Difference between revisions of "2005 AMC 12B Problems/Problem 9"
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+ | {{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? | ||
+ | |||
+ | <math>\textbf{(A) }\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\ 4 \qquad \textbf{(E) }\ 5 </math> | ||
== Solution == | == Solution == | ||
+ | To begin, we see that the remaining <math>30\%</math> of the students got <math>95</math> points. Assume that there are <math>20</math> students; 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 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{\textbf{(B)}\ 1}</math>. | ||
+ | |||
+ | == Solution 2== | ||
+ | The remaining <math>30\%</math> of the students got <math>95</math> points. | ||
+ | The mean is equal to <math>10\%\cdot70 + 25\%\cdot80 + 20\%\cdot85 + 15\%\cdot90 + 30\%\cdot95 = 86</math>. | ||
+ | The score greater than or equal to <math>50\%</math> of other scores is the median. Since <math>35\%</math> scored <math>80</math> or lower and the next <math>20\%</math> scored <math>85</math>, the median is <math>85</math>. The difference between the mean and the median is <math>\boxed{\textbf{(B)}\ 1}</math>. | ||
+ | |||
+ | ~mobius247 | ||
== 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}} | ||
+ | {{MAA Notice}} |
Latest revision as of 15:35, 16 December 2021
- The following problem is from both the 2005 AMC 12B #9 and 2005 AMC 10B #19, so both problems redirect to this page.
Contents
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. Assume that there are
students; 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 are both
. The mean is
. The difference between the mean and median, therefore, is
.
Solution 2
The remaining of the students got
points.
The mean is equal to
.
The score greater than or equal to
of other scores is the median. Since
scored
or lower and the next
scored
, the median is
. The difference between the mean and the median is
.
~mobius247
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.