What is the ratio of the mean proportional of 1.21 and 0.09 with the mean proportional of 0.16 and 0.36?

Correct — B. The mean proportional of a and b is √(ab).

First pair: √(1.21 × 0.09) = √1.21 × √0.09 = 1.1 × 0.3 = 0.33
Second pair: √(0.16 × 0.36) = √0.16 × √0.36 = 0.4 × 0.6 = 0.24

Ratio = 0.33 : 0.24
Divide both by 0.03: = 11 : 8 → option (b).

Why the others are wrong

• (a)
  13 : 8 — √1.21 is 1.1, not 1.3. A 13 in the first term needs 1.69 under the root, so this option gets the second half right and the first square root wrong.
• (c)
  11 : 13 — The 11 is right and the 8 has been replaced by 13. The second mean proportional is √(0.16 × 0.36) = 0.24, and 0.33 : 0.24 reduces to 11 : 8, so no 13 can appear.
• (d)
  13 : 12 — Neither number survives the arithmetic. 0.33 : 0.24 divides through by 0.03 to give exactly 11 : 8, and 13 : 12 is not that ratio in any reduced form.

b is the mean proportional between a and c when a : b = b : c. Cross-multiplying gives b² = ac, so b = √(ac) — the geometric mean, not the average.

The decimals here are chosen as exact squares: 1.21 = 1.1², 0.09 = 0.3², 0.16 = 0.4² and 0.36 = 0.6². No long division or root extraction is needed.

Ratios of decimals are cleared by dividing both sides by their common factor. 0.33 and 0.24 share 0.03, which takes them straight to 11 : 8.

Key facts

• The mean proportional (geometric mean) of a and b is √(ab).
• 1.21 = 1.1², 0.09 = 0.3², 0.16 = 0.4² and 0.36 = 0.6².
• √(1.21 × 0.09) = 0.33 and √(0.16 × 0.36) = 0.24.
• 0.33 : 0.24 reduces to 11 : 8.

Common traps

• Taking the arithmetic mean, (1.21 + 0.09)⁄2, instead of the geometric mean.
• Losing a decimal place: √0.09 is 0.3, not 0.03.
• Leaving the answer as 33 : 24 and looking for it among the options.

No directly related past PYQ was found.