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## Introduction to number and bar scales

The two kinds of scale we will be working with in this chapter are the number scale and the bar scale. The number scale is expressed as a ratio like 1 : 50. This simply means that 1 unit on the map represents 50 units on the ground. So 1 cm on the map will represent 50 cm on the ground, or 1 m on the map will represent 50 m on the ground. To use the number scale, you need to measure a distance on a map using your ruler, and then multiply that measurement by the “real” part of the scale ratio (50) given on the map, in order to get the real distance.

The bar scale is represented like this:

Each piece or segment of the bar represents a given distance, as labelled underneath. To use the bar scale, you need to measure how long one segment of the bar is on your ruler. You must then measure the distance on the map in centimetres; calculate how many segments of the bar graph it works out to be (the total distance measured; divided by the length of one segment); and then multiply it by the scale underneath. So, if 1 cm on the bar represents 10 m on ground, and the distance you measure on the map is 3 cm ( 3 cm $$\div$$ 1 cm length of segment = 3 segments) then the real distance on the ground is 3 $$\times$$ 10 m = 30 m.

### Example 1: Using the bar and number scales

#### Question

1. You are given a map with the number scale of 1 : 40. You measure a length (on the map) of 10 cm. What is this distance in real life?
2. You are given a map with the number scale of 1 : 500. You measure a distance on the map of 15 cm with your ruler. What is this distance in real life?
3. You are given the following bar scale:
You measure the distance on the map to be 15 cm. What is the actual distance?
4. You are given the following bar scale:
You measure the distance between two points on the map to be 11 cm. What is the distance on the ground?

1. Scale is 1 : 40.
10 cm $$\times$$ 40 = 400 cm = 4 m
The distance on the ground (in real life) is 4 m.
2. Scale is 1 : 500
Therefore actual distance is 15 cm $$\times$$ 500 = 7500 cm = 75 m.
3. 1 segment = 1,5 cm long, and represents 50 m.
15 cm $$\div$$ 1,5 cm (length of segment) = 10 so you have measured 10 segments in total.
10 segments = 10 $$\times$$ 50 m = 750 m
4. 1 segment = 2 cm long and represents 200 m. 11 cm $$\div$$ 2 cm (length of segment) = 5,5, so you have measured 5,5 segments in total.
5,5 segments = 5,5 $$\times$$ 200 m = 1100 m = 1,1 km.

### Exercise 1: Using the bar and number scales

You measure the distance between two building on a map to be of 5 cm. If the map has a number scale of 1: 100, what is the actual distance on the ground?

5 cm $$\times$$ 100 = 500 cm = 5 m

You are given a map with the number scale 1 : 20. you measure a distance of 12 cm on the map. What is the actual distance in real life?

12 cm $$\times$$ 20 = 240 cm = 2,4 m.

You measure a distance of 10 cm on a map with the following bar scale:

(1 cm = 15 m). What is the actual distance on the ground?

10 cm $$\div$$ 1 cm = 10 segments. 10 segments $$\times$$ 15 m = 150 m

You measure a distance of 15 cm on a map with the following bar scale:

(2 cm : 100 m) What is the actual distance on the ground?

15 cm $$\div$$ 2 cm = 7,5 segments. 7,5 segments $$\times$$ 100 m = 750 m.

## Using number and bar scales to measure distance

In the previous section about number and bar scales we only looked at how to calculate the actual length of an object or distance between two places when we know the length we measured on the map, and the scale used. However, number and bar scales are usually seen on maps and plans. In this section, we will learn how to measure the dimensions of objects and distance on scale maps and then how to use the number and bar scale to calculate the actual (real world) dimensions of those objects (like furniture and buildings).

### Example 2: Using the number scale to estimate distance

#### Question

Study the school map given below and answer the questions that follow:

1. Calculate the following real dimensions of the sports field in metres:
1. length
2. width
2. Calculate the length of the science classroom block in metres
3. Zuki walks from the tuckshop to his maths classroom, along the blue dotted line shown. Measure how far he walked in metres.

