Difference between revisions of "2010 AMC 12B Problems/Problem 10"
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{{duplicate|[[2010 AMC 12B Problems|2010 AMC 12B #10]] and [[2010 AMC 10B Problems|2010 AMC 10B #14]]}} | {{duplicate|[[2010 AMC 12B Problems|2010 AMC 12B #10]] and [[2010 AMC 10B Problems|2010 AMC 10B #14]]}} | ||
| − | == Problem | + | == Problem == |
The average of the numbers <math>1, 2, 3,\cdots, 98, 99,</math> and <math>x</math> is <math>100x</math>. What is <math>x</math>? | The average of the numbers <math>1, 2, 3,\cdots, 98, 99,</math> and <math>x</math> is <math>100x</math>. What is <math>x</math>? | ||
| Line 11: | Line 11: | ||
Using difference of squares: | Using difference of squares: | ||
<cmath>x=\frac{99\cdot50}{101\cdot99}=\frac{50}{101}</cmath> | <cmath>x=\frac{99\cdot50}{101\cdot99}=\frac{50}{101}</cmath> | ||
| − | Thus, the answer is <math>\boxed{\ | + | Thus, the answer is <math>\boxed{\textbf{(B) }\frac{50}{101}}</math>. |
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/vYXz4wStBUU?t=413 | ||
| + | |||
| + | ~IceMatrix | ||
== See also == | == See also == | ||
{{AMC12 box|year=2010|num-b=9|num-a=11|ab=B}} | {{AMC12 box|year=2010|num-b=9|num-a=11|ab=B}} | ||
| + | {{AMC10 box|year=2010|num-b=13|num-a=15|ab=B}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 17:06, 6 July 2023
- The following problem is from both the 2010 AMC 12B #10 and 2010 AMC 10B #14, so both problems redirect to this page.
Contents
Problem
The average of the numbers 
and 
is 
. What is ?
Solution
We first sum the first 
numbers: 
. Then, we know that the sum of the series is 
. There are 
terms, so we can divide this sum by and set it equal to :
![\[\frac{99\cdot50+x}{100}=100x \Rightarrow 99\cdot50=100^2x-x\Rightarrow \frac{99\cdot50}{100^2-1}=x\]](http://latex.artofproblemsolving.com/d/e/e/dee34095c95d4eb8d0faffde82a81b4071561d9e.png)
Using difference of squares:
![\[x=\frac{99\cdot50}{101\cdot99}=\frac{50}{101}\]](http://latex.artofproblemsolving.com/5/d/3/5d32530090ee83aa2d0b78c9093d51789be3b4f0.png)
Thus, the answer is 
.
Video Solution
https://youtu.be/vYXz4wStBUU?t=413
~IceMatrix
See also
| 2010 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
| 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 12 Problems and Solutions | |
| 2010 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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. 
