Difference between revisions of "1960 AHSME Problems/Problem 39"

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==Problem==
 
==Problem==
  
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To satisfy the equation <math>\frac{a+b}{a}=\frac{b}{a+b}</math>, <math>a</math> and <math>b</math> must be:
  
=See Also==
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<math> \textbf{(A)}\ \text{both rational}\qquad\textbf{(B)}\ \text{both real but not rational}\qquad\textbf{(C)}\ \text{both not real}\qquad </math>
{{AHSME 40p box|year=1960 |before=[[Problem 38]]|after=[[Problem 40]]}}
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<math> \textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not real} </math>
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==Solution==
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First, note that <math>a \neq 0</math> and <math>a \neq -b</math>.  Cross multiply both sides to get
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<cmath>a^2 + 2ab + b^2 = ab</cmath>
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Subtract both sides by <math>ab</math> to get
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<cmath>a^2 + ab + b^2 = 0</cmath>
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From the [[quadratic formula]],
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<cmath>a = \frac{-b \pm \sqrt{b^2 - 4b^2}}{2}</cmath>
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<cmath>a = \frac{-b \pm \sqrt{-3b^2}}{2}</cmath>
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If <math>b</math> is [[real]], then <math>\sqrt{-3b^2}</math> is imaginary because <math>-3b^2</math> is negative.
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If <math>b</math> is not real, where <math>b = m+ni</math> and <math>n \neq 0</math>, then <math>\sqrt{-3b^2}</math> evaluates to <math>\sqrt{-3m^2 - 6mni + 3n^2}</math>.  As long as <math>m \neq 0</math>, the expression can also be imaginary because a real number squared will be a real number.
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From these two points, the answer is <math>\boxed{\textbf{(E)}}</math>.
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==See Also==
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{{AHSME 40p box|year=1960|num-b=38|num-a=40}}
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[[Category:Intermediate Algebra Problems]]

Latest revision as of 19:21, 17 May 2018

Problem

To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be:

$\textbf{(A)}\ \text{both rational}\qquad\textbf{(B)}\ \text{both real but not rational}\qquad\textbf{(C)}\ \text{both not real}\qquad$ $\textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not real}$

Solution

First, note that $a \neq 0$ and $a \neq -b$. Cross multiply both sides to get \[a^2 + 2ab + b^2 = ab\] Subtract both sides by $ab$ to get \[a^2 + ab + b^2 = 0\] From the quadratic formula, \[a = \frac{-b \pm \sqrt{b^2 - 4b^2}}{2}\] \[a = \frac{-b \pm \sqrt{-3b^2}}{2}\] If $b$ is real, then $\sqrt{-3b^2}$ is imaginary because $-3b^2$ is negative. If $b$ is not real, where $b = m+ni$ and $n \neq 0$, then $\sqrt{-3b^2}$ evaluates to $\sqrt{-3m^2 - 6mni + 3n^2}$. As long as $m \neq 0$, the expression can also be imaginary because a real number squared will be a real number. From these two points, the answer is $\boxed{\textbf{(E)}}$.

See Also

1960 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 38
Followed by
Problem 40
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