AMC 10 · 2011 · #6
Grade 5 arithmeticPick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem hands us the END of the story (8 candies left) and asks for the START, so Tool #11 (Work Backwards) is the natural fit: undo each action in reverse order. To undo a give-away we add the candies back; to undo eating 1/3 we recover the whole from the 2/3 that remained. Tool #4 (Introduce a Variable) offers a forward equation as a cross-check, and Tool #6 (Guess and Check) lets us confirm the recovered start by running the story forward or by testing a choice.
Undo the last give-away
Read from the end: 8 candies were left on Day 3, so putting back the 4 he gave his sister leaves 12 right after Day 2's eating.
To reverse giving candies away, hand them back.
4.OA.A.3Work BackwardsSee what fraction survives eating
Eating always leaves , so those 12 candies are exactly two thirds of what he held that morning.
Eating one third always leaves two thirds behind.
Eating one third always leaves two thirds behind.
▸ Why?
The pile is exactly what was eaten plus what remains, so one names the other.
▸ Why?
A third is one of three equal shares, so two of those shares are what is left.
Recover each day's starting amount
Two thirds is 12, so a third is 6 and Day 2 began with 18; add back the brother's 2 to get 20, whose whole is 30 — choice (A).
Knowing two thirds of a pile lets you rebuild the whole pile.
5.NF.B.6Work BackwardsCheck by running the story forward
Replay 30 forward: eat 10, leave 20, give 2 to reach 18; eat 6, leave 12, give 4 to reach 8; eat that last 8. Every stage matches.
A start that replays into the exact ending must be the right start.
4.OA.A.3Guess And CheckStart from the end and undo each move: add back what was given away, and rebuild the whole pile from the two thirds that were left.
- Undo the last give-away
- See what fraction survives eating
- Recover each day's starting amount
- Check by running the story forward