Difference between revisions of "2002 AMC 10B Problems/Problem 25"
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==Solution 2== | ==Solution 2== | ||
We let <math>m</math> be the number of elements in the set and we let <math>n</math> be the average of the terms of the original list. Then we have <math>mn</math> is the sum of all the elements of the list. So we have two equations: <cmath>mn+15=(m+2)(n+1)=mn+m+2n+2</cmath> and <cmath>mn+16=(m+1)(n+2)=mn+2m+n+2.</cmath>Simplifying both equations and we get, | We let <math>m</math> be the number of elements in the set and we let <math>n</math> be the average of the terms of the original list. Then we have <math>mn</math> is the sum of all the elements of the list. So we have two equations: <cmath>mn+15=(m+2)(n+1)=mn+m+2n+2</cmath> and <cmath>mn+16=(m+1)(n+2)=mn+2m+n+2.</cmath>Simplifying both equations and we get, | ||
| − | + | <cmath>13=m+2n</cmath> | |
| − | 13 | + | <cmath>14=2m+n</cmath> |
| − | 14 | ||
| − | |||
Solving for <math>m</math> and <math>n</math>, we get <math>m=5</math> and <math>n=\boxed{\textbf{(A)}4}</math>. | Solving for <math>m</math> and <math>n</math>, we get <math>m=5</math> and <math>n=\boxed{\textbf{(A)}4}</math>. | ||
Revision as of 09:48, 30 December 2017
Contents
Problem
When 
is appended to a list of integers, the mean is increased by 
. When 
is appended to the enlarged list, the mean of the enlarged list is decreased by . How many integers were in the original list?
Solution
Let 
be the sum of the integers and 
be the number of elements in the list. Then we get the equations 
and 
. With a lot of algebra, the solution is found to be 
.
Solution 2
We let 
be the number of elements in the set and we let 
be the average of the terms of the original list. Then we have 
is the sum of all the elements of the list. So we have two equations: ![\[mn+15=(m+2)(n+1)=mn+m+2n+2\]](http://latex.artofproblemsolving.com/6/6/b/66b21d1fbe8f4cc78d1eb0fcc195630b089938aa.png)
and ![\[mn+16=(m+1)(n+2)=mn+2m+n+2.\]](http://latex.artofproblemsolving.com/a/f/4/af4fbe178484847c58081846dad4c976910bd24d.png)
Simplifying both equations and we get,
![\[13=m+2n\]](http://latex.artofproblemsolving.com/5/f/1/5f1613bc612071685ebd46c8791690ac0b36dfb6.png)
![\[14=2m+n\]](http://latex.artofproblemsolving.com/8/5/e/85e9e11be3059a8465cea31baf18907be815086f.png)
Solving for and , we get 
and 
.
See also
| 2002 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last problem | |
| 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 AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
