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Reduce a Fraction — How to Find, Examples & Rules

Reduce a Fraction

Simplify. That is, cancel out all common factors in the numerator and denominator until no common factors remain.

Key Formula

ab=a÷gcd⁡(a,b)b÷gcd⁡(a,b)\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}
Where:
  • aa = The numerator of the fraction
  • bb = The denominator of the fraction
  • gcd⁡(a,b)\gcd(a,b) = The greatest common divisor (largest factor shared by a and b)

Worked Example

Problem: Reduce the fraction 36/48 to its simplest form.
Step 1: Find the factors of the numerator and denominator. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
Step 2: Identify the greatest common divisor (GCD). The largest number that divides both 36 and 48 is 12.
gcd⁡(36,48)=12\gcd(36, 48) = 12
Step 3: Divide both the numerator and denominator by the GCD.
3648=36÷1248÷12=34\frac{36}{48} = \frac{36 \div 12}{48 \div 12} = \frac{3}{4}
Step 4: Confirm that 3 and 4 share no common factors other than 1. They don't, so the fraction is fully reduced.
Answer: 36/48 reduced to simplest form is 3/4.

Another Example

Problem: Reduce the fraction 18/24 by canceling common factors step by step.
Step 1: Both 18 and 24 are even numbers, so divide each by 2.
1824=18÷224÷2=912\frac{18}{24} = \frac{18 \div 2}{24 \div 2} = \frac{9}{12}
Step 2: Both 9 and 12 are divisible by 3, so divide each by 3.
912=9÷312÷3=34\frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4}
Step 3: 3 and 4 share no common factor other than 1, so the fraction is fully reduced.
Answer: 18/24 reduced to simplest form is 3/4.

Frequently Asked Questions

How do you know when a fraction is fully reduced?
A fraction is fully reduced when the numerator and denominator have no common factor other than 1. In other words, their greatest common divisor is 1. For example, 3/4 is fully reduced because no integer greater than 1 divides both 3 and 4.
Does reducing a fraction change its value?
No. Reducing a fraction produces an equivalent fraction with the same value. You are dividing both the numerator and denominator by the same nonzero number, which is the same as dividing the fraction by 1. For instance, 6/8 and 3/4 both equal 0.75.

Reducing a fraction vs. Finding an equivalent fraction

Reducing always makes the numerator and denominator smaller by dividing out common factors, moving toward the simplest form. Finding an equivalent fraction can go either direction — you might multiply both parts to get a larger equivalent (e.g., converting 1/2 to 3/6) or divide to get a smaller one. Reducing is specifically the process of simplifying to the lowest terms.

Why It Matters

Reduced fractions are easier to compare, add, subtract, and interpret. Many math problems and standardized tests require answers in simplest form. Working with smaller numbers also makes arithmetic less error-prone and helps you spot relationships between quantities more quickly.

Common Mistakes

Mistake: Dividing the numerator and denominator by different numbers instead of the same factor.
Correction: You must always divide both the numerator and denominator by the same number. Dividing by different numbers changes the value of the fraction.
Mistake: Stopping too early and not fully reducing the fraction (e.g., writing 6/8 as the answer instead of continuing to 3/4).
Correction: After each division, check whether the new numerator and denominator still share a common factor. Continue until their only common factor is 1, or use the GCD in a single step to guarantee you reach the simplest form.

Related Terms