Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 A photo with area 40 cm2 is enlarged so that every length is 1.5 times as long. What is the area of the enlarged photo?
Show worked solution
1.52 = 2.25, so the area is 40 × 2.25 = 90 cm2. 60 cm2 multiplies by 1.5 (a length factor); 120 cm2 multiplies by 2 × 1.5 = 3 instead of squaring; 135 cm2 multiplies by 1.53 = 3.375 (a volume factor).
2 A cube of side 2 cm and a cube of side 8 cm. Write, in simplest form, the ratio of their surface areas and the ratio of their volumes.
Show worked solution
Lengths 2 : 8 = 1 : 4. Surface areas 12 : 42 = 1 : 16. Volumes 13 : 43 = 1 : 64.
3 Two similar triangles have matching sides 6 cm and 8 cm. The smaller triangle has area 27 cm2. Find the area of the larger triangle, in cm2.
Show worked solution
k = \(\dfrac{4}{3}\) and k2 = \(\dfrac{16}{9}\), so the area is 27 × \(\dfrac{16}{9}\) = 3 × 16 = 48 cm2.
Level 2 · Core
4 Two similar jugs have surface areas in the ratio 9 : 49. What is the ratio of their capacities?
Show worked solution
Lengths √9 : √49 = 3 : 7, so capacities 33 : 73 = 27 : 343. 3 : 7 is the length ratio; 9 : 49 is the area ratio; 81 : 2401 squares the area ratio instead of cubing the length ratio.
5 Two similar statues have volumes 250 cm3 and 2000 cm3. The smaller one is 7 cm tall and has a surface area of 60 cm2. Find the height and the surface area of the larger one.
Show worked solution
Volume factor 2000 ÷ 250 = 8 = 23, so k = 2. Height 7 × 2 = 14 cm. Surface area 60 × 22 = 240 cm2.
6 A real roof has an area of 150 m2. A model of the house is built to scale 1 : 50. What is the area of the model's roof, in cm2?
Show worked solution
Model roof = 150 ÷ 2500 = 0.06 m2. 1 m2 = 10 000 cm2, so 0.06 m2 = 600 cm2. Converting with × 100 instead of × 10 000 gives 6, which is wrong.
Level 3 · Challenge
7 Three statements:
1 Doubling every dimension of a box doubles its capacity.
2 If a similar enlargement multiplies the area by 2.25, it multiplies the lengths by 1.5.
3 Two similar cones with volumes 8 cm3 and 125 cm3 have surface areas in the ratio 4 : 25.
Which statements are correct?
1 Doubling every dimension of a box doubles its capacity.
2 If a similar enlargement multiplies the area by 2.25, it multiplies the lengths by 1.5.
3 Two similar cones with volumes 8 cm3 and 125 cm3 have surface areas in the ratio 4 : 25.
Which statements are correct?
Show worked solution
Doubling lengths multiplies capacity by 23 = 8, not 2, so statement 1 is wrong (1 ✗). √2.25 = 1.5 (2 ✓). Lengths ∛8 : ∛125 = 2 : 5, so areas 4 : 25 (3 ✓). 2 and 3.
8 A model of a water tank is built to scale 1 : 20. The model needs 15 mL of paint and holds 2 L. How much paint does the real tank need, and how much does it hold?
Show worked solution
Paint: 15 × 400 = 6000 mL = 6 L. Capacity: 2 × 8000 = 16 000 L.
9 Two similar solids have surface areas 48 cm2 and 147 cm2. The smaller one has volume 128 cm3. Find the volume of the larger one, in cm3.
Show worked solution
48 : 147 = 16 : 49, so lengths 4 : 7 and volumes 64 : 343. Volume = 128 × \(\dfrac{343}{64}\) = 2 × 343 = 686 cm3.