If 20% of (P + Q) = 50% of (P − Q), then find the ratio P: Q.
- (a)3 : 7
- (b)7 : 3
- (c)2 : 5
- (d)5 : 2
Answer
Why
Correct — B. Turn the percentages into an equation and collect like terms.
As decimals: 0.2(P + Q) = 0.5(P − Q)
Multiply by 10: 2(P + Q) = 5(P − Q)
Expand: 2P + 2Q = 5P − 5Q
Collect: 2Q + 5Q = 5P − 2P, so 7Q = 3P
P : Q = 7 : 3 → option (b)
Check: 20% of (7 + 3) = 2 and 50% of (7 − 3) = 2.
Why the others are wrong
- (a)3 : 7 — 3 : 7 is Q : P, the ratio upside down. From 7Q = 3P, P is the larger. With P = 3 and Q = 7, P − Q is negative and the two sides cannot be equal.
- (c)2 : 5 — 2 : 5 copies 20% : 50% straight into the answer. With P = 2 and Q = 5, 20% of 7 is 1.4 while 50% of (2 − 5) is −1.5.
- (d)5 : 2 — 5 : 2 is the ratio of the brackets, (P + Q) : (P − Q), not of P to Q. With P = 5 and Q = 2, 20% of 7 is 1.4 but 50% of 3 is 1.5.
Concept
Write each percentage as a decimal times its bracket and the question becomes a linear equation in P and Q. Clear the decimals, expand, then gather the P terms on one side and the Q terms on the other.
A faster route is componendo-dividendo. The equation says (P + Q) : (P − Q) = 50 : 20 = 5 : 2, so P : Q = (5 + 2) : (5 − 2) = 7 : 3.
Note the swap. 20% of the first bracket equals 50% of the second, so the first bracket is the larger one: (P + Q) : (P − Q) is 50 : 20, not 20 : 50.
Key facts
- If a% of X = b% of Y, then X : Y = b : a.
- Componendo-dividendo: if (P + Q) : (P − Q) = m : n, then P : Q = (m + n) : (m − n).
- A ratio can be checked by putting its two numbers back into the original equation.
Study next
Common traps
- Writing (P + Q) : (P − Q) = 20 : 50. The bracket taken at the smaller percentage must be the larger quantity.
- Solving correctly to 7Q = 3P and then reading P : Q as 3 : 7.
18 Sep 2025, 09:00, Quant Q.3 uses 25% and 75%: (P + Q) : (P − Q) = 3 : 1, so P : Q = (3 + 1) : (3 − 1) = 2 : 1.
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