Map Scales and Distance

What a map scale means, the three ways it is written, and how to find an actual distance, a distance on the map, or the scale itself, on maps, floor plans and models.

Before you start

Units of length: $1\text{ km} = 1000\text{ m}$ and $1\text{ m} = 100\text{ cm}$ , so $1\text{ km} = 1000 \times 100\text{ cm} = 100\,000\text{ cm}\text{.}$
A ratio compares two quantities in the same unit, and then has no unit: $2\text{ m} : 40\text{ cm} = 200\text{ cm} : 40\text{ cm} = 5 : 1\text{.}$
Time taken = distance ÷ speed: 9 km at 6 km/h takes $9 \div 6 = 1.5$ h, which is 1 h 30 min.

What a Map Scale Says

A map is a drawing of the ground made smaller. Every distance on the ground is shrunk in the same way, so the map keeps the shape of what it shows. The scale of the map says how much smaller it is.

On a map with the scale $1 : 50\,000$ , a length of 1 cm on the map stands for $50\,000$ cm on the ground. Any unit works the same way: 1 mm on the map stands for $50\,000$ mm, and 3 cm stands for $3 \times 50\,000 = 150\,000$ cm.

Map scale

The scale of a map is the ratio map distance : actual distance, with both distances in the same unit.

The scale $1 : n$ means that 1 unit on the map represents $n$ of the same units on the ground. Every map distance is $\frac{1}{n}$ of the actual distance; $\frac{1}{n}$ is called the representative fraction of the map.

So the actual distance is directly proportional to the map distance: twice as long on the map means twice as long on the ground. The table is for the scale $1 : 50\,000$ , on which 1 cm stands for 0.5 km (Example 1 shows why). The actual distance divided by the map distance is 0.5 km per cm for every pair.

Map distance (cm)	$1$	$2$	$5$	$12$
Actual distance (km)	$0.5$	$1$	$2.5$	$6$
Actual ÷ map	$0.5$	$0.5$	$0.5$	$0.5$

Three Ways to Write a Scale

The same scale can be written in three ways. Each of these is the scale $1 : 50\,000$ .

Form	Example	What it says
A ratio $1 : n$	$1 : 50\,000$	The same units on both sides, so no unit is written
A statement	1 cm to 0.5 km, or 1 cm represents 500 m	The units on the two sides are different, and both are written
A scale bar	The bar drawn below	A line marked with actual distances, to measure along with a ruler

On the printed map, each section of this bar is 1 cm long and stands for 0.5 km. A scale bar stays right when a map is enlarged on a photocopier or zoomed in on a phone, because the bar grows with the map; a ratio printed on the map does not.

Same units before a ratio

In "1 cm to 2 km" the units are different, so it is not the ratio $1 : 2$ . Change both distances to one unit first: $2\text{ km} = 200\,000\text{ cm}\text{,}$ so the scale is $1 : 200\,000$ .

Changing from One Form to Another

Changing the units

$1\text{ km} = 1000\text{ m} = 100\,000\text{ cm}\text{.}$ Go through metres: centimetres ÷ 100 gives metres, and metres ÷ 1000 gives kilometres.

From $1 : n$ to a statement: 1 cm represents $n$ cm; change $n$ cm to m or km.

From a statement to $1 : n$ : change the actual distance to centimetres, then write map distance : actual distance and divide both terms by the map distance.

Example 1: From a ratio to a statement, and back

(a) Write the scale $1 : 50\,000$ in the form 1 cm to $k$ km. (b) Write the scale 1 cm to 2 km in the form $1 : n$ .

Solution

(a) 1 cm on the map represents $50\,000$ cm on the ground.

$50\,000\text{ cm} = 50\,000 \div 100\text{ m} = 500\text{ m}$

$500\text{ m} = 500 \div 1000\text{ km} = 0.5\text{ km}$

(b) Change 2 km to centimetres, so that both sides have the same unit:

$2\text{ km} = 2 \times 1000\text{ m} = 2000\text{ m}$

$2000\text{ m} = 2000 \times 100\text{ cm} = 200\,000\text{ cm}$

So the ratio is $1\text{ cm} : 200\,000\text{ cm} = 1 : 200\,000\text{.}$

(a) 1 cm to 0.5 km (or 1 cm represents 500 m). (b) $1 : 200\,000$ .

