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Primary Maths · Year 6Chapter 12 · Statistics: pie charts, line graphs and the mean
Year 6 · Chapter 12

Year 6 Pie Charts, Line Graphs and the Mean: worked examples and questions with answers

By the end of this chapter you will be able to

  • Read the per cent and the angle of a sector on a pie chart, and find the number in a sector when the total is known
  • Work out the angle of each sector from a table, and check that the angles add up to 360 degrees
  • Read values from a line graph, plot points on a scale and join them with straight lines
  • Solve problems with line graphs: comparing two lines, totals, the biggest rise and the average change for each hour
  • Find the mean of a set of values by dividing the total by the number of values
  • Use the mean to find a total or a missing value, and say what the mean does and does not tell us

You should already know

  • Fractions, and finding a fraction of an amount (Year 6, Chapter 4)
  • Per cent and its equivalents to fractions (Year 6, Chapter 6)
  • Reading and comparing a line graph, and the difference between two values (Year 5, Chapter 11)
  • Reading bar charts and tables, and tallies (Year 4, Chapter 11)

Where this leads

  • Using pie charts, line graphs, tables and averages together in the SATs revision chapter (Year 6, Chapter 13)

National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Interpret and construct pie charts and line graphs and use these to solve problems, Calculate and interpret the mean as an average.

Exercise P  Do you remember?

Non-calculator
1
FWhat is a quarter of 360 degrees? Give your answer in degrees.
2
FWork out 25 per cent of 80.
3
FWork out 9 + 6 + 13 + 8.
4
FWork out 96 ÷ 8.
5
FWrite 3/4 as a percentage. Give the number of per cent.
12.1

Reading a pie chart

Year 6
Interpret a pie chart: the sectors show parts of a whole, and the whole is 360 degrees or 100 per cent
Video lesson: Reading a pie chartComing soon

A pie chart shows how a whole is split into parts. The whole circle stands for the whole thing. Each part is a sector, a slice of the circle.

The whole circle is 360 degrees. It is also 100 per cent. The sectors always add up to the whole, so the per cents in a pie chart add up to 100, and the angles add up to 360.

A bigger sector means a bigger share. A sector that is half the circle is 50 per cent and 180 degrees. A sector that is a quarter is 25 per cent and 90 degrees. You do not need to measure the angles to read the chart. Read the key, which gives each sector its per cent.

Walk: 40%Bus: 25%Car: 20%Bike: 15%
How 100 pupils travel to school. The key gives the per cent for each sector.

Key ideas

  • The whole pie is 360 degrees, and it is 100 per cent.
  • The sectors add up to the whole: the per cents add to 100, and the angles add to 360.
  • Read the key to find the per cent of each sector.
  • A quarter of the pie is 25 per cent, and it is 90 degrees.
Worked example 12.1.1
What percentage of the pupils travel by car? (Use the pie chart of how the pupils travel.)
Walk: 40%Bus: 25%Car: 20%Bike: 15%

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Find the label in the key and read the per cent next to it.

  1. Find Car in the key. Each label in the key has its own sector.
    Car
  2. Read the per cent next to it. The per cent tells us the share of the whole pie that the car sector is.
    Car: 20%
  3. Write the answer. The per cent is out of 100.
    20

Check. All the per cents in the key add up to 100. ✓

Watch out. Read the per cent, not the size of the sector by eye.

Worked example 12.1.2
How many degrees is the sector for pasta in the lunch pie chart? (The whole pie is 360 degrees.)
Pasta: 35%Jacket potato: 25%Sandwiches: 20%Salad: 10%Wraps: 10%

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Change the per cent into a fraction, then take that fraction of 360 degrees.

  1. Write the per cent as a fraction. Per cent means out of 100.
    35/100 = 7/20
  2. Take that fraction of the whole pie. The whole pie is 360 degrees.
    7/20 × 360 = 126
  3. Write the answer. The sector is this many degrees.
    126

Check. 35 per cent of 360 degrees is 126 degrees. ✓

Watch out. Use 360 degrees for the whole pie. Do not use 100 when you want degrees.

Worked example 12.1.3
What fraction of the pupils chose swimming? Give your answer in its simplest form. (Use the sports pie chart.)
Football: 30%Netball: 20%Swimming: 25%Tennis: 15%Athletics: 10%

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Change the per cent to a fraction out of 100, then simplify it.

  1. Read the per cent for Swimming. Per cent means out of 100.
    25 per cent
  2. Write it as a fraction out of 100. The bottom number is 100 because it is per cent.
    25/100
  3. Simplify the fraction. Divide the top and bottom by the same number until you cannot.
    25/100 = 1/4

Check. 1/4 of the pie is 25 out of 100 per cent. ✓

Watch out. Simplify to the lowest terms. 25/100 is 1/4, not 1/5.

Watch out

Common mistake: reading a sector as a number of degrees when the question asks for a per cent. A sector of 90 degrees is a quarter of the pie, which is 25 per cent. Check the units in the question before you answer.

