X and Y are natural numbers other than 1, and Y is greater than X. Which of the following represents the largest number?

(a) XY ✓ UPSC's answer
(b) X/Y
(c) Y/X
(d) (X + Y)/XY

Why the answer is (a)

• Since X and Y are natural numbers greater than 1, the smallest possible values are X = 2 and Y = 3.

• For option (a) XY, the product is at least 2 * 3 = 6, which is always greater than 1.

• For option (b) X/Y, since Y > X, the fraction is strictly less than 1.

• For option (c) Y/X, the value can be close to 1 (e.g., 3/2 = 1.5) or larger, but it is always less than Y.

• For option (d) (X + Y)/XY, this simplifies to 1/X + 1/Y, which is always less than 1 for X, Y > 1.

• Comparing the magnitudes, XY grows multiplicatively while Y/X grows additively relative to X; for any natural numbers X, Y > 1 with Y > X, XY is always greater than Y/X (since XY > Y/X implies X^2Y > 1, which is true).

• Therefore, XY represents the largest number.

Why the other options are wrong

(b) X/Y: X/Y is less than 1 because the numerator X is smaller than the denominator Y.
(c) Y/X: Y/X is smaller than XY because dividing Y by X yields a value less than Y, whereas multiplying X by Y yields a value greater than Y.
(d) (X + Y)/XY: (X + Y)/XY simplifies to 1/X + 1/Y, which is always less than 1 for natural numbers greater than 1.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2018, held on 3 June 2018. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.