Larger number from ratio and geometric mean – UGC NET Question

Let the two numbers be 4x and 9x. Their geometric mean is √(4x × 9x) = √(36x²) = 6x, which is given as 12. Thus, 6x = 12, so x = 2. The numbers are 8 and 18, and the larger of the two is 18. This directly uses the definition of geometric mean as the square root of the product.

Option A:
Option A, 8, is the smaller number arising from the correct solution and does not answer the question asking for the larger number. Although 8 does participate in the correct ratio and geometric mean, it is not the required value.

Option B:
Option B, 12, is the geometric mean itself, not one of the original numbers. The geometric mean lies between the two numbers in magnitude and is not equal to either when the ratio is not 1:1. Therefore, 12 cannot be the larger number.

Option C:
Option C, 16, would produce a partner number of 64/9 ≈ 7.11 if we tried to keep the ratio 4:9, which is not an integer and also would not give a geometric mean of 12. So 16 does not fit the given constraints on the ratio and mean.

Option D:
Option D is correct because with x = 2, the numbers 8 and 18 satisfy both the ratio 4:9 and the geometric mean 12. Checking, √(8 × 18) = √144 = 12, which confirms that 18 is the correct larger number.