Exercise 12A  Practice: reading a pie chart

Non-calculator
1
FUse the travel pie chart. What percentage of the pupils walk to school?
Walk: 40%Bus: 25%Car: 20%Bike: 15%
2
FUse the travel pie chart. How many degrees is the sector for walking?
Walk: 40%Bus: 25%Car: 20%Bike: 15%
3
FUse the travel pie chart. How many degrees is the sector for cars?
Walk: 40%Bus: 25%Car: 20%Bike: 15%
4
FUse the travel pie chart. What fraction of the pupils travel by bus? Give your answer in its simplest form.
Walk: 40%Bus: 25%Car: 20%Bike: 15%
5
FUse the lunch pie chart. Which lunch is the most popular?
Pasta: 35%Jacket potato: 25%Sandwiches: 20%Salad: 10%Wraps: 10%
6
FUse the lunch pie chart. How many degrees is the sector for salad?
Pasta: 35%Jacket potato: 25%Sandwiches: 20%Salad: 10%Wraps: 10%
7
PSUse the sports pie chart. What fraction of the pupils chose tennis or athletics? Give your answer in its simplest form.
Football: 30%Netball: 20%Swimming: 25%Tennis: 15%Athletics: 10%
8
PSUse the sports pie chart. What percentage of the pupils did not choose football?
Football: 30%Netball: 20%Swimming: 25%Tennis: 15%Athletics: 10%
9
FUse the travel pie chart.   (a) What percentage of the pupils cycle?   (b) What percentage walk or take the bus?
Walk: 40%Bus: 25%Car: 20%Bike: 15%
(a)What percentage of the pupils cycle?
(b)What percentage walk or take the bus?
10
RUse the travel pie chart. Sam says more pupils walk to school than take the bus or cycle. Is Sam right? Explain your answer.
Walk: 40%Bus: 25%Car: 20%Bike: 15%

Watch out

Watch out: a pie chart does not show how many children there are. It shows the share of each sector. Two sectors with the same size have the same share, even if they have different names.

Exam tip

When you read a pie chart, check that the per cents add up to 100 before you use them.

12.2

Finding the numbers in a pie chart

Year 6
Interpret a pie chart with a known total: find the number in each sector from its angle or per cent
Video lesson: Finding the numbers in a pie chartComing soon

Sometimes a pie chart stands for a total number, such as 120 pupils. Then each sector shows a number of pupils, and you can find that number from the sector.

There are two ways to do it. If the chart gives the angle, write the angle as a fraction of 360, then take that fraction of the total. If the chart gives a per cent, write the per cent as a fraction of 100, then take that fraction of the total.

Check your answer by adding all the sector numbers. They must add up to the total.

Apples: 135°Bananas: 90°Oranges: 72°Grapes: 63°
The favourite fruit of 120 pupils. Each angle is the sector of the whole pie.

Key ideas

  • Angle method: the number = angle ÷ 360 × total.
  • Per cent method: the number = per cent ÷ 100 × total.
  • The numbers in all the sectors add up to the total.
  • A sector number is not the same as its angle. 135 degrees does not mean 135 pupils.
Worked example 12.2.1
How many of the 120 pupils chose apples? (The sector for apples is 135 degrees.)
Apples: 135°Bananas: 90°Oranges: 72°Grapes: 63°

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Turn the angle into a fraction of the whole pie, then find that fraction of the total.

  1. Write the sector as a fraction of the whole pie. The whole pie is 360 degrees, so divide the angle by 360.
    135 ÷ 360 = 3/8
  2. Take that fraction of the total. The total is 120, so take the same fraction of it.
    3/8 × 120 = 45
  3. Write the answer. Apples has this many.
    45

Check. 45 out of 120 is the same fraction as 135 degrees out of 360. ✓

Watch out. Divide the angle by 360 first. Do not multiply the angle by the total.

Worked example 12.2.2
A pie chart shows 40 children. The sector for cats is 25 per cent. How many children have a cat?
Dogs: 35%Cats: 25%Fish: 20%Rabbits: 10%None: 10%

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Change the per cent into a fraction, then take that fraction of the total.

  1. Write the per cent as a fraction. Per cent means out of 100.
    25/100 = 1/4
  2. Take that fraction of the total. The total is 40, so take 1/4 of it.
    1/4 × 40 = 10
  3. Write the answer. Cats has this many.
    10

Check. 10 is 1/4 of 40, so 25 per cent of 40 is 10. ✓

Watch out. Per cent is out of 100, not out of the total. Change it to a fraction first.

Worked example 12.2.3
A pie chart stands for 90 children. One sector is 80 degrees. How many children are in that sector?

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Turn the angle into a fraction of the whole pie, then find that fraction of the total.

  1. Write the sector as a fraction of the whole pie. The whole pie is 360 degrees, so divide the angle by 360.
    80 ÷ 360 = 2/9
  2. Take that fraction of the total. The total is 90, so take the same fraction of it.
    2/9 × 90 = 20
  3. Write the answer. The sector has this many.
    20

Check. 20 out of 90 is the same fraction as 80 degrees out of 360. ✓

Watch out. Divide the angle by 360 first. Do not multiply the angle by the total.

Watch out

Common mistake: taking the angle as the number of children. A sector of 135 degrees is a little more than a third of the pie, so it is a little more than a third of the total. Always divide the angle by 360 first.

