Difference between revisions of "1951 AHSME Problems/Problem 43"

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==Problem==
 
==Problem==
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Of the following statements, the only one that is incorrect is:
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<math> \textbf{(A)}\ \text{An inequality will remain true after each side is increased,} </math> <math>\text{ decreased, multiplied or divided zero excluded by the same positive quantity.} </math>
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<math> \textbf{(B)}\ \text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.} </math>
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<math> \textbf{(C)}\ \text{If the sum of two positive quantities is given, ther product is largest when they are equal.} </math>
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<math> \textbf{(D)}\ \text{If }a\text{ and }b\text{ are positive and unequal, }\frac{1}{2}(a^{2}+b^{2})\text{ is greater than }[\frac{1}{2}(a+b)]^{2}. </math>
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<math> \textbf{(E)}\ \text{If the product of two positive quantities is given, their sum is greatest when they are equal.} </math>
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==Solution==
 
==Solution==
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The answer is <math>\boxed{\textbf{(E)}}</math>.
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Quite the opposite of statement (E) is true--the sum <math>a+b</math> is <i>minimized</i> when <math>a=b</math>, but it approaches <math>\infty</math> when one of <math>a,b</math> gets arbitrarily small.
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== See Also ==
 
== See Also ==
 
{{AHSME 50p box|year=1951|num-b=42|num-a=44}}  
 
{{AHSME 50p box|year=1951|num-b=42|num-a=44}}  

Latest revision as of 11:01, 19 April 2014

Problem

Of the following statements, the only one that is incorrect is:

$\textbf{(A)}\ \text{An inequality will remain true after each side is increased,}$ $\text{ decreased, multiplied or divided zero excluded by the same positive quantity.}$

$\textbf{(B)}\ \text{The arithmetic mean of two unequal positive quantities is greater than their geometric mean.}$

$\textbf{(C)}\ \text{If the sum of two positive quantities is given, ther product is largest when they are equal.}$

$\textbf{(D)}\ \text{If }a\text{ and }b\text{ are positive and unequal, }\frac{1}{2}(a^{2}+b^{2})\text{ is greater than }[\frac{1}{2}(a+b)]^{2}.$

$\textbf{(E)}\ \text{If the product of two positive quantities is given, their sum is greatest when they are equal.}$

Solution

The answer is $\boxed{\textbf{(E)}}$. Quite the opposite of statement (E) is true--the sum $a+b$ is minimized when $a=b$, but it approaches $\infty$ when one of $a,b$ gets arbitrarily small.

See Also

1951 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 42
Followed by
Problem 44
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All AHSME Problems and Solutions

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