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

Correct Option (1)

: XY

Given that X and Y are natural numbers other than 1, and Y is greater than X. This implies X ≥ 2 and Y ≥ 3.

For the expression XY:

• Since X ≥ 2 and Y ≥ 3, the product XY will always be a natural number greater than or equal to 2 × 3 = 6.
• This value is inherently larger than any fraction less than 1 or any number marginally greater than 1, as demonstrated by comparing it with the other options.

Incorrect Options:

Let's analyze the other options based on the condition Y > X > 1 (implying X ≥ 2, Y ≥ 3):

Option 2: X/Y

• Since Y > X, the numerator X is smaller than the denominator Y.
• Therefore, the ratio X/Y will always be a proper fraction, meaning X/Y < 1.
• For example, if X=2, Y=3, then X/Y = 2/3 ≈ 0.67.
• Since XY ≥ 6, X/Y is clearly not the largest number.

Option 3: Y/X

• Since Y > X, the ratio Y/X will always be greater than 1.
• However, since X is a natural number greater than 1 (X ≥ 2), it implies that 1/X < 1.
• Thus, Y/X = Y × (1/X) < Y.
• Furthermore, since X ≥ 2, XY = X × Y ≥ 2Y.
• Therefore, Y/X < Y < XY.
• For example, if X=2, Y=3, then Y/X = 3/2 = 1.5. This is less than Y=3, and significantly less than XY=6.

Option 4: (X + Y) / XY

• This expression can be simplified as (X/XY) + (Y/XY) = 1/Y + 1/X.
• Since X ≥ 2, it follows that 1/X ≤ 1/2.
• Since Y ≥ 3, it follows that 1/Y ≤ 1/3.
• Therefore, 1/X + 1/Y ≤ 1/2 + 1/3 = 5/6.
• The maximum value of (X + Y) / XY is 5/6 (occurring when X=2, Y=3).
• As established, XY ≥ 6. Since (X + Y) / XY ≤ 5/6 and XY ≥ 6, it is evident that XY is the largest number among the given options.