Scale drawings and maps – Ratio and Proportions (Senior 1 Mathematics) | Ganzaa

Ratio and Proportions

Lessons

Senior 1 · Mathematics

Ratio and Proportions

5 lessons

Lesson 2 of 5

Scale drawings and maps

20 min

Let's begin

A classroom 8 m long will not fit on a sheet of paper, and neither will the road from Kigali to Muhanga. Architects and map-makers solve this the same way: they shrink every length by the same factor. Because every length shrinks equally, the drawing keeps the true shape, and one number, the scale, lets you turn any measurement on the paper back into a real distance.

Definition

Scale drawing: an accurate drawing of an object in which every length is the real length multiplied by the same factor. Angles are unchanged.

Scale: the ratio of a length on the drawing to the real length, in the same unit, written $1 : n$ . Example: a scale of $1 : 100$ means 1 cm on the plan stands for 100 cm (1 m) in real life.

Definition

Scale bar (linear scale): a line on a map divided into lengths marked with the real distances they stand for. Example: a bar on which each centimetre is marked "10 km".

The rule, and why it works

If the scale is $1 : n$ , every real length is $n$ times the length on the drawing:

\text{real length} = \text{drawing length} \times n

\text{drawing length} = \frac{\text{real length}}{n}

This is a direct proportion: double the length on the paper and the real length doubles too. Both lengths must be in the same unit before you use the ratio, so you will often need these conversions:

From	To centimetres
1 m	$100$ cm
1 km	$1,000$ m $= 100,000$ cm

Worked example

On a map with a scale of $1 : 50,000$ , two villages are 6 cm apart. How far apart are they in real life?

6 \times 50{,}000 = 300{,}000 \text{ cm} = 3{,}000 \text{ m} = 3 \text{ km}.

Worked example

A straight road is 2.4 km long. How long is it on a map with scale $1 : 20,000$ ?

Change to centimetres: $2.4 \text{ km} = 2.4 \times 100{,}000 = 240{,}000$ cm.
Divide by $n$ : $\dfrac{240{,}000}{20{,}000} = 12$ cm.

Worked example

On a builder's plan a wall 12 m long is drawn 6 cm long. What is the scale of the plan?

Same units: $6 \text{ cm} : 12 \text{ m} = 6 : 1{,}200$ . Divide both parts by 6: the scale is $1 : 200$ .

Worked example

On a map of Rwanda with scale $1 : 1,000,000$ , Kigali and Muhanga are about 4 cm apart. About how far apart are they?

4 \times 1{,}000{,}000 = 4{,}000{,}000 \text{ cm} = 40{,}000 \text{ m} = 40 \text{ km}.

This is the straight-line distance. The road is longer, because it winds between the hills.

Important

A scale bar stays correct when a map is enlarged, photocopied smaller or shown on a phone screen, because the bar grows or shrinks with the map. A written ratio such as $1 : 1,000,000$ is only true at the size the map was printed. On a screen, measure against the bar.

Warning

The units in a scale are the same on both sides. $1 : 50,000$ means 1 cm stands for 50,000 cm, not 50,000 m or 50,000 km. Convert the answer to metres or kilometres only at the end.

Tip

For areas, the scale works twice. At $1 : 100$ the classroom floor is $8 \times 6 = 48\ \text{cm}^2$ on the plan but $800 \times 600 = 480{,}000\ \text{cm}^2$ , which is $48\ \text{m}^2$ , in real life. That is $100^2 = 10{,}000$ times as big, not 100 times.

Corbettmaths: "Maps and Scales". Watch how each answer is found in centimetres first and changed to kilometres at the very end.

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Practice

Show the unit conversions in your working.

A drawing has scale $1 : 200$ . A line on it is 5 cm long. How long is the real object, in metres?
On a map with scale $1 : 25,000$ , a path measures 8 cm. How long is the path, in km?
A road 4.5 km long is drawn on a map with scale $1 : 50,000$ . How long is it on the map?
A football pitch 100 m long is drawn 20 cm long. Write the scale in the form $1 : n$ .
A model bus made at a scale of $1 : 40$ is 30 cm long. How long is the real bus?
A scale bar shows that 1 cm stands for 5 km. Write this scale as a ratio.
A plot of land 30 m by 20 m is drawn at a scale of $1 : 500$ . What are its measurements on the drawing?

Show answersHide answers

$5 \times 200 = 1{,}000$ cm $= 10$ m
$8 \times 25{,}000 = 200{,}000$ cm $= 2$ km
$450{,}000 \div 50{,}000 = 9$ cm
$20 : 10,000 = 1 : 500$
$30 \times 40 = 1{,}200$ cm $= 12$ m
$5 \text{ km} = 500{,}000$ cm, so the scale is $1 : 500,000$ .
$3{,}000 \div 500 = 6$ cm and $2{,}000 \div 500 = 4$ cm: 6 cm by 4 cm.

Sources and credits

Grade 7, Unit 1: Scale Drawings, IM 6–8 Math by Illustrative Mathematics, originally developed by Open Up Resources (© 2017–2019 Open Up Resources; adaptations © 2019 Illustrative Mathematics), provided by Kendall Hunt, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Maps and Scales, corbettmaths, embedded from YouTube.

The classroom plan, the Kigali–Muhanga sketch (distance worked out by Ganzaa from the towns' coordinates), the worked examples, practice items and quiz questions were written by Ganzaa.

1The rule, and why it works

The rule, and why it works