Difference between revisions of "2010 AMC 12B Problems/Problem 10"
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== Solution == | == Solution == | ||
We first sum the first <math>99</math> numbers: <math>\frac{99(100)}{2}=99\times50=4,950</math>. Then, we know that the sum of the series is <math>4,950+x</math>. Since the average is <math>100x</math>, and there are <math>100</math> terms, we also find the sum to equal <math>10,000x</math>. Setting equal - <math>10,000x=4,\,950+x \Rightarrow 9,999x=4,\,950 \Rightarrow x=\frac{4,950}{9,999} \Rightarrow x= \frac{50}{101}</math>. Thus, the answer is <math>\boxed{\text{B}}</math>. | We first sum the first <math>99</math> numbers: <math>\frac{99(100)}{2}=99\times50=4,950</math>. Then, we know that the sum of the series is <math>4,950+x</math>. Since the average is <math>100x</math>, and there are <math>100</math> terms, we also find the sum to equal <math>10,000x</math>. Setting equal - <math>10,000x=4,\,950+x \Rightarrow 9,999x=4,\,950 \Rightarrow x=\frac{4,950}{9,999} \Rightarrow x= \frac{50}{101}</math>. Thus, the answer is <math>\boxed{\text{B}}</math>. | ||
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== See also == | == See also == | ||
{{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}} | {{AMC12 box|year=2010|num-b=12|num-a=14|ab=B}} | ||
{{stub}} | {{stub}} | ||
Revision as of 12:53, 31 May 2011
Problem 10
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 
. Since the average is , and there are 
terms, we also find the sum to equal 
. Setting equal - 
. Thus, the answer is 
.
See also
| 2010 AMC 12B (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 AMC 12 Problems and Solutions | |
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