2007 AMC 8 Problems/Problem 13: Difference between revisions
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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> | ||
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== 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> | ||
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<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 05:55, 24 October 2025
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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
