2014 AMC 10A Problems/Problem 5

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The following problem is from both the 2014 AMC 12A #5 and 2014 AMC 10A #5, so both problems redirect to this page.

Problem

On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$ (Error compiling LaTeX. Unknown error_msg)


Solution 1

WLOG let there be $100$ students who took the test. We have $10$ students score $70$ points, $35$ students score $80$ points, $30$ students score $90$ points and $25$ students score $100$ points. The median is easy to find by simply eliminating members from each group. The median is $90$ points. The mean is just $\dfrac{700+2800+2700+2500}{100}=7+28+27+25=87$. The difference is $90-87=\boxed{\textbf{(C)}\ 3}$

Solution 2

The mean can solved by the following. 10% of 70 is 7, 35% of 80 is 28, 30% of 90 is 27, and (100% - 10% - 35% - 30%) = 25%. 25% of 100 is 25. 7 + 28 + 27 + 25 = 87.
The median can be solved by finding the score present at the 50% mark, which is 90.
90-87 equals 3, which is \boxed{\textbf{(C)}\ 3}$

See Also

2014 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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

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