Introduction to Fractions

A complete lesson on what fractions are, how to read and write them, equivalent fractions, simplifying, and how to add and subtract fractions with unlike denominators.

What Is a Fraction?

A fraction describes a part of a whole. It has two parts:

Numerator (top) — how many parts you have
Denominator (bottom) — how many equal parts the whole is divided into

$\frac{3}{4} \longrightarrow \text{3 parts out of 4}$

A fraction bar: the shaded portion represents the numerator, the total parts represent the denominator.

Types of Fractions

Type	Condition	Example
Proper	Numerator < Denominator	3/5
Improper	Numerator ≥ Denominator	7/4
Mixed number	Whole part + fraction	1¾

To convert: $1\frac{3}{4} = \frac{4 \times 1 + 3}{4} = \frac{7}{4}$

Equivalent Fractions

Multiply or divide both numerator and denominator by the same number:

$\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12}$

Simplifying

Divide both by their GCD:

$\frac{24}{36} \div \frac{12}{12} = \frac{2}{3}$

Adding and Subtracting

Find the LCD, rewrite, then add/subtract numerators:

1
Example: 1/4 + 1/6
LCD(4,6) = 12
2
Rewrite
$\frac{3}{12} + \frac{2}{12}$
3
Add
$\frac{5}{12}$

Tip Always simplify your final answer!

Which is greater: 5/8 or 3/5?

LCD = 40

$\frac{5}{8} = \frac{25}{40}, \quad \frac{3}{5} = \frac{24}{40}$

$\frac{5}{8} > \frac{3}{5}$

Real world: scaling a cake recipe

A recipe uses $\frac{3}{4}$ cup of sugar for 12 cookies. To make 24 cookies you double every ingredient: $\frac{3}{4} \times 2 = \frac{3}{2} = 1\frac{1}{2}$ cups. Fractions let chefs scale recipes up or down precisely.

Real world: splitting a pizza fairly

Eight friends share two pizzas equally. Each pizza is cut into 8 slices, so each person receives $\frac{8}{8} \div 4 = \frac{1}{4}$ of each pizza, or $\frac{2}{8} = \frac{1}{4}$ of the total. Simplifying fractions makes it easy to describe the fair share.

Real world: music time signatures

In a 3/4 time signature a bar holds 3 quarter-note beats. A half note fills $\frac{2}{4} = \frac{1}{2}$ of the bar. Musicians constantly add and compare fractions to keep rhythm.

Tip Whenever you scale a recipe, check whether the result simplifies —

\frac{6}{8}

cups is cleaner to measure as

\frac{3}{4}

cup.

Practice Problems

Convert 2 3/5 to an improper fraction.

Show answer

(5×2 + 3)/5 = 13/5.

Add 2/3 + 3/8.

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LCD = 24. 16/24 + 9/24 = 25/24 = 1 1/24.

Simplify 45/60.

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GCD(45,60) = 15. 45/60 = 3/4.