# Convert Real Measurements to Scale Diagrams

In this worksheet, students will apply scale factors to find the scale values of specific, real life elements or calculate the scale which has been used to create a scale diagram or model.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Ratio, Proportion and Rates of Change, Mensuration

Curriculum subtopic:   Ratio, Proportion and Rates of Change, Units and Measurement

Difficulty level:

### QUESTION 1 of 10

A lot of the time, it's not really the best idea to have things full size.

A map would be useless if it was full size, and a toy car doesn't really want to be the same size as a normal car, does it?

When we have this issue like this in maths, we use a scale diagram or a scale model.

What Is a Scale Diagram / Model?

A scale diagram is just a diagram where everything has been reduced by the same factor.

It could be half the size, a tenth of the size, or anything else, but every element must be reduced by exactly the same factor.

How Are Scales Written?

Scales are written as ratios, such as 1:100 or 1:50,000.

What Does a Scale Mean?

Scales are read left to right.

For example, the scale 1:100 would mean that every 1 unit of length on the scale is the same as 100 units in real life.

So an element that was 2 cm long on a scale diagram, would be 200 cm long in real life.

Let's look at this concept in action with some examples now.

e.g. A real car is 300 cm tall. A model of the car is created using a scale of 1:100. How tall would the model car be?

The scale tells us that every 1 cm on the model car will represent 100 cm in real life.

So all we need to do is to divide the real height by the scale factor, to find the height of the model:

300 ÷ 100 = 3 cm

So the height of the model car is 3 cm.

e.g. A model is made of a 2 m tall man. If the model is 4 cm tall, what scale has been used?

The first thing we should notice here is that the units used are different, so we need to make them the same before we start:

There are 100 cm in 1 m, so 2 m = 200 cm.

Now, we need to write these numbers as a ratio (remember that the model comes first):

4:200

Our final step is to simplify this ratio:

4:200 ÷ 4

1:50

In this activity, we will apply scale factors to find the scale values of specific, real life elements or calculate the scale which has been used to create a scale diagram or model.

A scale of 1:100 means that 2 cm on a model is 2 m in real life.

When is this statement true?

Always

Sometimes

Never

A car is 1.8 m wide.

If it is made into a model using a scale of 1:48, how wide will the model be?

Type your answer as a number below without any units, as these have already been provided for you.

Always

Sometimes

Never

A model railway is built to a scale of 1:500.

If the real track is 2.8 km long, how long will the model track be?

Type your answer as a number below without any units, as these have already been provided for you.

Always

Sometimes

Never

A model is made of a garden at a scale of 1:1000.

How tall will the model trees be if the real one are 1.2 m tall?

Type your answer as a number below without any units, as these have already been provided for you.

Always

Sometimes

Never

Match each pair of scale and real measurements on the left with the correct scale factor (expressed as a ratio) on the right.

## Column B

5 cm real; 1 cm model
1:5
14 cm real; 1.4 cm model
1:120
15 m real; 7.5 m model
1:2
0.48 m real; 0.4 cm model
1:10

A lorry is 7 m long.

If a model is made that is 3.5 cm long, what scale has been used?

## Column B

5 cm real; 1 cm model
1:5
14 cm real; 1.4 cm model
1:120
15 m real; 7.5 m model
1:2
0.48 m real; 0.4 cm model
1:10

The Burj Khalifa in Dubai is 828 metres tall.

If a model is made using a scale of 1:50, how tall will the model be?

## Column B

5 cm real; 1 cm model
1:5
14 cm real; 1.4 cm model
1:120
15 m real; 7.5 m model
1:2
0.48 m real; 0.4 cm model
1:10

Which of the measurements below have a scale factor of 1:150?

Select either 'Correct' or 'Incorrect' to indicate which measurements do match this scale factor and which don't.

A woman is 1.32 m tall.

If a model is made of her which is 2 cm tall, what scale has been used?

A building has a surface area of 1600 m2.

Would a model created with a scale of 1:1000 have a surface area of 1.6 m2?

Yes

No

• Question 1

A scale of 1:100 means that 2 cm on a model is 2 m in real life.

When is this statement true?

Sometimes
EDDIE SAYS
This statement may often be true, but it is not always true. The use of the scale 1:100 means that the real life object would be 100 times larger than the scale model, but the units of measurement could change.
• Question 2

A car is 1.8 m wide.

If it is made into a model using a scale of 1:48, how wide will the model be?

Type your answer as a number below without any units, as these have already been provided for you.

EDDIE SAYS
We know that the scale 1:48 means the model car will be 48 times smaller than the real car. As the model will be much smaller than the original, it is sensible to express our starting width in cm before we start. There are 100 centimetres in a metre, so: 1.8 × 100 = 180 cm Now we need to divide the width of the real car (in cm) by the scale factor to find the width of the model: 180 ÷ 48 = 3.75 cm
• Question 3

A model railway is built to a scale of 1:500.

