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AMC 8 · 2009 · #1

Grade 4 arithmetic
Archetype Arithmetic Expression Decomposition
multi-digit-arithmeticmental-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem

Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy?

Pick an answer.

(A)
3
(B)
4
(C)
7
(D)
11
(E)
14
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Toolkit + CCSS Solution

Understand

Restated: Bridget bought some apples. She gave half of them to Ann, then gave $3$ to Cassie, and kept the last $4$ for herself. How many apples did she buy?

Givens: Bridget gave half of the apples to Ann; After that, she gave $3$ apples to Cassie; She kept $4$ apples for herself; Answer choices: (A) $3$, (B) $4$, (C) $7$, (D) $11$, (E) $14$

Unknowns: The original number of apples Bridget bought

Understand

Restated: Bridget bought some apples. She gave half of them to Ann, then gave $3$ to Cassie, and kept the last $4$ for herself. How many apples did she buy?

Givens: Bridget gave half of the apples to Ann; After that, she gave $3$ apples to Cassie; She kept $4$ apples for herself; Answer choices: (A) $3$, (B) $4$, (C) $7$, (D) $11$, (E) $14$

Plan

Primary tool: #11 Work Backwards

Secondary: #7 Identify Subproblems

We know the final amount ($4$ apples kept) and the operations that led to it (gave half away, then gave $3$ more). Tool #11 (Work Backwards) is built for exactly this shape: start at the end and undo each step. Tool #7 (Identify Subproblems) helps name the two stages cleanly — "after Ann" and "after Cassie" — so each reversal is a single, simple operation.

Execute — Answer: E

#7 Identify Subproblems 4.OA.A.3 Step 1
  • Name the two stages so the reversal stays clean.
  • Let $A$ = apples Bridget had right after giving Ann's share, and let $T$ = the original total.
  • The order of events was $T \to A \to 4$.
$$T \xrightarrow{\text{give half to Ann}} A \xrightarrow{\text{give }3\text{ to Cassie}} 4$$

💡 Splitting the chain of events into named stages is the Tool #7 (subproblems) move — each arrow becomes one small undo step.

#11 Work Backwards 3.OA.D.8 Step 2
  • Undo the last step.
  • After Cassie got $3$, Bridget had $4$ left.
  • So right before giving Cassie those $3$ apples, Bridget had $4 + 3 = 7$ apples — that is $A$.
$$A = 4 + 3 = 7$$

💡 Reversing a "gave away $3$" event with a $+3$ addition is the first step of a two-step word problem.

#11 Work Backwards 3.OA.A.3 Step 3
  • Undo the first step.
  • $A = 7$ is what was left after giving half to Ann, so $7$ is exactly half of the original.
  • The original total is double that.
$$T = 2 \times A = 2 \times 7 = 14 \;\Rightarrow\; \textbf{(E)}$$

💡 Undoing "take half" is the same as multiplying by $2$ — a Grade 3 multiplication-within-$100$ idea.

[1] #7 4.OA.A.3 Name the two stages so the reversal stays clean. Let $A$ = apples Bridget had ri
[2] #11 3.OA.D.8 Undo the last step. After Cassie got $3$, Bridget had $4$ left. So right before
[3] #11 3.OA.A.3 Undo the first step. $A = 7$ is what was left after giving half to Ann, so $7$ i

Review

Reasonableness: Run the story forward with $T = 14$ to check: Bridget starts with $14$, gives half ($7$) to Ann and keeps $7$, gives $3$ to Cassie and keeps $7 - 3 = 4$. That matches the problem exactly, so $14$ is correct. The other choices fail fast: $3$, $4$, $7$, $11$ are not even, or do not leave $4$ after the two giveaways.

Alternative: Tool #6 (Guess and Check) on the choices: only an even number can be split in half, which kills (A) $3$, (C) $7$, (D) $11$. Try (B) $4$: half is $2$, then giving $3$ is impossible — fails. Try (E) $14$: half is $7$, minus $3$ leaves $4$ — matches. Answer (E).

CCSS standards used (min grade 4)

  • 3.OA.D.8 Solve two-step word problems using the four operations (Reversing the two events (subtract Cassie's $3$, then double to undo Ann's half) is a two-step word problem using addition and multiplication.)
  • 3.OA.A.3 Use multiplication and division within 100 to solve word problems (Doubling $7$ to recover the original total ($2 \times 7 = 14$) is multiplication within $100$ in a word-problem context.)
  • 4.OA.A.3 Solve multistep word problems with whole numbers, including problems with an unknown quantity (Treating the original number of apples as an unknown to be recovered through a chain of operations is exactly the Grade 4 multistep word-problem standard.)

⭐ This AMC 8 problem only needs Grade 4 "work backwards" reasoning — undo each step from the end — that you already know!

⭐ This AMC 8 problem only needs Grade 4 "work backwards" reasoning — undo each step from the end — that you already know!

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