Mathematics questions, chapter 4

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Scale Drawings

41 marks. Use Practice to check each answer as you go, or Exam to answer everything first. You may use a calculator. Show your working for Section B.

Section A: objective questions

12 marks

Choose the correct answer.

Question 1

Which statement describes a scale drawing of an object?

Every length is changed by a different amount and every angle stays the same
Every length is changed in the same proportion and every angle stays the same
Every length stays the same and every angle is changed in the same proportion
Every length and every angle is changed in the same proportion

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In a scale drawing all measurements of length are in the same proportion to the object, and the sizes of angles do not change.

For the teacher: Remembering, low difficulty, standard 4.1.1

Textbook p. 89

Question 2

A map has a scale of 1 : 20. What does this scale mean?

20 cm on the map is 1 cm on the ground
1 cm on the ground is 1 cm on the map
1 cm on the map represents 20 cm on the ground
1 cm on the map represents 1/20 cm on the ground

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Scale 1 : n means 1 unit on the drawing represents n units on the object. Here n = 20.

For the teacher: Understanding, low difficulty, standard 4.1.2

Textbook p. 89

Question 3

A scale drawing is drawn to the scale 1 : 14. Compared with the object, every length on the drawing is

4 times shorter
14 as long
4 cm longer
4 times as long

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n = 14 is less than 1, so the drawing is bigger. 1 unit on the drawing is only 14 unit on the object, so every length on the drawing is 4 times as long.

For the teacher: Understanding, low difficulty, standard 4.1.2

Textbook p. 91

Question 4

A map is drawn to the scale 1 : 250 000. What is the actual distance, in km, of a road that is 4 cm long on the map?

10 km
100 km
1 km
1 000 km

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4 × 250 000 = 1 000 000 cm = 10 km (1 km = 100 000 cm). 100 km comes from dividing by 10 000 instead of 100 000, and 1 000 km comes from reading 1 000 000 cm as metres.

For the teacher: Applying, low difficulty, standard 4.1.3

Textbook p. 93

Question 5

The scale of a map is 1 cm : 2.5 km. How long, in cm, is a distance of 17.5 km on the map?

43.75 cm
0.14 cm
7 cm
15 cm

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17.5 ÷ 2.5 = 7, so 17.5 km is 7 cm on the map. 43.75 comes from multiplying instead of dividing.

For the teacher: Applying, low difficulty, standard 4.1.3

Textbook p. 93

Question 6

A rectangular poster is 12 cm long and 8 cm wide. What are the length and width of its scale drawing, in cm, if the scale is 1 : 4?

48 and 32
3 and 2
16 and 12
4 and 8

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n = 4 > 1, so the drawing is smaller: 12 ÷ 4 = 3 cm and 8 ÷ 4 = 2 cm. 48 and 32 multiply instead of dividing.

For the teacher: Applying, low difficulty, standard 4.1.3

Textbook p. 94

Question 7

Diagram 1 shows a triangle and its scale drawing on the same grid of equal squares. State the scale used in the form 1 : n.

A right-angled triangle with a base of 4 squares and height 2 squares, and a drawing with a base of 6 squares and height 3 squares, on equal squares — Diagram 1

1 : 2
1 : 3
3 : 2
1 : 23

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Scale = drawing ÷ object = 6 ÷ 4 = 32 (also 3 ÷ 2). 3 : 2 = 1 : 23. 3 : 2 is not in the form 1 : n, and 1 : 32 would be the wrong way round.

For the teacher: Analysing, low difficulty, standard 4.1.3

Textbook p. 91

Question 8

Diagram 2 shows an object drawn on a grid with squares of 2 cm and its scale drawing on a grid with squares of 0.5 cm. Both use 3 squares by 3 squares. What is the scale?

