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Divide Fractions
by Fractions

Master the art of dividing fractions using the Keep-Change-Flip method, visual models, and real-world problem solving.

6th Grade Math

What Does Dividing Fractions Mean?

When you divide by a fraction, you are asking: "How many groups of this fraction fit inside that fraction?"

This concept appears everywhere: splitting food equally, measuring land, and calculating recipes. Let's build a toolkit to solve these problems confidently.

Seeing It with a Visual Model

Let's visualize \(\frac{1}{2} \div \frac{1}{4}\). How many quarter-sized pieces fit inside one half?

\(\frac{1}{2}\)
 other half 1 2 3 4 2 quarter-pieces fit in one half!

Why Does Flipping Work?

Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of a fraction is that fraction flipped upside down.

Reciprocal
The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). You swap the numerator and denominator.
Why It Works
Dividing by \(\frac{c}{d}\) undoes "\(\frac{c}{d}\) of something." Multiplying by \(\frac{d}{c}\) is the exact inverse operation.

The Keep-Change-Flip Rule

KEEP the first fraction
\(\frac{a}{b}\)
 CHANGE division to multiplication FLIP the second fraction
\(\frac{d}{c}\)
 Result:
\(\frac{a}{b} \div \frac{c}{d} = \frac{a \times d}{b \times c}\)

The General Formula

You can verify any answer by checking: does the divisor times the quotient equal the dividend?

Multiplying vs Dividing Fractions

Multiplying Dividing
Operation \(\frac{a}{b} \times \frac{c}{d}\) \(\frac{a}{b} \div \frac{c}{d}\)
Method Multiply straight across Flip second, then multiply
Formula \(\frac{ac}{bd}\) \(\frac{ad}{bc}\)
Result size Usually smaller Can be larger or smaller
Example \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\) \(\frac{2}{3} \div \frac{3}{4} = \frac{8}{9}\)

How to Apply the Method

1
Keep the first fraction exactly as it is.
2
Change the division sign (÷) to a multiplication sign (×).
3
Flip the second fraction (write its reciprocal).
4
Multiply numerator × numerator and denominator × denominator.
5
Simplify the result to lowest terms (divide top and bottom by their GCF).

Example 1: \(\frac{2}{3} \div \frac{3}{4}\)

1
Keep \(\frac{2}{3}\)
2
Change ÷ to ×
3
Flip \(\frac{3}{4}\) becomes \(\frac{4}{3}\)
4
Multiply: \(\frac{2}{3} \times \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9}\)

Example 2: Sharing Chocolate

Problem: Three people share \(\frac{1}{2}\) lb of chocolate equally. How much does each person get?

\(\frac{1}{2}\) lb
\(\frac{1}{6}\)
\(\frac{1}{6}\)
\(\frac{1}{6}\)
Each gets \(\frac{1}{6}\) lb

Example 3: Yogurt Servings

Problem: How many \(\frac{3}{4}\)-cup servings are in \(\frac{2}{3}\) of a cup of yogurt?

1
Set up: \(\frac{2}{3} \div \frac{3}{4}\)
2
Keep-Change-Flip: \(\frac{2}{3} \times \frac{4}{3}\)
3
Multiply: \(\frac{8}{9}\)

Example 4: Land Width

Problem: A rectangular strip of land has length \(\frac{3}{4}\) mi and area \(\frac{1}{2}\) square mi. How wide is it?

length = \(\frac{3}{4}\) mi
 w = ?
Area = \(\frac{1}{2}\) sq mi

Pro Tip: Cross-Simplify First

After flipping, look for common factors diagonally before multiplying. This keeps the numbers small and avoids simplifying at the end.

1
Problem: \(\frac{4}{9} \div \frac{2}{3}\)
2
Flip: \(\frac{4}{9} \times \frac{3}{2}\)
3
Cross-simplify: The 4 and 2 share factor 2. The 9 and 3 share factor 3. So: \(\frac{2}{3} \times \frac{1}{1}\)
4
Result: \(\frac{2}{3}\). No simplifying needed at the end!

What About Mixed Numbers?

If either number is a mixed number, convert it to an improper fraction first, then use Keep-Change-Flip as usual.

1
Problem: \(1\frac{1}{2} \div \frac{3}{4}\)
2
Convert: \(1\frac{1}{2} = \frac{3}{2}\)
3
Keep-Change-Flip: \(\frac{3}{2} \times \frac{4}{3}\)
4
Cross-simplify and multiply: \(\frac{1}{1} \times \frac{2}{1} = 2\)

Common Mistakes to Avoid

✕ Flipping the Wrong Fraction
Only flip the second fraction (the divisor). Flipping the first fraction gives a completely wrong answer.
✕ Dividing Across
Unlike addition, you cannot divide numerators and denominators separately. You must use Keep-Change-Flip.
✕ Forgetting to Simplify
Always reduce your final answer to lowest terms. For example, \(\frac{4}{6}\) should become \(\frac{2}{3}\).

Quick Reference Summary

Step What To Do Example: \(\frac{5}{6} \div \frac{2}{3}\)
1. Keep Keep first fraction \(\frac{5}{6}\)
2. Change ÷ becomes × \(\frac{5}{6}\ \times\)
3. Flip Reciprocal of second \(\frac{5}{6} \times \frac{3}{2}\)
4. Multiply Across top and bottom \(\frac{15}{12}\)
5. Simplify Reduce to lowest terms \(\frac{5}{4} = 1\frac{1}{4}\)

Question 1 of 5

What is \(\frac{3}{5} \div \frac{1}{2}\)?

Question 2 of 5

Three people share \(\frac{1}{2}\) lb of chocolate equally. How much chocolate does each person get?

Question 3 of 5

How many \(\frac{3}{4}\)-cup servings are in \(\frac{2}{3}\) of a cup of yogurt?

Question 4 of 5

A rectangular strip of land has length \(\frac{3}{4}\) mi and area \(\frac{1}{2}\) square mi. What is the width?

Question 5 of 5

Which expression is equivalent to \(\frac{7}{8} \div \frac{5}{6}\)?