Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monotonically increases till the age of 50 years and then it monotonically decreases. Let a_1 and a_2 (in years) denote the ages of the brothers and a_1 < a_2. Which one of the following statements is correct about the age when they attain the same weight? A. a_1 <= 50 <= a_2 B. a_1 < 50 < a_2 C. 30 < a_1 < a_2 D. Either a_1 = 50 or a_2 = 50

GATE 2025 · General Aptitude · Data Interpretation · medium

Answer: The correct condition is a_1 < 50 < a_2, which is option B.

1. Case: both brothers below 50: If both a_1 < a_2 are on the increasing branch (both < 50), then w(a_1) < w(a_2), so they cannot have the same weight.
2. Case: both brothers above 50: If both a_1 < a_2 are on the decreasing branch (both > 50), then w(a_1) > w(a_2), so they cannot have the same weight.
3. Only valid case: a_1 < 50 < a_2: When a_1 < 50 and a_2 > 50, the function takes value w* on the way up (at a_1) and on the way down (at a_2). This is the only configuration consistent with equal weights and a_1 < a_2.