Multiplying Fractions by Cancelling- Step-by-Step Method

What Is Cancelling in Fraction Multiplication?

When you multiply fractions, you multiply straight across: numerator times numerator, denominator times denominator. Nothing complicated there.

The problem is you often end up with ugly numbers that need reducing. Cancelling (also called cross-cancelling) lets you simplify before you multiply. Same answer, less work.

You don't need to wait until the end to simplify. You can do it in the middle.

How Cancelling Actually Works

Mathematically, you're dividing one factor from the numerator by one factor from the denominator. This works because multiplication and division are inverse operations—they cancel each other out.

The rule: any numerator can cancel with any denominator. You don't have to match them up in any particular order.

Step-by-Step: Cancelling When Multiplying Fractions

Here's the process:

• Step 1: Write both fractions with their full numerators and denominators
• Step 2: Look for numbers that divide evenly into both a numerator and a denominator
• Step 3: Divide both by that common factor
• Step 4: Repeat until no more cancelling makes sense
• Step 5: Multiply what's left across

Example 1: Basic Cancellation

Multiply ⅔ × ¾

Look at 2 (numerator of first fraction) and 4 (denominator of second). They share a factor of 2.

2 ÷ 2 = 1

4 ÷ 2 = 2

Now you have 1/3 × 3/2

Still see a common factor? 3 and 3. Cancel them.

3 ÷ 3 = 1

3 ÷ 3 = 1

Result: 1 × 1 = 1

Answer: 1/1 = 1

Do it the old way and you get 6/12, which reduces to 1/2. Wait—that's different. Let me redo this.

Actually, with proper cancelling:

2/3 × 3/4

Cancel 2 and 4: 2÷2=1, 4÷2=2

Cancel 3 and 3: 3÷3=1, 3÷3=1

1/1 × 1/2 = 1/2 ✓

Example 2: With Larger Numbers

Multiply 8/9 × 3/4

8 and 4 share a factor of 4:

8 ÷ 4 = 2

4 ÷ 4 = 1

3 and 9 share a factor of 3:

3 ÷ 3 = 1

9 ÷ 3 = 3

Result: 2/3 × 1/1 = 2/3

Skip cancelling and you'd get 24/36, then reduce. Same answer, more steps.

Example 3: Multiple Cancellations

Multiply 12/15 × 5/8 × 2/3

You can cancel across any numerator with any denominator, not just adjacent fractions.

12 and 8 share 4: 12÷4=3, 8÷4=2

15 and 5 share 5: 15÷5=3, 5÷5=1

3 and 3 share 3: 3÷3=1, 3÷3=1

Now you have: 3/3 × 1/2 × 1/1

3/3 = 1, so: 1 × 1/2 × 1 = 1/2

Answer: 1/2

Cancelling vs. Simplifying After: What's Better?

Here's a comparison if you're doing it the traditional way versus cancelling first:

The cancelling method keeps your numbers manageable. With 3+ fractions involved, this matters a lot.

Common Mistakes That Blow the Answer

Mistake 1: Cancelling across addition

You can only cancel when multiplying. ½ + ¼ does NOT let you cancel the 2 and 4. That's addition, not multiplication.

Mistake 2: Forgetting to cancel

Results in unnecessarily large numbers. You still get the right answer if you reduce at the end, but you're making extra work for yourself.

Mistake 3: Cancelling incorrectly

When you cancel, divide both the numerator and denominator by the same number. Don't divide one and forget the other.

Mistake 4: Cancelling a numerator with a denominator that doesn't exist

You can't cancel if there's no denominator to match. Make sure you're working with actual fractions, not whole numbers masquerading as fractions.

How to Get Started

Pick a problem from your homework or textbook. Don't reach for a calculator.

Step 1: Write out the problem with all numerators and denominators visible

Step 2: Find one numerator and one denominator that share a factor (look at small primes first: 2, 3, 5)

Step 3: Draw a line through both numbers and write the reduced values

Step 4: Check if anything else can cancel

Step 5: Multiply what's left

Practice with 10 problems. By problem 5, it'll feel automatic.

When Cancelling Doesn't Help Much

If all your numerators and denominators are prime numbers with no overlap, cancelling won't do anything. You'll multiply across and that's it.

Example: 3/7 × 5/11 × 2/13

No common factors exist. Multiplying gives 30/1001. No reduction needed because 30 and 1001 share no factors.

In these cases, just multiply straight through. Cancelling only helps when there's actually something to cancel.

The Bottom Line

Cancelling before you multiply fractions isn't some trick or shortcut—it's just simplifying early instead of late. Your answer ends up the same. The difference is you avoid wrestling with large numbers.

Do it every time. It'll save you minutes on tests and keep your scratch paper clean.