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 $B$ which is equal.

Then we could form equation $2x-1001 = 2007$

$2x = 3008$

$x = 1504$.

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

Solution 2

Let $x$ be the number of elements in $A$ not including the intersection. $2007-1001=1006$ total elements excluding the intersection. Since we know that $A=B$, we can find that $x=\frac{1006}2=503$. Now we need to add the intersection. $503+1001=\boxed{\textbf{(C)} 1504}$.

~savannahsolver