Exercise 12B  Practice: finding the numbers in each sector

Non-calculator
1
FThe pie chart shows the favourite fruit of 120 pupils. How many pupils chose apples?
Apples: 135°Bananas: 90°Oranges: 72°Grapes: 63°
2
FThe pie chart shows the favourite fruit of 120 pupils. How many pupils chose grapes?
Apples: 135°Bananas: 90°Oranges: 72°Grapes: 63°
3
FThe pie chart shows the favourite fruit of 120 pupils. How many pupils chose bananas?
Apples: 135°Bananas: 90°Oranges: 72°Grapes: 63°
4
FThe pie chart shows the pets of 40 children. How many children have a cat?
Dogs: 35%Cats: 25%Fish: 20%Rabbits: 10%None: 10%
5
FThe pie chart shows the pets of 40 children. How many children have a fish?
Dogs: 35%Cats: 25%Fish: 20%Rabbits: 10%None: 10%
6
FThe pie chart shows the pets of 40 children. How many children have a dog?
Dogs: 35%Cats: 25%Fish: 20%Rabbits: 10%None: 10%
7
FThe pie chart shows the pets of 40 children. How many children do not have a pet?
Dogs: 35%Cats: 25%Fish: 20%Rabbits: 10%None: 10%
8
FThe pie chart shows how 60 pupils travel to school. How many pupils travel by car?
Bus: 180°Walk: 72°Car: 60°Bike: 48°
9
FThe pie chart shows how 60 pupils travel to school.   (a) How many pupils travel by bus?   (b) How many pupils travel by bike?   (c) How many pupils do not travel by bus?
Bus: 180°Walk: 72°Car: 60°Bike: 48°
(a)how many travel by bus
(b)how many travel by bike
(c)how many do not travel by bus
10
RRuby says: "25 per cent of 40 is 15." Is Ruby right? Explain your answer, and say what the answer should be.
11
PSA pie chart stands for 60 children. One sector is 150 degrees. How many children are in that sector?

Watch out

Watch out: check that the sector numbers add up to the total. If they do not, one of your answers is wrong.

Exam tip

Work with the fraction first, then multiply by the total. It is easier to check each step this way.

12.3

Drawing a pie chart: the angles

Year 6
Construct pie charts: work out the angle of each sector from a frequency table
Video lesson: Drawing a pie chart: the anglesComing soon

To draw a pie chart from a table, you first need the angle of each sector. The table gives the number in each group, so the chart is in the same proportions as the table.

Follow these steps. First, find the total of all the numbers. Second, write each group as a fraction of the total. Third, multiply each fraction by 360 degrees. The angles of all the sectors must add up to 360. Then draw the sectors with a protractor, label each one and give the chart a title.

Water: 90°Milk: 60°Juice: 135°Smoothie: 75°
The favourite drink of 24 children, drawn from the table in the first example.

Key ideas

  • Angle of a sector = fraction of the total × 360 degrees.
  • The angles of all the sectors add up to 360 degrees.
  • Check that the total of the sector numbers is the total in the table.
  • Label every sector, so the reader knows what each one shows.
Worked example 12.3.1
A class is asked for its favourite drink. The table shows 24 children, and 6 chose water. Work out the angle for the water sector.
DrinkNumber
Water6
Milk4
Juice9
Smoothie5
Total24

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Find the fraction of the whole pie, then take that fraction of 360 degrees.

  1. Write the whole pie. A whole pie chart is a whole circle, and a whole circle is 360 degrees.
    whole pie = 360°
  2. Write the sector as a fraction of the total. Water is 6 out of 24, so divide 6 by 24.
    6 ÷ 24 = 1/4
  3. Multiply the fraction by 360 degrees. The fraction tells us the share of the whole circle.
    1/4 × 360 = 90
  4. Write the answer. The sector is this many degrees.
    90

Check. All the sector angles in a pie chart add up to 360 degrees. ✓

Watch out. Use 360 degrees for the whole pie, not 100. The 100 is only for percentages.

Worked example 12.3.2
Work out the angle of the fiction sector in a pie chart for 30 books. There are 15 fiction books.
Type of bookNumber
Fiction15
Non-fiction9
Poetry6
Total30

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Find the fraction of the whole pie, then take that fraction of 360 degrees.

  1. Write the whole pie. A whole pie chart is a whole circle, and a whole circle is 360 degrees.
    whole pie = 360°
  2. Write the sector as a fraction of the total. Fiction is 15 out of 30, so divide 15 by 30.
    15 ÷ 30 = 1/2
  3. Multiply the fraction by 360 degrees. The fraction tells us the share of the whole circle.
    1/2 × 360 = 180
  4. Write the answer. The sector is this many degrees.
    180

Check. All the sector angles in a pie chart add up to 360 degrees. ✓

Watch out. Use 360 degrees for the whole pie, not 100. The 100 is only for percentages.

Worked example 12.3.3
The four angles for the drinks are 90, 60, 135 and 75 degrees. Show that they add up to a whole pie.
DrinkNumber
Water6
Milk4
Juice9
Smoothie5
Total24

Key words. A pie chart shows a whole split into sectors. Each sector is a slice of the circle. The whole pie is 360 degrees, and it is also 100 per cent.

Method. Add the angles of all the sectors. They must make a whole pie.

  1. Work out the angle of each sector. Each angle is its share of the total, times 360 degrees.
    Water 90°; Milk 60°; Juice 135°; Smoothie 75°
  2. Add the angles. The angles of a whole pie add up to 360 degrees.
    90 + 60 + 135 + 75 = 360
  3. Write the answer. The angles make a whole turn, so the chart is correct.
    360

Check. 360 degrees is a whole turn. ✓

Watch out. If the total is not 360, one of the angles is wrong. Find it before you draw the chart.

Watch out

Common mistake: dividing the number by 360 instead of multiplying the fraction by 360. Write the fraction first, then multiply it by 360. For a sector of 6 out of 24, the fraction is 1/4, so the angle is 90 degrees, not 6 ÷ 360.

