AMC 10 · 2007 · #12
Grade 6 arithmeticPick an answer.
Percentages with no actual head count feel abstract, so replace the class with a convenient concrete size (Tool #9): pick 10 students, which turns 10% and 90% into a clean 1 junior and 9 seniors. An average is just a total divided by a count, so turn each average into a total number of points (Tool #7). The class total minus the seniors' total is exactly the one junior's score, recovered by working backwards (Tool #11).
Pick a convenient class size
A convenient class size makes both shares whole numbers.
Swapping vague percentages for a real head count of 10 makes every quantity something you can just add up.
6.RP.A.3Solve An Easier Related ProblemTurn averages into total points
Each average turns back into a total of points.
Turning each average back into a raw total lets you add and subtract groups of scores like ordinary numbers.
Turning each average back into a raw total lets whole groups of scores be added and subtracted like ordinary numbers.
▸ Why?
An average is a total shared over a count, so multiplying it back by the count returns the total.
▸ Why?
The class total is its members' scores added together, so removing one group leaves the rest.
Peel off the seniors to find the junior
Peeling the large group off leaves 93, choice (A).
The class total is the seniors' points plus the junior's, so subtracting the seniors leaves the junior's score alone.
4.NBT.B.4Work BackwardsTurn percentages into a real head count, change each average into a total number of points, then subtract the group you know to uncover the one you want.
- Pick a convenient class size
- Turn averages into total points
- Peel off the seniors to find the junior