2021 CSAT — Q70

Shares are proportional to $p, q, r$ (all positive). The maximum/minimum share corresponds to the largest/smallest of $p, q, r$.

Statement 1: if $p > q + r$, then since $q, r > 0$ we have $p > q$ and $p > r$, so $p$ is the largest $\Rightarrow$ $A$ gets the maximum. Correct (the condition is sufficient).

Statement 2: "$r < p + q$" does not force $r$ to be the smallest. Counterexample: $p = 1, q = 1, r = 1.5$. Here $r = 1.5 < p + q = 2$, yet $r$ is the largest, so $C$ gets the maximum, not the minimum. Incorrect.

Answer: (a) Only 1.

A part is largest iff it exceeds each other part — and “exceeds the sum of the others” ($p > q+r$) is a stronger condition that certainly does so, validating Statement 1. But “less than the sum of the others” ($r < p+q$) is weak: nearly every component satisfies it, so it can’t pin $r$ as smallest — one counterexample ($1,1,1.5$) demolishes Statement 2. The trap is symmetry-bias: students assume the second condition mirrors the first. Test the boundary with a quick counterexample before trusting a “min” claim.