Year 7 · Unit 5 Fractions · Lesson 2 of 7

More than one whole

A fraction can be bigger than 1. Count parts into whole boxes to write it as a mixed number, and go back again by counting the parts in the wholes.

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 1 of this unit) Write 12/16 in simplest form.
    Answer3/4. Divide the top and the bottom by 4, the greatest common factor.
  2. (Lesson 1 of this unit) What number goes in the box? 5/8 = ☐/32.
    Answer20. 32 ÷ 8 = 4, and 5 × 4 = 20.
  3. (Unit 1, lesson 3) How many 6s fit into 50, and how much is left over?
    Answer8 sixes make 48, and 2 is left over.
Words used in this lesson
improper fraction
a fraction whose top is at least as big as its bottom. For example: 9/4 is an improper fraction.
mixed number
a whole number and a fraction less than 1, added together. For example: 2 1/4 means 2 + 1/4.
whole-number part
how many full wholes there are. For example: in 5 2/9 the whole-number part is 5.
remainder
what is left over after dividing as far as whole numbers go. For example: 47 ÷ 9 has a remainder of 2.
proper fraction
a fraction less than 1: the top is less than the bottom. For example: 3/8 is a proper fraction.

Counting parts into wholes

A cook measures rice with a quarter-cup scoop and fills it 17 times. Predict first: is that more than 4 cups of rice, or fewer? Use the figure to build up to it. Choose quarters and slide the number of parts up to 17 yourself: each box is one cup, and a box is full when it holds 4 quarters.

quarters
9
9/4improper fraction
2 1/4mixed number
2full boxes

9 quarters fill 2 whole boxes (8 quarters) and 1 quarter is left over: 9/4 = 2 1/4.

Show me the steps
  1. 5 halves: 2 full boxes and 1 half left over, 5/2 = 2 1/2.
  2. 11 thirds: 3 full boxes and 2 thirds left over, 11/3 = 3 2/3.
  3. 16 quarters: 4 full boxes with nothing left over, 16/4 = 4.
  4. 23 fifths: 4 full boxes and 3 fifths left over, 23/5 = 4 3/5.
  5. 19 eighths: 2 full boxes and 3 eighths left over, 19/8 = 2 3/8.
  6. 24 tenths: 2 full boxes and 4 tenths left over, 24/10 = 2 4/10, which is 2 2/5 in simplest form.

A fraction can be more than 1. When the top is bigger than the bottom (or equal to it) it is an improper fraction, and it can be written as a mixed number: some whole numbers and a fraction less than 1.

Improper to mixed: divide the top (numerator) by the bottom (denominator). The answer is the whole number, the remainder is the new top, and the bottom stays the same. For 11/3: 11 ÷ 3 = 3 remainder 2, so 3 2/3.

Mixed to improper: each whole number holds d parts, so w wholes make w × d parts, and the extra parts are added: w r/d = (w × d + r)/d.

Back to the cook: 17 quarters fill 4 whole cups (16 quarters) with 1 quarter left over, so the rice is 4 1/4 cups: a little more than 4.

A common mistake: “3 2/7 means 3 × 2/7.” A mixed number is a sum: 3 2/7 = 3 + 2/7, which is a bit over 3. Three times 2/7 is 6/7, which is less than 1.

And back again, along the number line

To write a mixed number as an improper fraction, count how many parts are in the wholes, then add the extra parts. Move the sliders and read the two lines above the number line.

2
thirds
1
2 1/3mixed number
6parts in the whole numbers
7/3improper fraction

2 wholes are 2 × 3 = 6 thirds, and 1 more third makes 7 thirds: 2 1/3 = 7/3.

Show me the steps
  1. 1 2/3: 1 whole is 3 thirds, plus 2 thirds makes 5 thirds, so 5/3.
  2. 2 1/4: 2 wholes are 8 quarters, plus 1 quarter makes 9 quarters, so 9/4.
  3. 3 1/2: 3 wholes are 6 halves, plus 1 half makes 7 halves, so 7/2.
  4. 4 2/5: 4 wholes are 20 fifths, plus 2 fifths makes 22 fifths, so 22/5.
  5. 5 7/8: 5 wholes are 40 eighths, plus 7 eighths makes 47 eighths, so 47/8.

Worked examples

Write 47/9 as a mixed number.

  1. How many 9s fit in 47? 9 × 5 = 45, and 9 × 6 = 54 is too big. So 5 whole boxes.
  2. 47 − 45 = 2 parts are left over, and each part is a ninth.
  3. Check: 5 × 9 + 2 = 47. 5 2/9.

Write 4 5/8 as an improper fraction.

  1. 4 wholes are 4 × 8 = 32 eighths.
  2. Add the 5 extra eighths: 32 + 5 = 37 eighths.
  3. Check: 37 ÷ 8 = 4 remainder 5. 37/8.

Your go. The first step is given. Write 41/7 as a mixed number.

  1. 7 × 5 = 35 is the biggest multiple of 7 that is not more than 41, so there are 5 whole boxes.
  2. Now find the parts left over yourself, then check.
Check the answer

41 − 35 = 6 sevenths are left, so 41/7 = 5 6/7. Check: 5 × 7 + 6 = 41.

Check before you go on

1. Which mixed number is the same as 31/4?

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. Between which two whole numbers does 23/4 lie on the number line?

3. Which fraction is the same as 3 2/7?

Level 2 · Building

4. A recipe needs 2 3/4 cups of flour, and the only scoop holds a quarter of a cup. How many scoops are needed?

5. Nneka practises piano for 38 quarter-hours in one week. After she takes out the whole hours, how many quarter-hours are left over?

6. One of these conversions is wrong. Which one?

Level 3 · Stretch

7. What is the smallest improper fraction with a bottom number of 6 that is greater than 3? Give its top number.

8. The fraction n/8 is at least 5 and less than 6. How many whole numbers n can there be?

Something to think about

An improper fraction has a top number of 17 and its mixed number has a whole-number part of 2. What could its bottom number be? Is there more than one answer?

One way to think about it

17 ÷ d must be between 2 and 3. d = 6 gives 2 5/6, d = 7 gives 2 3/7 and d = 8 gives 2 1/8, so there are three answers. d = 5 gives 3 2/5, which is too big, and d = 9 gives 1 8/9, which is too small.

Exit ticket

9. Dev writes 5 3/4 = 8/4, because 5 + 3 = 8. Which reply is right?