Mathematics Standard • Year 12 • Ratios and rates • Lesson 3
Scale Reasoning, Mastery Challenge
Work backwards to find an unknown scale, handle the area version of a scale factor, redraw at a second scale, and estimate a real material order. Every question here needs a sentence of reasoning, not just a number.
1. Finding the scale
You are given both lengths and asked for the scale itself. 3 marks
Q1.1 A wall is known to be 18 m long. On a plan it measures 4.5 cm. Find the scale of the plan, written in the form 1:n.
2. Lengths scale by n, areas scale by n squared
Answer all three parts. Part (c) is where the marks separate. 4 marks
Q2.1 On a plan drawn at 1:200, a courtyard measures 3.5 cm by 2.0 cm.
(a) Find its real dimensions in metres. (b) Find its real area in square metres. (c) The courtyard measures 7 square centimetres on the plan. Explain why the real area is not 7 × 200 square centimetres, and show what it is instead.
3. The same room on two plans
An architect redraws a plan at a different scale. 4 marks
Q3.1 On plan A, drawn at 1:100, a room measures 5.5 cm by 4.0 cm. Plan B shows the same room at a scale of 1:250. Find the room's dimensions on plan B, and state which plan carries more detail.
4. Estimating a material order
A real order cannot be a fraction of a unit. Say what you rounded and why. 5 marks
Q4.1 On a house plan drawn at 1:100, three bedrooms measure 4.2 cm by 3.5 cm, 3.8 cm by 3.0 cm, and 3.2 cm by 3.0 cm. Carpet costs $54 per square metre and is sold only in whole square metres. Find the total carpet needed and the cost of the order.
5. Find the error
Identify the mistake, name it, and give the correct answer. 3 marks
Q5.1 A student measures a wall as 6.4 cm on a plan drawn at 1:50 and writes: "6.4 × 50 = 320, so the wall is 320 m long." Explain what has gone wrong and give the correct real length.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, Finding the scale
Put both into centimetres: 18 m = 1800 cm [1]. The scale is drawing : real = 4.5 : 1800, and dividing both terms by 4.5 gives 1 : 400 [1]. So the plan is drawn at 1:400 [1]. Check: 4.5 × 400 = 1800 cm = 18 m.
Q2.1, Lengths and areas
(a) 3.5 × 200 = 700 cm = 7 m, and 2.0 × 200 = 400 cm = 4 m [1].
(b) Area = 7 × 4 = 28 square metres [1].
(c) An area is a length multiplied by a length, so both lengths are multiplied by 200 and the area is multiplied by 200 × 200 = 40 000, not by 200 [1]. So the real area is 7 × 40 000 = 280 000 square centimetres, and since 1 square metre is 10 000 square centimetres, that is 280 000 ÷ 10 000 = 28 square metres, agreeing with part (b) [1].
Q3.1, The same room on two plans
First find the real size from plan A: 5.5 × 100 = 550 cm = 5.5 m, and 4.0 × 100 = 400 cm = 4.0 m [1]. Now divide by plan B's scale factor: 550 ÷ 250 = 2.2 cm and 400 ÷ 250 = 1.6 cm [2]. Plan A carries more detail, because a smaller scale factor shrinks the building less and so draws it larger [1]. Check: 2.2 × 250 = 550 cm = 5.5 m.
Q4.1, Estimating a carpet order
At 1:100 each centimetre reading becomes a metre reading. The rooms are 4.2 m by 3.5 m, 3.8 m by 3.0 m, and 3.2 m by 3.0 m [1].
Areas: 4.2 × 3.5 = 14.7, 3.8 × 3.0 = 11.4, and 3.2 × 3.0 = 9.6 square metres [2]. Total = 14.7 + 11.4 + 9.6 = 35.7 square metres [1].
Carpet is sold only in whole square metres, so the order must be rounded up to 36 square metres, because rounding down would leave part of a floor bare. Cost = 36 × 54 = $1944 [1]. Check: 1944 ÷ 54 = 36.
Q5.1, Find the error
The arithmetic is right but the unit is not [1]. The measurement was taken in centimetres, and a scale has no units, so 6.4 × 50 = 320 is 320 centimetres, not 320 metres [1]. Converting last gives 320 ÷ 100 = 3.2 m [1]. A sense check settles it immediately: a 320 m wall would be longer than three football fields.