Numerator
The top number. When multiplying, the numerators are multiplied together.
Free Math Guide
Multiplying and Dividing Fractions Explained
Learn how to multiply fractions, divide using the reciprocal, simplify before multiplying, and write each answer in simplest form.
Multiplying fractions is different from adding fractions: you do not need a common denominator. Multiply the numerators, multiply the denominators, and simplify. Division adds one important step: rewrite the problem as multiplication by the reciprocal of the second fraction.
Understanding the Parts of a Fraction
A fraction represents equal parts of a whole. The numerator is above the fraction bar, and the denominator is below it.
Denominator
The bottom number. When multiplying, the denominators are multiplied together.
38has numerator 3 and denominator 8
No Common Denominator Is Needed
Unlike addition and subtraction, fraction multiplication and division do not require matching denominators.
How to Multiply or Divide Fractions
Not every problem needs every step. Multiplication problems do not need rewriting, and simplification should only appear when common factors are present.
Rewrite Division as Multiplication
For division only, keep the first fraction, change division to multiplication, and flip the second fraction.
Simplify When Possible
Cancel common factors between any numerator and any denominator. Skip this step when nothing reduces.
Multiply Across
Multiply the remaining numerators together and multiply the remaining denominators together.
Check Simplest Form
If the final numerator and denominator still share a common factor, divide both by their greatest common factor.
How to Multiply Fractions
Multiply straight across: top times top and bottom times bottom. Simplify first when possible so the numbers stay small.
23×45=2 × 43 × 5=815
- Multiply the numerators: 2 × 4 = 8.
- Multiply the denominators: 3 × 5 = 15.
- Check whether 8/15 can be simplified.
- The only common factor of 8 and 15 is 1, so the answer is 8/15.
Multiplication Can Begin Immediately
If no numerator and denominator share a common factor, multiply across without adding an unnecessary simplification step.
How to Divide Fractions Using the Reciprocal
To divide by a fraction, multiply by its reciprocal. Create the reciprocal by switching the numerator and denominator of the second fraction.
34÷25=34×52
Keep the First Fraction
The first fraction remains exactly the same.
Change ÷ to ×
Rewrite the division symbol as multiplication.
Flip the Second Fraction
Switch only the numerator and denominator of the divisor.
Continue as Multiplication
Simplify when possible, then multiply straight across.
Flip Only the Second Fraction
Do not flip the first fraction. The reciprocal is taken only for the fraction you are dividing by.
Simplifying Before Multiplying
Before multiplying, look for common factors between a numerator and a denominator. Divide both numbers by the same factor.
23×910=11×35=35
- 2 and 10 share a factor of 2, so they become 1 and 5.
- 9 and 3 share a factor of 3, so they become 3 and 1.
- Multiply: 1 × 3 = 3 and 1 × 5 = 5.
- The answer is 3/5.
Cancellation Preserves the Product
You may divide a numerator and a denominator by the same common factor because that is equivalent to dividing by 1.
Different Ways to Simplify Before Multiplying
A student does not have to use one fixed cancellation pattern. Common factors may be removed within one fraction, across the two fractions, or through a combination of both.
Simplify Within a Fraction
78×104→78×52
The second fraction was reduced by dividing 10 and 4 by 2. Continue simplifying if another numerator and denominator still share a common factor.
Simplify Across the Fractions
78×104→74×54
The numerator 10 and denominator 8 were divided by 2. The product stays equivalent.
The Final Requirement
Whichever route you choose, no numerator should share a common factor greater than 1 with any denominator before you multiply.
Worked Fraction Examples
Multiplication With Cancellation
35×107=67
- 10 and 5 share a factor of 5.
- Reduce 10 to 2 and 5 to 1.
- Multiply 3 × 2 and 1 × 7.
- The answer is 6/7.
Division With No Cancellation
23÷18=23×81=163
- Keep 2/3.
- Change division to multiplication.
- Flip 1/8 to 8/1.
- No factors cancel, so multiply.
Division With Cancellation
78÷410=78×104=3516
- Flip 4/10 to 10/4.
- Reduce 10 and 8 by 2.
- Multiply 7 × 5 and 4 × 4.
- The answer is 35/16.
Product That Needs Final Reduction
23×38=624=14
This route is correct, but simplifying before multiplying would have avoided the larger intermediate fraction 6/24.
Special Fraction Cases
Multiplying by Zero
05×37= 0
Any number multiplied by zero equals zero.
Dividing by Zero
Division by zero is undefined. A zero fraction cannot be used as the divisor.
A Denominator of 1
91= 9
A fraction with denominator 1 is written as a whole number.
Improper Fractions
A simplified improper fraction is acceptable unless the directions specifically request a mixed number.
How to Check a Fraction Answer
Check the Reciprocal
For division, confirm that only the second fraction was flipped.
Check the Products
Verify that the numerators and denominators were multiplied separately.
Check Equivalent Simplification
Every cancellation must divide one numerator and one denominator by the same factor.
Check Simplest Form
Confirm that the final numerator and denominator share no common factor greater than 1.
Common Fraction Multiplication and Division Mistakes
Looking for a Common Denominator
Common denominators are needed for addition and subtraction, not for multiplication or division.
Flipping Both Fractions
When dividing, keep the first fraction and flip only the second.
Forgetting to Change Division to Multiplication
After taking the reciprocal, the operation must become multiplication.
Cancelling Two Numerators or Two Denominators
Cancellation must pair a numerator with a denominator.
Stopping Before All Factors Are Removed
Continue simplifying until no numerator shares a common factor greater than 1 with any denominator.
Simplifying When Nothing Reduces
If no common factor exists, move directly to multiplication instead of creating an unnecessary step.
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