Actual Distance and Map Distance

Finding a distance

With the scale $1 : n$ , and both distances in the same unit:

actual distance = map distance × $n$

map distance = actual distance ÷ $n$

With a statement such as 1 cm to $k$ km, multiply or divide by $k$ instead, and the answer is in km.

Example 2: Actual distance from the map

Two towns are 12 cm apart on a map with the scale $1 : 50\,000$ . What is the actual distance between them, in km?

Solution

1 cm represents $50\,000$ cm, so 12 cm represents $12 \times 50\,000 = 600\,000$ cm.

$600\,000\text{ cm} = 600\,000 \div 100\text{ m} = 6000\text{ m}$

$6000\text{ m} = 6000 \div 1000\text{ km} = 6\text{ km}$

Or, with the statement from Example 1, 1 cm to 0.5 km: 12 cm represents $12 \times 0.5 = 6$ km.

The towns are 6 km apart.

Example 3: Map distance from the actual distance

A straight stretch of park connector is 3.6 km long. How long is it on a map with the scale $1 : 20\,000$ ?

Solution

Change 3.6 km to centimetres:

$3.6\text{ km} = 3.6 \times 1000\text{ m} = 3600\text{ m}$

$3600\text{ m} = 3600 \times 100\text{ cm} = 360\,000\text{ cm}$

Map distance $= 360\,000 \div 20\,000 = 18$ cm.

Check: 1 cm represents $20\,000\text{ cm} = 200\text{ m} = 0.2\text{ km}\text{,}$ and $18 \times 0.2 = 3.6$ km.

The stretch is 18 cm long on the map.

To find the scale itself, write the map distance and the actual distance in one unit, as a ratio, and divide both terms by the map distance.

Example 4: Finding the scale

One of the runways at Changi Airport is 4 km long. On a map it is 16 cm long. Find the scale of the map in the form $1 : n$ .

Solution

$4\text{ km} = 4 \times 1000\text{ m} = 4000\text{ m} = 4000 \times 100\text{ cm} = 400\,000\text{ cm}$

map distance : actual distance $= 16 : 400\,000$

Divide both terms by 16: $16 \div 16 = 1$ and $400\,000 \div 16 = 25\,000\text{.}$

Check: $16 \times 25\,000 = 400\,000\text{.}$

The scale is $1 : 25\,000$ .

Reading a Scale Bar

Find the length of one section of the bar on the map and the distance it stands for. Then work out what 1 cm stands for, and use it as a statement.

Example 5: A scale bar

The scale bar of a map of a housing estate is drawn below. Each section is 2 cm long. (a) Write the scale in the form $1 : n$ . (b) On the map, the path from a block of flats to the MRT station measures 7.5 cm. How long is the path?

Solution

(a) 2 cm represents 200 m, so 1 cm represents $200 \div 2 = 100$ m.

$100\text{ m} = 100 \times 100\text{ cm} = 10\,000\text{ cm}\text{,}$ so the scale is $1 : 10\,000$ .

(b) 7.5 cm represents $7.5 \times 100 = 750$ m.

(a) $1 : 10\,000$ . (b) The path is 750 m long.

Which Map Shows More Detail?

A street map of one neighbourhood and a map of the whole of Singapore have very different scales. On the street map, 1 km of ground takes up a long line on the paper, so a footpath or a playground can be drawn. On the map of Singapore, the same 1 km is a short line.

Larger and smaller scales

The scale $1 : n$ is larger than the scale $1 : m$ when its representative fraction is larger, that is, when $\frac{1}{n} > \frac{1}{m}$ . This happens when $n$ is smaller than $m$ .

A map with a larger scale draws the same distance as a longer line, so it shows more detail, but a sheet of the same size covers less ground.

