If you have ever cut a cake or a pizza into slices, you have already dealt with fractions.

A fraction represents a part of a whole or a collection of objects. It is a way to divide something into equal parts or portions.

A fraction has two parts: the numerator and the denominator. The numerator is the number on top of the fraction, and it represents how many parts of the whole or collection are taken. The denominator is the number below the line, and it represents how many equal parts the whole is divided into.

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

For example, in the fraction $\frac{3}{4}$, $3$ is the numerator, and $4$ is the denominator. The line separating the numerator from the denominator is known as the vinculum.

You can read more about the vinculum here.

Fractions are used in many areas of mathematics and everyday life, such as measuring ingredients for a recipe.

What Is A Fraction?

A fraction is a mathematical representation of a part of a whole.

It is used to describe a portion or section of any quantity out of a whole. Fractions are commonly written in the form $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator.

Definition

A fraction is a numerical value that represents the quotient of two numbers, where the numerator represents the number of equal parts of the whole or collection that are taken, and the denominator represents the total number of equal parts that make up the whole or collection.

Parts of a Fraction

A fraction has two parts: the numerator and the denominator.

Equivalent Fractions

Equivalent fractions are fractions that have different numerators and denominators but represent the same value.

For example, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions because they both represent half of a whole.

You can read more about equivalent fractions here.

Proper and Improper Fractions

A proper fraction is a fraction where the numerator is less than the denominator. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Examples:

  • Proper: $\frac{1}{2}$
  • Improper: $\frac{3}{2}$

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. Adding or subtracting fractions with different denominators requires finding a common denominator.

You can learn how to add fractions here.

Unit Fraction

A unit fraction is a fraction where the numerator is $1$. For example, $\frac{1}{2}$.

Types of Fractions

There are three main types: proper, improper, and mixed fractions.

Proper Fractions

A proper fraction has a numerator less than the denominator. Example: $\frac{1}{4}$

Improper Fractions

An improper fraction has a numerator greater than or equal to the denominator. Example: $\frac{5}{3}$

Mixed Fractions

A mixed fraction combines a whole number and a proper fraction. Example: $2\frac{1}{3}$

To convert to an improper fraction:

$$ 2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} $$

Equivalent Fractions

Equivalent fractions represent the same value.

To find one:

$$ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} $$

More on equivalent fractions here

Operations with Fractions

Addition and Subtraction

Use common denominators:

  • Find LCM
  • Convert fractions
  • Add or subtract numerators

Learn more about adding fractions here

Multiplication and Division

Multiply:

$$ \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} $$

Divide:

$$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $$

Simplifying Fractions

Use the greatest common factor (GCF):

$$ \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} $$

Fractions in Real Life

Pizza

If you eat 2 slices out of 4, you ate:

$$ \frac{2}{4} = \frac{1}{2} $$

Cooking

Recipe needs $\frac{3}{4}$ cup flour → measure three-quarters of a cup.

Fractions Summary

  • A fraction has a numerator and denominator
  • Proper: numerator < denominator
  • Improper: numerator ≥ denominator
  • Mixed: whole number + proper fraction

Operations:

  • Add/Subtract → common denominators
  • Multiply → across numerators and denominators
  • Divide → multiply by reciprocal

Fractions are a foundational concept in math and everyday life.