Introduction to Fractions

What is a fraction?

A fraction describes a part of a whole. Imagine cutting a pizza into equal slices. A fraction tells you how many slices you have out of the total.

A fraction is written with two numbers:

• The numerator (top) — how many parts you have.
• The denominator (bottom) — how many equal parts make up the whole.

For example, 3/4 means the whole was cut into 4 equal parts and you have 3 of them.

The most important word is equal. The parts must all be the same size, or the fraction does not make sense.

Reading fractions

Fraction	Numerator	Denominator	Meaning
1/2	1	2	One of two equal parts (a half)
3/4	3	4	Three of four equal parts
5/8	5	8	Five of eight equal parts

If the numerator equals the denominator, you have the whole thing: 4/4 = 1.

Equivalent fractions

Different fractions can name the same value. These are called equivalent fractions.

A half of a pizza is the same amount of pizza whether you cut it into 2, 4, or 8 slices:

1/2 = 2/4 = 4/8

The rule: if you multiply (or divide) the numerator and denominator by the same number, the value stays the same.

$$ \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} $$

You can also write that inline as (1×3)/(2×3) = 3/6.

Simplifying fractions

A fraction is in its simplest form when the numerator and denominator share no common factor other than 1.

To simplify, divide both by their greatest common factor (GCF).

Example: simplify 8/12.

1. List factors. The GCF of 8 and 12 is 4.
2. Divide both: 8 ÷ 4 = 2 and 12 ÷ 4 = 3.
3. So 8/12 = 2/3.

You can check: 2/3 cannot be reduced further, because 2 and 3 share no common factor.

Comparing fractions

Which is bigger, 3/5 or 1/2? You cannot tell at a glance because the pieces are different sizes. The trick is to give them a common denominator.

1. A common denominator for 5 and 2 is 10.
2. Convert: 3/5 = 6/10 and 1/2 = 5/10.
3. Now compare numerators: 6 > 5, so 3/5 is larger.

Adding fractions with the same denominator

When the denominators already match, the pieces are the same size, so you simply add the numerators and keep the denominator.

2/7 + 3/7 = 5/7

You do not add the denominators, because that would change the size of each piece. (If denominators differ, first rewrite them as equivalent fractions with a common denominator, then add.)

A full worked example

Sam ate 2/8 of a chocolate bar and Maya ate 3/8. How much is left?

1. Together they ate 2/8 + 3/8 = 5/8.
2. The whole bar is 8/8, so what's left is 8/8 − 5/8 = 3/8.
3. Simplify if possible: 3/8 is already in simplest form.

So 3/8 of the bar remains.

Where this leads

Fractions are the foundation for decimals, percentages, ratios, and proportional reasoning. Once you are comfortable here, you'll be ready for Algebra Basics: Working with Variables, where fractions show up as coefficients and in solving equations.

Practise by splitting real things — pizzas, chocolate bars, a glass of juice — and naming the fraction you see.