Find the mean proportional of 12.96 and 0.16.

Correct — A. The mean proportional x sits in the middle of 12.96 : x :: x : 0.16, so cross-multiplying gives x² = the product.

x² = 12.96 × 0.16 = 2.0736
x = √2.0736

Root the two numbers separately rather than multiplying first:
√12.96 = 3.6 and √0.16 = 0.4
x = 3.6 × 0.4 = 1.44 → option (a).

Why the others are wrong

• (b)
  1.69 — 1.69 is 1.3², a familiar square with nothing to do with this pair. As an answer it squares to 2.8561, not to the required 2.0736.
• (c)
  1.96 — 1.96 squares to 3.8416, far above the product 2.0736 it has to match. Squaring the option is the fastest check available here.
• (d)
  1.21 — 1.21 squares to 1.4641, short of 2.0736. Averaging or halving the given numbers never yields a mean proportional — only the square root of their product does.

The mean proportional between a and b is the middle term of the continued proportion a : x :: x : b.

Cross-multiplying gives x² = ab, so x = √(ab) — the geometric mean, not the average. The average of 12.96 and 0.16 is 6.56, which is not on offer.

With decimals, root each number separately, because √(ab) = √a × √b. Both numbers here are exact squares, 3.6² and 0.4², so the arithmetic never runs past two decimal places.

Key facts

• The mean proportional between a and b is √(ab), the middle term of a : x :: x : b.
• √(ab) = √a × √b, so decimals are easier rooted separately than multiplied first.
• 12.96 = 3.6² and 0.16 = 0.4², so 12.96 × 0.16 = 2.0736 = 1.44².
• The mean proportional is the geometric mean, never the arithmetic mean (a + b)⁄2.

Common traps

• Averaging the two numbers instead of rooting their product
• Misplacing the decimal in √0.16, which is 0.4 and not 0.04
• Multiplying 12.96 × 0.16 by hand and losing a decimal place before rooting

No directly related past PYQ was found.