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

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Problem

Connie multiplies a number by $2$ and gets $60$ as her answer. However, she should have divided the number by $2$ to get the correct answer. What is the correct answer?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Connie's wrong calculation: (mystery number) $\times 2 = 60$; Connie's correct calculation should be: (mystery number) $\div 2$; Answer choices: (A) $7.5$, (B) $15$, (C) $30$, (D) $120$, (E) $240$

Unknowns: The correct answer — the result of dividing the mystery number by $2$

Understand

Restated: 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?

Plan

Primary tool: #8 Work Backward

Secondary: #3 Set Up an Equation

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 $2 \times n = 60$, solve for $n$, then compute $n \div 2$. The two tools point at the same plan — reverse the wrong step, then take the correct step.

Execute — Answer: B

#3 Set Up an Equation 3.OA.A.4 Step 1

Set up the equation for the wrong calculation.
Call the mystery number $n$.
Connie multiplied $n$ by $2$ and got $60$.

$$2 \times n = 60$$

💡 Grade 3 thinks of this as a missing-factor question: $2 \times \square = 60$.

#8 Work Backward 3.OA.B.6 Step 2

Work backward to recover $n$.
The inverse of multiplying by $2$ is dividing by $2$, so divide both sides by $2$.

$$n = 60 \div 2 = 30$$

💡 Division undoes multiplication. If doubling gave $60$, the original was half of $60$.

#8 Work Backward 3.OA.A.2 Step 3

Now do the calculation Connie was supposed to do: divide the mystery number $n = 30$ by $2$.

$$n \div 2 = 30 \div 2 = 15 \;\Rightarrow\; \textbf{(B)}$$

💡 Splitting $30$ into $2$ equal groups gives $15$ in each group.

[1] #3 3.OA.A.4 Set up the equation for the wrong calculation. Call the mystery number $n$. Conn

[2] #8 3.OA.B.6 Work backward to recover $n$. The inverse of multiplying by $2$ is dividing by $

[3] #8 3.OA.A.2 Now do the calculation Connie was supposed to do: divide the mystery number $n =

Review

Reasonableness: Check the chain: $30 \times 2 = 60$ (matches Connie's wrong result), and $30 \div 2 = 15$ (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 $60 = 4 \times 15$. 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.

Alternative: Tool #10 (Use a Related Problem): notice that the correct answer is what you get by dividing $n$ by $2$, while Connie's $60$ is what you get by multiplying $n$ by $2$. So the correct answer is $60$ divided by $4$ (because going from "$\times 2$" to "$\div 2$" multiplies the result by $\tfrac{1/2}{2} = \tfrac{1}{4}$). Then $60 \div 4 = 15$, giving (B) directly — no need to find $n$.

CCSS standards used (min grade 3)

3.OA.A.2 Interpret whole-number quotients of whole numbers (Reading $30 \div 2 = 15$ as splitting $30$ into $2$ equal groups of $15$.)
3.OA.A.4 Determine the unknown whole number in a multiplication or division equation (Writing the wrong calculation as $2 \times n = 60$ with $n$ unknown.)
3.OA.B.6 Understand division as an unknown-factor problem (Recovering $n$ by undoing the multiplication: $n = 60 \div 2 = 30$.)

⭐ When someone uses the wrong operation, undo it first to find the hidden number, then do the right operation. Here, $\times 2$ undone gives $30$, and the correct $\div 2$ gives $15$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 3)

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