Equivalent Fractions and Simplifying

Two fractions are equivalent when they represent the same amount, even though they use different numbers. For example, $\frac{1}{2}$ and $\frac{2}{4}$ and $\frac{3}{6}$ all represent the same quantity — half of something.

Understanding equivalent fractions is essential because you need this skill every time you add, subtract, or compare fractions with different denominators.

The Fundamental Rule

You can create an equivalent fraction by multiplying (or dividing) both the numerator and denominator by the same nonzero number. This works because you are really multiplying by a fancy form of 1:

$\frac{a}{b} = \frac{a \times n}{b \times n}$

Since $\frac{n}{n} = 1$, and multiplying by 1 does not change a value, the fraction stays equal.

Example 1: Build Equivalent Fractions of $\frac{2}{3}$

Multiply numerator and denominator by 2, 3, 4, and 5:

$\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12} = \frac{10}{15}$

All five fractions represent the same amount.

Example 2: Show That $\frac{3}{4} = \frac{15}{20}$

Check: $3 \times 5 = 15$ and $4 \times 5 = 20$. Both numerator and denominator were multiplied by 5, so the fractions are equivalent. ✓

Simplifying Fractions (Reducing to Lowest Terms)

A fraction is in lowest terms (or simplest form) when the numerator and denominator share no common factor other than 1.

To simplify a fraction, divide both the numerator and denominator by their Greatest Common Factor (GCF).

Steps:

1. Find the GCF of the numerator and denominator
2. Divide both by the GCF
3. The result is the simplified fraction

Example 3: Simplify $\frac{12}{18}$

Step 1: Find the GCF of 12 and 18.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• GCF = 6

Step 2: Divide both by 6:

$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$

Answer: $\frac{2}{3}$

Example 4: Simplify $\frac{35}{50}$

Step 1: Find the GCF of 35 and 50.

• Factors of 35: 1, 5, 7, 35
• Factors of 50: 1, 2, 5, 10, 25, 50
• GCF = 5

Step 2: Divide both by 5:

$\frac{35}{50} = \frac{35 \div 5}{50 \div 5} = \frac{7}{10}$

Answer: $\frac{7}{10}$

Example 5: Simplify $\frac{8}{9}$

Factors of 8: 1, 2, 4, 8. Factors of 9: 1, 3, 9. The only common factor is 1.

Answer: $\frac{8}{9}$ is already in lowest terms.

Step-by-Step Simplification (Divide by Small Primes)

If you do not spot the GCF right away, you can simplify in stages by dividing by small primes (2, 3, 5, 7…) until no more common factors remain.

Example 6: Simplify $\frac{36}{48}$

Both are even — divide by 2:

$\frac{36}{48} = \frac{18}{24}$

Still both even — divide by 2 again:

$\frac{18}{24} = \frac{9}{12}$

Both divisible by 3:

$\frac{9}{12} = \frac{3}{4}$

No more common factors. Answer: $\frac{3}{4}$

(You get the same result in one step: GCF of 36 and 48 is 12, and $\frac{36 \div 12}{48 \div 12} = \frac{3}{4}$.)

How to Check If Two Fractions Are Equivalent

Cross-multiply: if $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent, then $a \times d = b \times c$.

Example 7: Are $\frac{3}{8}$ and $\frac{9}{24}$ equivalent?

$3 \times 24 = 72 \qquad 8 \times 9 = 72$

The cross products are equal, so yes, the fractions are equivalent.

Example 8: Are $\frac{2}{5}$ and $\frac{5}{12}$ equivalent?

$2 \times 12 = 24 \qquad 5 \times 5 = 25$

The cross products are different, so no, these fractions are not equivalent.

Common Equivalent Fractions Reference

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Equivalent fractions represent the same value with different numbers
• Multiply or divide both numerator and denominator by the same number to create equivalent fractions
• To simplify, divide both parts by the GCF
• Cross-multiplication tests whether two fractions are equivalent
• A fraction is in lowest terms when the GCF of numerator and denominator is 1

Return to Arithmetic for more foundational math topics.