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Difference between revisions of "2001 AMC 10 Problems/Problem 10"

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== Solution 1==
 
== Solution 1==
  
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Look at the first two equations in the problem.
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The first two equations in the problem are <math>xy=24 </math> and <math>xz=48</math>. Since <math>xyz \ne 0</math>, we have <math>\frac{xy}{xz}=\frac{24}{48} \implies 2y=z </math>. We can substitute <math> z = 2y </math> into the third equation <math>yz = 72</math> to obtain <math> 2y^2=72 \implies y=6 </math> and <math> 2y=z=12 </math>. We replace <math>y</math> into the first equation to obtain <math> x=4 </math>.
  
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<math>xy=24 </math> and <math>xz=48</math>.
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Since we know every variable's value, we can substitute them in to find <math> x+y+z = 4+6+12 = \boxed{\textbf{(D) }22} </math>.
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We can say that <math> 2y=z </math>.
 
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Given <math>2y=z</math>, we can substitute <math> z </math> for <math> 2y </math> and find
 
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<math> 2y^2=72 </math>
 
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<math> y^2=36 </math>
 
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<math> y=6 </math>
 
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<math> 2y=z=12 </math>.
 
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We can replace y into the first equation.
 
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<math> 6x=24 </math>
 
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<math> x=4 </math>.
 
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Since we know every variable's value, we can substitute it in for <math> x+y+z = 4+6+12 = \boxed{\textbf{(D) }22} </math>.
 
  
 
== Solution 2 ==
 
== Solution 2 ==
  
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These equations are symmetric, and furthermore, they use multiplication. This makes us think to multiply them all. This gives <math>(xy)(yz)(xz) = (xyz)^2 = (24)(48)(72) = (24 \times 12)^2</math>.
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These equations are symmetric, and furthermore, they use multiplication. This makes us think to multiply them all. This gives <math>(xyz)^2 = (xy)(yz)(xz) = (24)(48)(72) = (24 \times 12)^2 \implies xyz = 288</math>. We divide <math>xyz = 288</math> by each of the given equations, which yields <math>x = 4</math>, <math>y = 6</math>, and <math>z = 12</math>. The desired sum is <math>4+6+12 = \boxed{22}</math>, so the answer is <math>\boxed{\textbf{(D)}}</math>.
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We square root:
 
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<math>xyz = 288</math>.
 
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Aha! We divide each of the given equations into this, yielding <math>x = 4</math>, <math>y = 6</math>, and <math>z = 12</math>. The desired sum is <math>4+6+12 = \boxed{22}</math>, so the answer is <math>\boxed{\textbf{(D)}}</math>.
 
  
 
== See Also ==
 
== See Also ==
  
 
{{AMC10 box|year=2001|num-b=9|num-a=11}}
 
{{AMC10 box|year=2001|num-b=9|num-a=11}}

Revision as of 22:44, 2 June 2011

Problem

If $x$, $y$, and $z$ are positive with $xy = 24$, $xz = 48$, and $yz = 72$, then $x + y + z$ is

$\textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24$

Solution 1

The first two equations in the problem are $xy=24$ and $xz=48$. Since $xyz \ne 0$, we have $\frac{xy}{xz}=\frac{24}{48} \implies 2y=z$. We can substitute $z = 2y$ into the third equation $yz = 72$ to obtain $2y^2=72 \implies y=6$ and $2y=z=12$. We replace $y$ into the first equation to obtain $x=4$.

Since we know every variable's value, we can substitute them in to find $x+y+z = 4+6+12 = \boxed{\textbf{(D) }22}$.

Solution 2

These equations are symmetric, and furthermore, they use multiplication. This makes us think to multiply them all. This gives $(xyz)^2 = (xy)(yz)(xz) = (24)(48)(72) = (24 \times 12)^2 \implies xyz = 288$. We divide $xyz = 288$ by each of the given equations, which yields $x = 4$, $y = 6$, and $z = 12$. The desired sum is $4+6+12 = \boxed{22}$, so the answer is $\boxed{\textbf{(D)}}$.

See Also

2001 AMC 10 (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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
All AMC 10 Problems and Solutions
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