the other by multiplying or dividing numerator and denominator by the same nonzero number (equivalent fractions).

With these basic ideas in place, we can now look more closely at different types of fractions and how to work with them in the following subchapters.