Scale
What is scale?
- In mathematics, scale can have many meanings, but in accurate drawings and constructions scale refers to a ratio
- Maps are usually drawn to a scale
- The scale chosen will depend on the area being mapped out
- A map of a classroom will use a large scale and can show a lot more detail
- A map of a country or a world map will use a small scale and can show a much bigger area but in less detail
How can scale be used to convert lengths?
- A ratio given for scale does not include units
- The ratio will work for any unit of length applied to both sides
- For example, the scale 1: 50000 could mean 1 cm = 50000 cm, 1 km = 50000 km or even 1 yard = 50000 yards
- However, if you’re measuring the length from a map it will be easiest to measure in cm
- The scale can be used to convert lengths found on a map to lengths in real life
- STEP 1
Use a ruler to measure the distance accurately on the map- For example, measuring a length from A to B as 5.8 cm
- STEP 2
Use the scale to find the actual distance in the same units (usually cm)- For example, if the scale is 1 : 150000 the actual distance = 5.8 cm × 150000
- STEP 3
Convert the actual distance to a more suitable unit- For example, 5.8 × 150000 = 870000 cm = 8.7 km
- STEP 1
- The scale can also be used to convert lengths found in real life into a length for a map or a scale drawing
- STEP 1
Convert the scale into a ratio of 1 cm : the units the actual distance is in- For example, if the distance is in km and the scale is 1 : 500000, convert 500000 cm into km
- 1 : 500000 = 1cm : 5 km
- STEP 2
Use this ratio to convert the actual distance to the scale distance- For example, if the actual distance = 20 km, the scale distance will be 20 ÷ 5 = 4 cm
- STEP 1
Exam Tip
- If you have many distances to find, or a very small scale, it may help to convert the scale to more suitable units first, just be careful not to mix them up!
- For example, the scale 1 : 500 000 000 can be converted to 1 cm = 5000 km
Worked example
A map is drawn where a length of 5 cm is equal to an actual distance of 0.6 km.
(a)
Write the scale on the map.
Convert both parts of the scale to the same units.
The answer needs to be in the form 1 : n so convert 0.6 km into cm using 1 m = 100 cm and 1 km = 1000 m.
The answer needs to be in the form 1 : n so convert 0.6 km into cm using 1 m = 100 cm and 1 km = 1000 m.
0.6 km = 0.6 × 1000 m = 600 m
600 m = 600 × 100 cm = 60 000 cm
Now the ratio has the same units 5 cm : 60 000 cm, you can remove the units.
5 : 60 000
Write in the form 1 : n by dividing both sides by 5.
1 : 12 000
(b)
The width of a park on the map is 17 mm.
Find the actual width of the park, giving your answer in metres.
Multiply 17 mm by 12 000 to find the actual width
17 mm × 12 000 = 204 000 mm
Convert from millimetres to metres by dividing by 1000
204 000 mm ÷ 1000
204 m
(c)
The distance from the mouth of the ocean to the first bridge over a river is 125 metres.
Find this distance on the map.
Convert 125 metres to cm
125 m = (125 × 100) cm = 12 500 cm
Divide by 12 000 to find the distance on the map
12 500 cm ÷ 12 000 = 1.0416... cm
Give your answer correct to at least 3 significant figures
1.04 cm
