AMC 10 · 2005 · #9
Grade 6 arithmeticPick an answer.
The problem gives percentages but no head count, so Tool #9 (Solve an Easier Related Problem) pins the count at a convenient 100 students; then every percentage becomes a plain number of students and the abstract question turns into counting real scores. Tool #7 (Identify Subproblems) splits the work into two independent pieces — find the median, find the mean — and only at the end subtracts them. Tool #3 (Eliminate Possibilities) uses the multiple-choice frame as a safety net: the difference is a small whole number, so a quick estimate can rule out the far-apart choices.
Turn percents into student counts
Choosing a hundred students turns percentages into head counts.
With no head count given, 100 students makes each percent an actual number of kids.
6.RP.A.3Solve An Easier Related ProblemLocate the median
Lining the scores up puts the middle inside one block.
Stack the scores in order and the middle of the pile is the median.
6.SP.B.5Identify SubproblemsCompute the mean
Weighting each score by its count gives the mean 86.
A weighted average pays each score by how many students earned it, then splits the pot evenly.
The mean pays each score by how many students earned it, then splits the whole pot evenly.
▸ Why?
An average is the total shared out over the count, so the total has to be built before it can be shared.
▸ Why?
Each percent is a count out of a hundred, so choosing a class of a hundred turns every percent into a head count.
Subtract to get the difference
Subtracting gives 1, choice (B).
Two measures of center for the same data are close, so their gap is a tiny whole number.
6.SP.A.3Eliminate PossibilitiesWhen only percents are given, pretend there are 100 people so each percent becomes a count; then the median is the middle score in order and the mean is the weighted average.
- Turn percents into student counts
- Locate the median
- Compute the mean
- Subtract to get the difference