Exercise 12C  Practice: working out the angles for a pie chart

Non-calculator
1
FThe table shows the favourite drink of 24 children. Work out the angle of the water sector, in degrees.
DrinkNumber
Water6
Milk4
Juice9
Smoothie5
Total24
2
FThe table shows the favourite drink of 24 children. Work out the angle of the juice sector, in degrees.
DrinkNumber
Water6
Milk4
Juice9
Smoothie5
Total24
3
FThe table shows the favourite drink of 24 children. Work out the angle of the smoothie sector, in degrees.
DrinkNumber
Water6
Milk4
Juice9
Smoothie5
Total24
4
FA class has 30 books in its library. The table shows the types. Work out the angle of the fiction sector, in degrees.
Type of bookNumber
Fiction15
Non-fiction9
Poetry6
Total30
5
FA class has 30 books in its library. The table shows the types. Work out the angle of the poetry sector, in degrees.
Type of bookNumber
Fiction15
Non-fiction9
Poetry6
Total30
6
FPupils voted for a class pet. The table shows 60 votes. Work out the angle of the hamster sector, in degrees.
PetNumber
Rabbit15
Hamster20
Guinea pig10
Fish15
Total60
7
FThe table shows the lunch choices of 40 children. Work out the angle of the sandwich sector, in degrees.
LunchNumber
Sandwich16
Jacket10
Pasta8
Salad6
Total40
8
FPupils voted for a class pet. The table shows 60 votes.   (a) Work out the angle of the guinea pig sector, in degrees.   (b) Work out the angle of the fish sector, in degrees.
PetNumber
Rabbit15
Hamster20
Guinea pig10
Fish15
Total60
(a)the angle of the guinea pig sector
(b)the angle of the fish sector
9
RA pie chart has a sector of 90 degrees for 15 children. Mia says the whole chart shows 60 children. Is she right? Explain your answer.
10
PSLee drew the jacket potato sector of a pie chart for 40 children as 100 degrees. The table shows 10 children chose a jacket potato. What should the angle be, in degrees?
LunchNumber
Sandwich16
Jacket10
Pasta8
Salad6
Total40
11
RA pie chart shows 36 pupils, and every sector is a whole number of pupils. What is the angle for one pupil, in degrees?

Watch out

Watch out: a sector for each group in the table needs its own angle. If you use the wrong total, every angle will be wrong. Check that the angles add up to 360 before you draw.

Exam tip

When you draw the chart with a protractor, start at the top, go clockwise, and draw the biggest sector first if you wish.

12.4

Reading and drawing line graphs

Year 6
Interpret and construct line graphs: read values from a scale, and plot points and join them with straight lines
Video lesson: Reading and drawing line graphsComing soon

A line graph shows how a value changes over time. A point is marked for each time, day, month or week. The points are joined with straight lines. The bottom of the graph shows the time, and the side shows the value and its units.

To read a value, find the label along the bottom, go straight up to the point, and then go across to the scale. To draw a line graph, choose a scale that fits all your values. Each square should be worth the same amount. Plot each point carefully, then join the points in order with a ruler.

If the graph has numbered points, you can write the point as coordinates. The first number is the point number from the left, and the second number is its value.

01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
The rainfall at a weather station in six months. Each point is one month.

Key ideas

  • Read the point, not the line between two points.
  • Each square on the scale is worth the same amount. Count the steps, not the lines.
  • To draw a point, count up the scale from zero to find the number of squares.
  • Join the points in order from left to right, with straight lines.
Worked example 12.4.1
How much rain fell in April? (Use the rainfall graph.)
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall

Key words. A line graph marks a point for each time or week, and joins the points with straight lines. The scale is the numbers up the side. The lines across the graph are the gridlines.

Method. Find the label, go up to the point, then go across to the scale.

  1. Find the time or week along the bottom. Find Apr on the horizontal line.
    Apr
  2. Go up to the line. Go straight up from the label to the point. Do not read the line between two points.
    the point for Apr
  3. Go across to the scale. Read the number on the scale level with the point. The scale is in mm.
    Rainfall in April = 20

Check. The point sits on a gridline, so the reading is exact. ✓

Watch out. Read the point itself. A line between two points only shows the change.

Worked example 12.4.2
The first point is 1, the second is 2, and so on. Write the coordinates of the point for June. (Use the rainfall graph.)
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall

Key words. A line graph marks a point for each time or week, and joins the points with straight lines. The scale is the numbers up the side. The lines across the graph are the gridlines.

Method. Use the position of the point for the across number and its value for the up number.

  1. Number the points from left to right. Start counting at 1 for the first point.
    Jun is point number 6
  2. Read the value on the scale. This is the up number, the value on the scale for that point.
    value = 70
  3. Write the coordinates. Write the point number first, then the value.
    (6, 70)

Check. Counting 6 points from the left reaches Jun, and its value is 70. ✓

Watch out. Write the position first, then the value. Do not swap them.

Worked example 12.4.3
Each square on the scale is worth £10. The point for Week 5 is £60. How many squares above zero should the point be drawn?
01020304050607080Week 1Week 2Week 3Week 4Week 5Money raised (£)

Key words. A line graph marks a point for each time or week, and joins the points with straight lines. The scale is the numbers up the side. The lines across the graph are the gridlines.

Method. Divide the value by the value of one square on the scale.

  1. Write the value to draw. The money raised in Week 5 is the value we need to plot.
    The money raised in Week 5 = 60
  2. Write the value of one square. Each square on the scale goes up by this much.
    1 square = 10
  3. Count the squares from zero. Divide the value by the value of one square.
    60 ÷ 10 = 6

Check. 6 squares of 10 make 60. ✓

Watch out. Count the steps on the scale, not the lines. Each square is worth the same amount.

