45. 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?

1. A. Value-I < Value-II < Value-III
2. B. Value-III < Value-II < Value-I
3. C. Value-I < Value-III < Value-II
4. D. Cannot be determined due to insufficient data

Answer: D

Explanation

We are given:
1. X + Y + Z = 37
2. X/Y = Y/Z => Y² = XZ (This means X, Y, Z are in geometric progression)
We need to compare X, Y, Z. Since X, Y, Z represent runs, they must be positive integers.

Let’s try to find integer solutions that satisfy both conditions:
Consider the case where Y=12.
If Y=12, then XZ = 12² = 144.
We also need X+Y+Z = 37 => X+12+Z = 37 => X+Z = 25.
Now we need two numbers X and Z whose sum is 25 and product is 144. These are the roots of the quadratic equation t² – 25t + 144 = 0.
(t – 9)(t – 16) = 0.
So, t = 9 or t = 16.
This gives two possible sets for (X, Z):
* Set 1: X=9, Z=16. Then (X, Y, Z) = (9, 12, 16). Here, X < Y < Z.
* Set 2: X=16, Z=9. Then (X, Y, Z) = (16, 12, 9). Here, Z < Y < X.

Since we found two valid sets of (X, Y, Z) that lead to different orderings of Value-I, Value-II, and Value-III, we cannot definitively determine the relationship between them. Therefore, the answer is "Cannot be determined due to insufficient data". This question tests problem-solving with multiple variables and recognizing when a unique solution is not possible.