If 25% of (P + Q) = 75% of (P − Q), then find the ratio P: Q.

Correct — B. Clear the percentages, then collect P and Q.

Write as fractions: 25% = 1⁄4, 75% = 3⁄4
So (P + Q) ÷ 4 = 3(P − Q) ÷ 4
Multiply both sides by 4: P + Q = 3P − 3Q

Collect terms: Q + 3Q = 3P − P
4Q = 2P, so P ÷ Q = 4 ÷ 2 = 2
P : Q = 2 : 1 → option (b)

Why the others are wrong

• (a)
  1 : 2 — 1 : 2 is Q : P, the ratio turned upside down. From 4Q = 2P, P is twice Q, so P : Q = 2 : 1.
• (c)
  1 : 3 — 1 : 3 fails the check. With P = 1 and Q = 3, 25% of (1 + 3) = 1, but 75% of (1 − 3) = −1.5.
• (d)
  3 : 1 — 3 : 1 is (P + Q) : (P − Q), the halfway ratio, not P : Q. Check P = 3, Q = 1: 25% of 4 = 1, but 75% of 2 = 1.5.

When a% of one quantity equals b% of another, the quantities are in the inverse ratio b : a. Here (P + Q) : (P − Q) = 75 : 25 = 3 : 1.

Now split the sum and the difference. Put P + Q = 3k and P − Q = k.
Adding gives 2P = 4k, so P = 2k. Subtracting gives 2Q = 2k, so Q = k.

Componendo-dividendo gives the same in one line: P ÷ Q = (3 + 1) ÷ (3 − 1) = 2.

Key facts

• If a% of X = b% of Y, then X : Y = b : a.
• If (P + Q) : (P − Q) = m : n, then P : Q = (m + n) : (m − n).
• 25% = 1⁄4 and 75% = 3⁄4.

Common traps

• Stopping at (P + Q) : (P − Q) = 3 : 1 and marking it as P : Q.
• Writing P : Q upside down after reaching 4Q = 2P.

No directly related past PYQ was found.