1958 AHSME Problems/Problem 44

Problem

Given the true statements: (1) If $a$ is greater than $b$, then $c$ is greater than $d$ (2) If $c$ is less than $d$, then $e$ is greater than $f$. A valid conclusion is:

$\textbf{(A)}\ \text{If }{a}\text{ is less than }{b}\text{, then }{e}\text{ is greater than }{f}\qquad \\ \textbf{(B)}\ \text{If }{e}\text{ is greater than }{f}\text{, then }{a}\text{ is less than }{b}\qquad \\ \textbf{(C)}\ \text{If }{e}\text{ is less than }{f}\text{, then }{a}\text{ is greater than }{b}\qquad \\ \textbf{(D)}\ \text{If }{a}\text{ is greater than }{b}\text{, then }{e}\text{ is less than }{f}\qquad \\ \textbf{(E)}\ \text{none of these}$

Solution

(A) is not valid because $a<b$ does not imply $c\le d$ (The inverse of Statement 1 is not necessarily true).

(B) is not valid because $e>f$ does not imply $c<d$ (The converse of Statement 2 is not necessarily true either).

In (C), $e<f$ does imply $c\ge d$ (contrapositive of Statement 2) but this does not tell us anything about $a$ and $b$.

In (D), $a>b$ does imply $c>d$, but the inverse of Statement 2 ($e\le f$) is not necessarily true.

So the answer is $\boxed{\textbf{(E) } \text{None of these}}$. (Incidentally, the wording of this problem appears to be wrongly assuming that $x\not<y$ implies $x>y$ (and vice versa) when in reality $x$ and $y$ could be equal. However, this does not change the fact that none of the four choices follow logically from the given statements.)

See Also

1958 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 43
Followed by
Problem 45
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
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS