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Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 7"

 
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Each card below covers up a number. The number written below each card is the sum of all the numbers covered by all of the other cards. What is the sum of all of the hidden numbers?
 
Each card below covers up a number. The number written below each card is the sum of all the numbers covered by all of the other cards. What is the sum of all of the hidden numbers?
  
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{{image}}
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<center>[[Image:Usc93.7.PNG]]</center>
  
 
<center><math> \mathrm{(A) \ }4.2 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }5.6 \qquad \mathrm{(D) \ }6.2  \qquad \mathrm{(E) \ }6.8  </math></center>
 
<center><math> \mathrm{(A) \ }4.2 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }5.6 \qquad \mathrm{(D) \ }6.2  \qquad \mathrm{(E) \ }6.8  </math></center>
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== Solution ==
 
== Solution ==
 
If we call the squares <math>a,b,c,d,e,f</math> (in order from left to right), we have:
 
If we call the squares <math>a,b,c,d,e,f</math> (in order from left to right), we have:
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<math>b+c+d+e+f=3</math>
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<math>b+c+d+e+f=3</math>,
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<math>a+c+d+e+f=8</math>
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<math>a+c+d+e+f=8</math>,
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<math>a+b+d+e+f=5</math>
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<math>a+b+d+e+f=5</math>,
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<math>a+b+c+e+f=6</math>
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<math>a+b+c+e+f=6</math>,
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<math>a+b+c+d+f=2</math>
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<math>a+b+c+d+f=2</math>,
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<math>a+b+c+d+e=4</math>
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<math>a+b+c+d+e=4</math>.
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Adding all the equations gives us <math>5(a+b+c+d+e)=28 \Longrightarrow a+b+c+d+e=5.6</math>.
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Adding all the equations gives us <math>5 (a + b + c + d + e) = 28 \Longrightarrow a + b + c + d + e = 5.6 </math>.
  
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== See also ==
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----
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* [[University of South Carolina High School Math Contest/1993 Exam]]
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* [[University of South Carolina High School Math Contest/1993 Exam/Problem 6|Previous Problem]]
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* [[University of South Carolina High School Math Contest/1993 Exam/Problem 8|Next Problem]]
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* [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]]
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[[Category:Introductory Algebra Problems]]

Latest revision as of 13:09, 12 October 2007

Problem

Each card below covers up a number. The number written below each card is the sum of all the numbers covered by all of the other cards. What is the sum of all of the hidden numbers?

Usc93.7.PNG
$\mathrm{(A) \ }4.2 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }5.6 \qquad \mathrm{(D) \ }6.2 \qquad \mathrm{(E) \ }6.8$

Solution

If we call the squares $a,b,c,d,e,f$ (in order from left to right), we have: $b+c+d+e+f=3$, $a+c+d+e+f=8$, $a+b+d+e+f=5$, $a+b+c+e+f=6$, $a+b+c+d+f=2$, $a+b+c+d+e=4$. Adding all the equations gives us $5 (a + b + c + d + e) = 28 \Longrightarrow a + b + c + d + e = 5.6$.

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