1. Use your ruler to measure the width of the sports field on the map. It is 5 cm wide.
Now use the number scale 1 : 500 to determine the actual width of the field:
5 cm $$\times$$ 500 = 2500 cm
(multiply your scaled measurement by the “real” number in the scale ratio)
2500 cm $$\div$$ 100 = 25 m
The field is 25 m wide
2. On the map, the field is 10 cm long.
10 $$\times$$ 500 = 5000 cm
5000 cm $$\div$$ 100 = 50 m
The field is 50 m long
1. On the map, the science classroom building is 5 cm long.
5 cm $$\times$$ 500 = 2500 cm
2500 cm $$\div$$ 100 = 25 m
The science classrooms are 25 m long
2. The blue dotted line is 6 cm long on the map.
6 $$\times$$ 500 = 3000 cm
3000 cm $$\div$$ 1000 = 30 m
Zuki walked 30 m from the tuckshop to his maths classroom.

### Note:

Although we say we are measuring distance on a map and calculating real distance, in reality we are approximating or estimating distance because the measurements you get using a scale and conversion are accurate only to the nearest metre or centimetre. For example, using the above measurement of how far Zuki walked from the tuckshop to his maths classroom, by our calculations he walked 30 m, whereas if we were to measure this distance exactly on the ground, using a measuring tape, he may actually have walked 30 m and 10 cm.

### Exercise 2: Using the number scale

Using the school map and scale given below, measure the drawings and then estimate the following real distances in metres:

1. The width and length of the school hall.

2. The width of the toilet block.

3. The distance between the science and maths buildings.

1. Width and length of the school hall on the map is 5 cm. 5 cm $$\times$$ 500 = 2500 cm = 25 m.

2. Width of the toilet block on the map is 2 cm. 2 cm $$\times$$ 500 = 1000 cm = 10 m.

3. Distance between the science and maths buildings on the map is 1 cm. 1 cm $$\times$$ 500 = 500 cm = 5 m.

### Example 3: Using the bar scale to estimate actual length

#### Question

Study the map of the room given below and answer the questions that follow:

1. Using a ruler and a calculator, calculate the following real lengths in metres:
1. The length of the room
2. the length of the couch

1. First measure the bar scale using your ruler.
1 cm on your ruler represents 20 cm on the ground.
Now measure the length of the couch.
It is 15 cm long. 15 cm $$\div$$ 1 cm (length of segment) = 15 segments,
15 segments $$\times$$ 20 cm = 300 cm = 3 m.
So the real length of the room is 3 m.
2. 1 cm on your ruler represents 20 cm on the ground.
On the map, the couch is 8 cm long. 8 cm $$\div$$ 1 cm (length of segment) = 8 segments.
8 segments $$\times$$ 20 = 160 cm =1,6 m
So the real length of the couch is 1,6 m.

### Exercise 3: Using the bar scale to estimate actual length

Using the diagram given below, calculate the real life dimensions for:

1. The length of the bookshelf.

2. The width and length of the chair

3. The length of each of the windows.

1. 1 cm on a ruler = 20 cm on the ground. Width of bookshelf is 7 cm. 7 cm $$\div$$ 1 cm (length of segment) = 7 segments of bar scale. 7 segments $$\times$$ 20 cm = 140 cm = 1,4 m. The bookshelf is 1,4 m wide.

2. 1 cm on a ruler = 20 cm on the ground. Width of chair is 3,5 cm. 3,5 cm $$\div$$ 1 cm (length of segment) = 3,5 segments of bar scale. 3,5 segments $$\times$$ 20 cm = 70 cm. Length of chair is 4 cm. 4 cm $$\div$$ 1 cm (length of segment) = 4 segments. 4 segments $$\times$$ 20 cm = 80 cm.

