2014 AMC 10A Problem 5

Percentages are easiest to handle if you imagine exactly $100$ students.

1. The groups are: $10$ students scored $70$, $35$ scored $80$, $30$ scored $90$, and the rest, $100 - 10 - 35 - 30 = 25$, scored $100$.

Mean.

1. Add up all the points: $10 \cdot 70 + 35 \cdot 80 + 30 \cdot 90 + 25 \cdot 100 = 700 + 2800 + 2700 + 2500 = 8700$.
2. Divide by $100$ students: the mean is $87$.

Median.

1. Line the students up from lowest score to highest. The median is the score in the middle of the line, around student number $50$.
2. Students $1$ to $10$ scored $70$. Students $11$ to $45$ scored $80$. Students $46$ to $75$ scored $90$.
3. Students $50$ and $51$ are both in the $90$ group, so the median is $90$.

Difference. $90 - 87 = 3$.

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

When data are given as percentages, pick a convenient number of people, such as $100$, and the percentages become counts. The mean uses every score, weighted by how many people got it. The median only asks where the middle person stands, so you add up the groups until you pass the halfway point. Here the $80$ group stops at $45\%$, just short of halfway, which pushes the median up to $90$.

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 $\boxed{\textbf{(C)}\ 3}$.