If p:q=4:1, find (5p+3q):(5p−3q).

Correct — A. Only the ratio is known, so write p = 4k and q = k.

Numerator: 5p + 3q = 20k + 3k = 23k
Denominator: 5p − 3q = 20k − 3k = 17k
Cancel k: 23k : 17k = 23 : 17 → option (a)

Why the others are wrong

• (b)
  17 : 23 — 17 : 23 is the ratio turned upside down, (5p − 3q) : (5p + 3q). The question puts the sum 5p + 3q first, so 23 leads.
• (c)
  7 : 17 — With p = 4, q = 1 the two terms are 23 and 17, so no 7 appears. Reading the ratio backwards (p = 1, q = 4) is what gives 17 and −7.
• (d)
  17 : 7 — 17 : 7 uses p = 1, q = 4, the ratio read backwards: 5 + 12 = 17 and 5 − 12 = −7, with the minus sign dropped. The stem makes p four times q.

Only the ratio p : q is fixed, not the values. Divide both terms by q and the expression depends on p⁄q alone:

(5p + 3q) : (5p − 3q) = (5 × p⁄q + 3) : (5 × p⁄q − 3)

With p⁄q = 4 this is (20 + 3) : (20 − 3) = 23 : 17. It works because every term holds exactly one p or one q, so q divides out of the top and the bottom alike.

Key facts

• p : q = 4 : 1 means p = 4k and q = k for some non-zero k.
• Dividing top and bottom by q turns (5p + 3q) : (5p − 3q) into (5 × p⁄q + 3) : (5 × p⁄q − 3).
• A ratio a : b and its reverse b : a are different answers unless a = b.

Common traps

• Writing the answer as (5p − 3q) : (5p + 3q), the reverse order, which gives 17 : 23.
• Taking p = 1 and q = 4 by reading the ratio from the wrong end.

No directly related past PYQ was found.