3. 1 cm on a ruler = 20 cm on the ground. Length of left hand window is 5 cm. 5 cm $$\div$$ 1 cm (length of segment) = 5 segments of bar scale. 5 segments $$\times$$ 20 cm = 100 cm = 1 m. Length of bottom window is is 7 cm. 7 cm $$\div$$ 1 cm (length of segment) = 7 segments of bar scale. 7 segments $$\times$$ 20 cm = 140 cm = 1,4 m.

By now you should understand how to use number and bar scales to measure real dimensions and distance on the ground when given a scale map. What happens if you resize a map though (for example you may want to make small photocopies of a map of your school, to hand out for an event taking place)? In the next example we will explore the effects on the number and bar scales when we resize maps.

### Example 4: Resizing and accuracy

#### Question

 Diagram 1 Diagram 2 Diagram 3 Diagram 4
1. Measure the width of the school bag in Diagram 1 and use the scale to calculate the width of the school bag.
2. Measure the school bag in Diagram 2 and use the scale to calculate the width of the school bag.
3. What do you notice about the answers for 1. and 2.?
4. Measure the width of the school bag in Diagram 3 and use the scale to calculate the width of the school bag.
5. Measure the school bag in Diagram 4 and use the scale to calculate the width of the school bag.
6. What do you notice about the answers for 4. and 5.?
7. Write a sentence to explain what you have learnt as a result of your calculations.

1. The measured width on the diagram is 3 cm. Therefore 3 cm $$\times$$ 15 = 45 cm. So the bag is 45 cm wide.
2. The measured width on the diagram is 5 cm. Therefore 5 cm $$\times$$ 15 = 75 cm. So the bag is now 75 cm wide!
3. These answers are very different. Which is correct? Is the bag 45 cm wide or 75 cm wide?
4. 1 segment = 1 cm long. The bag is 3 cm wide on the diagram, therefore 3 $$\times$$ 15 cm = 45 cm.
5. 1 segment = 1,5 cm long. The bag is 4,5 cm wide on the diagram. 4,5 $$\div$$ 1,5 = 3 segments. 3 segments $$\times$$ 15 cm = 45 cm
6. The answers for Questions 4 and 5 are the same!
7. When resizing scale diagrams using the number scale, we have to change the scale in order for it to remain accurate. (In the bigger diagram we would need to number scale to be 1 : 9 for the width of the bag to be 45 cm). When resizing diagrams using the bar scale, the length of the segments increases proportionally to the diagram, therefore the resized bar scale is also accurate and will give us the same answer.

If we resize a map that has a number scale on it, the number scale becomes incorrect. If a map is 10 cm wide when printed, and the number scale is 1 : 10 then 1 cm on the map represents 10 cm on the ground. However, if we reprint the map larger, and it is now 15 cm wide, our scale will still be 1 : 10 according to the map, but now 1,5 cm represents 10 cm on the ground (1,5 $$\times$$ 10 = 15 cm = width of map) so the answers to any scale calculations will now be wrong. When resizing maps that use the number scale, it is important to know that the scale changes with the map. This is a disadvantage to using the number scale.

If we resize a map that has a bar scale on it, the size of the bar scale will be resized with the map, and it will therefore remain accurate. This is an advantage to using the bar scale.

An advantage of the number scale is that we only have to measure one distance (we don't have to measure the length of one bar segment) and our calculations are usually fairly simple as a result. A disadvantage to using the bar scale is that we have to measure the length of one segment and measure the distance on the map, and our calculations can be more complicated because we have to calculate how many segments fit into the distance measured on the map.

## Drawing a scaled map when given real (actual) dimensions

We have learnt how to determine actual measurements when given a map and a scale. In this section we will look at the reverse process - how to determine scaled measurements when given actual dimensions, and draw an accurate two dimensional map. Remember that a scale drawing is exactly the same shape as the real (actual) object, just drawn smaller. In the next worked example we will look at how to draw a simple scaled map of a room.

