## What you need to know

### Pictographs

A pictograph (or pictogram) is a way of displaying data using pictures. It is a straightforward topic that once you know how the pictograms work then you should see they all work in the same way. Remember that the key given (or sometimes not given) is always key to the question.

Having a good knowledge of fractions will help with this topic.

## Example 1: Interpreting Pictographs

Riley recorded how much TV she watched from Monday to Friday one week and displayed her results using a.

a)How long did Riley spend watching TV on Thursday? Give your answer in hours and minutes.

b) Over the weekend (Saturday and Sunday), Riley watched 4 times as much TV as she did on Friday. How many hours of TV did she watch over the course of the weekend?

a) In order to be understood, all pictographs must have a key. A key is the thing that here is shown to the right of the table – it tells us how to read the pictures that are shown. In this case, one picture of a TV is worth 30 minutes. From that, we can gather than two TVs must be worth 60 minutes (60 is double 30).

So, looking at Thursday, we can see that there are 3 whole TVs and 1 half TV. Therefore, the total time spent watching TV on Thursday is

(3\times 30) + 15 = 105 \text{ minutes}

105 = 60 + 45 \equiv 1\text{ hour, }45\text{ minutes}

So, the total time is 1 hour and 45 minutes.

b) We need to work out how much TV she watched on Friday. There is one whole TV and one half-TV, so we get

\text{time spent watching TV on Friday }=30+15=45\text{ minutes}

During the weekend, she watched 4 times as much:

45 \times 4=180\text{ minutes}

180 = 60 \times 3 \equiv 3\text{ hours spent watching TV on the weekend}.

## Example 2: Constructing a Pictograph

Faye attends a big social football club every Sunday. For 5 weeks she counted the number of people who attended and recorded the results in the table below. Draw a pictograph of Faye’s data.

We’re going to use footballs as the picture – it’s good to use as obvious a picture as possible – so we now have to decide how many people each football will be worth. Here, we’re going to choose to have 1 football represent 20 people.

In weeks 2, 4, and 5, the problem is familiar. Considering whole footballs being worth 20, we know that half-footballs can be used to represent 10 people, so we get:

\text{Week 2: }60=20\times 3, \text{ so we will draw 3 footballs}However, we have to go one step further for weeks 1 and 3 and consider using a quarter of a football to represent 5 people. Doing this, we get

\text{Week 1: }55=40+15=(20\times 2)+(3\times 5), \text{ so we will draw 2 footballs and 3 quarters of a football}Now this is done, our completed pictograph (not forgetting the key) looks like.

### Example Questions

We are going to use an orange as the image for this picture (although you could use any symbol you like), so we now have to decide what our key should be. In this case, we are going to choose to make each image of an orange represent 2 oranges eaten. It is very important in your pictogram that you have a key that states that one orange image equals two oranges eaten.

(You could chose to make each image equal 1 orange, which is completely acceptable. However, it would be better to have each image represent multiple oranges in order make it easier for the person viewing the data to work out each category total.)

So, in week 1 there were 8 oranges eaten by Jenna’s family. 8\div2= 4, so we will have to draw 4 pictures for week 1.

In week 2 there were 9 oranges eaten. 9\div2 = 4.5, which means we will have to draw 4 whole oranges and one half-orange.

Continuing this process for the other two weeks, you should get a pictograph that looks like the below:

a) On Tuesday there are 4 whole pictures of shoes, and one quarter of a picture.

If one picture = 2km

then

\frac{1}{4} of a picture = 2\div 4=0.5\text{ km}

So, the distance walked on Tuesday is

(4\times 2)+ 0.5=8.5\text{ km}

b) 6km is her aim. Since one picture is worth 2km, then we need to find the days where there are 3 whole sets of trainers shown.

Therefore, we can see that there are 3 days – Tuesday, Thursday, and Friday – where she achieved her goal.

3) The pictogram below shows how much money was raised for charity by 6 members of a form group:

In the pictogram, each circle represents £20.

a) How many children raised more than £50?

b) What fraction of the children raised less than £60?

c) To the nearest pound, what was the mean amount of money raised?

a) We have been told that each circle represents £20 raised. Therefore every semi-circle represents £10 raised and every quarter-circle represents £5 raised.

For this question, we also need to know some common decimal facts, namely that \frac{1}{2}=0.5, \frac{1}{4}=0.25 and \frac{3}{4}=0.75.

All we need to do is look at each member of the form group individually to work out how much they have raised each:

Sally: 3 circles = 3\times\pounds20=\pounds60

Ahmed: 3\frac{1}{2} circles = 3.5\times\pounds20=\pounds70

Delaine: 1\frac{1}{4} circles = 1.25\times\pounds20=\pounds25

Priti: 1\frac{1}{4} circles = 1.25\times\pounds20=\pounds25

Annabelle: 4\frac{1}{2} circles = 4.5\times\pounds20=\pounds90

Derek: 2\frac{3}{4} circles = 2.75\times\pounds20=\pounds55

We can therefore see that Sally, Ahmed, Priti, Annabelle and Derek raised more than £50, so 5 people raised more than £50.

b) There were just 2 students who raised *less* than £60 and that was Delaine and Derek (Do not count Sally since £60 is not *less *than £60).

