Difference between revisions of "2016 AMC 12A Problems/Problem 11"
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<math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math> | <math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math> | ||
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+ | ==Problem== | ||
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+ | Each of the <math>100</math> 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 <math>42</math> students who cannot sing, <math>65</math> students who cannot dance, and <math>29</math> students who cannot act. How many students have two of these talents? | ||
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+ | <math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math> | ||
==Solution== | ==Solution== | ||
+ | Let <math>a</math> be the number of students that can only sing, <math>b</math> can only dance, and <math>c</math> can only act. | ||
+ | |||
+ | Let <math>ab</math> be the number of students that can sing and dance, <math>ac</math> can sing and act, and <math>bc</math> can dance and act. | ||
+ | |||
+ | From the information given in the problem, <math>a + ab + b = 29, b + bc + c = 42,</math> and <math>a + ac + c = 65</math>. | ||
+ | |||
+ | Adding these equations together, we get <math>2(a + b + c) + ab + bc + ac = 136</math>. | ||
+ | |||
+ | Since there are a total of <math>100</math> students, <math>a + b + c + ab + bc + ac = 100</math>. | ||
+ | |||
+ | Subtracting these equations, we get <math>a + b + c = 36</math>. | ||
+ | Our answer is <math>ab + bc + ac = 100 - (a + b + c) = 100 - 36 = \boxed{\textbf{(E)}\; 64}</math> | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2016|ab=A|num-b=24|num-a=26}} | {{AMC12 box|year=2016|ab=A|num-b=24|num-a=26}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 17:06, 4 February 2016
Contents
Problem
Each of the 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 students who cannot sing, students who cannot dance, and students who cannot act. How many students have two of these talents?
Problem
Each of the 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 students who cannot sing, students who cannot dance, and students who cannot act. How many students have two of these talents?
Solution
Let be the number of students that can only sing, can only dance, and can only act.
Let be the number of students that can sing and dance, can sing and act, and can dance and act.
From the information given in the problem, and .
Adding these equations together, we get .
Since there are a total of students, .
Subtracting these equations, we get .
Our answer is
See Also
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Problem 26 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.