Correct Option (D)
The problem provides two fundamental conditions regarding the runs scored by players X, Y, and Z:
- The total runs scored by the three players is 37: X + Y + Z = 37
- The ratio of runs scored by X to Y is equal to the ratio of runs scored by Y to Z: X/Y = Y/Z
From the ratio condition, cross-multiplication yields Y² = XZ.
We are seeking positive integer values for X, Y, and Z that satisfy both equations. Let's analyze the possibilities:
From X + Y + Z = 37, we can write X + Z = 37 - Y.
We need to find integer values for Y. Let's test integer values for Y. If Y = 12, then:
- X + Z = 37 - 12 = 25
- XZ = Y² = 12² = 144
We now need to find two positive integers X and Z whose sum is 25 and whose product is 144. These integers are 9 and 16.
This leads to two distinct valid scenarios for the runs scored:
- If X = 9 and Z = 16:
- Value-I (X) = 9
- Value-II (Y) = 12
- Value-III (Z) = 16
- In this case, the order is 9 < 12 < 16, which implies Value-I < Value-II < Value-III.
- If X = 16 and Z = 9:
- Value-I (X) = 16
- Value-II (Y) = 12
- Value-III (Z) = 9
- In this case, the order is 16 > 12 > 9, which implies Value-I > Value-II > Value-III (or Value-III < Value-II < Value-I).
Since both sets of values (X=9, Y=12, Z=16) and (X=16, Y=12, Z=9) are valid solutions that satisfy all the given conditions, and they result in different relative orderings of Value-I, Value-II, and Value-III, a unique relationship cannot be determined from the provided data.
Incorrect Options:
Options 1, 2, and 3 propose a definitive, singular order for Value-I, Value-II, and Value-III. However, as demonstrated, the given information permits at least two distinct sets of integer solutions that satisfy all conditions but yield conflicting orderings. Specifically:
- The solution (X=9, Y=12, Z=16) aligns with the ordering Value-I < Value-II < Value-III (Option 1).
- The solution (X=16, Y=12, Z=9) aligns with the ordering Value-III < Value-II < Value-I (Option 2).
Because the data does not lead to a unique and unambiguous ordering, any option asserting a specific fixed relationship is incorrect. Therefore, the relationship cannot be determined.
