2007 AMC 8 Problems/Problem 13

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 $\boxed{\textbf{(C)}\ 1504}$

Solution 2

First find the number of elements in $A$ without including the intersection. There are 2007 elements in total, so there are $1006$ elements in $A$ and $B$ excluding the intersection ( $2007-1001$ ). There are $503$ elements in set A after dividing $1006$ by $2$ . Add the intersection ( $1001$ ) to get $\boxed{\textbf{(C)}\ 1504}$

- spoamath321

2007 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2007 AMC 8 Problems/Problem 13

Contents

Problem

Solution

Solution 2

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