Proportional Reasoning with Scale Drawings — Challenge
Scaling changes every length by the same factor k — which multiplies area by k2 and volume by k3.
What these problems cover
These questions use 11 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Scale factor — The number every length is multiplied by.
- Similar figures — Same shape, different size: equal angles, proportional sides.
- Corresponding sides — Sides in matching positions in two similar figures.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Volume — Space inside, in cubic units.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Factor — A number that divides evenly into another.
- Scale — What one grid step is worth — and where the axis starts.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
Problem 1
A model car is built to a scale of 1:20. The model is 12 centimetres long. How long is the real car in centimetres?
- 2.4
- 60
- 120
- 240
- 300
Worked answer
240
A scale of 1:20 means every centimetre on the model stands for 20 centimetres on the real car.
So the real length is 12· 20 = 240 centimetres.
Problem 2
On a map, 1 centimetre represents 2.5 kilometres. Two towns are 6 centimetres apart on the map. How far apart are they in kilometres?
- 10
- 12.5
- 15
- 17.5
- 20
Worked answer
15
Each centimetre stands for 2.5 kilometres.
6· 2.5 = 15 kilometres.
Problem 3
Two rectangles are similar with a scale factor of 3. The smaller has an area of 8 square centimetres. What is the area of the larger in square centimetres?
- 24
- 36
- 48
- 64
- 72
Worked answer
72
Lengths multiply by 3, but area involves two lengths, so area multiplies by 32 = 9.
8· 9 = 72 square centimetres.
Multiplying the area by 3 to get 24 is the usual mistake: it scales only one dimension.
Problem 4
A triangle has sides 6, 8 and 10. A similar triangle has its shortest side equal to 15. What is the perimeter of the larger triangle?
- 40
- 48
- 54
- 60
- 72
Worked answer
60
The scale factor is the ratio of corresponding sides: 156 = 2.5.
The smaller triangle's perimeter is 6+8+10 = 24.
Perimeter is a length, so it scales by the same factor: 24· 2.5 = 60.
Problem 5
A drawing uses a scale of 1:50. A wall is 300 centimetres long in reality. How long is it on the drawing in centimetres?
- 4
- 5
- 6
- 15
- 250
Worked answer
6
Going from reality to the drawing divides by the scale, rather than multiplying.
30050 = 6 centimetres.
Problem 6
A photograph is 4 centimetres wide and 6 centimetres tall. It is enlarged so that its width becomes 10 centimetres. How tall is the enlargement in centimetres?
- 12
- 13
- 14
- 15
- 16
Worked answer
15
The scale factor is 104 = 2.5.
The height scales by the same factor: 6· 2.5 = 15 centimetres.
Adding the 6 centimetres of extra width to the height instead would give 12, which distorts the picture.
Problem 7
Two similar solids have a scale factor of 2. The smaller has a volume of 5 cubic centimetres. What is the volume of the larger in cubic centimetres?
- 10
- 20
- 40
- 80
- 160
Worked answer
40
Volume involves three lengths, so it scales by the cube of the scale factor: 23 = 8.
5· 8 = 40 cubic centimetres.
Doubling the volume to 10 scales only one dimension; multiplying by 4 scales two.
Problem 8
A model is built to a scale of 1:100. The model measures 3 metres. What does that represent in metres?
- 30
- 100
- 200
- 300
- 3000
Worked answer
300
Each metre on the model stands for 100 metres in reality.
3· 100 = 300 metres.
Problem 9
A blueprint uses a scale of 1 inch to 4 feet. A wall measures 5 inches on the blueprint. How many feet long is the wall?
- 9
- 16
- 20
- 24
- 80
Worked answer
20
Each inch on the blueprint stands for 4 feet.
5· 4 = 20 feet.
Problem 10
Two similar triangles have areas of 9 and 25 square centimetres. What is the ratio of their corresponding sides?
- 3:5
- 9:25
- 81:625
- 5:3
- 3:25
Worked answer
3:5
Areas of similar figures are in the ratio of the squares of the sides.
So the ratio of the sides is the ratio of the square roots: √9:√25 = 3:5.
The ratio 9:25 is the area ratio, not the side ratio.