Year 7 · Unit 5 Fractions · Lesson 6 of 7

How many fit?

Dividing by a fraction asks how many of it fit. Lay copies along a line, group parts, and see why that is the same as multiplying by the reciprocal. Then use the order of operations.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 5 of this unit) What is 1/2 of 2/7?
    Answer1/7. Multiply the tops (1 × 2) and the bottoms (2 × 7), then simplify 2/14.
  2. (Unit 1, lesson 6) Work out 3 + 4 × 2.
    Answer11. Multiplication comes before addition: 4 × 2 = 8, and 3 + 8 = 11.
  3. (Lesson 2 of this unit) Write 23/6 as a mixed number.
    Answer3 5/6. 6 × 3 = 18, and 23 − 18 = 5.
Words used in this lesson
reciprocal
a fraction turned upside down: the top and bottom swapped. For example: the reciprocal of 2/7 is 7/2.
divide
find how many times one number fits in another, or share equally. For example: 6 ÷ 2 = 3.
quotient
the result of a division. For example: the quotient of 12 ÷ 4 is 3.
copy
one piece of the same size. For example: four copies of 1/4 make a whole.
brackets
symbols that say what to do first. For example: in (1/2 + 1/4) ÷ 3/8 the brackets come first.

How many fit?

A tailor cuts ribbon into pieces that are 1/4 of a metre long. Predict first: from 3 metres of ribbon, will there be more than 3 pieces, or fewer? Set the figure to a length of 3 and a copy of 1/4 to find out, then try other lengths.

2
1/3
6copies that fit
2 × 3as a multiplication

Copies of 1/3 laid along 2: 6 copies fit exactly. 2 ÷ 1/3 = 2 × 3 = 6.

Show me the steps
  1. 3 ÷ 1/4: 12 copies of a quarter fit along 3.
  2. 2 ÷ 1/3: 6 copies of a third fit along 2.
  3. 3/2 ÷ 1/2: 3 copies of a half fit along 1 1/2.
  4. 3/4 ÷ 2/3: 1 full copy and then 1/8 of a copy, so 1 1/8.
  5. 5/6 ÷ 3/8: 2 full copies and 2/9 of another copy, so 2 2/9.

Dividing by a fraction asks how many of it fit. There are 4 quarters in 1, so there are 3 × 4 = 12 quarters in 3: 3 ÷ 1/4 = 3 × 4. In general, a ÷ (c/d) = a × (d/c). The fraction d/c is the reciprocal of c/d: the numerator (top) and the denominator (bottom) swapped over.

Why: one whole holds d parts of size 1/d. Grouped c at a time, that is d/c groups, so a whole holds d/c copies of c/d.

Dividing a positive number by a fraction between 0 and 1 gives a bigger answer than the number you started with.

Back to the ribbon: 3 metres holds 12 pieces of a quarter of a metre, many more than 3.

A common mistake: “Flip the first fraction.” Only the fraction you divide by is turned over: 3/5 ÷ 2/7 = 3/5 × 7/2, not 5/3 × 2/7. The first fraction stays as it is.

Grouping parts

Here is the same idea with parts. Cut each whole into d parts, then group the parts c at a time. The number of groups in one whole is d over c.

2
2/3
6parts in all
3groups of 2 parts
3the answer

2 wholes are 2 × 3 = 6 thirds. Grouped in twos, that is 3 groups: 2 ÷ 2/3 = 2 × 3/2 = 3.

Show me the steps
  1. 2 ÷ 2/3: 6 thirds in groups of 2 make 3 groups, so 3.
  2. 1 ÷ 3/4: 4 quarters in groups of 3 make 1 group with 1 quarter left, so 1 1/3.
  3. 3 ÷ 2/5: 15 fifths in groups of 2 make 7 groups with 1 fifth left, which is half a group, so 7 1/2.
  4. 2 ÷ 3/8: 16 eighths in groups of 3 make 5 groups with 1 eighth left, so 5 1/3.
  5. 4 ÷ 5/6: 24 sixths in groups of 5 make 4 groups with 4 sixths left, so 4 4/5.

Worked examples

Work out 3/8 ÷ 9/16.

  1. Turn the second fraction over and multiply: 3/8 × 16/9.
  2. Simplify first: 3 and 9 share 3, and 16 and 8 share 8. That leaves 1/1 × 2/3.
  3. 2/3. Check: 2/3 × 9/16 = 18/48 = 3/8.

Work out 2 1/4 ÷ 3/4.

  1. Improper fraction first: 2 1/4 = 9/4.
  2. 9/4 ÷ 3/4 = 9/4 × 4/3 = 36/12.
  3. 3. Check by counting: there are 9 quarters, and 3 quarters fit 3 times.

Several operations

The order of operations from Unit 1 still holds: brackets first, then multiplying and dividing, then adding and subtracting.

Work out 5/6 − 1/2 × 2/3.

  1. Multiply before subtracting: 1/2 × 2/3 = 1/3.
  2. Now 5/6 − 1/3 = 5/6 − 2/6 = 1/2.

Your go. The first step is given. Work out 1/2 + 2/3 × 3/4.

  1. Multiply before adding: 2/3 × 3/4 = 1/2.
  2. Now add it to the 1/2 yourself, then check.
Check the answer

1/2 + 1/2 = 1. If you had added first you would have got 7/6 × 3/4, which is 7/8: the order matters.

Check before you go on

1. Which is the same as 3/5 ÷ 2/7?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. How many one-fifth-litre cups can be filled from 4 litres of juice?

3. The reciprocal of a fraction is 9/4. What is the bottom number of the fraction itself?

Level 2 · Building

4. Work out 5/6 ÷ 1/12. What is the answer?

5. Three quarters of a pizza is shared equally among 6 people. Each person gets a fraction of the whole pizza. What is the bottom number of that fraction in simplest form?

6. A rectangle has an area of 5/6 square metres and a length of 5/4 metres. Its width is a fraction of a metre. What is the bottom number of the width in simplest form?

Level 3 · Stretch

7. Work out 1 1/2 + 3/4 ÷ 1/2.

8. A ribbon 7 1/2 metres long is cut into pieces of 1 1/4 metres. How many pieces are there?

Something to think about

Is 6 ÷ 1/3 the same as 6 ÷ 3? Work both out, and say what question each one is asking.

One way to think about it

6 ÷ 3 = 2 asks “share 6 into 3 equal groups: how big is each?” 6 ÷ 1/3 = 18 asks “how many thirds fit in 6?” They are different questions. Dividing by a number less than 1 gives more than you started with, because each copy is smaller than a whole.

Exit ticket

9. Why does dividing a number by 1/2 give twice the number?