Difference between revisions of "1950 AHSME Problems/Problem 4"

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We can use difference of two squares to expand <math>a^2-b^2=(a-b)(a+b)</math> and factor to get <math>ab-b^{2}=b(a-b)</math> and <math>a^2-ab=a(a-b)</math>
 
We can use difference of two squares to expand <math>a^2-b^2=(a-b)(a+b)</math> and factor to get <math>ab-b^{2}=b(a-b)</math> and <math>a^2-ab=a(a-b)</math>
 
  <math> \dfrac{a^{2}-b^{2}}{ab}  -  \dfrac{ab-b^{2}}{ab-a^{2}} =\dfrac{(a-b)(a+b)}{ab}-\dfrac{b(a-b)}{a(a-b)}</math>
 
  <math> \dfrac{a^{2}-b^{2}}{ab}  -  \dfrac{ab-b^{2}}{ab-a^{2}} =\dfrac{(a-b)(a+b)}{ab}-\dfrac{b(a-b)}{a(a-b)}</math>
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We can further factor to get <math>\dfrac{(a-b)(a+b)}{ab}-\dfrac{b(a-b)}{a(a-b)}</math> If we assume b is not equal to a, this is equal to <math>\dfrac{(a-b)(a+b)}{ab}-\dfrac{b}{a}=\dfrac{(a-b)(a+b)}{ab}-\dfrac{b^2}{ab}=\dfrac{(a-b)(a+b)-b^2}{ab}=\dfrac{a^2-2b^2}{ab}</math>
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The answer is
  
 
==See Also==
 
==See Also==
  
 
{{AHSME box|year=1950|num-b=3|num-a=5}}
 
{{AHSME box|year=1950|num-b=3|num-a=5}}

Revision as of 13:47, 28 October 2011

Problem

Reduced to lowest terms, $\frac{a^{2}-b^{2}}{ab}$ - $\frac{ab-b^{2}}{ab-a^{2}}$ is equal to:

$\textbf{(A)}\ \frac{a}{b}\qquad\textbf{(B)}\ \frac{a^{2}-2b^{2}}{ab}\qquad\textbf{(C)}\ a^{2}\qquad\textbf{(D)}\ a-2b\qquad\textbf{(E)}\ \text{None of these}$

Solution

We can use difference of two squares to expand $a^2-b^2=(a-b)(a+b)$ and factor to get $ab-b^{2}=b(a-b)$ and $a^2-ab=a(a-b)$

$\dfrac{a^{2}-b^{2}}{ab}  -  \dfrac{ab-b^{2}}{ab-a^{2}} =\dfrac{(a-b)(a+b)}{ab}-\dfrac{b(a-b)}{a(a-b)}$

We can further factor to get $\dfrac{(a-b)(a+b)}{ab}-\dfrac{b(a-b)}{a(a-b)}$ If we assume b is not equal to a, this is equal to $\dfrac{(a-b)(a+b)}{ab}-\dfrac{b}{a}=\dfrac{(a-b)(a+b)}{ab}-\dfrac{b^2}{ab}=\dfrac{(a-b)(a+b)-b^2}{ab}=\dfrac{a^2-2b^2}{ab}$

The answer is

See Also

1950 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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