Dividing Fractions
Dividing fractions uses the Keep-Change-Flip (invert and multiply) rule: keep the first fraction, change division to multiplication, and flip the second fraction to its reciprocal. In Grade 6 Saxon Math Course 1, a/b ÷ c/d = a/b × d/c. For example, ½ ÷ ¼ = ½ × 4/1 = 4/2 = 2, meaning half a pizza divided into quarter-pizza portions gives 2 portions. This rule converts every fraction division problem into a multiplication problem that follows the familiar numerator-times-numerator pattern.
Key Concepts
Property To divide by a fraction, multiply by its reciprocal. This is often remembered as 'invert and multiply'. $$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $$.
Examples $$ \frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3} $$ $$ \frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{5}{6} $$ $$ \frac{4}{7} \div \frac{2}{3} = \frac{4}{7} \times \frac{3}{2} = \frac{12}{14} = \frac{6}{7} $$.
Explanation Dividing fractions seems tricky, but it’s just asking 'how many of the second fraction fit into the first?' The secret is to flip the second fraction (find its reciprocal) and multiply. This works because you first find how many of your divisor sized pieces fit into one whole, and then you multiply to see how many fit into your original amount.
Common Questions
What is the Keep-Change-Flip rule for dividing fractions?
Keep the first fraction unchanged. Change ÷ to ×. Flip the second fraction (use its reciprocal). Then multiply.
Calculate 3/4 ÷ 3/8.
Keep 3/4, change to ×, flip 3/8 to 8/3: (3/4) × (8/3) = 24/12 = 2.
Why does dividing by a fraction equal multiplying by its reciprocal?
Because (a/b) ÷ (c/d) = (a/b) × (d/c) is a property proven from the definition of division and multiplicative inverse.
What is 2 ÷ 1/3?
2/1 × 3/1 = 6. Dividing 2 by 1/3 asks how many thirds fit in 2 — the answer is 6.
Can you cancel before multiplying when dividing fractions?
Yes. Once you rewrite as multiplication, cancel common factors between any numerator and any denominator before multiplying to keep numbers small.