Example 6: Comparing two maps

Map A has the scale $1 : 25\,000$ and map B has the scale $1 : 200\,000$ . (a) How long is a 5 km road on each map? (b) Which map has the larger scale? Which shows more detail?

Solution

(a) $5\text{ km} = 5 \times 100\,000\text{ cm} = 500\,000\text{ cm}$

Map A: $500\,000 \div 25\,000 = 20$ cm. Map B: $500\,000 \div 200\,000 = 2.5$ cm.

The road is $20 \div 2.5 = 8$ times as long on map A, because $200\,000 \div 25\,000 = 8\text{.}$

(b) $25\,000$ is smaller than $200\,000$ , so $\frac{1}{25\,000}$ is larger than $\frac{1}{200\,000}$ .

(a) 20 cm on map A and 2.5 cm on map B. (b) Map A has the larger scale, and it shows more detail.

Journeys, Floor Plans and Models

For a journey, find the actual distance from the map first, then use the speed. Scales are also used for anything drawn or built smaller than life: floor plans of flats are often drawn to $1 : 100$ or $1 : 50$ , and model cars are often made to $1 : 18$ or $1 : 43$ .

Example 7: A journey

A cyclist rides along a park connector that measures 15 cm on a map with the scale $1 : 40\,000$ . She rides at an average speed of 12 km/h. How long does the ride take, in minutes?

Solution

1 cm represents $40\,000$ cm. $40\,000\text{ cm} = 40\,000 \div 100\text{ m} = 400\text{ m} = 0.4\text{ km}$

The route is $15 \times 0.4 = 6$ km long.

Time $= 6 \div 12 = 0.5$ h, and $0.5\text{ h} = 0.5 \times 60\text{ min} = 30\text{ min}\text{.}$

The ride takes 30 minutes.

Example 8: A floor plan

The floor plan of a bedroom in an HDB flat is drawn to the scale $1 : 50$ . On the plan the room is 7 cm long and 6 cm wide. (a) Find the actual length and width of the room, in metres. (b) A bed is 1.9 m long. How long is it on the plan?

Solution

(a) 1 cm on the plan represents 50 cm.

Length: $7 \times 50 = 350$ cm, and $350\text{ cm} = 350 \div 100\text{ m} = 3.5\text{ m}\text{.}$

Width: $6 \times 50 = 300$ cm, and $300\text{ cm} = 300 \div 100\text{ m} = 3\text{ m}\text{.}$

(b) $1.9\text{ m} = 1.9 \times 100\text{ cm} = 190\text{ cm}\text{,}$ and $190 \div 50 = 3.8$ cm.

(a) The room is 3.5 m long and 3 m wide. (b) The bed is 3.8 cm long on the plan.

Example 9: A model car

A model of a car is made to the scale $1 : 18$ . The model is 25 cm long. (a) How long is the real car, in metres? (b) The real car is 1.8 m wide. How wide is the model?

Solution

(a) 1 cm on the model represents 18 cm on the car: $25 \times 18 = 450$ cm, and $450\text{ cm} = 4.5\text{ m}$ .

(b) $1.8\text{ m} = 1.8 \times 100\text{ cm} = 180\text{ cm}\text{,}$ and $180 \div 18 = 10$ cm.

(a) The real car is 4.5 m long. (b) The model is 10 cm wide.

Common Mistakes

Mixing units in a ratio: 1 cm to 5 km is $1 : 500\,000$ , not $1 : 5$ .
Changing km to cm by multiplying by 1000 only. $1\text{ km} = 1000\text{ m} = 100\,000\text{ cm}\text{:}$ multiply by 1000, then by 100.
Dividing by $n$ to find an actual distance. The ground is larger than the map, so the actual distance is the larger number: multiply.
Calling $1 : 200\,000$ a larger scale than $1 : 25\,000$ because $200\,000$ is the larger number. The larger scale has the smaller $n$ .

Practice Questions

Try each question before you open its solution, and write every change of unit in full.

