Dividing Fractions with Canceling
Dividing fractions with canceling combines the Keep-Change-Flip rule with pre-multiplication simplification. In Grade 6 Saxon Math Course 1, students must first rewrite the division as multiplication (flip the second fraction), then cancel common factors between any numerator and any denominator before multiplying. For (3/4) ÷ (9/8) = (3/4) × (8/9): cancel 3 with 9 (÷3) and 4 with 8 (÷4) to get (1/1) × (2/3) = 2/3. Canceling before multiplying keeps numbers small and avoids simplifying a large fraction at the end.
Key Concepts
Property A division problem must be rewritten as a multiplication problem before you can cancel. To divide, multiply the first fraction by the reciprocal of the second fraction. $$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $$.
Examples Example 1: $$\frac{5}{8} \div \frac{5}{4} = \frac{5}{8} \times \frac{4}{5} = \frac{\stackrel{1}{\cancel{5}}}{\underset{2}{\cancel{8}}} \times \frac{\stackrel{1}{\cancel{4}}}{\cancel{5} 1} = \frac{1}{2}$$ Example 2: $$\frac{8}{9} \div \frac{2}{3} = \frac{8}{9} \times \frac{3}{2} = \frac{\stackrel{4}{\cancel{8}}}{\underset{3}{\cancel{9}}} \times \frac{\stackrel{1}{\cancel{3}}}{\underset{1}{\cancel{2}}} = \frac{4}{3} = 1\frac{1}{3}$$ Example 3: $$3\frac{1}{3} \div 1\frac{2}{3} = \frac{10}{3} \div \frac{5}{3} = \frac{10}{3} \times \frac{3}{5} = \frac{\stackrel{2}{\cancel{10}}}{\underset{1}{\cancel{3}}} \times \frac{\stackrel{1}{\cancel{3}}}{\cancel{5} 1} = 2$$.
Explanation Remember: you cannot cancel in a division problem directly! It's an exclusive club for multiplication only. To get your problem on the guest list, you have to use the 'keep, change, flip' method. Flip the second fraction to find its reciprocal, change the sign to multiply, and then you're officially invited to the canceling party.
Common Questions
Why must you rewrite as multiplication before canceling?
Canceling is only valid in multiplication. Division must first be converted to multiplication via Keep-Change-Flip, then you cancel across numerators and denominators.
Calculate (3/4) ÷ (9/8) with canceling.
Rewrite: (3/4) × (8/9). Cancel 3 and 9 by 3: 1/4 × 8/3. Cancel 4 and 8 by 4: 1/1 × 2/3 = 2/3.
Can you cancel before flipping the second fraction?
No. The canceling step happens after the problem is rewritten as multiplication. Canceling in the original division form is not valid.
What does canceling mean in fraction multiplication?
Dividing a numerator and any denominator by their GCF, simplifying before multiplying and keeping numbers smaller.
Calculate (5/6) ÷ (10/3) with canceling.
Rewrite: (5/6) × (3/10). Cancel 5 and 10 by 5: 1/6 × 3/2. Cancel 6 and 3 by 3: 1/2 × 1/2 = 1/4.