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Daily Maths

Daily Maths

Maps and scales

Maps And Scales Paper 1 + 2 10 min read

On this page
  1. In this note
  2. Formula bank
  3. Distances on a map
  4. Writing a scale as 1 : n
  5. Areas on maps: square the scale
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 1.11 (map scales) and 4.2 (scale drawings)
  • 1.11Use a map scale (\(1:n\) or "1 cm represents ...") to find real and map distances.
  • 1.11Write a scale in words as \(1:n\).
  • 1.11Use scales for areas (square the scale) and for models.

Before you start: ratio and unit conversion. Drawing to scale: Constructions and Scale Drawings.

Formula bank

Map → realLearn itreal = map \(\times n\)do it in cm, then change units.Use it when you measured on the map.
Real → mapLearn itmap = real (in cm) \(\div n\)Use it when you know the real distance.
Key factLearn it1 km = 100 000 cm1000 m, each 100 cm.Use it when every map question.
Area scaleLearn it1 cm² stands for (what 1 cm stands for)\(^{2}\)1 : 50 000: 1 cm = 0.5 km, so 1 cm² = 0.25 km².Use it when areas on maps.

Distances on a map

A scale of 1 : 50 000 means 1 unit on the map stands for 50 000 of the SAME unit in real life. The map is real life with every length divided by 50 000.

Map ↔ realmap length(cm)real length(cm → m → km)× n, then convert unitsconvert to cm, then ÷ n
Scale 1 : n. Always do the scaling in centimetres.

General methodHow to answer any map distance question

  1. Map → real: map length (cm) × n = real length in cm. Then ÷ 100 for m or ÷ 100 000 for km.
  2. Real → map: change the real length to cm (km × 100 000), then ÷ n.
  3. Give the answer in the unit the question asks.

Example Scale 1 : 50 000. Two towns are 7.4 cm apart on the map. Real distance in km?

× 50 000 to get cm, then ÷ 100 000 for km.

  1. \(7.4\times50\,000=370\,000\) cm
  2. \(370\,000\div100\,000=3.7\)

Answer 3.7 km

Try it yourself

1
Scale 1 : 20 000. Map 6 cm. Real distance in km?
Show answer

Answer 1.2 km

2
Scale 1 : 25 000. A road is 4.5 km. Length on the map in cm?
Show answer

Answer 18 cm

3
Scale 1 : 200 000. Real 8 km. Map distance in cm?
Show answer

Answer 4 cm

Writing a scale as 1 : n

General methodHow to turn any scale in words into 1 : n

  1. Write both sides in cm ("2 cm represents 5 km" → 2 : 500 000).
  2. Divide both sides by the first number so it becomes 1.

Example "4 cm represents 1 km". Write the scale as 1 : n.

Both in cm, then ÷ 4.

  1. 4 : 100 000
  2. ÷ 4

Answer 1 : 25 000

Try it yourself

1
"1 cm represents 2 km" as 1 : n. n =
Show answer

Answer 200 000

2
"6 cm represents 1.5 km" as 1 : n. n =
Show answer

Answer 25 000

Models work the same way in any unit: a 1 : 40 model of a 4.2 m car is \(420\div40=10.5\) cm long.

Areas on maps: square the scale

On a 1 : 50 000 map, 1 cm stands for 0.5 km. A 1 cm by 1 cm square stands for 0.5 km by 0.5 km, so 1 cm² = 0.25 km²: the length scale is squared.

Area scale1 cm × 1 cmeach cm → 0.5 km0.25 km²0.5 km × 0.5 km
Both sides are scaled, so the area is scaled twice.

General methodHow to change any area on a map

  1. Find what 1 cm stands for, in the unit you want (km or m).
  2. Square it: that is what 1 cm² stands for.
  3. Map → real: multiply by it. Real → map: divide by it.

Example A forest is 12 cm² on a 1 : 50 000 map. Real area in km²?

1 cm = 0.5 km, so 1 cm² = 0.25 km².

  1. \(0.5^{2}=0.25\)
  2. \(12\times0.25=3\)

Answer 3 km²

Try it yourself

1
Scale 1 : 250 000. A lake is 8 cm² on the map. Real area in km²?
Hint
1 cm = 2.5 km.
Show answer

Answer 50 km²

2
A park of 20 km² on a 1 : 200 000 map. Map area in cm²?
Hint
1 cm² = 4 km².
Show answer

Answer 5 cm²

Ways they test it

Type 1 Map to real

"Scale 1 : 20 000. A path is 13.5 cm on the map. Real length in km." (2 marks)

Spot it a measurement on the map.

  1. × n.
  2. Change cm to km (÷ 100 000).

For example 13.5 cm at 1 : 20 000

  1. \(13.5\times20\,000=270\,000\) cm

Answer 2.7 km

Type 2 Real to map

"Two villages are 18 km apart. How far on the map?"

Spot it a real distance.

  1. km → cm.
  2. ÷ n.

