# 1970 AHSME Problems/Problem 17

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## Problem

If $r>0$, then for all $p$ and $q$ such that $pq\ne 0$ and $pr>qr$, we have $\text{(A) } -p>-q\quad \text{(B) } -p>q\quad \text{(C) } 1>-q/p\quad \text{(D) } 1

## Solution

If $r>0$ and $pr>qr$, we can divide by the positive number $r$ and not change the inequality direction to get $p>q$. Multiplying by $-1$ (and flipping the inequality sign because we're multiplying by a negative number) leads to $-p < -q$, which directly contradicts $A$. Thus, $A$ is always false.

If $p < 0$ (which is possible but not guaranteed), we can divide the true statement $p>q$ by $p$ to get $1 > \frac{q}{p}$. This contradicts $D$. Thus, $D$ is sometimes false, which is bad enough to be eliminated.

If $(p, q, r) = (3, 2, 1)$, then the condition that $pr > qr$ is satisfied. However, $-p = -3$ and $q = 2$, so $-p > q$ is false for at least this case, eliminating $B$.

If $(p, q, r) = (2, -3, 1)$, then $pr > qr$ is also satisfied. However, $-\frac{q}{p} = 1.5$, so $1 > -\frac{q}{p}$ is false, eliminating $C$.

All four options do not follow from the premises, leading to $\fbox{E}$ as the correct answer.

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