Dividing a Mixed Number by a Fraction
Dividing a Mixed Number by a Fraction is a Grade 6 math skill from Big Ideas Math, Course 1, Chapter 2: Fractions and Decimals. The procedure: convert the mixed number to an improper fraction, then multiply by the reciprocal of the divisor (keep-change-flip). Example: 2 1/4 ÷ 3/4 = 9/4 × 4/3 = 36/12 = 3. Another: 1 3/5 ÷ 2/5 = 8/5 × 5/2 = 40/10 = 4. This skill extends fraction division and is essential for solving proportion problems and unit rate calculations in Grade 6 and 7.
Key Concepts
Property To divide a mixed number by a fraction, first convert the mixed number to an improper fraction. Then, multiply the improper fraction by the reciprocal of the divisor. $$A \frac{b}{c} \div \frac{d}{e} = \frac{A \times c + b}{c} \times \frac{e}{d}$$.
Examples $3 \frac{1}{2} \div \frac{1}{4} = \frac{7}{2} \div \frac{1}{4} = \frac{7}{2} \times \frac{4}{1} = \frac{28}{2} = 14$ $2 \frac{2}{5} \div \frac{3}{4} = \frac{12}{5} \div \frac{3}{4} = \frac{12}{5} \times \frac{4}{3} = \frac{48}{15} = \frac{16}{5} = 3 \frac{1}{5}$.
Explanation This process combines your knowledge of mixed numbers and fraction division. The first step is always to convert the mixed number dividend into an improper fraction. Once you have two fractions, you apply the rule for division by multiplying by the reciprocal of the second fraction. Finally, simplify your answer and convert it back into a mixed number if it is an improper fraction.
Common Questions
How do you divide a mixed number by a fraction?
Step 1: Convert the mixed number to an improper fraction. Step 2: Flip the divisor fraction (find its reciprocal). Step 3: Multiply the two fractions. Step 4: Simplify. Example: 2 1/4 ÷ 3/4 = 9/4 × 4/3 = 36/12 = 3.
What is the keep-change-flip rule for division?
When dividing fractions, Keep the first fraction as-is, Change the division sign to multiplication, Flip the second fraction (take its reciprocal). Example: 9/4 ÷ 3/4 → Keep 9/4, Change to ×, Flip 3/4 to 4/3 → 9/4 × 4/3 = 36/12 = 3.
Why does dividing by a fraction equal multiplying by its reciprocal?
Division asks 'how many of this fit into that?' Dividing by 1/2 asks how many halves fit into the number. A number always contains twice as many halves as wholes, so dividing by 1/2 doubles the number — same as multiplying by 2 (the reciprocal of 1/2).
What is a worked example of mixed number divided by fraction?
1 3/5 ÷ 2/5: Convert 1 3/5 = 8/5. Keep 8/5, change ÷ to ×, flip 2/5 to 5/2: 8/5 × 5/2 = 40/10 = 4. Check: 4 × 2/5 = 8/5 = 1 3/5. Correct.
When do Grade 6 students learn to divide mixed numbers by fractions?
This skill is in Big Ideas Math, Course 1, Chapter 2: Fractions and Decimals, as part of comprehensive Grade 6 fraction operations including multiplication and all forms of division.
What mistakes do students make when dividing mixed numbers by fractions?
Common errors: forgetting to convert the mixed number to an improper fraction first, flipping the wrong fraction (should flip the divisor, not the dividend), and forgetting to simplify the final answer.