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Dividing
Fractions

Master the art of dividing fractions by fractions using the Keep-Change-Flip method, and learn to interpret quotients in real-world contexts.

Pre-Algebra

What Is Fraction Division?

When you divide fractions, you are asking: "How many groups of the divisor fit into the dividend?" For example, \( \frac{3}{4} \div \frac{1}{2} \) asks: how many halves fit inside three-fourths?

Seeing It Visually

Let's visualize \( \frac{3}{4} \div \frac{1}{2} \). We shade \( \frac{3}{4} \) of a bar, then see how many \( \frac{1}{2} \)-sized pieces fit inside.

Dividend: \( \frac{3}{4} \)
\( \frac{1}{4} \)
\( \frac{1}{4} \)
\( \frac{1}{4} \)
\( \frac{1}{4} \)
Divisor pieces: \( \frac{1}{2} \) each
1st half
2nd half
1.5 halves fit in \( \frac{3}{4} \) → Answer: \( \frac{3}{2} \)

Understanding the Reciprocal

The reciprocal (or multiplicative inverse) of a fraction is created by flipping the numerator and denominator. When you multiply a number by its reciprocal, the product is always \( 1 \).

Original
\( \frac{2}{3} \)
Reciprocal
\( \frac{3}{2} \)
Product
\( \frac{2}{3} \times \frac{3}{2} = 1 \)

The Keep-Change-Flip Rule

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

Why Does Flipping Work?

1
Division is the inverse of multiplication. Asking \( a \div b \) is the same as asking "what times \( b \) gives \( a \)?"
2
Dividing by \( \frac{c}{d} \) means multiplying by the number that "undoes" \( \frac{c}{d} \): its reciprocal \( \frac{d}{c} \).
3
Proof: Writing as a complex fraction gives \( \frac{\frac{a}{b}}{\frac{c}{d}} \). We can multiply the numerator and denominator by the reciprocal \( \frac{d}{c} \) to simplify, which results in \( \frac{a}{b} \times \frac{d}{c} \).

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
Result \( \frac{a \times c}{b \times d} \) \( \frac{a \times d}{b \times c} \)
Example \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \) \( \frac{2}{3} \div \frac{4}{5} = \frac{10}{12} = \frac{5}{6} \)

How to Apply the Method

1
Keep the first fraction exactly as it is.
2
Change the division sign (\( \div \)) to a multiplication sign (\( \times \)).
3
Flip the second fraction (write its reciprocal).
4
Multiply numerators together and denominators together.
5
Simplify the result to lowest terms (or convert to a mixed number).

Example 1: Basic Division

Compute \( \frac{3}{5} \div \frac{2}{7} \)

1
Keep: \( \frac{3}{5} \)
2
Change: \( \div \) becomes \( \times \)
3
Flip: \( \frac{2}{7} \) becomes \( \frac{7}{2} \)
4
Multiply: \( \frac{3}{5} \times \frac{7}{2} = \frac{21}{10} \)
5
Simplify: \( \frac{21}{10} = 2\frac{1}{10} \)

Example 2: Simplify First

Compute \( \frac{4}{9} \div \frac{8}{3} \)

1
Rewrite: \( \frac{4}{9} \times \frac{3}{8} \)
2
Cross-cancel before multiplying: \( 4 \) and \( 8 \) share a factor of \( 4 \): so \( \frac{\cancel{4}^{\,1}}{9} \times \frac{3}{\cancel{8}^{\,2}} \). Also \( 3 \) and \( 9 \) share a factor of \( 3 \): \( \frac{1}{\cancel{9}^{\,3}} \times \frac{\cancel{3}^{\,1}}{2} \).
3
Multiply: \( \frac{1 \times 1}{3 \times 2} = \frac{1}{6} \)

Example 3: Real-World Context

A recipe calls for \( \frac{3}{4} \) cup of flour. Each scoop holds \( \frac{1}{8} \) cup. How many scoops do you need?

1
Set up: \( \frac{3}{4} \div \frac{1}{8} \)
2
Keep-Change-Flip: \( \frac{3}{4} \times \frac{8}{1} \)
3
Multiply: \( \frac{3 \times 8}{4 \times 1} = \frac{24}{4} = 6 \)

Dividing with Mixed Numbers

To divide mixed numbers, first convert them to improper fractions, then apply Keep-Change-Flip.

1
Problem: \( 2\frac{1}{3} \div 1\frac{1}{2} \)
2
Convert: \( \frac{7}{3} \div \frac{3}{2} \)
3
Keep-Change-Flip: \( \frac{7}{3} \times \frac{2}{3} \)
4
Multiply: \( \frac{14}{9} = 1\frac{5}{9} \)

Whole Numbers \( \div \) Fractions

Any whole number can be written as a fraction over 1. For example, \( 5 = \frac{5}{1} \). Then apply the same rule.

Start
\( 5 \div \frac{2}{3} \)
Rewrite
\( \frac{5}{1} \div \frac{2}{3} \)
KCF
\( \frac{5}{1} \times \frac{3}{2} \)
\( \frac{15}{2} \)

Interpreting the Quotient

In context, the quotient of a fraction division problem tells you how many groups of the divisor are contained in the dividend, or the size of each group when sharing equally.

Measurement
"How many \( \frac{1}{4} \)-lb servings are in \( \frac{3}{2} \) lbs?" Answer: the number of groups.
Partitive
"If \( \frac{2}{3} \) of a tank uses \( \frac{1}{2} \) gallon, how much per full tank?" Answer: the size of the whole.

Common Mistakes

✕ Flipping the Wrong Fraction
Students flip the first fraction instead of the second. Remember: Keep the first, Flip the second.
✕ Forgetting to Convert Mixed Numbers
You must convert mixed numbers to improper fractions before applying Keep-Change-Flip.
✕ Dividing Straight Across
Unlike multiplication, you cannot simply divide numerators and denominators. You must flip first.

Summary Checklist

Step Action Example for \( 3 \div \frac{1}{4} \)
1 Write both as fractions \( 3 = \frac{3}{1} \)
2 Keep the first fraction \( \frac{3}{1} \)
3 Change \( \div \) to \( \times \) \( \frac{3}{1} \times \)
4 Flip the second fraction \( \frac{3}{1} \times \frac{4}{1} \)
5 Multiply and simplify \( \frac{12}{1} = 12 \)

Question 1 of 5

What is \( \frac{2}{5} \div \frac{3}{4} \)?

Question 2 of 5

What is \( 6 \div \frac{3}{5} \)?

Question 3 of 5

A ribbon is \( \frac{7}{8} \) yard long. You cut pieces that are each \( \frac{1}{4} \) yard. How many pieces can you cut?

Question 4 of 5

What is \( 2\frac{1}{4} \div \frac{3}{8} \)?

Question 5 of 5

Which expression is equivalent to \( \frac{5}{6} \div \frac{2}{9} \)?