Difference between revisions of "2007 AMC 8 Problems/Problem 13"

Revision as of 21:45, 20 April 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 $\boxed{\textbf{(C)}\ 1504}$

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

Video Solution by WhyMath

https://youtu.be/3LtGb3KjhoU

~savannahsolver

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 36: / Line 36: @@
 - spoamath321
+==Video Solution by WhyMath==
+https://youtu.be/3LtGb3KjhoU
+~savannahsolver
 ==See Also==
 {{AMC8 box|year=2007|num-b=12|num-a=14}}
 {{MAA Notice}}

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