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Difference between revisions of "2010 AMC 8 Problems/Problem 20"

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==Problem 20==
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==Problem==
 
In a room, <math>2/5</math> of the people are wearing gloves, and <math>3/4</math> of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?  
 
In a room, <math>2/5</math> of the people are wearing gloves, and <math>3/4</math> of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?  
  
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==Solution==
 
==Solution==
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Let <math>x</math> be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of <math>LCM(4,5)=20</math>. Since we are trying to find the minimum <math>x</math>, we must use the smallest possible value for the number of people in the room. Similarly, as we are trying to minimize <math>x</math>, we can assume that everyone in the room must be wearing either a hat or a glove or both. That is, there are no people present who wearing neither of the two items. Thus, we can say that there are <math>20</math> people in the room, all of which are wearing at least a hat or a glove.
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Let <math>x</math> be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of 4 and 5. Since we are trying to find the minimum <math>x</math>, we must use the ''least'' common multiple. <math>lcm(4,5) = 20</math>. Thus, we can say that there are <math>20</math> people in the room, all of which are wearing at least a hat or a glove. (Any people wearing neither item would unnecessarily increase the number of people in the room.)
  
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It follows that there are <math>\frac{2}{5}\cdot 20 = 8</math> people wearing gloves and <math>\frac{3}{4}\cdot 20 = 15</math> people wearing hats. Then by applying the Principle of Inclusion Exclusion (PIE), the total number of people in the room wearing either a hat or a glove or both is  <math>8+15-x = 23-x</math>. Since we know that this equals <math>20</math>, it follows that <math>23-x = 20</math>, which implies that <math>x=3</math>. Thus, <math>\boxed{\textbf{(A)}\ 3}</math> is the correct answer.
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It follows that there are <math>\frac{2}{5}\cdot 20 = 8</math> people wearing gloves and <math>\frac{3}{4}\cdot 20 = 15</math> people wearing hats. Then by applying the Principle of Inclusion and Exclusion (PIE), the total number of people in the room wearing either a hat or a glove or both is  <math>8+15-x = 23-x</math>, where <math>x</math> is the number wearing both. Since everyone in the room is wearing at least one item (see above), <math>23-x = 20</math>, and so <math>x=\boxed{\textbf{(A)}\ 3}</math>.
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==Video by MathTalks==
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https://www.youtube.com/watch?v=KSYVsSJDX-0&feature=youtu.be
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==Video Solution by WhyMath==
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https://youtu.be/Ym28CYMKIW8
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~savannahsolver
  
 
==See Also==
 
==See Also==
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{{AMC8 box|year=2011|num-b=19|num-a=21}}
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{{AMC8 box|year=2010|num-b=19|num-a=21}}
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{{MAA Notice}}

Latest revision as of 20:47, 23 October 2024

Problem

In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?

$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$

Solution

Let $x$ be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of 4 and 5. Since we are trying to find the minimum $x$, we must use the least common multiple. $lcm(4,5) = 20$. Thus, we can say that there are $20$ people in the room, all of which are wearing at least a hat or a glove. (Any people wearing neither item would unnecessarily increase the number of people in the room.)

It follows that there are $\frac{2}{5}\cdot 20 = 8$ people wearing gloves and $\frac{3}{4}\cdot 20 = 15$ people wearing hats. Then by applying the Principle of Inclusion and Exclusion (PIE), the total number of people in the room wearing either a hat or a glove or both is $8+15-x = 23-x$, where $x$ is the number wearing both. Since everyone in the room is wearing at least one item (see above), $23-x = 20$, and so $x=\boxed{\textbf{(A)}\ 3}$.

Video by MathTalks

https://www.youtube.com/watch?v=KSYVsSJDX-0&feature=youtu.be

Video Solution by WhyMath

https://youtu.be/Ym28CYMKIW8

~savannahsolver

See Also

2010 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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 AJHSME/AMC 8 Problems and Solutions

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