AMC 10 · 2014 · #5
Grade 6 arithmeticPick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 four groups are the whole class, so their shares add to 100% — subtract the three known ones to get 25%.
"The rest" always means whatever is left over after the named parts — the complement up to the whole.
6.RP.A.3Change Focus Count The ComplementPick a convenient class size
The head count can't change the answer, so take a class of 20 — every share is then a whole number of students.
Turning percents into a real count of 20 students makes the middle student easy to locate.
6.RP.A.3Introduce A VariableCompute the mean
Multiply each score by how many students got it, add, then divide by 20: the mean is 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
Line the 20 scores up: two 70s, then seven 80s, so the 10th and 11th both land in the 90 group — the median is 90.
The median sits where the running count first crosses the halfway point — here, inside the 90 group.
The middle score sits where the running count first crosses the halfway point.
▸ Why?
The middle of an ordered list is the entry with as many below it as above it.
▸ Why?
The average, by contrast, weighs every score by how often it happened, so the two need not agree.
Take the difference
Subtract the smaller center from the larger: 90 minus 87 leaves a gap of 3.
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