Equivalent Fractions

Two fractions are equivalent when they represent the same value. Multiply or divide both the numerator and denominator by the same non-zero number.

What Are Equivalent Fractions?

Two fractions are equivalent if they represent the same part of a whole.

$\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$

Each bar shows exactly half shaded — the fractions are equivalent.

How to Find Equivalent Fractions

Multiply both numerator and denominator by the same number:

$\frac{3}{5} \times \frac{2}{2} = \frac{6}{10}$

Divide both by the same number (simplify):

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

Note You can always multiply or divide by the same number — you are really multiplying by 1 in disguise.

Simplest Form

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

Simplest Form

\frac{a}{b} \text{ is simplified when } \gcd(a,b) = 1

GCD = Greatest Common Divisor

Simplify 18/24

GCD(18, 24) = 6

$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

Comparing Fractions

To compare $\frac{2}{3}$ and $\frac{3}{5}$ , rewrite with a common denominator (LCD = 15):

$\frac{2}{3} = \frac{10}{15}, \quad \frac{3}{5} = \frac{9}{15}$

So $\frac{2}{3} > \frac{3}{5}$ .

Real-World Applications

Real world: map scales

A map uses the scale $\dfrac{1}{50\,000}$ — 1 cm on the map represents 50,000 cm (500 m) in real life. The equivalent fraction $\dfrac{2}{100\,000}$ describes the same scale written differently. Equivalent fractions let cartographers express scales flexibly across units.

Real world: shop discounts

A shop offers " $\dfrac{1}{4}$ off" every item. The cashier system records this as a 25% discount, i.e. $\dfrac{25}{100}$ . These are equivalent fractions: $\dfrac{1}{4} = \dfrac{25}{100} = 0.25$ .

Real world: comparing nutrition labels

Cereal A has 12 g of sugar per 48 g serving ( $\dfrac{12}{48}$ ). Cereal B has 5 g per 20 g serving ( $\dfrac{5}{20}$ ).

Simplify both: $\dfrac{12}{48} = \dfrac{1}{4}$ and $\dfrac{5}{20} = \dfrac{1}{4}$ . Both cereals contain exactly the same proportion of sugar — equivalent fractions reveal this instantly.

Tip

Where do you see this? Map scales, currency exchange rates, cooking measurements ("3/4 cup = 12 tablespoons"), nutrition percentages, and music time signatures all use equivalent fractions.

Practice Problems

Write two fractions equivalent to 3/7.

Show answer

6/14 and 9/21 (multiply numerator and denominator by 2, then by 3).

Simplify 20/36 to its simplest form.

Show answer

GCD(20, 36) = 4, so 20/36 = 5/9.