2014 AMC 10A Problem 5

The remaining share of students is $100\% - 10\% - 35\% - 30\% = 25\%$, and they scored $100$.

Mean. Weight each score by its share:
$$0.10 \cdot 70 + 0.35 \cdot 80 + 0.30 \cdot 90 + 0.25 \cdot 100 = 7 + 28 + 27 + 25 = 87.$$

Median. Order the scores from lowest to highest and accumulate the percentages: the $70$s cover $0$–$10\%$, the $80$s cover $10$–$45\%$, and the $90$s cover $45$–$75\%$. The middle of the data, the $50\%$ mark, falls inside the $90$ group, so the median is $90$.

The difference is $90 - 87 = 3$.

The answer is $\boxed{\textbf{(C)}\ 3}$.

With data given as percentages, imagine $100$ students (or $20$): the mean is a weighted average, and the median is found by adding up the percentages until you pass $50\%$. Here the $80$ group ends at $45\%$, so the median jumps up to $90$ even though $80$ is the most common score; median and mode need not agree.

• Taking the median as the middle of the listed values (85) or averaging the four scores instead of weighting by the percentages.
• Stopping the cumulative count at $45\%$ and declaring the median to be $80$, which gives a difference of $7$ (not even a choice, a useful sanity check).