Watch out

Common mistake: reading the value from the bottom instead of the side. Go up from the time to the point, then across to the scale. Another mistake is to count the lines as one unit each, when each step on the scale may be worth 10 or 20.

Exercise 12D  Practice: reading and drawing line graphs

Non-calculator
1
FUse the rainfall graph. How much rain fell in March, in mm?
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
2
FUse the rainfall graph. In which month was the rainfall 50 mm?
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
3
FUse the money graph. How much money was raised in Week 4, in pounds?
01020304050607080Week 1Week 2Week 3Week 4Week 5Money raised (£)Money raised
4
FUse the battery graph. What percentage of the charge was left at 12:00?
02040608010008:0010:0012:0014:0016:00Battery (%)Battery
5
FYou are drawing the money graph on a grid. Each square on the scale is worth £10. The point for Week 2 is £50. How many squares above zero should the point be drawn?
01020304050607080Week 1Week 2Week 3Week 4Week 5Money raised (£)
6
PSYou are drawing the battery graph. Each square on the scale is 20 per cent. The point for 14:00 is 40 per cent. How many squares above zero should the point be drawn?
02040608010008:0010:0012:0014:0016:00Battery (%)
7
PSNumber the months 1 for January, 2 for February, and so on. Write the coordinates of the point for March, as (month number, rainfall in mm).
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
8
FUse the rainfall graph.   (a) Which month has the highest point?   (b) Which month has the lowest point?
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
(a)the month with the highest point
(b)the month with the lowest point
9
FUse the rainfall graph. How many more millimetres of rain fell in January than in February?
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
10
RUse the rainfall graph. Sam says the rainfall went down in every month from January to June. Is Sam right? Explain how you know.
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall

Watch out

Watch out: check the scale before you read. A scale may go up in steps of 2, 5, 10 or 20, so one square is not always one unit.

Exam tip

When you draw a line graph, use a ruler for the lines and check each point against the table before you join them.

12.5

Solving problems with line graphs

Year 6
Use line graphs to solve problems: compare two lines, find sums and differences, and find the biggest rise and the rate of change
Video lesson: Solving problems with line graphsComing soon

Line graphs help you solve problems. Often a question asks you to compare two lines, to find the difference between two points, or to find the total of several points. Read each value carefully before you calculate.

A rise is how much a value goes up between two points. The biggest rise is the pair of neighbouring points where the line goes up the most. To find the average change for each hour, divide the total change by the number of hours. Count the gaps between the times, not the points.

When two lines are on the same graph, they are equal at any point where they meet at the same height.

051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
The points scored by two teams in five quiz rounds. Each team has its own line.

Key ideas

  • Read both values before you compare, and use the same round for both teams.
  • The total is found by adding all the values you need.
  • A rise is the value at the end minus the value at the start.
  • Average change per hour = total change ÷ number of hours.
Worked example 12.5.1
In round 5, how many more points did the Red team score than the Blue team? (Use the quiz graph.)
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team

Key words. The change is how much a value has gone up or down. To find it, take the first value away from the second value.

Method. Read both values, then take the smaller one away from the bigger one.

  1. Read the first value. Red team is 60 points.
    Red team = 60
  2. Read the second value. Blue team is 55 points.
    Blue team = 55
  3. Find the difference. Red team is the bigger, so take the smaller value away.
    60 − 55 = 5
  4. Write the answer.
    5

Check. 55 + 5 = 60, so the difference is right. ✓

Watch out. Subtract the smaller value from the bigger value. The answer is never negative.

Worked example 12.5.2
What is the total number of points scored by the Blue team in rounds 1 to 3? (Use the quiz graph.)
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team

Key words. A line graph marks a point for each time or week, and joins the points with straight lines. The scale is the numbers up the side. The lines across the graph are the gridlines.

Method. Read every value you need, then add them all together.

  1. Read each value you need. Find each point on the graph, one at a time.
    Round 1 40, Round 2 30, Round 3 45
  2. Add the values. Add them all together to find the total.
    40 + 30 + 45 = 115
  3. Write the answer. This is the total for the Blue team in rounds 1 to 3.
    115

Check. Add them in a different order to check: the total is still 115. ✓

Watch out. Make sure you have read every point, including the last one.

Worked example 12.5.3
The water in the rain barrel rose from 08:00 to 11:00. On average, how many litres did it rise for each hour? (Use the barrel graph.)
05101520253035404550556008:0009:0010:0011:0012:00LitresWater

Key words. The change is how much a value has gone up or down. To find it, take the first value away from the second value.

Method. Divide the total change by the number of hours.

  1. Find the total change. The water level changed by this much.
    30 litres
  2. Write the number of hours. Count the gaps between the times, one hour for each gap.
    3 hours
  3. Divide the change by the hours. This gives the average change for each hour.
    30 ÷ 3 = 10
  4. Write the answer. This is the change for each hour, on average.
    10

Check. 10 × 3 = 30, so the answer is right. ✓

Watch out. Count the gaps, not the points. Five points make four gaps between them.

Watch out

Common mistake: counting the gaps wrong. From 08:00 to 12:00 there are four hours, so there are four gaps. Count the gaps, not the points on the graph.

