Similarity and Scale — Solutions

← Back to Shape and Measurement › General Maths

1. Q1 — Scale factor of similar rectangles

   Fluency
   • k = 10 ÷ 4 = 2.5 (check: 15 ÷ 6 = 2.5 ✓). Scale factor k = 2.5.

2. Q2 — Using scale factor to find lengths and areas

   Fluency
   • (a) New length = 3 × 8 = 24 cm.
   • (b) Area ratio = k² = 9. New area = 9 × 24 = 216 cm².

3. Q3 — Map scale distance

   Fluency
   • 6.4 × 50 000 = 320 000 cm = 3200 m = 3.2 km.

4. Q4 — Similar triangles: scale factor and side lengths

   Understanding
   • (a) Perimeter of small = 21 cm. k = 63 ÷ 21 = 3.
   • (b) Sides: 5 × 3 = 15 cm, 7 × 3 = 21 cm, 9 × 3 = 27 cm. (Check: 15+21+27 = 63 ✓)

5. Q5 — Scale factor and volume

   Understanding
   • Volume ratio = k³ = 2.5³ = 15.625. Enlarged volume = 15.625 × 48 = 750 cm³.

6. Q6 — Photo enlargement

   Understanding
   • (a) k = 24 ÷ 15 = 1.6.
   • (b) New width = 1.6 × 10 = 16 cm.
   • (c) Area ratio = k² = 1.6² = 2.56. (Check: 384 ÷ 150 = 2.56 ✓)

7. Q7 — Similar cylinders: dimensions and volume

   Understanding
   • (a) r_large = 1.5 × 4 = 6 cm. h_large = 1.5 × 6 = 9 cm.
   • (b) SA ratio = k² = 1.5² = 2.25 : 1.
   • (c) Volume ratio = k³ = 1.5³ = 3.375 : 1.
   • (d) V_small = 96π cm³. V_large = 3.375 × 96π = 324π ≈ 1017.88 cm³.

8. Q8 — Model car at scale 1:24

   Understanding
   • (a) Length: 19 cm. Width: 7.5 cm. Height: 6 cm.
   • (b) SA ratio (model : real) = (1/24)² = 1 : 576.
   • (c) Volume ratio = 24³ = 13 824. Real tank = 12 mL × 13 824 ≈ 166 L.

9. Q9 — Similar pyramids: finding scale factor from volumes

   Problem Solving
   • (a) 64 : 216 = 8 : 27.
   • (b) k = ³√(27/8) = 3/2 = 1.5.
   • (c) SA ratio = k² = 2.25, i.e. 4 : 9 (small to large).
   • (d) Large base = 1.5 × 4 = 6 cm.

10. Q10 — Similar spherical tanks and painting cost

    Problem Solving
    • (a) r_small = (2/3) × 1.5 = 1.0 m.
    • (b) SA_large = 9π ≈ 28.27 m². SA_small = 4π ≈ 12.57 m².
    • (c) V_large = 4.5π ≈ 14.14 m³. V_small = (4/3)π ≈ 4.19 m³.
    • (d) Cost ratio = k² = (2/3)² = 4/9. Small cost = (4/9) × $8100 = $3 600.