Type 3 Scale in words → 1 : n

"2 cm represents 5 km. Write the scale as 1 : n." (2 marks)

Spot it "represents".

  1. Same units.
  2. Make the first part 1.

Type 4 Distance, then time

"9.6 cm at 1 : 50 000, walked at 4 km/h. How long, in hours and minutes?"

Spot it a map and a speed.

  1. Real distance in km.
  2. time = distance ÷ speed.
  3. Decimal hours × 60.

For example 9.6 cm, 1 : 50 000, 4 km/h

  1. 4.8 km
  2. \(4.8\div4=1.2\) h

Answer 1 hour 12 minutes

Type 5 Area

"A lake is 6.5 cm² on a 1 : 40 000 map. Real area in m²?" (3 marks)

Spot it cm² or km² in a map question.

  1. What 1 cm stands for (in m).
  2. Square it, then multiply.

For example 1 cm = 400 m

  1. 1 cm² = 160 000 m²
  2. \(6.5\times160\,000\)

Answer 1 040 000 m²

Mistakes to avoid

  • ✗ 1 km = 1000 cm✓ 1 km = 100 000 cm1000 m, each 100 cm.
  • ✗ Scale '2 : 5' for 2 cm ↔ 5 km✓ 1 : 250 000Same units, first part 1.
  • ✗ Map area × n✓ map area × (what 1 cm stands for)²Area scales in two directions.
  • ✗ Answer left in cm✓ convert to the unit asked270 000 cm is 2.7 km.

Summary

  • Scale 1 : n: × n to real, ÷ n to the map, in cm.
  • 1 km = 100 000 cm.
  • Words → 1 : n: same units, then divide by the first part.
  • Areas: (what 1 cm stands for)².

If you see …

  • "represents"write as 1 : n in the same units.
  • cm² on a mapsquare the length scale.
  • a speed after a map distancereal km first, then time = distance ÷ speed.

Practice

Practice: easy, medium, hard

Easy

1
Scale 1 : 10 000. 3 cm on the map is how many metres?
Show working
  1. \(3\times10\,000=30\,000\) cm = 300 m

Answer 300 m

2
A model is 1 : 50. The real height is 3 m. Model height in cm?
Show working
  1. \(300\div50=6\)

Answer 6 cm

3
Scale 1 : 200 000. Real 8 km. Map distance in cm?
Show working
  1. \(800\,000\div200\,000=4\)

Answer 4 cm

Medium

1
On a map 3 cm represents 1.2 km. Write the scale as 1 : n. n =
Guide me

What is it asking? n

What do we know? 3 cm ↔ 1.2 km

Plan same units, then ÷ 3

Show working
  1. 3 : 120 000
  2. ÷ 3: 1 : 40 000

Answer 40 000

2
Same map: a road is 4.4 km long. Length on the map in cm?
Guide me

What is it asking? map length

What do we know? 1 : 40 000

Plan km → cm, then ÷ 40 000

Show working
  1. \(440\,000\div40\,000=11\)

Answer 11 cm

3
Scale 1 : 25 000. A route is 5.2 cm on the map, cycled at 13 km/h. How many minutes?
Guide me

What is it asking? time in minutes

What do we know? map length, scale, speed

Plan real km, ÷ speed, × 60

Show working
  1. \(5.2\times25\,000=130\,000\) cm = 1.3 km
  2. \(1.3\div13=0.1\) h = 6 min

Answer 6 minutes

Hard

1
A lake is 15 cm² on a 1 : 25 000 map. Real area in km²?
Guide me

What is it asking? real area

What do we know? 1 cm = 0.25 km

Plan square, then × 15

Show working
  1. \(0.25^{2}=0.0625\)
  2. \(15\times0.0625=0.9375\)

Answer 0.9375 km²

PartAnswer / workingMarksGuidance
1 cm² → 0.0625 km²M1
0.9375A1accept 0.938
Total: 2 marks
2
Scale 1 : 50 000. A walk is 9.6 cm on the map, at 4 km/h. Time in hours and minutes?
Guide me

What is it asking? the time

What do we know? 9.6 cm, 1 : 50 000, 4 km/h

Plan km, then hours, then minutes

Show working
  1. \(9.6\times50\,000=480\,000\) cm = 4.8 km
  2. \(4.8\div4=1.2\) h
  3. \(0.2\times60=12\) min

Answer 1 hour 12 minutes

PartAnswer / workingMarksGuidance
4.8 kmM1
1.2 hM1
1 h 12 minA1
Total: 3 marks
3
A lake is 6.5 cm² on a 1 : 40 000 map. Real area in m²?
Guide me

What is it asking? area in m²

What do we know? 1 cm = 400 m

Plan \(400^{2}\), then × 6.5

Show working
  1. \(400^{2}=160\,000\)
  2. \(6.5\times160\,000=1\,040\,000\)

Answer 1 040 000 m²

PartAnswer / workingMarksGuidance
400 mM1
\(160\,000\text{ m}^2\)M1
1 040 000A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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