If 28.9 : x :: x : 36.1, and x > 0, then find the value of x.

Correct — D. In a : x :: x : b the middle term repeats, so x is the mean proportional and x² = ab.

x² = 28.9 × 36.1
As tenths: (289 ⁄ 10) × (361 ⁄ 10)
289 = 17² and 361 = 19²
x = (17 × 19) ⁄ 10 = 323 ⁄ 10 = 32.3 → option (d)

Squaring back confirms it: 32.3² = 1043.29, and 28.9 × 36.1 = 1043.29.

Why the others are wrong

• (a)
  38.3 — 38.3² = 1466.89, far above the required 1043.29. It also exceeds 36.1, and a mean proportional always lies between the two extremes.
• (b)
  30.4 — 30.4² = 924.16, short of 1043.29 by about 119. It sits below both 32.3 and the smaller extreme's fair share of the product.
• (c)
  35 — 35² = 1225, not 1043.29. A pair such as 25 and 49 would give 35, but 28.9 and 36.1 do not.

a : x :: x : b is a continued proportion. Cross-multiplying gives x² = ab, so x is the geometric mean of the two extremes, not their average.

The arithmetic here is meant to be done by recognition, not by long multiplication. 28.9 and 36.1 are 289 and 361 shifted one decimal place, and both are perfect squares — 17² and 19².

That makes the product a perfect square over 100, so its root ends in a single decimal place. Seeing 289 and 361 saves the whole computation.

Key facts

• In a : x :: x : b, x is the mean proportional and x² = ab.
• 289 = 17² and 361 = 19², so 28.9 × 36.1 = (32.3)².
• The mean proportional of two positive numbers always lies between them.

Common traps

• Taking the arithmetic mean instead of the geometric one. Here (28.9 + 36.1) ⁄ 2 = 32.5, which is not printed, so the slip lands on the right option by luck and stays undetected.
• Reading the statement as 28.9 : 36.1 :: x : x, which makes x meaningless.
• Attempting √1043.29 digit by digit instead of splitting 289 and 361 into squares.

No directly related past PYQ was found.