Exercise 12E  Practice: solving problems with line graphs

Non-calculator
1
FUse the quiz graph. In round 4, how many more points did the Red team score than the Blue team?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
2
FUse the quiz graph. In which round did both teams score the same number of points?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
3
FUse the quiz graph. What is the total number of points scored by the Red team in all five rounds?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
4
PSUse the quiz graph.   (a) What is the total number of points scored by the Blue team in all five rounds?   (b) How many more points did the Blue team score than the Red team, over all five rounds?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
(a)the Blue team total
(b)how many more the Blue team scored in total
5
PSUse the quiz graph. Between which two rounds did the Red team score the biggest rise?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
6
FUse the barrel graph. How many litres did the water level rise from 08:00 to 12:00?
05101520253035404550556008:0009:0010:0011:0012:00LitresWater
7
FUse the barrel graph. What was the average rise in litres for each hour from 08:00 to 12:00?
05101520253035404550556008:0009:0010:0011:0012:00LitresWater
8
FUse the barrel graph. Between which two times did the water level rise the most?
05101520253035404550556008:0009:0010:0011:0012:00LitresWater
9
PSUse the quiz graph.   (a) In round 3, how many more points did the Blue team score than the Red team?   (b) In how many of the five rounds did the Red team score more points than the Blue team?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
(a)how many more the Blue team scored in round 3
(b)the number of rounds the Red team scored more
10
RUse the barrel graph. Ali says the water filled at the same speed every hour. Is Ali right? Explain how you know.
05101520253035404550556008:0009:0010:0011:0012:00LitresWater

Watch out

Watch out: a line that goes up is not always going up steadily. Check each pair of neighbouring points before you say a line rises the same amount each time.

Exam tip

Write down each value you read before you calculate. It is easy to mix up two lines on the same graph.

12.6

The mean as an average

Year 6
Calculate and interpret the mean as an average: add the values, count them, and divide the total by the count
Video lesson: The mean as an averageComing soon

The mean is an average. It tells you the value that each one would have if the values were shared out equally. For example, if four children score 6, 8, 7 and 9 in a test, the mean is the score each child would get if all the points were shared equally.

To find the mean, follow three steps. First, add all the values to find the total. Second, count how many values there are. Third, divide the total by the number of values. Every value counts, even a zero.

The mean is always between the smallest value and the biggest value. It does not have to be one of the values in the list.

0246810Amira6Noah8Priya7Jack9Test scores (out of 10)
Four test scores. The mean is 7.5, which is between the lowest score of 6 and the highest score of 9.

Key ideas

  • Mean = total ÷ the number of values.
  • Count every value, including a zero.
  • The mean is between the smallest and the biggest value.
  • The mean is also called the average.
Worked example 12.6.1
Find the mean of the four test scores 6, 8, 7 and 9.

Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.

Method. Add all the values, then share the total equally between the number of values.

  1. Write the values. These are all the values. Count every one, even a zero.
    6, 8, 7, 9
  2. Add the values to find the total. The total is what all the values make together.
    6 + 8 + 7 + 9 = 30
  3. Count the values. Count how many numbers there are. Do not count the total.
    There are 4 values
  4. Divide the total by the number of values. Sharing the total equally between the values gives the mean.
    30 ÷ 4 = 7.5

Check. 7.5 × 4 = 30, so the mean is right. ✓

Watch out. Divide by how many values there are, not by the biggest value. Do not forget to count a zero.

Worked example 12.6.2
Ali read 3, 5, 4, 2 and 6 books in five weeks. Find the mean number of books read in one week.

Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.

Method. Add all the values, then share the total equally between the number of values.

  1. Write the values. These are all the values. Count every one, even a zero.
    3, 5, 4, 2, 6
  2. Add the values to find the total. The total is what all the values make together.
    3 + 5 + 4 + 2 + 6 = 20
  3. Count the values. Count how many numbers there are. Do not count the total.
    There are 5 values
  4. Divide the total by the number of values. Sharing the total equally between the values gives the mean.
    20 ÷ 5 = 4

Check. 4 × 5 = 20, so the mean is right. ✓

Watch out. Divide by how many values there are, not by the biggest value. Do not forget to count a zero.

Worked example 12.6.3
A team scored 0, 2, 3 and 1 goals in four matches. Find the mean number of goals in a match.

Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.

Method. Add all the values, then share the total equally between the number of values.

  1. Write the values. These are all the values. Count every one, even a zero.
    0, 2, 3, 1
  2. Add the values to find the total. The total is what all the values make together.
    0 + 2 + 3 + 1 = 6
  3. Count the values. Count how many numbers there are. Do not count the total.
    There are 4 values
  4. Divide the total by the number of values. Sharing the total equally between the values gives the mean.
    6 ÷ 4 = 1.5

Check. 1.5 × 4 = 6, so the mean is right. ✓

Watch out. Divide by how many values there are, not by the biggest value. Do not forget to count a zero.

Watch out

Common mistake: dividing by the wrong number. Divide by how many values there are, not by the biggest value, and do not forget a zero. Also, the mean is not the middle number when the values are put in order. That middle number is a different average.

Exercise 12F  Practice: finding the mean

Non-calculator
1
FFour children score 4, 7, 9 and 6 in a quiz. Find the mean score.
2
FRuby ran 12 km, 15 km and 9 km in three weeks. Find her mean distance for each week, in km.
3
FA team scored 3, 0, 5, 4 and 8 goals in five matches. Find the mean number of goals in a match.
4
FLeo read for 20, 25, 15, 30, 10 and 20 minutes on six days. Find the mean number of minutes he read each day.
5
FThe number of children at a football club on four Saturdays was 18, 22, 20 and 24. Find the mean number of children each Saturday.
6
FFive pupils threw a ball 5 m, 6 m, 4 m, 7 m and 8 m. Find the mean distance, in metres.
7
FThe scores of two groups are shown.   (a) Group X: 2, 4, 6 and 8. Find the mean.   (b) Group Y: 1, 3 and 5. Find the mean.
(a)the mean of group X
(b)the mean of group Y
8
RMo says that if the mean of four numbers is 10, the four numbers add up to 40. Is Mo right? Explain your answer.
9
RIs the mean always one of the numbers in the list? Use the numbers 2, 4 and 9 to explain.
10
PSSam scores 8, 6, 9, 7 and 5 in five spelling tests. Find his mean score.

