Difference between revisions of "1985 AHSME Problems/Problem 1"

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<math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \  } 17 \qquad \mathrm{(D) \  } 18 \qquad \mathrm{(E) \  }19  </math>
 
<math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \  } 17 \qquad \mathrm{(D) \  } 18 \qquad \mathrm{(E) \  }19  </math>
  
==Solution==
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==Solution 1==
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We have <cmath>2x+1=82x=7x=72,</cmath> so <cmath>4x+1=4(72)+1=2(7)+1=(A) 15.</cmath>
  
===Solution 1===
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==Solution 2==
From <math> 2x+1=8 </math>, we subtract <math> 1 </math> from both sides to get <math> 2x=7 </math>. We can divide both sides now by <math> 2 </math> to get <math> x=\frac{7}{2} </math>. Now we can substitute this into <math> 4x+1 </math> to get <math> 4x+1=4\left(\frac{7}{2}\right)+1=(2)(7)+1=14+1=15, \boxed{\text{A}} </math>.
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From <math>2x = 7</math> (as above), we can directly compute <cmath>\begin{align*}4x &= 2(2x) \ &= 2(7) \ &= 14,\end{align*}</cmath> so <math>4x+1 = 14+1 = \boxed{\text{(A)} \ 15}</math>.
 
 
===Solution 2===
 
We proceed from <math> 2x=7 </math> as above. Notice that <math> 4x=2(2x)=2(7)=14 </math>, so <math> 4x+1=14+1=15, \boxed{\text{A}} </math>.
 
  
 
==See Also==
 
==See Also==
 
{{AHSME box|year=1985|before=First Problem|num-a=2}}
 
{{AHSME box|year=1985|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:21, 19 March 2024

Problem

If $2x+1=8$, then $4x+1=$

$\mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \  } 17 \qquad \mathrm{(D) \  } 18 \qquad \mathrm{(E) \  }19$

Solution 1

We have \begin{align*}2x+1 = 8 &\iff 2x = 7 \\ &\iff x = \frac{7}{2},\end{align*} so \begin{align*}4x+1 &= 4\left(\frac{7}{2}\right)+1 \\ &= 2(7)+1 \\ &= \boxed{\text{(A)} \ 15}.\end{align*}

Solution 2

From $2x = 7$ (as above), we can directly compute \begin{align*}4x &= 2(2x) \\ &= 2(7) \\ &= 14,\end{align*} so $4x+1 = 14+1 = \boxed{\text{(A)} \ 15}$.

See Also

1985 AHSME (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
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

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