AMC 10 · 2014 · #5
Grade 6 arithmeticPick an answer.
The scores come as a distribution (percent of students at each score), not a plain list. Tool #15 says: lay the data out in an organized table — score, share of students, and running total — because that single table hands you both answers. The mean is the weighted sum straight from the shares; the median is wherever the running total first passes the halfway mark. First, Tool #16 fills the gap: the four groups must total 100%, so the unstated group is the complement of the others. Then Tool #4 makes the counting concrete — instead of juggling percentages, pick one convenient class size so every group becomes a whole number of students and the median is easy to point at.
Find the missing group
The missing group is 25 percent.
"The rest" always means whatever is left over after the named parts — the complement up to the whole.
The rest always means whatever is left over after the named parts.
▸ Why?
Everyone is in one named group or in none of them, so the unnamed group is the opposite of all the rest.
▸ Why?
The class is exactly its groups put together, so the shares have to fill it completely.
Pick a convenient class size
A convenient class size makes every count whole.
Turning percents into a real count of 20 students makes the middle student easy to locate.
6.RP.A.3Introduce A VariableCompute the mean
The mean comes to 87.
A weighted average just counts each score as many times as it actually occurred.
6.SP.B.5Organize Information In More WaysFind the median
Running totals put the median at 90.
The median sits where the running count first crosses the halfway point — here, inside the 90 group.
6.SP.B.5Organize Information In More WaysTake the difference
The gap is 3, choice (C).
The low 70 scores drag the mean below the median, so the gap between them is what we report.
6.NS.B.3Organize Information In More WaysA pile of percentages is really an ordered data list — build the table once and the mean pops out as a weighted average while the median is just the score sitting at the halfway mark.
- Find the missing group
- Pick a convenient class size
- Compute the mean
- Find the median
- Take the difference