If the mean proportional between p and q is 12, then the possible values of p and q, respectively, are:

Correct — B. The mean proportional m between p and q is the middle term of p : m = m : q, so cross-multiplying gives m² = pq.

Here m = 12, so the product pq must be 12² = 144.

Test the products: 3 × 28 = 84, 3 × 38 = 114, 24 × 16 = 384.
Only 24 × 6 = 144 → option (b).

Why the others are wrong

• (a)
  3, 28 — 3 × 28 = 84, so the mean proportional of this pair is √84 ≈ 9.17, not 12.
• (c)
  3, 38 — 3 × 38 = 114, giving a mean proportional of about 10.68. The product has to land on exactly 144.
• (d)
  24, 16 — 24 × 16 = 384, whose square root is about 19.6. The first number is right and the second is not — 144 ⁄ 24 = 6.

"b is the mean proportional between a and c" means a : b = b : c. Cross-multiplying gives b² = ac, so b = √(ac) — the geometric mean of the two terms.

This question inverts it: the mean is fixed at 12 and you are asked which pair multiplies to 144. A proportion question becomes a one-line product check.

Keep it apart from the third proportional, where a : b = b : c has c unknown and c = b² ⁄ a.

Key facts

• The mean proportional between a and c is √(ac), the geometric mean.
• If the mean proportional is m, the product of the two outer terms is m².
• The third proportional to a and b is b² ⁄ a, which is a different quantity.

Common traps

• Averaging the two numbers, which gives the arithmetic mean instead
• Reading 12 as one of the two terms rather than the middle term
• Choosing 24, 16 because 24 appears in the correct pair as well

No directly related past PYQ was found.