Mean proportional between two squares – UGC NET Question

The mean proportional m between two positive numbers a and b satisfies a:m = m:b. This implies m² = ab. For a = 25 and b = 100, we have m² = 25 × 100 = 2500, so m = √2500 = 50. Thus, 50 is the mean proportional between 25 and 100.

Option A:
Option A is correct because substituting m = 50 gives 25:50 = 1:2 and 50:100 = 1:2, so the two ratios are equal. This verifies the defining property of the mean proportional. No other option satisfies a:m = m:b for the given pair of numbers.

Option B:
Option B, 40, yields m² = 1600, which is significantly smaller than 25 × 100. The resulting ratios 25:40 and 40:100 simplify to different forms (5:8 and 2:5), so they are not equal. Hence 40 does not fulfill the condition for a mean proportional.

Option C:
Option C, 60, gives m² = 3600, which is larger than 2500 and breaks the equality of the two ratios. Using m = 60 leads to 25:60 ≠ 60:100, showing that 60 cannot be the correct geometric mean.

Option D:
Option D, 75, further exaggerates this mismatch; 25:75 simplifies to 1:3 whereas 75:100 simplifies to 3:4. These ratios do not agree, so 75 does not serve as a mean proportional between 25 and 100.