AMC 10 · 2016 · #8
Grade 6 rate-ratioPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The end state is the clean number we know exactly: 0 coins after the third crossing. Tool #11 (Work Backwards) starts there and undoes one crossing at a time. To undo a crossing, run its two steps in reverse order: first add back the 40 toll, then halve to undo the doubling. Three reverse steps walk us back to the starting amount. (Tool #4, Introduce a Variable, gives the same answer through a single equation and is shown as the check.)
See the repeating cycle
Each crossing is the same machine—double, then subtract 40—run three times ending at 0, so undoing it three times returns to the start.
Three identical crossings means one reverse step, used three times, retraces the whole trip.
4.OA.C.5Work BackwardsBuild the reverse step
To undo one crossing, reverse both actions in opposite order: add 40 back, then divide by 2.
Undoing actions means peeling them off in the reverse order you put them on, using the opposite operation each time.
6.EE.B.7Work BackwardsUndo the third crossing
Undo the last crossing: add 40 to the final 0, then halve, leaving 20 just before the third crossing.
Reversing the last crossing tells you the money he carried onto that crossing.
6.EE.B.7Work BackwardsUndo the second and first crossings
Repeat: before the second crossing (20+40)/2 = 30, before the first (30+40)/2 = 35, the starting amount (C).
Each reverse step rewinds one more crossing until you reach the beginning.
6.EE.B.7Work BackwardsWhen you know the ending, run the steps in reverse with the opposite operations to find where it all began.
- See the repeating cycle
- Build the reverse step
- Undo the third crossing
- Undo the second and first crossings