In ΔPQR, if 4 ∠ P = 5 ∠ Q = 20 ∠ R, then the value of ∠ Q is:

Correct — D. Equal products mean the angles are inversely proportional to their coefficients, not directly.

Let 4∠P = 5∠Q = 20∠R = k.
∠P = k⁄4, ∠Q = k⁄5, ∠R = k⁄20

Angle sum: k⁄4 + k⁄5 + k⁄20 = 180°
Over 20: (5k + 4k + k)⁄20 = 180°
10k⁄20 = 180°, so k⁄2 = 180° and k = 360°

∠Q = 360 ÷ 5 = 72° → option (d).

Check: ∠P = 90°, ∠Q = 72°, ∠R = 18°, and 90 + 72 + 18 = 180°.

Why the others are wrong

• (a)
  45° — No angle of this triangle is 45°; the three are 90°, 72° and 18°. 45° is simply 180° ÷ 4, which treats the coefficient 4 as a share of the angle sum rather than a multiplier on ∠P.
• (b)
  90° — 90° is ∠P, the largest angle here, because the smallest coefficient produces the biggest angle. The question asks for ∠Q, not ∠P.
• (c)
  36° — 36° is 180° ÷ 5, the same misreading applied to ∠Q's coefficient. The coefficients scale the angles, they do not partition 180° between them.

The statement 4∠P = 5∠Q = 20∠R says three products are equal, so set the whole chain equal to one constant k.

Each angle then becomes k divided by its own coefficient, which puts the angles in the ratio 1⁄4 : 1⁄5 : 1⁄20, or 5 : 4 : 1 once the denominators are cleared.

The larger the coefficient, the smaller the angle. That inversion is the whole point of the item.

Feed the three expressions into the 180° angle sum, solve for k, and read off the angle the question names.

Key facts

• Setting 4∠P = 5∠Q = 20∠R = k gives ∠P = k⁄4, ∠Q = k⁄5 and ∠R = k⁄20.
• The angles are in the ratio 5 : 4 : 1, so 10 parts make 180° and one part is 18°.
• The triangle's angles are 90°, 72° and 18°, which makes it right-angled at P.

Common traps

• Reading the coefficients 4, 5 and 20 as the ratio of the angles themselves.
• Solving correctly for k and then reporting ∠P instead of ∠Q.
• Dividing 180° by one coefficient, which manufactures 45° or 36°.

No directly related past PYQ was found.