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Difference between revisions of "1970 AHSME Problems/Problem 17"

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== Solution ==
 
== Solution ==
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If <math>r>0</math> and <math>pr>qr</math>, we can divide by the positive number <math>r</math> and not change the inequality direction to get <math>p>q</math>.  Multiplying by <math>-1</math> (and flipping the inequality sign because we're multiplying by a negative number) leads to <math>-p < -q</math>, which directly contradicts <math>A</math>.
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If <math>r>0</math> and <math>pr>qr</math>, we can divide by the positive number <math>r</math> and not change the inequality direction to get <math>p>q</math>.  Multiplying by <math>-1</math> (and flipping the inequality sign because we're multiplying by a negative number) leads to <math>-p < -q</math>, which directly contradicts <math>A</math>.  Thus, <math>A</math> is always false.
  
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If <math>p < 0</math> (which is possible but not guaranteed), we can divide by <math>p</math> to get <math>1 > \frac{q}{p}</math>.  This contradicts <math>D</math>.
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If <math>p < 0</math> (which is possible but not guaranteed), we can divide the true statement <math>p>q</math> by <math>p</math> to get <math>1 > \frac{q}{p}</math>.  This contradicts <math>D</math>.  Thus, <math>D</math> is sometimes false, which is bad enough to be eliminated.
  
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If <math>(p, q, r) = (3, 2, 1)</math>, then the condition that <math>pr > qr</math> is satisfied.  However, <math>-p = -3</math> and <math>q = 2</math>, so <math>-p > q</math> is false, eliminating <math>B</math>.
+
If <math>(p, q, r) = (3, 2, 1)</math>, then the condition that <math>pr > qr</math> is satisfied.  However, <math>-p = -3</math> and <math>q = 2</math>, so <math>-p > q</math> is false for at least this case, eliminating <math>B</math>.
  
 
If <math>(p, q, r) = (2, -3, 1)</math>, then <math>pr > qr</math> is also satisfied.  However, <math>-\frac{q}{p} = 1.5</math>, so <math>1 > -\frac{q}{p}</math> is false, eliminating <math>C</math>.
 
If <math>(p, q, r) = (2, -3, 1)</math>, then <math>pr > qr</math> is also satisfied.  However, <math>-\frac{q}{p} = 1.5</math>, so <math>1 > -\frac{q}{p}</math> is false, eliminating <math>C</math>.

Latest revision as of 01:37, 17 December 2021

Problem

If $r>0$, then for all $p$ and $q$ such that $pq\ne 0$ and $pr>qr$, we have

$\text{(A) } -p>-q\quad \text{(B) } -p>q\quad \text{(C) } 1>-q/p\quad \text{(D) } 1<q/p\quad \text{(E) None of these}$

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.

See also

1970 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

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