Dividing a Fraction by a Fraction

Master the final piece of fraction division: dividing one fraction by another!

1. Visual Model: How Many Fit?

The Big Idea

When you divide one fraction by another, you're asking: "How many of the second fraction fit in the first?"

Example 1: $\frac{1}{2} \div \frac{1}{4}$

Question: Raju has $\frac{1}{2}$ of a cake. He cuts it into equal pieces. Each piece is $\frac{1}{4}$ of the cake. How many pieces of cake does he get?

Think: How many quarters fit in half? 2!

Remember the Rule:

Dividing by

\frac{1}{4}

is the same as multiplying by 4

\frac{1}{2}

\frac{1}{4}

\frac{1}{2}

× 4

Working:

\frac{1}{2}

\frac{1}{4}

\frac{1}{2}

× 4

\frac{1 × 4}{2 × 1}

\frac{4}{2}

= 2 (simplified)

Answer:2

Raju gets 2 pieces of cake.

Example 2: $\frac{2}{3} \div \frac{1}{6}$

Question: Hassan has $\frac{2}{3}$ of a pancake. He cuts it into equal pieces. Each piece is $\frac{1}{6}$ of a pancake. How many pieces of pancake does he get?

Remember the Rule:

Dividing by

\frac{1}{6}

is the same as multiplying by 6

\frac{2}{3}

\frac{1}{6}

\frac{2}{3}

× 6

Working:

\frac{2}{3}

\frac{1}{6}

\frac{2}{3}

× 6

\frac{2 × 6}{3 × 1}

\frac{12}{3}

= 4 (simplified)

Answer:4

Hassan gets 4 pieces of pancake.

Bar Model Insight: If we divide a bar into sixths, $\frac{2}{3}$ covers 4 of those sixths!

Example 3: $\frac{3}{4} \div \frac{1}{8}$

Remember the Rule:

Dividing by

\frac{1}{8}

is the same as multiplying by 8

\frac{3}{4}

\frac{1}{8}

\frac{3}{4}

× 8

Working:

\frac{3}{4}

\frac{1}{8}

\frac{3}{4}

× 8

\frac{3 × 8}{4 × 1}

\frac{24}{4}

= 6 (simplified)

Answer:6

2. The Reciprocal Rule

The Golden Rule for Dividing Fractions

$\frac{2}{3} \div \frac{2}{9} = \frac{2}{3} \times \frac{9}{2} = 3$

Keep - Change - Flip!

Keep

First fraction stays the same

Change

Division to multiplication

Flip

Second fraction upside down: this is its reciprocal

Worked Example: $\frac{2}{3} \div \frac{2}{9}$

Remember the Rule:

Dividing by

\frac{2}{9}

is the same as multiplying by

\frac{9}{2}

\frac{2}{3}

\frac{2}{9}

\frac{2}{3}

\frac{9}{2}

Working:

\frac{2}{3}

\frac{2}{9}

\frac{2}{3}

\frac{9}{2}

\frac{2 × 9}{3 × 2}

\frac{18}{6}

= 3 (simplified)

Answer:3

Another Example: $\frac{4}{10} \div \frac{2}{5}$

Remember the Rule:

Dividing by

\frac{2}{5}

is the same as multiplying by

\frac{5}{2}

\frac{4}{10}

\frac{2}{5}

\frac{4}{10}

\frac{5}{2}

Working:

\frac{4}{10}

\frac{2}{5}

\frac{4}{10}

\frac{5}{2}

\frac{4 × 5}{10 × 2}

\frac{20}{20}

= 1 (simplified)

Answer:1

3. Simplifying Before Multiplying

Pro Tip: Simplify First!

You can simplify across the numerator and denominator before multiplying. This makes the calculation much easier!

Example: $\frac{4}{9} \div \frac{2}{3}$

Step 1: Set up as multiplication

$\frac{4}{9} \times \frac{3}{2}$

Step 2: Simplify across!

4 and 2 share a common factor of 2: $4 \div 2 = 2$ and $2 \div 2 = 1$

9 and 3 share a common factor of 3: $9 \div 3 = 3$ and $3 \div 3 = 1$

$\frac{2}{3} \times \frac{1}{1} = \frac{2}{3}$

4. Practice Problems

Problem 1: $\frac{1}{3} \div \frac{1}{6}$

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{6}

is the same as multiplying by 6

\frac{1}{3}

\frac{1}{6}

\frac{1}{3}

× 6

Working:

\frac{1}{3}

\frac{1}{6}

\frac{1}{3}

× 6

\frac{1 × 6}{3 × 1}

\frac{6}{3}

= 2 (simplified)

