Numbers, units and arithmetic
Numbers, units and arithmetic

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

Free course

Numbers, units and arithmetic

3.13 More division with fractions

How about other fractions? What is 6 ÷ ? This means how many are there in 6? Consider six squares, divided into quarters, and count how many groups of three quarters there are.

Activity 51


Three quarters is 3 × ? So dividing by is the same as dividing by 3 and by . Dividing by is multiplying by 4. So dividing by is the same as dividing by 3 and multiplying by 4, which is multiplying by . So 6 ÷ = 6 × = = 8.

The general rule for dividing by a fraction can be stated as follows.

To divide by a fraction, turn the fraction upside down and multiply.

Does this also work for dividing by ? Yes, because multiplying by n is the same as multiplying by . It also works for division of whole numbers: dividing by 4 (= ) is the same as multiplying by .

Example 23



  • Using the rule above,

  • This is

  • (divide top and bottom by 3).

  • Alternatively divide top and bottom by 3 by cancelling by 3 to give

  • So .

Turning a fraction upside down is called finding its reciprocal.

The reciprocal of is . The reciprocal of is , i.e. 6.

So an alternative formulation of the rule for dividing by a number can be stated as follows.

Division by a number is the same as multiplication by its reciprocal.


Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to University-level study, we offer two introductory routes to our qualifications. You could either choose to start with an Access module, or a module which allows you to count your previous learning towards an Open University qualification. Read our guide on Where to take your learning next for more information.

Not ready for formal University study? Then browse over 1000 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371