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

See Also

2007 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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All AJHSME/AMC 8 Problems and Solutions

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