the ratio of areas of two triangles is 4 : 3 and the ratio of their heights in 3:4, then the ratio of their bases is

Area of a triangle is half the base times the height, so Area = ½ × base × height. In a ratio the ½ cancels, so Area₁ : Area₂ = (b₁ × h₁) : (b₂ × h₂). Here the area ratio is 4 : 3 and the height ratio h₁ : h₂ = 3 : 4.

Taking the area relation: (b₁/b₂) × (h₁/h₂) = 4/3, so (b₁/b₂) × (3/4) = 4/3. Therefore b₁/b₂ = (4/3) × (4/3) = 16/9. The bases are in ratio 16 : 9, which is option B.

Why the others fail:

• A (7:16) does not come from multiplying the given ratios correctly.
• C (9:16) is the reciprocal of the right answer, which is what you get if you flip the height ratio while multiplying.
• D (9:8) has no link to 4:3 and 3:4 once the product is worked out.

Core concept: area is a product of two quantities, so ratios combine by multiplication, and a ratio can be isolated by dividing out the other ratio. This is basic mensuration, but the same proportional reasoning recurs in similar-triangle questions and in Mains answers on scaling of plot areas.

Trap: candidates multiply the height ratio in the wrong direction and land on 9:16. Keep it fixed: base ratio = area ratio divided by height ratio.