Proportional Reasoning with Scale Drawings — Practice
Scaling changes every length by the same factor k — which multiplies area by k2 and volume by k3.
What these questions 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 drawing uses scale 1 : 50. A line measures 6 cm. How long is the real object in cm?
Worked answer
300
6 × 50 = 300 cm, which is 3 m.
Problem 2
A figure is enlarged by scale factor 3. A side of 5 cm becomes how long?
Worked answer
15
5 × 3 = 15 cm
Problem 3
Two similar triangles have corresponding sides 6 and 15. Find the scale factor from the smaller to the larger.
Worked answer
2.5
156 = 2.5
Problem 4
On a map, 1 cm represents 4 km. How many km does 7 cm represent?
Worked answer
28
7 × 4 = 28 km
Problem 5
A 4 in by 6 in photo is enlarged by factor 3. How long is the longer side now?
Worked answer
18
6 × 3 = 18 in
Problem 6
Triangle ABC has sides 6, 8, 10. A similar triangle has shortest side 9. Find its perimeter.
Worked answer
36
k = 96 = 1.5
Sides become 9, 12, 15
Perimeter = 36. (Or 24 × 1.5 = 36.)
Problem 7
A 3 × 4 rectangle is scaled by factor 2. Find the new area.
Worked answer
48
New rectangle: 6 × 8 = 48
Check: original area 12, times 22 = 4, gives 48 ✓
Problem 8
If every length is multiplied by 4, the area is multiplied by what?
- 4
- 8
- 16
- 64
Worked answer
16
k2 = 42 = 16
Problem 9
A 5 ft person casts a 4 ft shadow. At the same moment a tree casts a 32 ft shadow. How tall is the tree?
Worked answer
40
54 = h32
4h = 160
h = 40 ft
Problem 10
A floor plan at 1 : 40 shows a room 8.5 cm by 6 cm. Find the real area in square metres.
Worked answer
8.16
8.5 × 40 = 340 cm = 3.4 m
6 × 40 = 240 cm = 2.4 m
3.4 × 2.4 = 8.16 m2
Problem 11
A model is built at 1 : 25. The real car is 4.5 m long. How long is the model in cm?
Worked answer
18
4.5 m = 450 cm
450 ÷ 25 = 18 cm
Problem 12
A 6 × 9 rectangle is similar to one whose shorter side is 10. Find its longer side.
Worked answer
15
k = 106 = 53
9 × 53 = 15
Problem 13
A model car is 112 scale. The real hood has area 3.6 m2. Find the model hood's area in m2.
Worked answer
0.025
3.6 ÷ 144 = 0.025 m2
Problem 14
A cube's edge is tripled. Its volume is multiplied by what?
Worked answer
27
k3 = 33 = 27
A 2-cube holds 8; a 6-cube holds 216, which is 27 times as much.
Problem 15
Two similar figures have areas 25 cm2 and 64 cm2. The smaller has perimeter 20 cm. Find the larger perimeter.
Worked answer
32
k2 = 6425, so k = 85
Perimeter is a length, so it scales by k
20 × 85 = 32 cm
Problem 16
A drawing uses scale 1 : 250. A field is 300 m long. How long is it on the drawing, in cm?
Worked answer
120
300 m = 30,000 cm
30000 ÷ 250 = 120 cm