Cross canceling, also known as simplifying before multiplying, is a technique used when multiplying fractions to make the arithmetic easier. It involves simplifying fractions diagonally by dividing the numerator of one fraction and the denominator of the other by their greatest common factor (GCF).
Here's a breakdown of how to cross cancel:
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Set up the problem: Write out the multiplication problem with the two fractions side by side. For example: (2/5) * (15/8)
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Identify common factors diagonally: Look for common factors between the numerator of the first fraction and the denominator of the second fraction, and between the denominator of the first fraction and the numerator of the second fraction. In our example:
- 2 and 8 share a common factor of 2.
- 5 and 15 share a common factor of 5.
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Divide by the greatest common factor (GCF): Divide both the numerator and denominator by their respective GCFs.
- 2 ÷ 2 = 1. Replace the 2 in (2/5) with 1.
- 8 ÷ 2 = 4. Replace the 8 in (15/8) with 4.
- 5 ÷ 5 = 1. Replace the 5 in (2/5) with 1.
- 15 ÷ 5 = 3. Replace the 15 in (15/8) with 3.
The problem now looks like this: (1/1) * (3/4)
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Multiply the simplified fractions: Multiply the new numerators and the new denominators.
- 1 * 3 = 3
- 1 * 4 = 4
Therefore, (1/1) (3/4) = 3/4. So (2/5) (15/8) = 3/4
Why does cross canceling work?
Cross canceling is essentially simplifying the fractions before multiplying, which is mathematically equivalent to simplifying the result after multiplying. It leverages the associative and commutative properties of multiplication.
Example:
Let's work through the example again: (2/5) * (15/8)
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Identify GCFs:
- GCF(2, 8) = 2
- GCF(5, 15) = 5
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Divide:
- 2/2 = 1
- 8/2 = 4
- 5/5 = 1
- 15/5 = 3
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Rewrite:
(1/1) * (3/4) -
Multiply:
(1 3) / (1 4) = 3/4
In summary, cross canceling is a shortcut for simplifying fractions when multiplying. It involves dividing diagonally by common factors before multiplying the numerators and denominators together.