Therefore, of the 6 students, 2 of them raised less than £60, so we can write this as the following fraction:

\dfrac{2}{6}

This fraction can be simplified. Since both 2 and 6 are multiples of 2, if we divide both top and bottom by 2, we will have the fraction in its simplest form:

\dfrac{1}{3}

c) We have already worked out how much money each individual student raised. If we add up these amounts up, we will have a combined total of money raised:

\pounds60+\pounds70+\pounds25+\pounds65+\pounds90+\pounds55 = \pounds365

Since there are 6 students, the mean amount raised will be the combined total divided by 6:

\pounds365\div6 = \pounds60\text{ to the nearest pound}

4) The pictogram shows some information about the number of types of butterflies in a butterfly farm:

In total, there are 60 Purple Emperors.

There are 3 times as many Red Admirals as there are Purple Emperors.

The number of Silver-studded Blue butterflies is \frac{2}{3} the number of Red Admirals.

There are 100% more Black Hairstreaks than there are Silver-studded Blues.

The number of Wood Whites is 37.5% of the number of Black Hairstreaks.

Complete the pictograph above.

The first thing we need to do in this question is to work out how many butterflies are represented by each butterfly image in the pictograph. We are told that there are 60 Purple Emperors, and this is shown with 1\frac{1}{2} butterflies.

Therefore, each butterfly image in the pictograph must represent:

1\text{ image}=60\div1.5 = 40\text{ butterflies}

We are told that there are 3 times as many Red Admirals as there are Purple Emperors. If there are 60 Purple Emperors, then there must be 3\times60 = 180 Red Admirals.

We are told that the number of Silver-studded Blue butterflies is \frac{2}{3} the number of Red Admirals. If there are 180 Red Admirals, then the number of Silver-studded Blues can be calculated as follows:

180\times\dfrac{2}{3} = 120\text { Silver-studded Blues}

We are told that there are 100% more Black Hairstreaks than there are Silver-studded Blue butterflies. This means that the number of Blasck Hairstreaks is double the number of Silver-studded Blues. Since there are 120 Silver-studded Blues, then there must be 2\times120 = 240 Black Hairstreaks.

Finally, we are told that the number of Wood Whites is 37.5% of the number of Black Hairstreaks. Since there are 240 Black Hairstreaks, then the number of Wood Whites can be calculated as follows:

240\times0.375 = 90\text{ Wood Whites}

Since we know know exactly how many butterflies there are of each species, we now need to work out how many butterfly images to draw to represent each species total.

One butterfly image represents 40 butterflies, so we can calculate the number of butterfly images we need to draw for each species as follows:

\text{Red Admiral: }180\div40=4.5\text{ butterfly images}

\text{Silver-studded Blue: }120\div40=3\text{ butterfly images}

\text{Black Hairstreak: }240\div40=6\text{ butterfly images}

\text{Wood White: }90\div40=2.25\text{ butterfly images}

Therefore, your final pictograph should look like the below:

5) Gary has been tracking the amount of guitar practice he does over a 4-week period.

The ratio of guitar practice he does for week 1 to week 2 is 7 : 8.

The ratio of guitar practice he does for week 2 to week 3 is 2 : 1.

The ratio of guitar practice he does for week 3 to week 4 is 2 : 3.

Gary does 24 hours of guitar practice in week 4.

Complete the below pictogram to represent the above information

The key piece of information that we have is that in week 4, Gary does 24 hours of guitar practice.

We can use this key piece of information to help us solve the statement: the ratio of guitar practice he does for week 3 to week 4 is 2 : 3. (This is the only statement we can try to work out at the moment, since the only known value we have is the week 4 value.)

If the ratio of guitar practice is 2 : 3 for week 3 to week 4, then we will need to find an equivalent ratio for x: 24 where x represents the week 3 value. Since the week 4 figure is 8 times greater than the figure given in the ratio (24\div3=8), then we will have an equivalent ratio if we also multiply the week 3 ratio figure by 8.

Since 2\times8=16, Gary therefore does 16 hours of guitar practice in week 3.

Since we now know the week 3 value, we can work out the week 2 value.

If the ratio of guitar practice is 2 : 1 for week 2 to week 3, then we will need to find an equivalent ratio for x : 16 where x is the week 2 value. Since the week 3 figure is 16 times greater than the figure given in the ratio, then we will have an equivalent ratio if we also multiply the week 2 ratio figure by 16.

Since 16\times2=32, Gary therefore does 32 hours of guitar practice in week 2.

Since we now know the week 2 value, we can work out the week 1 value.

If the ratio of guitar practice is 7 : 8 for week 1 to week 2, then we will need to find an equivalent ratio for x : 32 where x is the week 1 value. Since the week 2 figure is 4 times greater than the figure given in the ratio (32\div8=4), then we will have an equivalent ratio if we also multiply the week 1 ratio figure by 4.

Since 4\times7=28, Gary therefore does 28 hours of guitar practice in week 1.

Now that we have that total number of hours for weeks 1 – 4 (28 hours, 32 hours, 16 hours and 24 hours), we need to work out how to show this on the pictogram. Since all of the above numbers are divisible by 4, then it would be logical make your pictogram image represent 4 hours of practice.

For week 1, you would need 28\div4=7 \text{ complete images}

For week 2, you would need 32\div4=8 \text{ complete images}

For week 3, you would need 16\div4=4 \text{ complete images}

For week 4, you would need 24\div4=6 \text{ complete images}

Your final pictogram should be similar to the below:

(The image doesn’t have to be a guitar; it can be anything of your choice! Keep it simple and use a circle if you like!)

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