Difference between revisions of "2017 AMC 10B Problems/Problem 13"
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==Solution 2 (Subtraction)== | ==Solution 2 (Subtraction)== | ||
− | 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. Since we know that there are 9 students taking at least 2 classes, there must be 12 - 9 = | + | The total number of classes taken is <math>10 + 13 + 9 = 32</math>. Each student is taking at least one class so let's subtract the <math>20</math> classes ( <math>1</math> per each of the <math>20</math> students) from <math>32</math> classes to get <math>12</math>. <math>12</math> classes is the total number of extra classes taken by the students who take <math>2</math> or <math>3</math> classes. Since we know that there are <math>9</math> students taking at least <math>2</math> classes, there must be <math>12 - 9 = \boxed{\textbf{(C) } 3}</math> students that are taking all <math>3</math> classes. |
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==See Also== | ==See Also== | ||
{{AMC10 box|year=2017|ab=B|num-b=12|num-a=14}} | {{AMC10 box|year=2017|ab=B|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 17:59, 17 January 2021
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 1
By PIE (Property of Inclusion/Exclusion), we have
Number of people in at least two sets is
So,
which gives
Solution 2 (Subtraction)
The total number of classes taken is . Each student is taking at least one class so let's subtract the
classes (
per each of the
students) from
classes to get
.
classes is the total number of extra classes taken by the students who take
or
classes. Since we know that there are
students taking at least
classes, there must be
students that are taking all
classes.
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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