2017 AMC 10B Problems/Problem 13
Problem
There are 
students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 
students taking yoga, 
taking bridge, and 
taking painting. There are students taking at least two classes. How many students are taking all three classes?
Solution
By PIE (Property of Inclusion/Exclusion), we have
Number of people in at least two sets is 
So, 
which gives 
Solution 1 (System of Equations)
The total number of classes taken is 10 + 13 + 9 = 32 . Each student is taking at least one class so let's subtract the 20 classes ( 1 per each of the 20 students) from 32 classes to get 12 . 12 classes is the total number of extra classes taken by the students who take 2 or 3 classes.
Now let's set up our system of equations: Let x be equal to the number of students taking 2 classes and let y be equal to the number of students taking 3 classes.
x + y = 9
x + 2 y = 12
(Note: We know there are 9 total students taking either 2 or 3 classes and we already subtracted one class per each of the 20 students (the 9 students are included) from the total number of classes so it is only 1 x and 2 y.)
Solving for this system of equations we get, y = 3. Therefore the 
See Also
| 2017 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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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