AMC 10 · 2011 · #6

Easy mode Grade 5

Problem

Casper starts with a bag of candies. On the first day he eats $\frac{1}{3}$ of them, then gives $2$ candies to his brother.

On the second day he eats $\frac{1}{3}$ of what is left, then gives $4$ candies to his sister. On the third day he eats his last $8$ candies.

How many candies did Casper have at the start?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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.

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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

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

#11 Work Backwards 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.

$$8 + 4 = 12$$

💡 To reverse giving candies away, hand them back.

#11 Work Backwards 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.

$$1 - \frac{1}{3} = \frac{2}{3}$$

💡 Eating one third always leaves two thirds behind.

#11 Work Backwards 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).

$$\tfrac{2}{3}\,x = 12 \Rightarrow x = 18;\quad 18 + 2 = 20;\quad \tfrac{2}{3}\,y = 20 \Rightarrow y = 30 \;\Rightarrow\; \textbf{(A)}$$

💡 Knowing two thirds of a pile lets you rebuild the whole pile.

#6 Guess and Check 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.

$$30 \xrightarrow{-10} 20 \xrightarrow{-2} 18 \xrightarrow{-6} 12 \xrightarrow{-4} 8 \xrightarrow{-8} 0$$

💡 A start that replays into the exact ending must be the right start.

[1] #11 4.OA.A.3 Read the story from the end. On Day 3 Casper eats his final $8$ candies, so he h

[2] #11 5.NF.A.2 Each day Casper eats $\frac{1}{3}$ of that day's candies. Whatever he does not e

[3] #11 5.NF.B.6 If $\frac{2}{3}$ of a day's candies equals $12$, then $\frac{1}{3}$ of them is h

[4] #6 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.3 Solve 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.2 Solve 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.6 Solve 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.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Undo the last give-away

See what fraction survives eating

Recover each day's starting amount

Check by running the story forward

Review

CCSS standards used (min grade 5)

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