Q. 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 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?
(a) Value-I < Value-II < Value-III
(b) Value-III < Value-II < Value-I
(c) Value-I < Value-III < Value-II
(d) Cannot be determined due to insufficient data
Correct Answer : d)
UPSC Prelims 2025 CSAT
Explanation :
1. Let X, Y, Z be the runs scored by players X, Y, Z respectively.
We know
X + Y + Z = 37
X/Y = Y/Z.
2. From X/Y = Y/Z ⇒ X·Z = Y².
A standard integer‐solution parametrization is
X = d·r²,
Y = d·r·s,
Z = d·s²,
where d, r, s are positive integers.
3. Then
X + Y + Z = d (r² + r·s + s²) = 37.
Since 37 is prime, the only way to factor is d = 1 and r² + r·s + s² = 37.
4. Solve r² + r·s + s² = 37 in positive integers (r, s):
– Try s = 3 ⇒ r² + 3r + 9 = 37 ⇒ r² + 3r − 28 = 0 ⇒ r = 4 (positive root).
– Or by symmetry r = 3, s = 4 also works (3² + 3·4 + 4² = 9 + 12 + 16 = 37).
5. Thus two possible triples (d = 1):
a) (r, s) = (4, 3) ⇒ (X, Y, Z) = (16, 12, 9) ⇒ X > Y > Z.
b) (r, s) = (3, 4) ⇒ (X, Y, Z) = ( 9, 12,16) ⇒ X < Y < Z.
6. These give different orderings of Value-I (X), Value-II (Y), Value-III (Z), so we cannot decide uniquely which is smallest, middle or largest.
Therefore the data are insufficient to fix the ordering; the correct choice is (d).
