AMC 10 · 2005 · #19
Grade 6 arithmeticPick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
The four listed groups total 70%, so the remaining 30% scored 95; take 100 students and each percent becomes a head count.
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
Lined up in order, positions 1–35 hold the 70s and 80s and 36–55 hold the 85s, so the 50th and 51st scores are both 85.
Stack the scores in order and the middle of the pile is the median.
6.SP.B.5Identify SubproblemsCompute the mean
Weighted total 70(10)+80(25)+85(20)+90(15)+95(30)=8600, so the mean is 8600÷100=86.
A weighted average pays each score by how many students earned it, then splits the pot evenly.
A weighted average pays each score by how many students earned it, then splits the pot evenly.
▸ Why?
An average is a total shared over a count, so the total has to be built before it can be shared.
▸ Why?
Each percent is a count out of a hundred, so a class of a hundred turns every percent into a head count.
Subtract to get the difference
Mean 86 minus median 85 gives 1; the two centers nearly coincide, which rules out 2, 4, and 5 — 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