A triangle on 2 cm squares and the same triangle on 0.5 cm squares, both 3 squares wide — Diagram 2

1 : 4
4 : 1
1 : 1.5
1 : 14

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The numbers of squares are the same, so scale = grid size of drawing ÷ grid size of object = 0.5 ÷ 2 = 14, that is 1 : 4. 4 : 1 and 1 : 14 both show a bigger drawing, which is the wrong way round here, because the drawing is smaller than the object.

For the teacher: Analysing, low difficulty, standard 4.1.3

Textbook p. 92

Question 9

A classroom is 4.5 m long. How long, in cm, is the classroom on a plan with the scale 1 : 50?

225 cm
9 cm
0.9 cm
90 cm

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4.5 m = 450 cm and 450 ÷ 50 = 9 cm. 225 comes from multiplying 4.5 by 50; 0.9 comes from using 1 m = 10 cm; 90 divides by 5 instead of 50.

For the teacher: Applying, low difficulty, standard 4.1.5

Textbook p. 99

Question 10

A rectangle of area 40 cm² is drawn to the scale 1 : 2. What is the area of the scale drawing?

20 cm²
5 cm²
10 cm²
80 cm²

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Lengths are halved, so area is multiplied by 12 × 12 = 14: 40 ÷ 4 = 10 cm². 20 cm² forgets that area depends on two lengths.

For the teacher: Analysing, medium difficulty, standard 4.1.5

Textbook p. 101

Question 11

The distance between two towns on a map is 6 cm. The scale is 1 cm : 20 km. A car travels from one town to the other at 80 km/h. How long does the journey take?

1 hour
1 hour 15 minutes
4 hours
1 hour 30 minutes

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Distance = 6 × 20 = 120 km. Time = 120 ÷ 80 = 1.5 hours = 1 hour 30 minutes.

For the teacher: Applying, medium difficulty, standard 4.1.5

Textbook p. 99

Question 12

Faizal says, "If I draw a rectangle to the scale 1 : 2, the area of the drawing is half the area of the object." Which is a correct evaluation?

He is wrong. Length and width are both halved, so the area is 14 of the object's area
He is right. The scale is 1 : 2, so the area is 12
He is wrong. The area is 2 times the object's area
He is wrong. The area is 13 of the object's area

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Area is length × width and both are halved: 12 × 12 = 14. Areas change by the square of the length factor.

For the teacher: Evaluating, high difficulty, standard 4.1.5

Textbook p. 101

Section B: structured questions

29 marks

Answer all questions. Show your working.

Question 13

Diagram 3 shows a right-angled triangle, the object, and two drawings, X and Y, on a grid of 1 cm squares. One of the drawings is a scale drawing of the object.

Object triangle with legs 4 cm and 3 cm; drawing X with legs 8 cm and 6 cm; drawing Y with legs 8 cm and 3 cm — Diagram 3

[1 mark] (a)Which drawing is a scale drawing of the object?

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Drawing X [1]1

Tick each point you got.

[1 mark] (b)Give a reason for your answer.

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All sides are 2 times longer (8 ÷ 4 = 6 ÷ 3 = 2), the same proportion [1]1

Tick each point you got.

[2 marks] (c)State the scale used in the form 1 : n.

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Scale = 8 : 4 = 2 : 1 [1]1
= 1 : 12 [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 4.1.1

Textbook p. 88

Question 14

Diagram 4 shows a right-angled triangle ABC on a grid of 1 cm squares and its scale drawing A'B'C' on a grid of 1.5 cm squares. Both drawings are 4 squares wide and 3 squares high.

Triangle ABC on 1 cm squares and triangle A'B'C' on 1.5 cm squares, each with 4 squares along the base and 3 squares high — Diagram 4

[2 marks] (a)Find the scale used, in the form 1 : n.

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Scale = 1.5 : 1 = 3 : 2 [1]1
= 1 : 23 [1]1

Tick each point you got.

[1 mark] (b)Is A'B'C' larger or smaller than ABC? Give a reason using the value of n.

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Larger, because n = 23 is less than 1 [1]1

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[1 mark] (c)The hypotenuse BC of the object is 5 cm. Find the length of the hypotenuse B'C'.

