Difference between revisions of "2005 AMC 12B Problems/Problem 9"
Flying2828 (talk | contribs) |
|||
| Line 1: | Line 1: | ||
| + | {{duplicate|[[2005 AMC 12B Problems|2005 AMC 12B #9]] and [[2005 AMC 10B Problems|2005 AMC 10B #19]]}} | ||
== Problem == | == Problem == | ||
| − | On a certain math exam, <math>10%</math> of the students got <math>70</math> points, <math>25%</math> got <math>80</math> points, <math>20%</math> got <math>85</math> points, <math>15%</math> got <math>90</math> points, and the rest got <math>95</math> points. What is the difference between the mean and the median score on this exam? | + | On a certain math exam, <math>10\%</math> of the students got <math>70</math> points, <math>25\%</math> got <math>80</math> points, <math>20\%</math> got <math>85</math> points, <math>15\%</math> got <math>90</math> points, and the rest got <math>95</math> points. What is the difference between the mean and the median score on this exam? |
| + | |||
| + | <math>\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5 </math> | ||
== Solution == | == Solution == | ||
| − | + | To begin, we see that the remaining <math>30\%</math> of the students got <math>95</math> points. The smallest case is that there are <math>20</math> students, because <math>\gcd\ (10,25,20,15,30)=5</math>, and <math>5\%^{-1}=20</math>. We see that <math>2</math> students got <math>70</math> points, <math>5</math> students got <math>80</math> points, <math>4</math> students got <math>85</math> points, <math>3</math> students got <math>90</math> points, and <math>6</math> students got <math>95</math> points. The median is <math>85</math>, since the <math>10^{\text{th}}</math> and <math>11^{\text{th}}</math> terms in an ordered sequence are both <math>85</math>. The mean is <math>\dfrac{70\,(2)+80\,(5)+85\,(4)+90\,(3)+95\,(6)}{20}=\dfrac{1720}{20}=86</math>. The difference between the mean and median, therefore, is <math>\boxed{\mathrm{(B)}\ 1}</math>. | |
== See also == | == See also == | ||
| − | + | {{AMC10 box|year=2005|ab=B|num-b=18|num-a=20}} | |
| − | + | {{AMC12 box|year=2005|ab=B|num-b=8|num-a=10}} | |
| − | |||
Revision as of 14:51, 16 July 2011
- The following problem is from both the 2005 AMC 12B #9 and 2005 AMC 10B #19, so both problems redirect to this page.
Problem
On a certain math exam, 
of the students got 
points, 
got 
points, 
got 
points, 
got 
points, and the rest got 
points. What is the difference between the mean and the median score on this exam?
Solution
To begin, we see that the remaining 
of the students got points. The smallest case is that there are 
students, because 
, and 
. We see that 
students got points, 
students got points, 
students got points, 
students got points, and 
students got points. The median is , since the 
and 
terms in an ordered sequence are both . The mean is 
. The difference between the mean and median, therefore, is 
.
See also
| 2005 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 | ||
| 2005 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 8 |
Followed by Problem 10 |
| 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 | |
