30 is the mean proportional of 18 and A. Find the value of A.

Correct — C. The mean proportional m of a and b is the middle term of a : m = m : b, so m² = a × b.

Here m = 30 and a = 18:
30² = 18 × A
900 = 18A
A = 900 ⁄ 18 = 50 → option (c).

Check the proportion: 18 : 30 = 3 : 5 and 30 : 50 = 3 : 5, so the two ratios match.

Why the others are wrong

• (a)
  60 — 18 × 60 = 1080, and √1080 ≈ 32.9, not 30. The product must land on 900 exactly.
• (b)
  40 — 18 × 40 = 720, whose square root is about 26.8 — short of the required 30.
• (d)
  45 — 18 × 45 = 810 and √810 ≈ 28.5. It is the closest of the wrong options, but still not 900.

'Mean proportional' names the middle term of a continued proportion a : m = m : b. Cross-multiplying gives m² = ab, so m = √(ab) — the geometric mean, not the average.

When the mean term is known and one end is missing, rearrange to b = m²⁄a. That also makes A here the third proportional to 18 and 30, since the third proportional to a and b is b²⁄a.

Key facts

• The mean proportional of a and b is √(ab), the geometric mean.
• Given the mean proportional m and one term a, the other term is m²⁄a.
• The third proportional to a and b is b²⁄a, so 50 is the third proportional to 18 and 30.
• Geometric and arithmetic means differ: √(18 × 50) = 30 while (18 + 50)⁄2 = 34.

Common traps

• Averaging instead of multiplying, which gives 42 rather than 50.
• Treating 30 as an extreme rather than the middle term, which would give √540 ≈ 23.2.
• Dividing 900 by 30 instead of by 18 and answering 30.

No directly related past PYQ was found.