AMC 10 · 2021 · #7
Grade 7 probabilityPick an answer.
There is only one set of data here — the five enrollments 50, 20, 20, 5, 5 — and the entire problem is that t and s tally it two different ways. That makes Tool #15 (Organize Information in More Ways) the engine: build the teacher list, then rebuild the very same classes as a student list and watch the numbers change. Tool #16 (Change Focus) names the switch precisely — stop looking at the 5 teachers and look at the 100 students, which is what turns a plain average into a weighted one. Tool #8 (Analyze the Units) keeps the two divisions straight, since t divides by 5 teachers while s divides by 100 students. Tool #14 (Extreme Principle) settles the sign of the answer before any arithmetic, by testing the all-equal case and a lopsided case. Tool #3 (Eliminate Possibilities) then cashes that sign in against the five choices.
Tally the classes by teacher
By teacher it is a plain average.
Every teacher gets one vote, so t is nothing more than the 100 students shared evenly over 5 rooms.
6.SP.A.3Organize Information In More WaysRe-tally the same classes by student
By student, big classes weigh more.
Sampling students instead of teachers makes a class get counted once for every person sitting in it, so crowded classes speak louder.
Sampling students instead of teachers makes a class get counted once for every person sitting in it.
▸ Why?
Each class is counted once per member, so a crowded class contributes many times over.
▸ Why?
An average weights each value by how often it appears, which is exactly what that recount changes.
Average over the hundred students
Take the average over the hundred students.
Each class size is weighted by how many people actually experience it, which is what turns s into a weighted average instead of a plain one.
6.SP.B.5Analyze The UnitsDecide the sign before finishing
The sign can be predicted in advance.
A crowded class has more students on hand to report its own crowdedness, so the student-side average always gets dragged upward.
7.SP.B.4Extreme PrincipleSubtract and match a choice
Subtracting gives negative thirteen point five.
The gap is negative because s counts the same rooms from the crowded side, so subtracting it from t lands below zero.
7.NS.A.3Eliminate PossibilitiesAsking the teachers and asking the students are two different tallies of the same five classes: a class of n students is counted once by its teacher but n times by its students, so the student-side average is the bigger one — here 20 against 33.5, a gap of 13.5.
- Tally the classes by teacher
- Re-tally the same classes by student
- Average over the hundred students
- Decide the sign before finishing
- Subtract and match a choice