Question 1

Write each scale in the form 1 cm to $k$ km. (a) $1 : 100\,000$ (b) $1 : 25\,000$ (c) $1 : 300\,000$

ShowHide solution

In each, 1 cm represents $n$ cm. Divide by 100 for metres, then by 1000 for kilometres.

(a) $100\,000\text{ cm} = 1000\text{ m} = 1\text{ km}$

(b) $25\,000\text{ cm} = 250\text{ m} = 0.25\text{ km}$

(a) 1 cm to 1 km (b) 1 cm to 0.25 km (c) 1 cm to 3 km

Question 2

Write each scale in the form $1 : n$ . (a) 1 cm to 5 km (b) 1 cm represents 250 m (c) 2 cm represents 1 km

ShowHide solution

(a) $5\text{ km} = 5000\text{ m} = 5000 \times 100\text{ cm} = 500\,000\text{ cm}\text{,}$ so the scale is $1 : 500\,000$ .

(b) $250\text{ m} = 250 \times 100\text{ cm} = 25\,000\text{ cm}\text{,}$ so the scale is $1 : 25\,000$ .

(a) $1 : 500\,000$ (b) $1 : 25\,000$ (c) $1 : 50\,000$

Question 3

On a map of Singapore with the scale $1 : 250\,000$ , the main island measures 20 cm from east to west. About how wide is the island, in km?

ShowHide solution

1 cm represents $250\,000\text{ cm} = 2500\text{ m} = 2.5\text{ km}\text{.}$

20 cm represents $20 \times 2.5 = 50$ km.

The main island is about 50 km wide from east to west.

Question 4

The scale of a map is 1 cm to 0.8 km. (a) Two bus interchanges are 6 km apart. How far apart are they on the map? (b) Write the scale in the form $1 : n$ .

ShowHide solution

(a) Map distance $= 6 \div 0.8 = 7.5$ cm. Check: $7.5 \times 0.8 = 6$ km.

(b) $0.8\text{ km} = 0.8 \times 1000\text{ m} = 800\text{ m} = 800 \times 100\text{ cm} = 80\,000\text{ cm}\text{,}$ so the scale is $1 : 80\,000$ .

(a) 7.5 cm (b) $1 : 80\,000$

Question 5

A straight road 2.4 km long is 8 cm long on a map. Find the scale of the map in the form $1 : n$ .

ShowHide solution

$2.4\text{ km} = 2400\text{ m} = 2400 \times 100\text{ cm} = 240\,000\text{ cm}$

map distance : actual distance $= 8 : 240\,000$

Divide both terms by 8: $240\,000 \div 8 = 30\,000\text{.}$ Check: $8 \times 30\,000 = 240\,000\text{.}$

The scale is $1 : 30\,000$ .

Question 6

The scale bar of a map is drawn below. Each section is 1.5 cm long. (a) Write the scale in the form $1 : n$ . (b) Two towns are 11.5 cm apart on the map. How far apart are they?

ShowHide solution

(a) 1.5 cm represents 3 km, so 1 cm represents $3 \div 1.5 = 2$ km.

$2\text{ km} = 2000\text{ m} = 2000 \times 100\text{ cm} = 200\,000\text{ cm}\text{,}$ so the scale is $1 : 200\,000$ .

(b) 11.5 cm represents $11.5 \times 2 = 23$ km.

(a) $1 : 200\,000$ (b) The towns are 23 km apart.

Question 7

Map P has the scale $1 : 20\,000$ , map Q has the scale $1 : 50\,000$ and map R has the scale $1 : 10\,000$ . (a) How long is a 1 km path on each map? (b) List the maps from the largest scale to the smallest. Which map shows the most detail?

ShowHide solution

(a) $1\text{ km} = 100\,000\text{ cm}\text{.}$

P: $100\,000 \div 20\,000 = 5$ cm. Q: $100\,000 \div 50\,000 = 2$ cm. R: $100\,000 \div 10\,000 = 10$ cm.

