Find the mean proportional of 36 and 100.

Correct — D. The mean proportional of a and b is the x that makes a : x = x : b, so x² = ab and x = √(ab) — the geometric mean, not the ordinary average.

x = √(36 × 100)
= √3600
= 60 → option (d).

Faster still, split the root: √(36 × 100) = √36 × √100 = 6 × 10 = 60.

Check by squaring back: 60² = 3600 = 36 × 100.

Why the others are wrong

• (a)
  55 — Square it back and the check fails at once: 55² = 3,025, well short of the 3,600 that 36 × 100 requires.
• (b)
  70 — 70² = 4,900, far past 3,600. It is not the arithmetic mean either — that would be (36 + 100)⁄2 = 68 — so no averaging route lands here.
• (c)
  65 — 65² = 4,225, not 3,600. Every mean proportional must satisfy x² = ab, and squaring is the one-line test that eliminates this option.

"Mean proportional" is the old name for the geometric mean. Between a and b it is the single term that makes a continued proportion a : x :: x : b.

Cross-multiplying that proportion gives x² = ab, so x = √(ab), taking the positive root by convention.

It is not the arithmetic mean. For 36 and 100 the geometric mean is 60 while the arithmetic mean is 68, and the geometric mean is the smaller of the two whenever the numbers differ.

Key facts

• Mean proportional of a and b = √(ab).
• √(36 × 100) = √36 × √100 = 6 × 10 = 60.
• The geometric mean of two distinct positive numbers is always less than their arithmetic mean.
• Third proportional is different: for a and b it is b²⁄a, not √(ab).

Common traps

• Averaging the two numbers, which gives 68 rather than 60
• Halving the product instead of taking its square root
• Confusing the mean proportional with the third proportional, b²⁄a

No directly related past PYQ was found.