If the real track is 2.8 km long, how long will the model track be?

Type your answer as a number below without any units, as these have already been provided for you.

EDDIE SAYS
We know that the scale 1:500 means the model track will be 500 times smaller than the real track. As the model will be much smaller than the original, it is sensible to express our starting length in metres before we start. There are 1000 metres in a kilometre, so: 2.8 × 1000 = 2800 m Now we need to divide the length of the real track (in m) by the scale factor to find the length of the model: 2800 ÷ 500 = 5.6 m
• Question 4

A model is made of a garden at a scale of 1:1000.

How tall will the model trees be if the real one are 1.2 m tall?

Type your answer as a number below without any units, as these have already been provided for you.

EDDIE SAYS
We know that the scale 1:1000 means the model garden will be 1000 times smaller than the real garden. As the model will be much smaller than the original, it is sensible to express our starting height in cm before we start. There are 100 centimetres in a metre, so: 1.2 × 100 = 120 cm Now we need to divide the height of the real trees (in cm) by the scale factor to find the height of the model trees: 120 ÷ 1000 = 0.12 cm
• Question 5

Match each pair of scale and real measurements on the left with the correct scale factor (expressed as a ratio) on the right.

## Column B

5 cm real; 1 cm model
1:5
14 cm real; 1.4 cm model
1:10
15 m real; 7.5 m model
1:2
0.48 m real; 0.4 cm model
1:120
EDDIE SAYS
Did you remember the process we followed in the Introduction? Let's work through one pair of measurements as an example: 0.48 m real; 0.4 cm model 1) Make sure the measurements are expressed in the same units: There are 100 cm in 1 m, so 0.48 m = 48 cm 2) Write the two measurements as a ratio: 0.4:48 3) Cancel or multiply the ratio to its simplest form. Note, we cannot have a ratio which uses a decimal so, in this case, we need to multiply to convert the decimal to a whole number: 0.4:48 × 2.5 1:120 Can you use this example to work independently and find the other matches? If you are not feeling totally confident with this process, review the Introduction now before you move on to tackle the rest of this activity.
• Question 6

A lorry is 7 m long.

If a model is made that is 3.5 cm long, what scale has been used?

EDDIE SAYS
Let's follow our process again. 1) Make sure the measurements are expressed in the same units: There are 100 cm in 1 m, so 7 m = 700 cm 2) Write the two measurements as a ratio: 3.5:700 3) Cancel the ratio to its simplest form: 3.5:700 ÷ 3.5 1:200 Remember that we read the ratio from left to right, so we must write the model ratio on the left for our answer to be correct.
• Question 7

The Burj Khalifa in Dubai is 828 metres tall.

If a model is made using a scale of 1:50, how tall will the model be?

EDDIE SAYS
We know that the scale 1:50 means the model building will be 50 times smaller than the real one. We need to divide the height of the real building (in cm) by the scale factor to find the height of the model: 828 ÷ 50 = 16.56 m As there are 100 centimetres in a metre, we can multiply our answer by 100 to express this height in cm: 16.56 × 100 = 1656 cm
• Question 8

Which of the measurements below have a scale factor of 1:150?

Select either 'Correct' or 'Incorrect' to indicate which measurements do match this scale factor and which don't.

EDDIE SAYS
We can quickly tell if each pair of measurements fit this scale by dividing one of the numbers by the other to see if this creates the target of 150. However, in order to do this, both measurements must be expressed using the same units. Let's work through each, one at a time. 15 ÷ 1 = 15, which does not match 150 so this is an incorrect answer 150 ÷ 1 = 150, which does match so this is a correct answer We need to convert our '150 m' into cm before we start here. There are 100 cm in 1 m, so 150 m = 1500 cm 1500 ÷ 100 = 15, which does not match either so this is also an incorrect answer Were you able to locate the one correct answer here?
• Question 9

A woman is 1.32 m tall.

If a model is made of her which is 2 cm tall, what scale has been used?

EDDIE SAYS
Let's follow our process again. 1) Make sure the measurements are expressed in the same units: There are 100 cm in 1 m, so 1.32 m = 132 cm 2) Write the two measurements as a ratio: 2:132 3) Cancel the ratio to its simplest form: 2:132 ÷ 2 1:66
• Question 10

A building has a surface area of 1600 m2.

Would a model created with a scale of 1:1000 have a surface area of 1.6 m2?

No
EDDIE SAYS
This was a really tricky final question, so well done if you got it right! This is really testing how carefully you read and understood the Introduction. Scale factors can only be applied to lengths and other measurements expressed in 2 dimensions. As soon as we move on to working with area or volume, we need to find area or volume scale factors, which are more complex. If you are ready for a challenge, you could try an activity now which focuses on these concepts.
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