The ratio of the circumferences of two circular flower beds is 3:5. If the smaller flower bed has a radius of 9 m, what is the radius of the larger flower bed?

Correct — A. Circumference is 2πr, so the 2π cancels and circumferences are in the same ratio as the radii.

r₁ : r₂ = 3 : 5
Smaller radius r₁ = 9, so 9 : r₂ = 3 : 5
Cross-multiply: 3 × r₂ = 9 × 5 = 45
r₂ = 45 ÷ 3 = 15 m → option (a)

Why the others are wrong

• (b)
  17 m — 17 m gives radii of 9 : 17, which does not reduce to 3 : 5. Scaling 3 : 5 so the first term is 9 multiplies both terms by 3, giving 9 : 15.
• (c)
  19 m — 19 m gives radii of 9 : 19. Because circumference follows the radius, the beds' circumferences would also be 9 : 19, not 3 : 5.
• (d)
  21m — 21m gives 9 : 21 = 3 : 7. That makes the larger bed 7⁄3 of the smaller, where 3 : 5 makes it 5⁄3 of 9, which is 15 m.

Circumference is 2πr, a fixed multiple of the radius. So a ratio of circumferences is the same ratio of radii, and of diameters.

Area behaves differently: it is πr², so an area ratio is the square of the radius ratio. Radii of 3 : 5 give circumferences of 3 : 5 but areas of 9 : 25.

Key facts

• Circumference = 2πr = πd.
• Ratio of circumferences = ratio of radii = ratio of diameters.
• Ratio of areas = (ratio of radii)².

Common traps

• Squaring the ratio as if circumferences behaved like areas
• Setting 9 against the 5 instead of the 3: the smaller bed takes the smaller term

No directly related past PYQ was found.