Answer:2

Problem 2: $\frac{5}{6} \div \frac{1}{12}$

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{12}

is the same as multiplying by 12

\frac{5}{6}

\frac{1}{12}

\frac{5}{6}

× 12

Working:

\frac{5}{6}

\frac{1}{12}

\frac{5}{6}

× 12

\frac{5 × 12}{6 × 1}

\frac{60}{6}

= 10 (simplified)

Answer:10

Problem 3: $\frac{2}{9} \div \frac{1}{4}$

ShowHide solution

$\frac{2}{9} \div \frac{1}{4} = \frac{2}{9} \times \frac{4}{1}$

$= \frac{2 \times 4}{9 \times 1}$

$= \frac{8}{9}$

Problem 4 (Word Problem)

Jiahao had $\frac{2}{3}$ of a pizza. He gave each friend $\frac{1}{3}$ of the whole pizza until it was all given away. How many friends did he give the pieces of pizza to?

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{3}

is the same as multiplying by 3

\frac{2}{3}

\frac{1}{3}

\frac{2}{3}

× 3

Working:

\frac{2}{3}

\frac{1}{3}

\frac{2}{3}

× 3

\frac{2 × 3}{3 × 1}

\frac{6}{3}

= 2 (simplified)

Answer:2

Jiahao gave pizza to 2 friends.

Problem 5 (Word Problem)

Salleh has $\frac{5}{9}$ kg of popcorn. He packs it into packets. Each packet has a mass of $\frac{1}{9}$ kg. How many packets of popcorn does Salleh pack?

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{9}

is the same as multiplying by 9

\frac{5}{9}

\frac{1}{9}

\frac{5}{9}

× 9

Working:

\frac{5}{9}

\frac{1}{9}

\frac{5}{9}

× 9

\frac{5 × 9}{9 × 1}

\frac{45}{9}

= 5 (simplified)

Answer:5

Salleh packs 5 packets.

Problem 6 (Challenge)

A jug contains $\frac{2}{5}$ litre of orange juice. Audrey pours the juice equally into some glasses such that each glass contains $\frac{1}{10}$ litre of juice. How many glasses does she use?

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{10}

is the same as multiplying by 10

\frac{2}{5}

\frac{1}{10}

\frac{2}{5}

× 10

Working:

\frac{2}{5}

\frac{1}{10}

\frac{2}{5}

× 10

\frac{2 × 10}{5 × 1}

\frac{20}{5}

= 4 (simplified)

Answer:4

Audrey uses 4 glasses.

Key Takeaways

The Question: "How many of the second fraction fit in the first?"

The Rule: $\frac{2}{3} \div \frac{2}{9} = \frac{2}{3} \times \frac{9}{2} = 3$

Remember: Keep - Change - Flip!

Pro Tip: Simplify before multiplying to make calculations easier!

Always: Express your answer in the simplest form!

Dividing a Fraction by a Fraction

1. Visual Model: How Many Fit?

The Big Idea

Example 1: 12÷14\frac{1}{2} \div \frac{1}{4}21​÷41​

Example 2: 23÷16\frac{2}{3} \div \frac{1}{6}32​÷61​

Example 3: 34÷18\frac{3}{4} \div \frac{1}{8}43​÷81​

2. The Reciprocal Rule

The Golden Rule for Dividing Fractions

Worked Example: 23÷29\frac{2}{3} \div \frac{2}{9}32​÷92​

Another Example: 410÷25\frac{4}{10} \div \frac{2}{5}104​÷52​

3. Simplifying Before Multiplying

Pro Tip: Simplify First!

Example: 49÷23\frac{4}{9} \div \frac{2}{3}94​÷32​

4. Practice Problems

Problem 1: 13÷16\frac{1}{3} \div \frac{1}{6}31​÷61​

Problem 2: 56÷112\frac{5}{6} \div \frac{1}{12}65​÷121​

Problem 3: 29÷14\frac{2}{9} \div \frac{1}{4}92​÷41​

Problem 4 (Word Problem)

Problem 5 (Word Problem)

Problem 6 (Challenge)

Key Takeaways

Example 1: $\frac{1}{2} \div \frac{1}{4}$

Example 2: $\frac{2}{3} \div \frac{1}{6}$

Example 3: $\frac{3}{4} \div \frac{1}{8}$

Worked Example: $\frac{2}{3} \div \frac{2}{9}$

Another Example: $\frac{4}{10} \div \frac{2}{5}$

Example: $\frac{4}{9} \div \frac{2}{3}$

Problem 1: $\frac{1}{3} \div \frac{1}{6}$

Problem 2: $\frac{5}{6} \div \frac{1}{12}$

Problem 3: $\frac{2}{9} \div \frac{1}{4}$