In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of number of runs scored by X to the number of runs scored by Y is equal to the ratio of number of runs scored by Y to number of runs scored by Z.
Value-I = Runs scored by X
Value-II = Runs scored by Y
Value-III = Runs scored by Z
Which one of the following is correct?

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:

1. 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.
2. 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.