AMC 10 · 2016 · #10
Grade 7 countingPick an answer.
Instead of hunting for the whole seating chart, tool #4 (Introduce a Variable) measures each person by a single signed number: how far they moved, with right as + and left as -. The key fact is that nobody enters or leaves the row, so every seat someone leaves is taken by someone else, which forces all five signed moves to add up to 0. Tool #1 (Draw a Diagram) keeps the left-right number line straight in mind, and once Ada's move is known, tool #3 (Eliminate Possibilities) uses the end-seat clue to rule out one of the two ends and pin down where she started.
Turn each move into a signed number
Each move becomes a signed number.
Right and left are opposite directions, so a + and - sign captures each move in one number.
6.NS.C.5Introduce A VariableAll five moves add to zero
All five must add to zero.
Rearranging people inside a fixed set of seats can't push the group net-left or net-right, so the signed moves must cancel to zero.
Rearranging people inside a fixed set of seats cannot push the group net-left or net-right.
▸ Why?
Every seat vacated is a seat filled, so what one person gains another gives up.
▸ Why?
The signed moves therefore add to nothing, which is exactly what a total of zero means.
Solve for Ada's move
So her own move is one left.
Whatever the others push to the right, Ada's move has to pull back the same amount to keep the balance at zero.
7.NS.A.1Introduce A VariableUse the end-seat clue to find the start
The end-seat clue gives 2, choice (B).
Moving left lands you on a smaller number, so the only end seat she could reach by stepping left is seat 1.
6.NS.C.6Eliminate PossibilitiesTag each move + for right and - for left; since no one leaves the row, all the moves cancel to zero, so Ada's must be -1, putting her start at seat 2.
- Turn each move into a signed number
- All five moves add to zero
- Solve for Ada's move
- Use the end-seat clue to find the start