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Difference between revisions of "2007 AMC 8 Problems/Problem 13"
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The answer is <math>\boxed{\textbf{(C)}\ 1504}</math>
 
The answer is <math>\boxed{\textbf{(C)}\ 1504}</math>
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== Solution 2 ==
 
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First find the number of elements in <math>A</math> without including the intersection. There are 2007 elements in total, so there are <math>1006</math> elements in <math>A</math> and <math>B</math> excluding the intersection (<math>2007-1001</math>). There are <math>503</math> elements in set A after dividing <math>1006</math> by <math>2</math>. Add the intersection (<math>1001</math>) to get <math>\boxed{\textbf{(C)}\ 1504}</math>
 
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- spoamath321
 
  
 
==Video Solution by WhyMath==
 
==Video Solution by WhyMath==

Revision as of 12:08, 8 July 2021

Problem

Sets $A$ and $B$, shown in the Venn diagram, have the same number of elements. Their union has $2007$ elements and their intersection has $1001$ elements. Find the number of elements in $A$.

[asy] defaultpen(linewidth(0.7)); draw(Circle(origin, 5)); draw(Circle((5,0), 5)); label("$A$", (0,5), N); label("$B$", (5,5), N); label("$1001$", (2.5, -0.5), N);[/asy]

$\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510$

Solution

Let $x$ be the number of elements in $A$ and $B$.

Since the union is the sum of all elements in $A$ and $B$,

and $A$ and $B$ have the same number of elements then,

$2x-1001 = 2007$

$2x = 3008$

$x = 1504$.

The answer is $\boxed{\textbf{(C)}\ 1504}$

Video Solution by WhyMath

https://youtu.be/3LtGb3KjhoU

~savannahsolver

See Also

2007 AMC 8 (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 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

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