Arithmetic
How to Simplify Fractions Step by Step
A fraction is in simplest form when its numerator and denominator share no positive factor other than 1. Divide both by their greatest common divisor to reduce the fraction without changing its value.
Reviewed
Why reducing a fraction preserves its value
The numerator and denominator describe how many equal parts are selected and how many make one whole. Dividing both by the same nonzero number only changes the names and sizes of the parts; it does not change the amount represented.
For example, 12/18 and 2/3 mark the same point on a number line. Each group of six eighteenth-sized parts can be treated as one third-sized part.
a/b = (a / g) / (b / g), where g = gcd(|a|, |b|)A dependable reduction method
Find the greatest common divisor of the absolute values of the numerator and denominator. Divide both numbers by it, then place any negative sign in the numerator. If the greatest common divisor is 1, the fraction is already reduced.
- List factors or use the Euclidean algorithm to find the greatest common divisor.
- Divide the numerator and denominator by that divisor.
- Check that the new numerator and denominator have no common factor greater than 1.
- Verify the value by cross-multiplying the original and reduced fractions.
Handling zero and negative signs
A zero numerator reduces to 0 as long as the denominator is nonzero. A zero denominator is never allowed because division by zero is undefined.
The fractions -3/5, 3/-5, and -(3/5) have the same value. Writing the sign in the numerator gives a consistent final form.
Worked examples
Reduce 84/126
Both numbers are divisible by several common factors, so use the greatest one.
gcd(84, 126) = 4284 / 42 = 2126 / 42 = 3
Result: 84/126 = 2/3
Reduce -45/60
Keep the sign while reducing the absolute values.
gcd(45, 60) = 15-45 / 15 = -360 / 15 = 4
Result: -45/60 = -3/4
Common mistakes
- Dividing only the numerator or only the denominator changes the value.
- Stopping after one common factor even though the new numbers still share another factor.
- Treating a denominator of zero as a fraction that can be reduced.
Where this idea is used
- Reporting exact answers without unnecessarily large numbers.
- Comparing fractions and finding equivalent ratios.
- Preparing a fraction for addition, multiplication, or conversion to a mixed number.
Questions about simplifying fractions
Can every fraction be simplified?
Every valid fraction can be written in lowest terms, but a fraction whose numerator and denominator are already coprime will not get smaller.
Does simplifying change a fraction's decimal value?
No. Equivalent fractions produce the same quotient, aside from ordinary limits in decimal rounding.
What happens if the numerator is larger than the denominator?
Reduce it in the same way. You may then leave it as an improper fraction or convert it to a mixed number.