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5 × 1.5 = 7.5 cm [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 4.1.3

Textbook p. 92

Question 15

Hui Min draws a scale drawing of a right-angled triangle with sides of 6 cm, 8 cm and 10 cm. She uses the scale 1 : 2.

[2 marks] (a)Find the lengths of the three sides of her drawing.

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6 ÷ 2 = 3 cm and 8 ÷ 2 = 4 cm [1]1
10 ÷ 2 = 5 cm [1]1

Tick each point you got.

[1 mark] (b)State the size of the largest angle in her drawing.

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90° (angles do not change in a scale drawing) [1]1

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[2 marks] (c)Find the area of the object and the area of the drawing, in cm².

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Object: 12 × 6 × 8 = 24 cm² [1]1
Drawing: 12 × 3 × 4 = 6 cm² [1]1

Tick each point you got.

[1 mark] (d)Write the ratio of the area of the drawing to the area of the object in its simplest form.

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6 : 24 = 1 : 4 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 4.1.4

Textbook p. 96

Question 16

Diagram 5 shows the scale drawing of the L-shaped floor of a hall. The scale used is 1 : 200.

L-shaped floor plan: 6 cm long and 5 cm wide with a 3 cm by 2 cm rectangle missing from one corner; scale 1 : 200 — Diagram 5

[1 mark] (a)Find the actual length of the longest side of the floor, in m.

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6 × 200 = 1 200 cm = 12 m [1]1

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[2 marks] (b)Find the actual area of the floor, in m².

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Large rectangle: 12 × 10 = 120 m²; missing corner: 6 × 4 = 24 m² [1]1
120 − 24 = 96 m² [1]1

Tick each point you got.

[2 marks] (c)The floor is covered with square tiles of side 40 cm. Each tile costs RM8. Find the total cost of the tiles.

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96 ÷ 0.16 = 600 tiles [1]1
600 × RM8 = RM4 800 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 4.1.5

Textbook p. 99

Question 17

Diagram 6 shows triangle X, which is the scale drawing of triangle Y with the scale 1 : n. The area of X is 72 cm² and the area of Y is 8 cm². The base of Y is 4 cm.

Triangle X with area 72 square centimetres and triangle Y with area 8 square centimetres and base 4 cm — Diagram 6

[1 mark] (a)Find the ratio of the area of X to the area of Y.

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72 : 8 = 9 : 1 [1]1

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[1 mark] (b)Hence find the ratio of the length of a side of X to the corresponding side of Y.

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√9 : √1 = 3 : 1 [1]1

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[1 mark] (c)Find the value of n.

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3 : 1 = 1 : 13, so n = 13 [1]1

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[1 mark] (d)Find the length of the base of X.

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4 × 3 = 12 cm [1]1

Tick each point you got.

For the teacher: Analysing, high difficulty, standard 4.1.5

Textbook p. 101

Question 18

Hazim wants to draw a plan of a rectangular field, 120 m long and 80 m wide, on a sheet of paper. The drawing must fit inside a space of 27 cm by 19 cm. He is choosing between the scales 1 : 400 and 1 : 500.

[2 marks] (a)Find the length and width of the plan, in cm, if the scale is 1 : 500.

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12 000 ÷ 500 = 24 cm [1]1
8 000 ÷ 500 = 16 cm [1]1

Tick each point you got.

[2 marks] (b)Which scale should Hazim choose? Give a reason.

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1 : 500 [1]1
1 : 400 gives 30 cm by 20 cm, which does not fit in 27 cm by 19 cm [1]1

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[2 marks] (c)A mower cuts the grass of the field at 800 m² per hour. How long does it take to cut the whole field?

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Area = 120 × 80 = 9 600 m² [1]1
9 600 ÷ 800 = 12 hours [1]1

Tick each point you got.

For the teacher: Creating, medium difficulty, standard 4.1.4

Textbook p. 103

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