What are fractions — the definition
Fractions represent a part of a whole or a part of a group. A fraction is written as one number over another, separated by a line called a vinculum:
numerator / denominator
- The denominator (bottom number) shows how many equal parts the whole has been divided into.
- The numerator (top number) shows how many of those parts you have.
So ¾ means the whole has been cut into 4 equal parts, and you have 3 of them.
There are three types of fractions:
- Proper fractions — the numerator is smaller than the denominator: ½, ¾, ⅝. The value is less than 1.
- Improper fractions — the numerator is equal to or larger than the denominator: 5/4, 7/3. The value is 1 or more.
- Mixed numbers — a whole number combined with a proper fraction: 1¾, 2⅔. These are just another way of writing improper fractions.
To convert an improper fraction to a mixed number: divide the numerator by the denominator. The quotient is the whole number part; the remainder becomes the new numerator. So 7/3 = 2 remainder 1 = 2⅓.
What does simplest form mean? It means the numerator and denominator share no common factors other than 1. So 4/6 simplifies to 2/3 by dividing both numbers by 2.
Equivalent fractions and how to compare them
Equivalent fractions are values that look different but represent the same amount. For example:
½ = 2/4 = 3/6 = 4/8
All of these represent the same amount — half. To create them, multiply or divide both the numerator and denominator by the same number.
They are essential for comparing and calculating. To compare two values, the easiest method is to convert them to a common denominator — the same denominator. To compare ⅔ and ¾:
- ⅔ = 8/12
- ¾ = 9/12
Now it is clear that ¾ is larger. The same idea applies when adding and subtracting them.
Fractions, decimals, and percentages are three ways of expressing the same value: - ½ = 0.5 = 50% - ¼ = 0.25 = 25% - ¾ = 0.75 = 75%
Being able to move between these forms is a key numeracy skill that students use throughout secondary school and adult life.
How to add, subtract and multiply
Adding and subtracting
If the denominators are the same, simply add or subtract the numerators and keep the denominator:
⅜ + ⅛ = 4/8 = ½
If the denominators are different, first find a common denominator:
½ + ⅓ → convert to 3/6 + 2/6 = 5/6
Multiplying
Multiply numerators together and denominators together:
⅔ × ¾ = (2×3)/(3×4) = 6/12 = ½
Dividing
To divide, multiply by the reciprocal (flip the second value upside down):
½ ÷ ¼ = ½ × 4/1 = 4/2 = 2
These rules apply at every level of mathematics, from primary school arithmetic to university-level calculus.
For parents helping at home, using physical objects — folding paper, cutting fruit, or measuring cooking ingredients — makes the concept concrete and memorable. The For parents guide includes more ideas for maths support at home.
Frequently asked questions
- Why are fractions hard for many students?
- This topic can be confusing because they break two rules that students have learned for whole numbers: a larger denominator does not mean a larger fraction (¼ is smaller than ½), and multiplying can give a smaller result (½ × ½ = ¼). Building understanding with physical objects before introducing abstract notation helps enormously.
- What is the difference between a fraction and a ratio?
- A fraction compares a part to a whole — ¾ means 3 out of 4 parts. A ratio compares two quantities to each other — 3:1 means 3 of one thing for every 1 of another. Both use the same numerator/denominator notation but describe different relationships. A fraction can be expressed as a ratio, but not all ratios are fractions.
- What are fractions used for in real life?
- They appear in cooking (½ cup, ¾ teaspoon), construction (measurements in inches or centimetres), finance (interest rates, discounts), time (quarter past, half an hour), and sport (winning ratios, statistics). They are also the foundation of algebra, probability, and every scientific calculation involving division.
- At what school age are fractions introduced?
- Most curricula introduce simple fractions like ½ and ¼ around age 6–8 (grades 1–2) using physical objects. The full notation, equivalent forms, and calculations with different denominators are typically taught from age 8–11 (grades 3–5). Operations — multiplying and dividing — usually follow in grades 5–6.