Multiply Fractions Cross Cancel: How It Works

Yes, you can cross-cancel when multiplying fractions. Cross-cancellation is a shortcut that simplifies fractions before you multiply, making the calculation easier and the final answer simpler. To use it, you divide any numerator and any denominator by a common factor before performing the multiplication.

Formula

Without cross-cancelling, the product of two fractions is:

(a/b) × (c/d) = (a × c) / (b × d)

Where:

• a = numerator of the first fraction
• b = denominator of the first fraction (b ≠ 0)
• c = numerator of the second fraction
• d = denominator of the second fraction (d ≠ 0)

Cross-cancelling applies the same rules but simplifies first. You divide a numerator and a denominator by their greatest common divisor (GCD) before multiplying. This works because dividing both a numerator and a denominator by the same number is equivalent to dividing the entire fraction by 1.

Worked Example

Using the default values from the calculator: (3/4) × (2/9).

1. Write the multiplication: (3 × 2) / (4 × 9).
2. Cross-cancel: 3 and 9 share a factor of 3. Divide both by 3: 3 becomes 1, 9 becomes 3. The problem is now (1 × 2) / (4 × 3).
3. Cross-cancel again: 2 and 4 share a factor of 2. Divide both by 2: 2 becomes 1, 4 becomes 2. The problem is now (1 × 1) / (2 × 3).
4. Multiply straight across: (1 × 1) = 1, (2 × 3) = 6. The answer is 1/6.

The calculator gives the same result: 1/6.

Why Cross-Cancelling Works

Cross-cancelling is valid because of the commutative property of multiplication. When you multiply fractions, you are multiplying all numerators together and all denominators together. Dividing a numerator and a denominator by the same number before multiplying is the same as dividing the entire fraction by that number (which equals 1) after multiplying. For example, (3/4) × (2/9) = (3×2)/(4×9) = 6/36 = 1/6. Cross-cancelling just does the division earlier, avoiding large numbers.

You can cancel any numerator with any denominator, not just the ones that are diagonally across from each other. For instance, in (2/3) × (3/4), you can cancel the 3 in the first denominator with the 3 in the second numerator. The result is (2/1) × (1/4) = 2/4 = 1/2. The key is that you must divide a numerator and a denominator by the same factor.

Common Mistakes and How to Avoid Them

• Cancelling within a single fraction: You cannot cancel a numerator with its own denominator unless you are simplifying that fraction alone. For example, in (2/3) × (4/5), you cannot cancel 2 and 3 because they are in the same fraction. Only cross-cancel between different fractions.
• Forgetting to simplify after multiplying: Even with cross-cancelling, the final answer may still be reducible. Always check if the resulting fraction can be simplified further.
• Mixing up numerators and denominators: Cross-cancelling only works between a numerator and a denominator. You cannot cancel a numerator with another numerator or a denominator with another denominator.
• Not dividing by the greatest common factor: You can cancel by any common factor, but using the greatest common factor saves steps. For example, in (4/9) × (3/8), you could cancel 4 and 8 by 2, then 3 and 9 by 3, but using the GCF directly is faster.

When to Use Cross-Cancelling

Cross-cancelling is always optional. It is most helpful when you are multiplying two or more fractions and the numbers are large. It reduces the size of the numbers you multiply, which lowers the chance of arithmetic errors and makes mental math easier. It is also useful when you are multiplying mixed numbers (after converting them to improper fractions) or when you are solving equations that involve fraction multiplication.

However, cross-cancelling is not necessary. You can always multiply straight across and simplify at the end. The final answer will be the same either way. Some people prefer to multiply first and simplify later, especially when they are using a calculator. The choice is a matter of personal preference and the specific numbers involved.

Cross-Cancelling with More Than Two Fractions

Cross-cancelling works with any number of fractions. For example, (2/3) × (4/5) × (5/6). You can cancel the 5 in the second denominator with the 5 in the third numerator, and the 3 in the first denominator with the 6 in the third denominator (dividing by 3). You can also cancel the 2 in the first numerator with the 4 in the second numerator? No, that is not allowed because both are numerators. You can cancel 2 with 6 (a denominator) if you wish. The rule remains: any numerator with any denominator.

When multiplying three or more fractions, cross-cancel systematically. Write all numerators in a row and all denominators in a row, then cancel common factors across the two rows. This approach prevents missing a possible cancellation.

Frequently Asked Questions

Can you cross-cancel when adding or subtracting fractions?

No. Cross-cancelling only applies to multiplication (and division, when you convert to multiplication by the reciprocal). For addition or subtraction, you must find a common denominator. Cross-cancelling there would change the value of the fractions.

What if there is no common factor between any numerator and denominator?

If no numerator and denominator share a common factor greater than 1, then cross-cancelling is not possible. You simply multiply straight across and simplify the result if possible. For example, (2/3) × (5/7) has no common factors, so the product is 10/21, which is already in simplest form.

Does cross-cancelling work with mixed numbers?

Yes, but you must first convert mixed numbers to improper fractions. For example, 1 1/2 × 2/3 becomes (3/2) × (2/3). Then you can cross-cancel: the 2 in the first denominator cancels with the 2 in the second numerator, and the 3 in the first numerator cancels with the 3 in the second denominator, leaving 1/1 × 1/1 = 1.

Is cross-cancelling the same as simplifying?

Cross-cancelling is a form of simplifying that happens before multiplication. Simplifying a fraction means dividing its numerator and denominator by a common factor. Cross-cancelling does that between two different fractions. The result is the same as simplifying the final product, but it often involves smaller numbers.

Why is cross-cancelling taught in schools?

It is taught because it reduces the size of the numbers you have to multiply, which makes calculations faster and less error-prone, especially when working without a calculator. It also reinforces the concept of equivalent fractions and the idea that you can divide by 1 in any form.

Can cross-cancelling be used with variables?

Yes, the same rules apply to algebraic fractions. For example, (x/2) × (4/y) can be cross-cancelled: 2 and 4 share a factor of 2, giving (x/1) × (2/y) = 2x/y. This is a common technique in algebra to simplify expressions before multiplying.

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in a | Numerator of first fraction | - | 3 | 0
in b | Denominator of first fraction | - | 4 | 0
in c | Numerator of second fraction | - | 2 | 0
in d | Denominator of second fraction | - | 9 | 0
out result | Simplified product | - | (a / gcd(a, d)) * (c / gcd(c, b)) / ((b / gcd(c, b)) * (d / gcd(a, d))) | 0