Two cylinders have the radii of their bases in the ratio 4 : 5 and their heights are in the ratio 5 : 7. Find the ratio of their curved surface areas.

Correct — D. Curved surface area of a cylinder is 2πrh, so comparing two CSAs needs only r × h from each — the 2π cancels.

r₁ : r₂ = 4 : 5
h₁ : h₂ = 5 : 7

CSA₁ : CSA₂ = (4 × 5) : (5 × 7)
= 20 : 35
Divide both sides by 5 → 4 : 7 → option (d).

Why the others are wrong

• (a)
  5 : 4 — 5 : 4 is the radius ratio turned upside down. Inverting a ratio answers the reciprocal question, and nothing here is being compared the other way round.
• (b)
  4 : 5 — 4 : 5 is the radius ratio alone, used as if the two heights were equal. They are 5 : 7, so the height factor has to be multiplied in first.
• (c)
  1 : 2 — 1 : 2 appears nowhere in the working. The product ratio is 20 : 35, whose only reduction is 4 : 7; a ratio is divided by a common factor, never rounded to a neater pair.

For a cylinder, curved surface area = 2πrh, total surface area = 2πr(r + h) and volume = πr²h.

When ratios replace numbers, write each quantity as the product of its ratio terms and let the constants cancel. CSA is linear in both r and h, so CSA₁ : CSA₂ = (r₁h₁) : (r₂h₂).

Volume would square the radius ratio instead. Total surface area cannot be found from these two ratios at all, because r + h needs the actual sizes, not just their proportions.

Key facts

• Curved surface area of a cylinder = 2πrh.
• Given only ratios, CSA₁ : CSA₂ = (r₁ × h₁) : (r₂ × h₂), since 2π cancels.
• (4 × 5) : (5 × 7) = 20 : 35 = 4 : 7.
• The volume ratio for the same two cylinders would be (16 × 5) : (25 × 7) = 80 : 175 = 16 : 35.

Common traps

• Adding the two ratios instead of multiplying the terms.
• Using πr²h and squaring the radius ratio, which answers the volume question.
• Reporting 4 : 5 because the radius ratio is printed first in the question.

No directly related past PYQ was found.