(b) The smaller $n$ is, the larger the scale: R ( $10\,000$ ), then P ( $20\,000$ ), then Q ( $50\,000$ ). The path is longest on map R.

(a) P: 5 cm, Q: 2 cm, R: 10 cm. (b) R, P, Q; map R shows the most detail.

Question 8

On a map with the scale $1 : 200\,000$ , the North–South MRT Line measures 22.5 cm along its track from end to end. (a) About how long is the line, in km? (b) A train runs the whole line at an average speed of 36 km/h, stops included. How many minutes does it take?

ShowHide solution

(a) 1 cm represents $200\,000\text{ cm} = 2000\text{ m} = 2\text{ km}\text{,}$ so the line is $22.5 \times 2 = 45$ km long.

(b) Time $= 45 \div 36 = 1.25$ h, and $1.25\text{ h} = 1.25 \times 60\text{ min} = 75\text{ min}\text{.}$

(a) About 45 km. (b) 75 minutes (1 h 15 min).

Question 9

On a floor plan with the scale $1 : 50$ , a kitchen is 6.4 cm long and 4.8 cm wide. (a) Find its actual length and width, in metres. (b) A refrigerator is 0.7 m wide. How wide is it on the plan?

ShowHide solution

(a) 1 cm represents 50 cm. Length: $6.4 \times 50 = 320$ cm $= 3.2$ m. Width: $4.8 \times 50 = 240$ cm $= 2.4$ m.

(b) $0.7\text{ m} = 0.7 \times 100\text{ cm} = 70\text{ cm}\text{,}$ and $70 \div 50 = 1.4$ cm.

(a) 3.2 m long and 2.4 m wide. (b) 1.4 cm.

Question 10

A model of a six-car MRT train is made to the scale $1 : 120$ . The model is 115 cm long. (a) How long is the real train, in metres? (b) Each door of the real train is 1.44 m wide. How wide is a door on the model?

ShowHide solution

(a) $115 \times 120 = 13\,800$ cm, and $13\,800\text{ cm} = 13\,800 \div 100\text{ m} = 138\text{ m}\text{.}$

(b) $1.44\text{ m} = 1.44 \times 100\text{ cm} = 144\text{ cm}\text{,}$ and $144 \div 120 = 1.2$ cm.

(a) 138 m (b) 1.2 cm

Question 11

Two places are 6 cm apart on a map with the scale $1 : 50\,000$ . How far apart are they on a map with the scale $1 : 20\,000$ ?

ShowHide solution

Actual distance $= 6 \times 50\,000 = 300\,000$ cm.

On the second map: $300\,000 \div 20\,000 = 15$ cm.

Check: the second map has the larger scale, so the distance on it should be longer, and 15 cm is longer than 6 cm.

They are 15 cm apart on the second map.

Question 12

A map says "1 cm represents 4 km". Priya writes its scale as $1 : 4000$ , because $1\text{ km} = 1000\text{ m}$ . What is her mistake, and what is the scale?

ShowHide solution

She changed 4 km to 4000 m but left the map distance in centimetres, so her ratio compares centimetres with metres.

In one unit: $4\text{ km} = 4000\text{ m} = 4000 \times 100\text{ cm} = 400\,000\text{ cm}\text{.}$

Her ratio mixes centimetres and metres. The scale is $1 : 400\,000$ .

Key Points

The scale $1 : n$ means 1 unit on the map represents $n$ of the same units on the ground; both terms are in the same unit.
A scale is written as a ratio ( $1 : 50\,000$ ), as a statement (1 cm to 0.5 km) or as a scale bar.
$1\text{ km} = 1000\text{ m} = 100\,000\text{ cm}\text{.}$
Actual distance = map distance × $n$ , and map distance = actual distance ÷ $n$ , in the same unit.
To find a scale, write map distance : actual distance in one unit and divide both terms by the map distance.
A larger scale has a smaller $n$ : it draws the same distance as a longer line and shows more detail.
For a journey, find the actual distance first, then time = distance ÷ speed.