Watch out

Watch out: the mean can be a decimal, such as 7.5 or 1.5. Do not round it to a whole number unless the question asks you to.

Exam tip

Check your mean by multiplying it by the number of values. You should get the total back.

12.7

Using the mean to solve problems

Year 6
Use the mean to find a total or a missing value, to combine two means, and to say what a mean does and does not tell us
Video lesson: Using the mean to solve problemsComing soon

If you know the mean and the number of values, you can find the total. Multiply the mean by the number of values. This works because the mean is the total shared equally.

You can also find a missing value. Work out the total from the mean, then take away the values you know. What is left is the missing value.

When two groups are put together, do not simply average the two means. First find each group total, add the totals, and divide by the number of values in both groups.

A mean is one number, so it does not show every value. A single very high or very low value can change the mean a lot.

Key ideas

  • Total = mean × the number of values.
  • Missing value = total from the mean − the values you know.
  • Two groups together: add the two totals, then divide by all the values.
  • A mean does not show each value. It shows the average.
Worked example 12.7.1
Six friends have a mean of 9 stickers each. How many stickers do they have altogether?

Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.

Method. Multiply the mean by the number of values to get the total.

  1. Write the mean and the number of values. The mean is the average of the values.
    mean 9, number of values 6
  2. Multiply the mean by the number of values. Sharing the total equally gave the mean, so multiplying gives the total back.
    9 × 6 = 54
  3. Write the answer. This is the total of all the values.
    54

Check. 54 ÷ 6 = 9, so the mean is right. ✓

Watch out. Multiply to find the total from a mean. Divide only when you need the mean from a total.

Worked example 12.7.2
Five numbers have a mean of 8. Four of them are 9, 6, 7 and 10. Find the fifth number.

Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.

Method. Find the total from the mean, then take away the values you know.

  1. Work out the total from the mean. The total is the mean times the number of values.
    8 × 5 = 40
  2. Add the values you know. These values are already in the list.
    9 + 6 + 7 + 10 = 32
  3. Take the known total away from the total. The missing value is what is left over.
    40 − 32 = 8
  4. Write the answer. This is the missing value.
    8

Check. 32 + 8 = 40, and 40 ÷ 5 = 8. ✓

Watch out. Find the total first. Do not take away the mean from one value.

Worked example 12.7.3
Class A has 5 children with a mean score of 12. Class B has 5 children with a mean score of 15. Find the mean score of all 10 children.

Key words. The mean is the same as the average. Add all the values to get the total, then share the total equally between the number of values.

Method. Turn each mean back into a total, add the totals, then divide by the number of values altogether.

  1. Find the total of the first group. Multiply its mean by its number of values.
    12 × 5 = 60
  2. Find the total of the second group. Multiply its mean by its number of values.
    15 × 5 = 75
  3. Add the two totals. Together the two groups have this total.
    60 + 75 = 135
  4. Count everybody and divide. There are 5 + 5 = 10 values altogether.
    135 ÷ 10 = 13.5

Check. 13.5 × 10 = 135, so the combined mean is right. ✓

Watch out. Do not just average the two means. That only works when both groups have the same number of values.

Watch out

Common mistake: averaging two means without weighting them. If the groups have different numbers of values, the mean of all the values is not the average of the two means. Always add the totals first.

Exercise 12G  Practice: using the mean to solve problems

Non-calculator
1
FEight bags have a mean mass of 7 kg each. What is the total mass of the eight bags, in kg?
2
FFour children have a mean test score of 7. Three of their scores are 5, 8 and 9. What is the fourth score?
3
FFive numbers have a mean of 6. Four of them are 2, 4, 7 and 9. Find the fifth number.
4
RSam says that if the mean of five numbers is 6, one of the numbers must be 6. Is Sam right? Use the numbers 3, 4, 5, 8 and 10 to explain.
5
PSClass 6A has 5 children with a mean score of 8. Class 6B has 5 children with a mean score of 14. Find the mean score of all 10 children.
6
FRuby reads for a mean of 25 pages each day over 6 days. How many pages does she read altogether?
7
PSFive numbers have a mean of 12, so their total is 60. One of the numbers is 20, and it is changed to 10. What is the new mean?
8
PSFive numbers are 6, 9, 7, 10 and 8.   (a) Find the mean.   (b) The 8 is changed to 13. Find the new mean.
(a)the mean of the five numbers
(b)the new mean after the 8 is changed to 13
9
RThree numbers have a mean of 4. Could their total be 10? Explain your answer.
10
PSFour children have a mean height of 140 cm. Three of them are 136 cm, 142 cm and 150 cm tall. How tall is the fourth child, in cm?

Watch out

Watch out: a mean tells you the average, not every value. Two sets of values can have the same mean and be very different.

Exam tip

When you find a missing value, check it by putting it back into the list and working out the mean again.

