Dividing a Whole Number by a Fraction

Learn to answer: "How many fractional pieces fit in a whole number?"

1. The Counting Model: How Many Fit?

The Big Idea

When you divide a whole number by a fraction, you're asking: "How many of these fractional pieces fit in the whole?"

When you divide by a proper fraction (a fraction less than 1), the answer is LARGER than the whole number you started with!

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

Question: Raju has 1 pizza. He wants to give $\frac{1}{4}$ of the pizza to each of his friends. How many friends can he give the pizza to?

Think: How many quarters are there in 1 whole? 4!

Remember the Rule:

Dividing by

\frac{1}{4}

is the same as multiplying by 4

1 ÷

\frac{1}{4}

= 1 × 4

Working:

1 ÷

\frac{1}{4}

= 1 × 4

\frac{1 × 4}{1 × 1}

\frac{4}{1}

= 4

Answer:4

Raju can give the pizza to 4 friends.

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

Question: Anna has 2 pizzas. She gives $\frac{1}{4}$ of a pizza to each of her friends. How many friends can she give the pizzas to?

Remember the Rule:

Dividing by

\frac{1}{4}

is the same as multiplying by 4

2 ÷

\frac{1}{4}

= 2 × 4

Working:

2 ÷

\frac{1}{4}

= 2 × 4

\frac{2 × 4}{1 × 1}

\frac{8}{1}

= 8

Answer:8

Anna can give the pizzas to 8 friends.

Notice: Dividing by $\frac{1}{4}$ is the same as multiplying by 4!

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

Question: A group of children share 4 bars of chocolate equally. Each child gets $\frac{1}{3}$ of a chocolate bar. How many children are there?

Remember the Rule:

Dividing by

\frac{1}{3}

is the same as multiplying by 3

4 ÷

\frac{1}{3}

= 4 × 3

Working:

4 ÷

\frac{1}{3}

= 4 × 3

\frac{4 × 3}{1 × 1}

\frac{12}{1}

= 12

Answer:12

There are 12 children.

2. Dividing by Non-Unit Fractions

What About $\frac{2}{3}$ or $\frac{3}{4}$ ?

The same idea applies! We still ask "how many fit?" but now each portion is larger.

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

Question: A group of children share 4 bars of chocolate equally. Each child gets $\frac{2}{3}$ of a chocolate bar. How many children are there?

Remember the Rule:

Dividing by

\frac{2}{3}

is the same as multiplying by

\frac{3}{2}

4 ÷

\frac{2}{3}

= 4 ×

\frac{3}{2}

Working:

4 ÷

\frac{2}{3}

= 4 ×

\frac{3}{2}

\frac{4 × 3}{1 × 2}

\frac{12}{2}

= 6 (simplified)

Answer:6

There are 6 children.

Think: How many two-thirds are there in 4 wholes? 6!

Example 5: $6 \div \frac{3}{4}$

Remember the Rule:

Dividing by

\frac{3}{4}

is the same as multiplying by

\frac{4}{3}

6 ÷

\frac{3}{4}

= 6 ×

\frac{4}{3}

Working:

6 ÷

\frac{3}{4}

= 6 ×

\frac{4}{3}

\frac{6 × 4}{1 × 3}

\frac{24}{3}

= 8 (simplified)

Answer:8

3. The Reciprocal Rule

The Formula

$4 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6$

To divide by a fraction, multiply by its reciprocal (flip the fraction)!

Examples of Reciprocals

Reciprocal of $\frac{1}{3}$

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

Reciprocal of $\frac{2}{3}$

$\frac{3}{2}$

Reciprocal of $\frac{1}{4}$

$\frac{4}{1} = 4$

Reciprocal of $\frac{3}{5}$

$\frac{5}{3}$

4. Practice Problems

Problem 1

A jug contains 3 litres of juice. Each glass holds $\frac{1}{5}$ litre. How many glasses can be filled?

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{5}

is the same as multiplying by 5

3 ÷

\frac{1}{5}

= 3 × 5

Working:

3 ÷

\frac{1}{5}

= 3 × 5

\frac{3 × 5}{1 × 1}

\frac{15}{1}

= 15

Answer:15

15 glasses can be filled.

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

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{6}

is the same as multiplying by 6

5 ÷

\frac{1}{6}

= 5 × 6

Working:

5 ÷

\frac{1}{6}

= 5 × 6

\frac{5 × 6}{1 × 1}

\frac{30}{1}

= 30

Answer:30

Problem 3

John has 4 litres of juice. He fills bottles that each hold $\frac{4}{5}$ litre. How many bottles can he fill?

ShowHide solution

Remember the Rule:

Dividing by

\frac{4}{5}

is the same as multiplying by

\frac{5}{4}

4 ÷

\frac{4}{5}

= 4 ×

\frac{5}{4}

Working:

4 ÷

\frac{4}{5}

= 4 ×

\frac{5}{4}

\frac{4 × 5}{1 × 4}

\frac{20}{4}

= 5 (simplified)

Answer:5

John can fill 5 bottles.

Problem 4 (Challenge!)

Ramli took $\frac{1}{12}$ h to complete a puzzle. How many such puzzles did he complete in $\frac{3}{4}$ h?

ShowHide solution

Remember the Rule:

Dividing by

\frac{1}{12}

is the same as multiplying by 12

\frac{3}{4}

\frac{1}{12}

\frac{3}{4}

× 12

Working:

\frac{3}{4}

\frac{1}{12}

\frac{3}{4}

× 12

\frac{3 × 12}{4 × 1}

\frac{36}{4}

= 9 (simplified)

Answer:9

Ramli completed 9 puzzles.

Key Takeaways

Dividing by a proper fraction (less than 1) gives a LARGER answer (opposite to dividing by a whole number greater than 1!)

Think "How many fit?" - How many of the fraction fit in the whole number?

The Rule: $4 \div \frac{2}{3} = 4 \times \frac{3}{2} = 6$ (flip and multiply!)

Dividing by $\frac{1}{n}$ is the same as multiplying by n.

Dividing a Whole Number by a Fraction

1. The Counting Model: How Many Fit?

The Big Idea

Example 1: 1÷141 \div \frac{1}{4}1÷41​

Example 2: 2÷142 \div \frac{1}{4}2÷41​

Example 3: 4÷134 \div \frac{1}{3}4÷31​

2. Dividing by Non-Unit Fractions

What About 23\frac{2}{3}32​ or 34\frac{3}{4}43​?

Example 4: 4÷234 \div \frac{2}{3}4÷32​

Example 5: 6÷346 \div \frac{3}{4}6÷43​

3. The Reciprocal Rule

The Formula

Examples of Reciprocals

4. Practice Problems

Problem 1

Problem 2: 5÷165 \div \frac{1}{6}5÷61​

Problem 3

Problem 4 (Challenge!)

Key Takeaways

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

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

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

What About $\frac{2}{3}$ or $\frac{3}{4}$ ?

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

Example 5: $6 \div \frac{3}{4}$

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