In order to draw a map you need two pieces of information. Firstly you need to know the actual measurements of everything that has to go onto the map. Secondly you need to know what scale you have to use. The scale will depend on the original measurements, how much detail the map has to show and the size of the map. If you want to draw a map, or plan, of a room in your house on a sheet of A4 paper and include detail of the furniture you would not use a scale of 1: 10 000 (this scale means that 1 cm in real life is equal to 10 000 cm or 1 km in real life).

In Grade 10 the scale will be given to you.

### Example 5: Drawing scaled maps

#### Question

Draw a scaled map of a room that has real dimensions 3 m by 4,5 m. Use a number scale of 1 : 50.

The scale of 1: 50 means that 1 unit on your drawing will represent 50 units in real life so 1 cm on your drawing will represent 50 cm in real life.

• The width of the room is 3 m.

• Convert 3 m to cm:
3 m $$\times$$ 100 = 300 cm
• Use the scale to calculate the scaled width on the map:
300 cm $$\div$$ 50 cm = 6 cm
(Divide the actual, real measurement of the room by the 'real number' from the scale)
• The length of the room is 4,5 m.

• Convert 4,5 m to cm:
4,5 $$\times$$ 100 = 450 cm
• Use the scale to calculate the scaled length on the map:
450 cm $$\div$$ 50 cm = 9 cm
• The scaled measurements are 6 cm and 9 cm. We can now draw this on our plan. Don't forget to include the scale on your map!

### Example 6: Drawing a scaled map

#### Question

In this worked example we will add some furniture to the room in the previous example.

The room has the same dimensions (3 m $$\times$$ 4,5 m) and the scale to be used is still 1 : 50.

Draw the following items using the dimensions provided:

1. A couch 2 m $$\times$$ 1,2 m
2. A window 2 m long
3. A table 1,5 m wide and 2 m long.

You may arrange the furniture in the room in any way which you think is sensible.

The scale of 1: 50 means that 1 unit on your drawing will represent 50 units in real life so 1 cm on your drawing will represent 50 cm in real life.

The scaled dimensions of the room are the same as in the previous worked example: 6 cm $$\times$$ 9 cm.

1. The width of the couch is 1,2 m.
1,2 m = 120 cm
120 cm $$\div$$ 50 = 2,4 cm
The length of the couch is 2 m
2 m = 200 cm
200 $$\div$$ 50 = 4 cm.
So the scaled dimensions of the couch are 2,4 cm and 4 cm
2. The length of the window is 2 m
2 m = 200 cm
200 $$\div$$ 50 = 4 cm.
So the scaled dimension for the length of the window is 2 cm
3. The width of the table is 1 m.
1 m = 100 cm
100 cm $$\div$$ 50 = 2 cm
The length of the table is 1,5 m
1,5 m = 150 cm
150 $$\div$$ 50 = 3 cm.
So the scaled dimensions of the table are 2 cm and 3 cm

### Exercise 4: Drawing a scaled map

The bedroom in the picture is 3,5 m by 4 m. It has a standard sized single bed of 92 cm by 188 cm. The bedside table is 400 mm square. Draw a floor plan to show the layout of the room. Use the number scale 1 : 50.

Room real measurements:

width 3,5 m = 350 cm

length 4 m = 400 cm

Scale drawing:

350 $$\div$$ 50 = 7 cm

400 $$\div$$ 40 = 8 cm

Bed real measurements:

Width = 92 cm

Length = 188 cm

Scale drawing:

92 cm $$\div$$ 50 = 1,84 cm

188 cm $$\div$$ 50 = 3,76 cm

Bedside table real measurements:

400 mm

400 mm $$\div$$ 50 = 8 mm

Scale drawing:

### Exercise 5: drawing scaled maps

Divide into groups in order to measure and draw an accurate, scaled floor plan of your classroom. Use a scale of 1 : 50. You will need to measure all the large objects (e.g. desks, windows, the blackboard) in the classroom, calculate what their scaled dimensions will be and then draw them carefully on your floor plan. Can you think of a different or better way to arrange the furniture in your classroom?