2017 AMC 10B Problems/Problem 13

Problem

There are $20$ 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 $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution 1

By PIE (Property of Inclusion/Exclusion), we have

$|A_1 \cup A_2 \cup A_3| = \sum |A_i| - \sum |A_i \cap A_j| + |A_1 \cap A_2 \cap A_3|.$ Number of people in at least two sets is $\sum |A_i \cap A_j| - 2|A_1 \cap A_2 \cap A_3| = 9.$ So, $20 = (10 + 13 + 9) - (9 + 2x) + x,$ which gives $x = \boxed{\textbf{(C) } 3}.$

Solution 2 (Subtraction)

The total number of classes taken among the 20 students 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. Since we know that there are $9$ students taking at least $2$ classes, there must be $12 - 9 = \boxed{\textbf{(C) } 3}$ students that are taking all $3$ classes.

Solution 3 (Algebra)

Total class count is 32. Assume there are $a$ students taking one class, $b$ students taking two classes, ad $c$ students taking three classes. Because there are $20$ students total, $a+b+c = 20$ . Because each student taking two classes is counted twice, and each student taking three classes is counted thrice in the total class count, $a+2b+3c = 10+13+9 = 32$ . There are $9$ students taking two or three classes, so $b+c = 9$ . Solving this system of equations gives us $c=\boxed{\textbf{(C) 3}}$ .

Solution 4 (Venn Diagrams and Algebraic Substitution)

Let us assign the following variables and put them in our Venn Diagram: $a$ which designates the number of people taking exactly Bridge and Yoga. $b$ which designates the number of people taking exactly Bridge and Painting. $c$ which designates the number of people that took all $3$ classes or what we want to find. $d$ which designates the number of people taking exactly Yoga and Painting.

Note: The Venn Diagram is linked here: https://artofproblemsolving.com/wiki/index.php/File:IMG_20220704_192122676_2.jpg#filelinks

Let's now recall what information we have given: There are exactly $9$ people that are taking at least $2$ classes meaning in other words, $9$ people total are taking strictly $2$ classes or strictly all the available classes meaning that $a+b+c+d=9$ .

Let's now start filling out the Venn Diagram: Strictly taking Bridge, no other classes: We know in total, the number is $13$ , however this includes the people taking other classes too meaning we'd need to do some subtraction. From our Venn Diagram we see that we'd need to subtract the following variables to get our wanted outcome here, $a, b, c$ . Giving our answer as $13-(a+b+c)$ .

However, this equation seems complicated as it has $3$ different variables, so to make this look a lot less complicated we can use our earlier equation: $a+b+c+d=9$ to see that $a+b+c=9-d$ . This means that this can also be written as $13-(9-d)=4+d$ .

Strictly taking Yoga Only: The total number of people is $10$ , but this would also count people taking other classes too along with it, so we need to subtract this overcount which is visible in the Venn Diagram giving us: $10-(a+c+d)$ .

Again, we can use substitution to see that $a+c+d=9-b$ . This simplifies our equation to $10-(9-b)=1+b$ .

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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