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Difference between revisions of "2016 AMC 10B Problems/Problem 1"

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==Problem==
 
==Problem==
  
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What is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \frac{1}{2}</math>?
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What is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \tfrac{1}{2}</math>?
  
 
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20</math>
 
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20</math>
  
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==Solution==
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Factorizing the numerator, <math>\frac{\frac{1}{a}\cdot(2+\frac{1}{2})}{a}</math> then becomes <math>\frac{\frac{5}{2}}{a^{2}}</math> which is equal to <math>\frac{5}{2}\cdot 2^2</math> which is <math>\boxed{\textbf{(D) }10}</math>.
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==Solution 2==
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Substituting <math>\frac{1}{2}</math> for <math>a</math> in <math>\frac{\frac{1}{a}\cdot(2+\frac{1}{2})}{a}</math> gives us <math>\boxed{\textbf{(D) }10}</math>. ~peelybonehead
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==Video Solution (CREATIVE THINKING)==
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https://youtu.be/2erUXM5pD2g
  
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~Education, the Study of Everything
  
  
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==Solution==
 
  
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<math>\frac{frac{1}{a}*(2+frac{1}{2})}{a}</math> then becomes <math>frac{frac{5}{2}}{a^{2}}</math> which is equal to <math>frac{5}{2}*2^{2}</math> = <math>{(D)}\ 10\qquad\textbf{(E)}\ 20</math>
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==Video Solution==
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https://youtu.be/1IZ3oj1iGf0
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~savannahsolver
  
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Solution by ngeorge
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==See Also==
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{{AMC10 box|year=2016|ab=B|before=-|num-a=2}}
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{{MAA Notice}}

Latest revision as of 22:13, 21 November 2023

Problem

What is the value of $\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}$ when $a= \tfrac{1}{2}$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$

Solution

Factorizing the numerator, $\frac{\frac{1}{a}\cdot(2+\frac{1}{2})}{a}$ then becomes $\frac{\frac{5}{2}}{a^{2}}$ which is equal to $\frac{5}{2}\cdot 2^2$ which is $\boxed{\textbf{(D) }10}$.

Solution 2

Substituting $\frac{1}{2}$ for $a$ in $\frac{\frac{1}{a}\cdot(2+\frac{1}{2})}{a}$ gives us $\boxed{\textbf{(D) }10}$. ~peelybonehead

Video Solution (CREATIVE THINKING)

https://youtu.be/2erUXM5pD2g

~Education, the Study of Everything


Video Solution

https://youtu.be/1IZ3oj1iGf0

~savannahsolver

See Also

2016 AMC 10B (Problems • Answer Key • Resources)
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Problem 2
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All AMC 10 Problems and Solutions

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