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Difference between revisions of "2016 AMC 12A Problems/Problem 11"

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An easier way to solve the problem:
 
An easier way to solve the problem:
 
Since <math>42</math> students cannot sing, there are <math>100-42=58</math> students who can.
 
Since <math>42</math> students cannot sing, there are <math>100-42=58</math> students who can.
 +
 
Similarly <math>65</math> students cannot sing, there are <math>100-65=35</math> students who can.
 
Similarly <math>65</math> students cannot sing, there are <math>100-65=35</math> students who can.
 +
 
And <math>29</math> students cannot sing, there are <math>100-29=71</math> students who can.
 
And <math>29</math> students cannot sing, there are <math>100-29=71</math> students who can.
  
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There are no students who have all <math>3</math> talents, nor those who have none <math>(0)</math>, so only <math>1</math> or <math>2</math> talents are viable.
 
There are no students who have all <math>3</math> talents, nor those who have none <math>(0)</math>, so only <math>1</math> or <math>2</math> talents are viable.
  
−
Thus, there are <math>164-100=\boxed{\text{bf(E) }64}</math> students who have <math>2</math> of <math>3</math> talents.
+
Thus, there are <math>164-100=\boxed{\textbf{(E) }64}</math> students who have <math>2</math> of <math>3</math> talents.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2016|ab=A|num-b=10|num-a=12}}
 
{{AMC12 box|year=2016|ab=A|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:16, 5 February 2016

Problem

Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents?

$\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64$

Solution

Let $a$ be the number of students that can only sing, $b$ can only dance, and $c$ can only act.

Let $ab$ be the number of students that can sing and dance, $ac$ can sing and act, and $bc$ can dance and act.

From the information given in the problem, $a + ab + b = 29, b + bc + c = 42,$ and $a + ac + c = 65$.

Adding these equations together, we get $2(a + b + c) + ab + bc + ac = 136$.

Since there are a total of $100$ students, $a + b + c + ab + bc + ac = 100$.

Subtracting these equations, we get $a + b + c = 36$.

Our answer is $ab + bc + ac = 100 - (a + b + c) = 100 - 36 = \boxed{\textbf{(E)}\; 64}$

Solution 2

An easier way to solve the problem: Since $42$ students cannot sing, there are $100-42=58$ students who can.

Similarly $65$ students cannot sing, there are $100-65=35$ students who can.

And $29$ students cannot sing, there are $100-29=71$ students who can.

Therefore, there are $58+35+71=164$ students in all ignoring the overlaps between $2$ of $3$ talent categories. There are no students who have all $3$ talents, nor those who have none $(0)$, so only $1$ or $2$ talents are viable.

Thus, there are $164-100=\boxed{\textbf{(E) }64}$ students who have $2$ of $3$ talents.

See Also

2016 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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