Canceling Common Factors
Canceling common factors before multiplying fractions simplifies the calculation by dividing a numerator of one fraction and any denominator by a shared factor before performing the multiplication. In Grade 6 Saxon Math Course 1 (Chapter 7: Fractions and Geometric Concepts), students recognize that any factor appearing in both the numerator and denominator can be divided out (equals 1 when divided by itself). For (5 x 4) / (8 x 15): factor as (5 x 4) / (8 x 15), cancel 5 with 15 (giving 3) and 4 with 8 (giving 2), leaving 1 / (2 x 3) = 1/6. This avoids large intermediate numbers.
Key Concepts
Property Any prime factor that appears in both the numerator and the denominator can be canceled. This is because any number divided by itself, such as $ \frac{5}{5} $, is equal to 1.
Examples $$ \frac{48}{400} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3}{2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5} = \frac{\cancel{2\cdot2\cdot2\cdot2} \cdot 3}{\cancel{2\cdot2\cdot2\cdot2} \cdot 5 \cdot 5} = \frac{3}{25} $$ $$ \frac{125}{500} = \frac{5 \cdot 5 \cdot 5}{2 \cdot 2 \cdot 5 \cdot 5 \cdot 5} = \frac{\cancel{5\cdot5\cdot5}}{2 \cdot 2 \cdot \cancel{5\cdot5\cdot5}} = \frac{1}{4} $$ $$ \frac{3 \cdot \cancel{5} \cdot \cancel{5} \cdot \cancel{5}}{2 \cdot 2 \cdot 2 \cdot \cancel{5} \cdot \cancel{5} \cdot \cancel{5}} = \frac{3}{2 \cdot 2 \cdot 2} = \frac{3}{8} $$.
Explanation Think of common factors in a fraction as a tug of war. A '5' on top pulls against a '5' on the bottom, and they cancel each other out! This works because $ \frac{5}{5} $ is just 1. Wiping out these matching pairs makes big fractions simple without changing their overall value.
Common Questions
What is canceling common factors in fraction multiplication?
Before multiplying, divide a numerator of one fraction and a denominator of either fraction by a shared factor. This simplifies the calculation without changing the result.
Simplify (3/8) x (4/9) by canceling before multiplying.
Cancel 3 and 9 (divide both by 3: gives 1 and 3). Cancel 4 and 8 (divide both by 4: gives 1 and 2). Result: 1/(2 x 3) = 1/6.
Why does canceling not change the final answer?
You are dividing a numerator and a denominator by the same factor, which equals dividing the overall expression by 1. The value is unchanged.
Can you cancel a factor from the same fraction or must it be diagonal?
You can cancel any numerator with any denominator in the entire multiplication, not just within the same fraction.
Multiply (5/6) x (3/10) using canceling.
Cancel 5 and 10 (divide by 5: gives 1 and 2). Cancel 3 and 6 (divide by 3: gives 1 and 2). Result: (1 x 1)/(2 x 2) = 1/4.