The mean proportional of 2 and 32 is _____.

Correct — B. The mean proportional of two numbers is the middle term x in the proportion 2 : x :: x : 32, so cross-multiplying gives x² = 2 × 32.

x² = 2 × 32 = 64
x = √64 = 8

Proportion questions take the positive root, so the mean proportional is 8 → option (b).

Why the others are wrong

• (a)
  16 — 16² = 256, not 64. Sixteen is the mean proportional of 8 and 32, since 8 × 32 = 256 — a pair the question never gives you.
• (c)
  12 — 12² = 144, well past the 64 the pair produces. A mean proportional must square back to the product of the two numbers, and nothing about 2 and 32 yields 12.
• (d)
  6 — 6² = 36, not 64. Sitting somewhere between 2 and 32 is not the test — the test is x² = 2 × 32.

A proportion a : x :: x : b repeats the same term in both middle places. Cross-multiplying gives x² = ab, so x = √(ab). That x is the mean proportional, known in other chapters as the geometric mean.

It is not the arithmetic mean (a + b)⁄2, which for 2 and 32 would be 17. The geometric mean never exceeds the arithmetic mean, and the two are equal only when the numbers are, so 8 < 17 is that inequality at work.

The neighbouring term to know is the third proportional: in a : b :: b : c the last term is c = b²⁄a.

Key facts

• The mean proportional (geometric mean) of a and b is √(ab), the repeated middle term of a : x :: x : b.
• √(2 × 32) = √64 = 8.
• The third proportional to a and b is b²⁄a, a different quantity from the mean proportional.
• For 2 and 32 the arithmetic mean is 17 and the geometric mean is 8, matching the rule that AM is never below GM.

Common traps

• Answering with the arithmetic mean 17, which is not among the four options and so signals the wrong formula.
• Taking the negative root: x = −8 also satisfies x² = 64, but a mean proportional is taken positive.
• Confusing mean proportional with third proportional, which for 2 and 32 would be 512.

No directly related past PYQ was found.