Difference between revisions of "2007 AMC 8 Problems/Problem 13"
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the number of elements in <math>A</math>. | the number of elements in <math>A</math>. | ||
| − | < | + | <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> | ||
<math>\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510</math> | <math>\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510</math> | ||
| Line 11: | Line 17: | ||
== Solution == | == Solution == | ||
| − | Let <math>x</math> be the number of elements in <math>A</math> and <math>B</math>. | + | Let <math>x</math> be the number of elements in <math>A</math> and <math>B</math> which is equal. |
| − | |||
| − | |||
| − | |||
| − | |||
| + | Then we could form equation | ||
<math>2x-1001 = 2007</math> | <math>2x-1001 = 2007</math> | ||
| Line 23: | Line 26: | ||
<math>x = 1504</math>. | <math>x = 1504</math>. | ||
| − | The answer is <math>\boxed{C}</math> | + | The answer is <math>\boxed{\textbf{(C)}\ 1504}</math> |
| + | |||
| + | ==Solution 2== | ||
| + | Let <math>x</math> be the number of elements in <math>A</math> not including the intersection. <math>2007-1001=1006</math> total elements excluding the intersection. Since we know that <math>A=B</math>, we can find that <math>x=\frac{1006}2=503</math>. Now we need to add the intersection. <math>503+1001=\boxed{\textbf{(C)} 1504}</math>. | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/3LtGb3KjhoU | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://www.youtube.com/watch?v=6F9x1XBOAeo | ||
| + | |||
| + | ==Video Solution by AliceWang== | ||
| + | https://youtu.be/ThBO09fGBgM | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2007|num-b=12|num-a=14}} | {{AMC8 box|year=2007|num-b=12|num-a=14}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 13:33, 8 July 2024
Contents
Problem
Sets 
and 
, shown in the Venn diagram, have the same number of elements.
Their union has 
elements and their intersection has 
elements. Find
the number of elements in .
![[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]](http://latex.artofproblemsolving.com/b/f/d/bfdb09b66054973cff11a173317d646fe1d66de1.png)
Solution
Let 
be the number of elements in and which is equal.
Then we could form equation
.
The answer is 
Solution 2
Let be the number of elements in not including the intersection. 
total elements excluding the intersection. Since we know that 
, we can find that 
. Now we need to add the intersection. 
.
Video Solution by WhyMath
~savannahsolver
Video Solution
https://www.youtube.com/watch?v=6F9x1XBOAeo
Video Solution by AliceWang
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
