AMC 8 · 2005 · #1

Grade 3 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

linear-equations-one-varmulti-digit-arithmeticmental-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticmental-arithmetic

📏 Short solution 💡 2 insights

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Problem

Connie multiplied some number by 2 and got 60. She was supposed to divide that same number by 2 instead. What answer should she have gotten?

Pick an answer.

(A)

7.5

(B)

(C)

(D)

120

(E)

240

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backward

We don't know the mystery number, but we do know what happened after Connie multiplied it by 2. Tool #8 (Work Backward) says: undo the wrong operation to recover the original number, then apply the right one. Tool #3 (Set Up an Equation) gives us the matching algebra: write 2 × n = 60, solve for n, then compute n ÷ 2. The two tools point at the same plan — reverse the wrong step, then take the correct step.

1STEP 1

Call the mystery number n; Connie's wrong step multiplied it by 2 to get 60.

2 × n = 60

2STEP 2

Undo the ×2 with its inverse — divide both sides by 2 — to recover n = 30.

n = 60 ÷ 2 = 30

3STEP 3

Now do Connie's intended step — divide n by 2: 30 ÷ 2 = 15 → (B).

n ÷ 2 = 30 ÷ 2 = 15 → (B)

Answer

Check the chain: 30 × 2 = 60 (matches Connie's wrong result), and 30 ÷ 2 = 15 (the correct answer). The wrong answer 60 should be four times the correct answer, because multiplying by 2 instead of dividing by 2 scales up by a factor of 4 — and indeed 60 = 4 × 15. Choice (C) 30 is a trap (that's the mystery number, not the answer); (D) 120 doubles again, (E) 240 quadruples — both go the wrong direction.

💡Key takeaway

When someone uses the wrong operation, undo it first to find the hidden number, then do the right operation. Here, × 2 undone gives 30, and the correct ÷ 2 gives 15.

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