Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2008 AMC 8 Problems/Problem 11"
(→‎Solution 2 (Venn Diagram))
Line 15: Line 15:
 
<asy>
 
<asy>
 
draw(circle((0,0),5));
 
draw(circle((0,0),5));
−
draw(circle((4,0),5));
+
draw(circle((5,0),5));
 +
label("$x$",(2.3,1.5),S);
 +
label("$20$",(7,0),S);
 +
label("$26$",(-2,0),S);
 
</asy>
 
</asy>
 +
 +
Let <math>x</math> be the number of students with both a dog and a cat.
 +
 +
Therefore, we have
 +
 +
<cmath>26+20-x = 39</cmath>
 +
<cmath>46-x = 39</cmath>
 +
<math></math>x = \boxed{\textbf{(A)} ~7}$.
 +
 +
~MrThinker
 +
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2008|num-b=10|num-a=12}}
 
{{AMC8 box|year=2008|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 10:34, 4 September 2022

Problem

Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat?

$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ 39\qquad \textbf{(E)}\ 46$

Solution 1

The union of two sets is equal to the sum of each set minus their intersection. The number of students that have both a dog and a cat is $20+26-39 = \boxed{\textbf{(A)}\ 7}$.

Solution 2 (Venn Diagram)

We create a diagram: [asy] draw(circle((0,0),5)); draw(circle((5,0),5)); label("$x$",(2.3,1.5),S); label("$20$",(7,0),S); label("$26$",(-2,0),S); [/asy]

Let $x$ be the number of students with both a dog and a cat.

Therefore, we have

\[26+20-x = 39\] \[46-x = 39\] $$ (Error compiling LaTeX. Unknown error_msg)x = \boxed{\textbf{(A)} ~7}$.

~MrThinker

See Also

2008 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_AMC_8_Problems/Problem_11&oldid=177943"