Chapter summary

SkillMethod
Pie chartThe whole pie is 360 degrees, and it is 100 per cent. The sectors add up to the whole.
Angle of a sectorFraction of the whole, times 360 degrees. Per cent can be used the same way.
Number in a sectorAngle ÷ 360 × total, or per cent ÷ 100 × total.
Drawing a pie chartWrite each group as a fraction of the total, times 360. The angles must add up to 360.
Line graphRead the point, go across to the scale. Join the points in order with straight lines.
Change and rateChange = end value minus start value. Rate for each hour = change ÷ number of hours.
MeanMean = total ÷ number of values. Count every value, including a zero.
Totals and missing valuesTotal = mean × number of values. Missing value = total minus the values you know.
Two groupsAdd the two totals, then divide by the number of values in both groups.

Can you…?

  • I can read the per cent and the angle of a sector on a pie chart.
  • I can find the number in a sector from its angle or per cent, when the total is known.
  • I can work out the angle of each sector from a table, and check that the angles add up to 360.
  • I can read values from a line graph, and plot points on a scale.
  • I can solve problems with line graphs: differences, totals, the biggest rise and the average change for each hour.
  • I can find the mean of a set of values, and count every value, including a zero.
  • I can use the mean to find a total or a missing value, and explain what the mean does and does not tell us.

Key terms

Pie chart
A circle split into sectors, where each sector shows a part of the whole.
Sector
A slice of a pie chart, with its own angle and its own share of the whole.
Per cent
Out of 100. A whole pie chart is 100 per cent.
Line graph
A graph that marks a point for each time or week, and joins the points with straight lines.
Scale
The numbers up the side of a graph, which show the value of each point.
Gridline
A line across a graph that helps you read a value exactly.
Mean
The average: the total of the values divided by the number of values.
Total
All the values added together.
Rate
How much something changes for each unit, such as each hour.
Average
A single value that represents a set of values. In this chapter, the average is the mean.

Exercise 12Z  Check-up

Non-calculator
1
The pie chart shows how the pupils travel to school. How many degrees is the sector for the bus? (Walk 40, bus 25, car 20, bike 15 per cent.)
Walk: 40%Bus: 25%Car: 20%Bike: 15%
2
The pie chart shows the favourite fruit of 120 pupils. How many pupils chose bananas? The sector for bananas is 90 degrees.
Apples: 135°Bananas: 90°Oranges: 72°Grapes: 63°
3
A pie chart stands for 30 children. One sector is 48 degrees. How many children are in that sector?
4
12 of the 40 children in a club chose football. What percentage of the children is this? Give your answer as a number.
5
Use the rainfall graph. How much rain fell in June, in mm?
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
6
Use the rainfall graph. In which month was the rainfall 40 mm?
01020304050607080JanFebMarAprMayJunRainfall (mm)Rainfall
7
Use the quiz graph. How many points did the Blue team score in round 3?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
8
Find the mean of 8, 10, 12, 14 and 6.
9
Four numbers have a mean of 8. Three of them are 6, 9 and 11. What is the fourth number?
10
Find the mean of each set of numbers.   (a) 3, 6 and 9.   (b) 2, 8 and 5.
(a)the mean of 3, 6 and 9
(b)the mean of 2, 8 and 5

Total 21 marks. Allow about 21 minutes.

Exercise 12X  SATs-style questions

Non-calculator
1
FThe pie chart shows how the pupils travel to school. 75 pupils are in the survey. How many of them walk to school? The sector for walking is 40 per cent.
Walk: 40%Bus: 25%Car: 20%Bike: 15%
[1 mark]
2
FUse the quiz graph. How many points did the Blue team score in round 4?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
[1 mark]
3
FFind the mean of 6, 8, 10 and 12.[1 mark]
4
PSSixty pupils were asked how they travel to school. 10 of them travel by car. Work out the angle of the car sector in a pie chart. Show your working. Give the angle in degrees.[2 marks]
5
PSThe pie chart shows the pets of 40 children. The sector for dogs is 35 per cent. How many children have a dog? Show your working.
Dogs: 35%Cats: 25%Fish: 20%Rabbits: 10%None: 10%
[2 marks]
6
PSUse the barrel graph. How many litres did the water level rise from 09:00 to 11:00? Show your working.
05101520253035404550556008:0009:0010:0011:0012:00LitresWater
[2 marks]
7
PSFive numbers have a mean of 7. Four of them are 3, 9, 5 and 8. Find the fifth number. Show your working.[2 marks]
8
PSThe graph shows the money raised each week, in pounds.   (a) Find the total money raised in the five weeks.   (b) Find the mean money raised each week.   (c) Which weeks raised more than the mean?[3 marks]
(a)the total money raised in the five weeks
(b)the mean money raised each week
(c)the weeks that raised more than the mean
9
PSUse the quiz graph.   (a) In round 5, how many more points did the Red team score than the Blue team?   (b) In how many of the five rounds did the Red team score more points than the Blue team?
051015202530354045505560Round 1Round 2Round 3Round 4Round 5PointsRed teamBlue team
[3 marks]
(a)how many more points the Red team scored in round 5
(b)the number of rounds the Red team scored more
10
RClass A has 5 children with a mean score of 12. Class B has 5 children with a mean score of 15.   (a) Find the total score of Class A.   (b) Find the total score of Class B.   (c) Find the mean score of all 10 children.   (d) Sam says the answer to (c) is (12 + 15) ÷ 2. Explain whether Sam is right in general.[4 marks]
(a)the total score of Class A
(b)the total score of Class B
(c)the mean score of all 10 children
(d)whether Sam is right in general
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