AMC 10 · 2011 · #6
Grade 5 arithmeticOn Halloween Casper ate 31 of his candies and then gave 2 candies to his brother. The next day he ate 31 of his remaining candies and then gave 4 candies to his sister. On the third day he ate his final 8 candies. How many candies did Casper have at the beginning?
Pick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Casper starts with some candies. Day 1: he eats $\frac{1}{3}$ of them, then gives $2$ away. Day 2: he eats $\frac{1}{3}$ of what is left, then gives $4$ away. Day 3: he eats his last $8$ candies. Find how many he had at the very start.
Givens: Each day he eats $\frac{1}{3}$ of the candies he has at the start of that day; After eating on Day 1 he gives away $2$; after eating on Day 2 he gives away $4$; On Day 3 exactly $8$ candies are left, and he eats all of them; Answer choices: (A) $30$, (B) $39$, (C) $48$, (D) $57$, (E) $66$
Unknowns: The number of candies Casper had at the beginning
Understand
Restated: Casper starts with some candies. Day 1: he eats $\frac{1}{3}$ of them, then gives $2$ away. Day 2: he eats $\frac{1}{3}$ of what is left, then gives $4$ away. Day 3: he eats his last $8$ candies. Find how many he had at the very start.
Givens: Each day he eats $\frac{1}{3}$ of the candies he has at the start of that day; After eating on Day 1 he gives away $2$; after eating on Day 2 he gives away $4$; On Day 3 exactly $8$ candies are left, and he eats all of them; Answer choices: (A) $30$, (B) $39$, (C) $48$, (D) $57$, (E) $66$
Plan
Primary tool: #11 Work Backwards
Secondary: #4 Introduce a Variable, #6 Guess and Check
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 $\frac{1}{3}$ we recover the whole from the $\frac{2}{3}$ 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.
Execute — Answer: A
4.OA.A.3 Step 1 Undo the last give-away
- Read the story from the end.
- On Day 3 Casper eats his final $8$ candies, so he had $8$ at the start of Day 3.
- Those $8$ are what was left after he gave $4$ to his sister on Day 2.
- Undoing that gift means putting the $4$ back: before the gift he had $8 + 4 = 12$.
- So right after he finished eating on Day 2, he had $12$ candies.
💡 To reverse giving candies away, hand them back.
5.NF.A.2 Step 2 See what fraction survives eating
- Each day Casper eats $\frac{1}{3}$ of that day's candies.
- Whatever he does not eat stays, and the part that stays is $1 - \frac{1}{3} = \frac{2}{3}$.
- So the $12$ candies left after eating on Day 2 are exactly $\frac{2}{3}$ of the number he had at the start of Day 2.
- That link between $\frac{2}{3}$ and $12$ is the key we will use to step backward through each eating.
💡 Eating one third always leaves two thirds behind.
5.NF.B.6 Step 3 Recover each day's starting amount
- If $\frac{2}{3}$ of a day's candies equals $12$, then $\frac{1}{3}$ of them is half of $12$, which is $6$, so the whole ($\frac{3}{3}$) is $3 \times 6 = 18$.
- Casper began Day 2 with $18$ candies.
- Those $18$ are what was left after he gave $2$ to his brother on Day 1, so before that gift he had $18 + 2 = 20$.
- Now $20$ is $\frac{2}{3}$ of his original pile: $\frac{1}{3}$ is half of $20$, which is $10$, so the whole start is $3 \times 10 = 30$.
- Casper began with $30$ candies, which is choice (A).
💡 Knowing two thirds of a pile lets you rebuild the whole pile.
4.OA.A.3 Step 4 Check by running the story forward
- Test the recovered start of $30$ by playing the story forward.
- Day 1: eat $\frac{1}{3}$ of $30 = 10$, leaving $20$; give $2$ away, leaving $18$.
- Day 2: eat $\frac{1}{3}$ of $18 = 6$, leaving $12$; give $4$ away, leaving $8$.
- Day 3: eat the final $8$.
- Everything matches the problem exactly, so $30$ is correct.
💡 A start that replays into the exact ending must be the right start.
4.OA.A.3 Read the story from the end. On Day 3 Casper eats his final $8$ candies, so he h 5.NF.A.2 Each day Casper eats $\frac{1}{3}$ of that day's candies. Whatever he does not e 5.NF.B.6 If $\frac{2}{3}$ of a day's candies equals $12$, then $\frac{1}{3}$ of them is h 4.OA.A.3 Test the recovered start of $30$ by playing the story forward. Day 1: eat $\frac Review
Reasonableness: The forward replay lands on $0$ candies after Day 3 with the right amounts left at every stage ($20$, $18$, $12$, $8$), so the answer $30$ is consistent with the whole story. It is also the smallest choice, which fits a shrinking pile that must stay a multiple of $3$ before each eating. Bigger choices like $66$ would leave far more than $8$ at the end.
Alternative: Set up a forward equation with a variable. Let $x$ be the start. After Day 1: $\frac{2}{3}x - 2$. After Day 2 eating: $\frac{2}{3}\left(\frac{2}{3}x - 2\right)$, then $-4$ leaves the final $8$: $\frac{2}{3}\left(\frac{2}{3}x - 2\right) - 4 = 8$. Solving gives $\frac{2}{3}x - 2 = 18$, so $\frac{2}{3}x = 20$ and $x = 30$. You could also just test the choices; only $30$ replays to exactly $8$ candies on Day 3.
CCSS standards used (min grade 5)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Undoing each give-away by adding candies back ($8+4=12$, $18+2=20$) and replaying the story forward to check the answer.)5.NF.A.2Solve word problems involving addition and subtraction of fractions (Finding the fraction that survives eating: $1 - \frac{1}{3} = \frac{2}{3}$ remains each day.)5.NF.B.6Solve real-world problems involving multiplication of fractions and mixed numbers (Recovering each day's whole amount from the $\frac{2}{3}$ that remained ($\frac{2}{3}$ of the pile $=12 \Rightarrow 18$, and $=20 \Rightarrow 30$).)
⭐ Start 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.
⭐ Start 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.
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