Numerator
The numerator is the top number. It tells how many equal parts are being counted.
Free Math Guide
Simplifying Fractions Explained
Learn how to reduce fractions by finding the greatest common factor, dividing the numerator and denominator by the same number, and checking that the answer is in simplest form.
Simplifying a fraction means rewriting it with smaller numbers without changing its value. You do this by dividing the numerator and denominator by the same common factor. The most efficient method is to divide both numbers by their greatest common factor.
Understanding the Parts of a Fraction
A fraction represents equal parts of a whole. The top number is called the numerator, and the bottom number is called the denominator.
Denominator
The denominator is the bottom number. It tells how many equal parts make one whole.
6
8
means 6 out of 8 equal parts
The Denominator Cannot Be Zero
A fraction cannot have zero as its denominator because division by zero is undefined.
What Does It Mean to Simplify a Fraction?
Simplifying a fraction means rewriting it with the smallest possible whole-number numerator and denominator while keeping the same value.
6
8
=
3
4
The fractions 6/8 and 3/4 represent the same amount. The second fraction uses smaller numbers, so it is the simplified form.
Simplifying Does Not Change the Value
Dividing the numerator and denominator by the same nonzero number creates an equivalent fraction.
Understanding Common Factors
A factor is a whole number that divides evenly into another number. A common factor divides evenly into both the numerator and denominator.
Factors of 12
1, 2, 3, 4, 6, 12
Each of these numbers divides evenly into 12.
Factors of 18
1, 2, 3, 6, 9, 18
Each of these numbers divides evenly into 18.
Common Factors of 12 and 18
The factors shared by 12 and 18 are 1, 2, 3, and 6.
Finding the Greatest Common Factor
The greatest common factor, or GCF, is the largest whole number that divides evenly into both the numerator and denominator.
GCF(12, 18) = 6
Since 6 is the greatest common factor, dividing both 12 and 18 by 6 simplifies the fraction in one step.
List the Factors
Write the factors of both numbers and select the largest factor they share.
Test Divisibility
Check large possible factors first to find the greatest number that divides both values evenly.
Why Use the GCF?
Dividing by the GCF reduces the fraction completely in one step. A smaller common factor may require several steps.
How to Simplify a Fraction
Use these steps to reduce a fraction to simplest form.
Find the Greatest Common Factor
Determine the largest whole number that divides evenly into both the numerator and denominator.
Divide the Numerator
Divide the numerator by the greatest common factor.
Divide the Denominator
Divide the denominator by the same greatest common factor.
Check Simplest Form
Confirm that the new numerator and denominator share no common factor greater than 1.
Simplifying 12/18
First find the greatest common factor of 12 and 18. Their greatest common factor is 6.
12
18
÷
6
6
=
2
3
- The factors of 12 are 1, 2, 3, 4, 6, and 12.
- The factors of 18 are 1, 2, 3, 6, 9, and 18.
- The greatest common factor is 6.
- Divide the numerator: 12 ÷ 6 = 2.
- Divide the denominator: 18 ÷ 6 = 3.
- The simplified fraction is 2/3.
- The only common factor of 2 and 3 is 1.
Final Answer
The fraction 12/18 simplifies to 2/3.
Simplifying in Several Steps
You can simplify a fraction by dividing by smaller common factors more than once. This method works, but it may take longer than dividing by the GCF.
24
36
=
12
18
=
6
9
=
2
3
- Divide 24 and 36 by 2 to get 12/18.
- Divide 12 and 18 by 2 to get 6/9.
- Divide 6 and 9 by 3 to get 2/3.
- The fraction 2/3 is in simplest form.
Faster Method
The GCF of 24 and 36 is 12, so you could divide both numbers by 12 and reach 2/3 immediately.
Fractions Already in Simplest Form
A fraction is already simplified when the numerator and denominator share no common factor greater than 1.
Example: 5/7
5
7
The only common factor of 5 and 7 is 1, so 5/7 is already in simplest form.
Example: 8/15
8
15
The factors of 8 and 15 have only 1 in common, so the fraction cannot be reduced.
A GCF of 1 Means Simplest Form
When the greatest common factor of the numerator and denominator is 1, the fraction is fully simplified.
Simplifying Improper Fractions
An improper fraction has a numerator that is greater than or equal to its denominator. Simplify it using the same GCF process.
18
12
÷
6
6
=
3
2
- The greatest common factor of 18 and 12 is 6.
- Divide 18 by 6 to get 3.
- Divide 12 by 6 to get 2.
- The simplified improper fraction is 3/2.
Simplified Form Versus Mixed Number Form
The fraction 3/2 is simplified. It may also be written as the mixed number 1 1/2 when the problem asks for a mixed number.
Special Fraction Cases
A Zero Numerator
0
8
= 0
Any fraction with a numerator of zero and a nonzero denominator equals zero.
A Denominator of 1
7
1
= 7
A fraction with a denominator of 1 is equal to its numerator as a whole number.
Numerator Equals Denominator
9
9
= 1
A nonzero number divided by itself equals 1.
Denominator of Zero
A denominator of zero is not allowed because division by zero is undefined.
How to Check a Simplified Fraction
Check the GCF
The simplified numerator and denominator should have a greatest common factor of 1.
Check Equivalent Values
The original and simplified fractions should represent the same value.
Check Both Divisions
Make sure you divided the numerator and denominator by the same number.
Multiply Back
Multiply the simplified numerator and denominator by the GCF to recover the original fraction.
2
3
×
6
6
=
12
18
Common Simplifying Fractions Mistakes
Dividing Only the Numerator
You must divide both the numerator and denominator by the same number to keep the fraction equivalent.
Dividing by Different Numbers
Dividing the numerator and denominator by different numbers changes the value of the fraction.
Using a Number That Is Not a Common Factor
The divisor must divide evenly into both the numerator and denominator.
Stopping Too Early
A fraction is not fully simplified if the new numerator and denominator still share a factor greater than 1.
Thinking the Numbers Must Become Smaller by Subtraction
Simplifying fractions uses division, not subtraction. Subtracting the same number from the numerator and denominator does not preserve the fraction's value.
Changing a Fraction That Is Already Simplified
If the GCF is 1, the fraction is already in simplest form.
Forgetting Special Cases
Write 0 divided by a nonzero number as 0, and write a fraction with denominator 1 as a whole number when appropriate.
Ready to Practice?
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