4.4 Dividing Fractions

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: reciprocals, dividing by whole numbers and by fractions, and mixed numbers. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Find the reciprocal of a whole number or a fraction
• Divide a fraction by a whole number and a whole number by a fraction
• Divide one fraction by another by multiplying by the reciprocal
• Divide mixed numbers and solve "how many fit?" problems

Key words

• Reciprocal: Flip the fraction upside down. The reciprocal of 3/5 is 5/3; of 4 is 1/4.
• Dividend: The number being divided.
• Divisor: The number you divide by.
• Quotient: The answer to a division.

Reciprocals

A number times its reciprocal is always 1. To find the reciprocal of a fraction, swap the numerator and denominator. A whole number n is n/1, so its reciprocal is 1/n.

• Reciprocal; 3/5 5/3; 7 1/7; 1/8 8; 2 1/2 = 5/2 2/5

Worked example: checking

3/5 × 5/3 = 15/15 = 1 ✓

Answer: the reciprocal of 3/5 is 5/3

Dividing a fraction by a whole number

Dividing by 2 is the same as finding half, or multiplying by 1/2. So multiply the denominator by the whole number.

Worked example: fraction ÷ whole number

3/4 ÷ 2 = 3/4 × 1/2 = 3/8
6/7 ÷ 3 = 6/7 × 1/3 = 6/21 = 2/7

Answer: 3/8 and 2/7

Dividing by a fraction

"3 ÷ 1/4" asks: how many quarters fit into 3? There are 4 quarters in each whole, so 12. Notice that 3 ÷ 1/4 = 3 × 4.

Key fact: To divide by a fraction, multiply by its reciprocal. Keep the first fraction, change ÷ to ×, flip the second fraction.

Worked example: fraction ÷ fraction

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
7/8 ÷ 1/4 = 7/8 × 4/1 = 28/8 = 3 1/2

Answer: 5/6 and 3 1/2

Mixed numbers and problems

Change mixed numbers to improper fractions first. In word problems, "how many ... fit into ..." means divide.

Worked example: mixed numbers

2 1/4 ÷ 3/8 = 9/4 × 8/3 = 72/12 = 6

Answer: 6

Worked example: how many bottles?

How many 3/4 litre bottles can be filled from 6 litres of water?
6 ÷ 3/4 = 6 × 4/3 = 24/3 = 8

Answer: 8 bottles

Common mistakes

• Flipping the first fraction instead of the second.
• Forgetting to change ÷ to × after flipping.
• Dividing mixed numbers without changing them to improper fractions first.
• Expecting the answer to be smaller: dividing by a fraction less than 1 gives a bigger answer.

Revision questions

Try these, then check your answers below each one.

Question 1. Write down the reciprocal of 4/9. (Type your answer like 2/5.)

Answer: 9/4

Working: The reciprocal of 4/9 is 9/4.

Question 2. Work out 4 ÷ 1/3.

Answer: 12

Working: There are 3 thirds in each whole: 4 × 3 = 12.

Question 3. Work out 3/5 ÷ 3. (Type your answer like 2/5.)

Answer: 1/5

Working: 3/5 × 1/3 = 3/15 = 1/5.

Related lessons

• Previous lesson: 4.3 Multiplying Fractions
• Next lesson: 4.5 Fractions and Decimals
• In this chapter: 4.1 Working with Fractions
• In this chapter: 4.2 Operations with Fractions
• In this chapter: 4.6 Fractions and Percentages

Continue in the full course

This lesson is section 4.4 (EGCMF0404) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.