Introduction to Fractions

A fraction represents a part of a whole. When you cut a pizza into 8 equal slices and eat 3, you have eaten $\frac{3}{8}$ of the pizza. The fraction tells you two things: how many parts you are talking about (the top number) and how many equal parts the whole was divided into (the bottom number).

Parts of a Fraction

Every fraction has two numbers separated by a fraction bar:

$\frac{\text{numerator}}{\text{denominator}}$

• Numerator (top number): how many parts you have
• Denominator (bottom number): how many equal parts make up the whole
• Fraction bar: means “divided by” — so $\frac{3}{4}$ also means $3 \div 4$

Example 1: Identify the Parts of $\frac{5}{8}$

• Numerator = 5 (you have 5 parts)
• Denominator = 8 (the whole is divided into 8 equal parts)
• This means 5 out of 8 equal parts

Types of Fractions

Proper Fractions

A proper fraction has a numerator that is smaller than the denominator. Its value is always less than 1.

$\frac{1}{2}, \quad \frac{3}{4}, \quad \frac{7}{10}$

Improper Fractions

An improper fraction has a numerator that is equal to or larger than the denominator. Its value is 1 or greater.

$\frac{5}{3}, \quad \frac{8}{8}, \quad \frac{11}{4}$

$\frac{8}{8} = 1$ (the numerator equals the denominator, so you have the whole thing).

Mixed Numbers

A mixed number combines a whole number with a proper fraction.

$2\frac{1}{3}, \quad 5\frac{3}{4}, \quad 1\frac{7}{8}$

$2\frac{1}{3}$ means 2 whole units plus $\frac{1}{3}$ of another unit.

Every improper fraction can be written as a mixed number, and every mixed number can be written as an improper fraction. You will learn how to convert between them in Mixed Numbers and Improper Fractions.

Fractions on the Number Line

Fractions live between (and on) the whole numbers on a number line. To place $\frac{3}{4}$ on a number line, divide the space between 0 and 1 into 4 equal parts, then count 3 parts from 0.

This visual confirms that $\frac{3}{4}$ is between 0 and 1, closer to 1 — which makes sense because 3 is close to 4.

Fractions That Equal Whole Numbers

Any fraction where the numerator is a multiple of the denominator equals a whole number:

$\frac{4}{4} = 1, \quad \frac{6}{3} = 2, \quad \frac{15}{5} = 3$

To find the whole number, divide the numerator by the denominator.

Fractions Mean Division

The fraction bar is another way to write division. This is one of the most important ideas in arithmetic:

$\frac{3}{4} = 3 \div 4 = 0.75$

This connection between fractions and division is why fractions and decimals are interchangeable — every fraction can be written as a decimal, and most decimals can be written as fractions.

Unit Fractions

A unit fraction has 1 as its numerator:

$\frac{1}{2}, \quad \frac{1}{3}, \quad \frac{1}{4}, \quad \frac{1}{5}, \quad \frac{1}{8}$

Unit fractions are building blocks — any fraction is just a count of unit fractions. For example, $\frac{3}{4}$ is three copies of $\frac{1}{4}$.

An important pattern: larger denominators make smaller fractions. Think of splitting a pizza — the more slices you cut, the smaller each slice gets.

$\frac{1}{2} > \frac{1}{3} > \frac{1}{4} > \frac{1}{5} > \frac{1}{8} > \frac{1}{10}$

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A fraction has a numerator (top) and a denominator (bottom)
• Proper fractions are less than 1; improper fractions are 1 or greater
• Mixed numbers combine a whole number with a proper fraction
• The fraction bar means division — $\frac{a}{b} = a \div b$
• Larger denominators make smaller unit fractions

Return to Arithmetic for more foundational math topics.