Mathematics Standard • Year 11 • Module 2 • Lesson 12
Scale Drawings and Maps — Problem Set
Apply scale, area-scale-factor and floor-plan reasoning to real Australian house-block, park-map and renovation scenarios.
Problem 1 — Valuing a house block from a scale drawing
A real-estate scale drawing of a rectangular house block uses a scale of 1:500. The block on the drawing measures 6.4 cm × 4.0 cm. Land in this suburb is selling for $850 per m².
Set up: What are we solving for?
(i) Find the actual dimensions of the block in metres. 1 mark
(ii) Find the actual area of the block in m². 1 mark
(iii) Estimate the value of the block at $850 per m². State the answer to the nearest dollar with a clear conclusion sentence. 2 marks
Stuck? Revisit lesson § Worked Example 2 — Area Scale Factor.Problem 2 — L-shaped apartment living area
A floor plan of an apartment uses scale 1:80. The combined living/dining area appears as an L-shape formed by a 7 cm × 5 cm rectangle with a 3 cm × 3 cm corner section removed.
Set up: What are we solving for?
(i) Find the drawn area of the L-shape in cm². 1 mark
(ii) Find the actual area of the L-shape in m². 2 marks
(iii) Floorboards cost $89 per m² supplied and laid. Estimate the cost of floorboarding the living area. 2 marks
Stuck? Revisit lesson § Worked Example 2. The area scale factor for 1:80 is 80² = 6 400.Problem 3 — Bushwalking map (scale bar)
A bushwalking map of a section of the Blue Mountains has a scale bar showing 2 cm on the map = 1 km on the ground.
Set up: What are we solving for?
(i) Express the scale as a ratio 1:n. 1 mark
(ii) A walking trail measures 11.4 cm on the map. Find its actual length in km. 1 mark
(iii) A protected wetland appears on the map as an irregular shape with area 18 cm². Find the actual area in km². 2 marks
(iv) A hiker walks the 11.4 cm trail at an average pace of 4 km/h. Estimate the walking time in hours and minutes. 1 mark
Stuck? Revisit lesson § Practice Q7 — area scale factor uses n², not n.Problem 4 — Carpet for three bedrooms (floor plan)
A 1:50 floor plan shows three bedrooms.
Master: 8 cm × 7 cm rectangle.
Bedroom 2: 7 cm × 6 cm rectangle.
Bedroom 3: 6 cm × 5 cm rectangle.
Set up: What are we solving for?
(i) Find the actual dimensions of each bedroom in metres. 2 marks
(ii) Find the total carpeted area in m² across all three bedrooms. 1 mark
(iii) Carpet costs $62/m² and the installer requires a 7% wastage allowance on top of the floor area. Find the total carpet bill to the nearest dollar. 2 marks
Stuck on (iii)? Multiply total m² by 1.07 to add the 7% wastage, then by $62.Problem 5 — Finding an unknown scale (site plan)
A council site plan shows a sports oval. A surveyor measures the long boundary fence in real life as 175 m. On the site plan, the same boundary fence is drawn as 7 cm.
Set up: What are we solving for?
(i) Express the scale as a ratio 1:n. 2 marks
(ii) A grandstand on the plan covers a 4 cm × 1.5 cm rectangle. Use the scale you found to determine the actual area of the grandstand in m². 2 marks
(iii) The same oval is to be reprinted at scale 1:1 000 (so it fits on A4 paper). What length, in cm, will the 175 m boundary fence appear at on the reprint? 2 marks
Stuck? Revisit lesson § Worked Example 3 — Finding the Scale. Convert the actual length to cm before dividing.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Valuing the block
Set up. Convert drawn dims to actual metres, compute area, multiply by $/m².
(i) Length = 6.4 × 500 = 3 200 cm = 32 m. Width = 4.0 × 500 = 2 000 cm = 20 m. Actual block: 32 m × 20 m.
(ii) Area = 32 × 20 = 640 m².
(iii) Value = 640 × $850 = $544,000. Conclusion: at $850/m² the block is worth about $544,000.
Problem 2 — L-shaped living area
Set up. Drawn area first, then × 80² to get actual cm², then ÷10 000 → m².
(i) Drawn area = 7 × 5 − 3 × 3 = 35 − 9 = 26 cm².
(ii) Actual = 26 × 80² = 26 × 6 400 = 166 400 cm² = 16.64 m².
(iii) Cost = 16.64 × $89 = $1,480.96 ≈ $1,481.
Problem 3 — Bushwalking map
Set up. Convert the scale-bar relationship into 1:n, then apply length scale n and area scale n².
(i) 2 cm = 1 km = 100 000 cm → 1 cm = 50 000 cm. Scale = 1:50 000.
(ii) Length = 11.4 ÷ 2 × 1 km = 5.7 km.
(iii) 1 cm = 0.5 km, so 1 cm² = 0.25 km². Actual = 18 × 0.25 = 4.5 km².
(iv) T = 5.7 ÷ 4 = 1.425 h = 1 h + 0.425 × 60 min = 1 h 25.5 min ≈ 1 h 26 min.
Problem 4 — Three-bedroom carpet
Set up. Scale 1:50 means each drawn cm = 50 cm = 0.5 m, so drawn dims × 0.5 give actual m.
(i) Master: 4 m × 3.5 m. Bedroom 2: 3.5 m × 3 m. Bedroom 3: 3 m × 2.5 m.
(ii) Areas: 14 m² + 10.5 m² + 7.5 m² = 32 m².
(iii) With wastage: 32 × 1.07 = 34.24 m². Cost = 34.24 × $62 = $2,122.88 ≈ $2,123.
Problem 5 — Finding an unknown scale
Set up. Convert actual length to cm, divide by drawn length to find n.
(i) 175 m = 17 500 cm. n = 17 500 ÷ 7 = 2 500. Scale = 1:2 500.
(ii) Actual grandstand = (4 × 2 500) cm × (1.5 × 2 500) cm = 10 000 cm × 3 750 cm = 100 m × 37.5 m. Area = 3 750 m².
(iii) At 1:1 000, drawn = 17 500 ÷ 1 000 = 17.5 cm. (Larger drawn length — because 1:1 000 is a larger scale than 1:2 500.)