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|)Why dividing both parts preserves the amount
The same half can be partitioned into two quarters or three sixths. The number of pieces changes, but the shaded share does not.
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.
Worked example 1
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
Try a reduction without guessing
Find the greatest common divisor first, then divide once.
1. Reduce 150/210.
Hint: The greatest common divisor is 30.
Answer: 150/210 = 5/7
2. Reduce 0/24.
Hint: Zero divided by any nonzero denominator is zero.
Answer: 0/24 = 0
3. Can 18/24 be reduced by 6?
Hint: Check that 6 divides both 18 and 24.
Answer: Yes. 18/24 = 3/4
Checks before calling a fraction simplest
- 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.
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.
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.