AMC 8 · 2009 · #1

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

multi-digit-arithmeticmental-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

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?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

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.

1STEP 1

Name two stages: T = the original total, A = what's left after Ann's half, giving the chain T → A → 4.

T → A → 4

2STEP 2

Undo the last event: before giving Cassie 3, Bridget had 4 + 3 = 7 apples — that is A.

A = 4 + 3 = 7

3STEP 3

A = 7 is only half the original, so double it: T = 2 × 7 = 14 → (E).

T = 2 × A = 2 × 7 = 14 → (E)

Answer

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.

💡Key takeaway

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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