How to divide fractions: keep, change, flip

Dividing asks "how many of these fit into that?" How many quarters fit into 3? Twelve, because each whole holds 4 quarters: $3 \div \tfrac{1}{4} = 3 \times 4 = 12$. Dividing by $\tfrac{1}{4}$ is the same as multiplying by 4, and flipping $\tfrac{1}{4}$ gives $\tfrac{4}{1}$.

Convert to improper fractions first:

Write the whole number over 1:

• $\tfrac{3}{4} \div 6 = \tfrac{3}{4} \times \tfrac{1}{6} = \tfrac{3}{24} = \tfrac{1}{8}$
• $6 \div \tfrac{3}{4} = \tfrac{6}{1} \times \tfrac{4}{3} = \tfrac{24}{3} = 8$

A ribbon is $4\tfrac{1}{2}$ m long. How many pieces of $\tfrac{3}{4}$ m can be cut from it?

$4\tfrac{1}{2} \div \tfrac{3}{4} = \tfrac{9}{2} \times \tfrac{4}{3} = \tfrac{36}{6} = 6$ pieces.

"How many … in …" and "share … into pieces of …" usually mean divide.

• Flipping the first fraction instead of the second.
• Flipping both fractions.
• Forgetting to convert mixed numbers.
• Dividing tops and bottoms directly. It can work ($\tfrac{6}{8} \div \tfrac{3}{4} = \tfrac{2}{2}$) but often gives awkward fractions; keep, change, flip always works.

(a) $\tfrac{3}{5} \div \tfrac{1}{2}$, (b) $\tfrac{7}{8} \div \tfrac{7}{4}$, (c) $3\tfrac{1}{3} \div \tfrac{5}{6}$. Answers: $\tfrac{6}{5}$, $\tfrac{1}{2}$, 4. Check with the fraction calculator.

When the fractions have the same denominator, you can just divide the numerators:

It works because you're asking how many $\tfrac{2}{7}$s fit into $\tfrac{6}{7}$, which is the same as how many 2s fit into 6. Keep, change, flip gives the same answer: $\tfrac{6}{7} \times \tfrac{7}{2} = \tfrac{42}{14} = 3$.

Draw a bar for each whole and split it into the fraction you're dividing by. For $2 \div \tfrac{1}{3}$, split two bars into thirds and count them: there are 6. That's why $2 \div \tfrac{1}{3} = 2 \times 3 = 6$. Drawing a model is a good way to check that an answer is sensible, and it's how many primary schools first teach fraction division.

The method doesn't change when there are letters:

Cancel common factors before multiplying to keep it simple: 9 and 3 share 3, and 2 and 4 share 2, which gives $\tfrac{x}{1} \times \tfrac{3}{2} = \tfrac{3x}{2}$. For more algebra practice, see expanding brackets.