Mathematics Standard • Year 12 • Ratios and rates • Lesson 3
Map Scales, Skill Drill
Build fluency in the two directions of a scale: plan length to real length by multiplying, and real length back to plan length by dividing. Then apply the same rule to map scales.
1. Quick recall
Answer in the space provided. 1 mark each
Q1.1 A scale of 1:n means that 1 unit on the drawing represents ____________ units in reality.
Q1.2 To go from a plan length out to a real length you ____________ by the scale factor.
Q1.3 To go from a real length back to a plan length you ____________ by the scale factor.
2. Worked example, plan length to real length
Follow each line. Every step has its reason.
Problem. A wall measures 4.5 cm on a plan drawn at 1:200. Find its real length.
Step 1, decide the direction.
plan → real, so multiply by 200
Reason: the scale says 1 unit on the plan stands for 200 of the same units in reality.
Step 2, multiply, staying in centimetres.
4.5 × 200 = 900 cm
Reason: the scale has no units, so it works in whatever unit you measured in.
Step 3, convert last, then check.
900 ÷ 100 = 9 m, and 9 m = 900 cm, 900 ÷ 200 = 4.5 cm
Reason: converting before you apply the scale invites you to divide by 100 twice.
3. Plan length to real length
All of these come from a floor plan drawn at a scale of 1:100. Give each answer in metres. 1 mark each
Q3.1 A wall measuring 2.5 cm on the plan.
Q3.2 A room measuring 7.3 cm across on the plan.
Q3.3 A driveway measuring 12.0 cm on the plan.
4. Real length to plan length
Convert to centimetres first, then divide. Give each answer in centimetres. 2 marks each
Q4.1 A fence 9 m long, drawn at a scale of 1:300.
Q4.2 A boundary 24 m long, drawn at a scale of 1:400.
5. Map scales
Same rule, larger numbers. Show the centimetre answer before converting. 2 marks each
Q5.1 On a 1:25 000 map, two points are 4 cm apart. Find the real distance in kilometres.
Q5.2 On a 1:50 000 map, a road is 9.2 cm long. Find its real length in kilometres.
Q5.3 On a 1:10 000 map, a walking track is 3.5 cm long. Find its real length in metres.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, What the scale means
n units. The scale carries no units of its own, so one of anything on the drawing stands for n of the same thing in reality.
Q1.2, Plan out to real
Multiply by the scale factor.
Q1.3, Real back to plan
Divide by the scale factor.
Q3.1, 2.5 cm at 1:100
2.5 × 100 = 250 cm = 2.5 m. At 1:100 the centimetre reading becomes the metre reading, which is why the scale is so common on house plans.
Q3.2, 7.3 cm at 1:100
7.3 × 100 = 730 cm = 7.3 m.
Q3.3, 12.0 cm at 1:100
12.0 × 100 = 1200 cm = 12 m.
Q4.1, 9 m at 1:300
9 m = 900 cm, and 900 ÷ 300 = 3 cm. Check: 3 × 300 = 900 cm = 9 m. Dividing 9 by 300 without converting first gives 0.03 cm, which is the usual slip.
Q4.2, 24 m at 1:400
24 m = 2400 cm, and 2400 ÷ 400 = 6 cm. Check: 6 × 400 = 2400 cm = 24 m.
Q5.1, 4 cm on a 1:25 000 map
4 × 25 000 = 100 000 cm = 1000 m = 1 km.
Q5.2, 9.2 cm on a 1:50 000 map
9.2 × 50 000 = 460 000 cm = 4600 m = 4.6 km. Check: 4.6 km = 460 000 cm and 460 000 ÷ 50 000 = 9.2 cm.
Q5.3, 3.5 cm on a 1:10 000 map
3.5 × 10 000 = 35 000 cm = 350 m, which is 0.35 km. Read the unit the question asks for before you stop converting.