AMC 10 · 2002 · #6

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

order-of-operationslinear-equations-one-var work-backwards ↑ Prerequisites: order-of-operations

📏 Medium solution 💡 2 insights

Problem

Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

138

AMC 10 2002 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: The correct instructions were: take a certain number, subtract 3, then divide by 9. Cindy did the wrong operations instead — she subtracted 9 and then divided by 3 — and got 43. Find what she would have gotten if she had followed the correct instructions.

Givens: The correct procedure is: subtract 3 from the number, then divide by 9; Cindy's wrong procedure was: subtract 9 from the number, then divide by 3; Cindy's wrong procedure gave a result of $43$; Both procedures start from the same original number; Answer choices: (A) $15$, (B) $34$, (C) $43$, (D) $51$, (E) $138$

Unknowns: The result of applying the correct procedure to the original number

Understand

Plan

Primary tool: #11 Work Backwards

Secondary: #7 Identify Subproblems, #13 Convert to Algebra

The correct answer depends on the original number, which is hidden — but we are handed the END of Cindy's wrong procedure (the result $43$). Tool #11 (Work Backwards) is built for exactly this: run her two operations in reverse, undoing each with its opposite, to recover the starting number. Tool #7 (Identify Subproblems) then makes the job two clean stages: first recover the number, then feed that number through the correct procedure. The tempting trap is to answer $138$ (choice E), which is only the original number — not the corrected result the question actually asks for.

Execute — Answer: A

#11 Work Backwards 4.OA.A.3 Step 1

Recover the original number

Cindy took the number, subtracted 9, then divided by 3 to get 43.
To find the number, undo those steps in reverse order using opposite operations.
The last thing she did was divide by 3, so undo it by multiplying: $43 \times 3 = 129$.
Before that she subtracted 9, so undo it by adding: $129 + 9 = 138$.
The original number is $138$.

$$43 \times 3 = 129,\quad 129 + 9 = 138$$

💡 Running a chain of operations backwards with their opposites lands you exactly where you started.

#7 Identify Subproblems 5.NBT.B.6 Step 2

Apply the correct procedure

Now put $138$ through the instructions Cindy was supposed to follow: first subtract 3, giving $138 - 3 = 135$; then divide by 9, giving $135 \div 9 = 15$.
So the correct answer is $15$.

$$138 - 3 = 135,\quad 135 \div 9 = 15$$

💡 Once the starting number is known, the correct result is just a matter of doing the right two operations in order.

#7 Identify Subproblems 4.OA.A.3 Step 3

Match to a choice

The correct result is $15$, which is choice (A).
Note that $138$ appears as choice (E) to catch anyone who stops at the recovered number, and $43$ is choice (C) to catch anyone who reports Cindy's wrong result.

$$15 = \textbf{(A)}$$

💡 The question asks for the corrected answer, not the original number or the wrong answer.

[1] #11 4.OA.A.3 Cindy took the number, subtracted 9, then divided by 3 to get 43. To find the nu

[2] #7 5.NBT.B.6 Now put $138$ through the instructions Cindy was supposed to follow: first subtr

[3] #7 4.OA.A.3 The correct result is $15$, which is choice (A). Note that $138$ appears as choi

Review

Reasonableness: Check the recovered number forward through Cindy's wrong steps: $138 - 9 = 129$, and $129 \div 3 = 43$ — exactly her stated result, so $138$ is correct. Then the correct steps give $(138 - 3) \div 9 = 135 \div 9 = 15$, and $135$ divides evenly by $9$, as clean whole-number arithmetic should. The corrected answer $15$ is much smaller than the wrong answer $43$, which makes sense: the correct procedure divides by the larger divisor $9$ instead of $3$.

Alternative: Use algebra (tool #13). Let the number be $x$. Cindy's wrong procedure gives $\frac{x-9}{3} = 43$, so $x - 9 = 129$ and $x = 138$. Substitute into the correct procedure: $\frac{138-3}{9} = \frac{135}{9} = 15$, again choice (A).

CCSS standards used (min grade 5)

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Reversing Cindy's two-step procedure with opposite operations to recover the number $138$, and interpreting which quantity the question actually asks for.)
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors (Carrying out the correct procedure on $138$: subtracting 3 and dividing $135 \div 9 = 15$.)

⭐ When you know the end result, undo each step with its opposite to get back to the start — then read the question again to answer what it truly asks.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Recover the original number

Apply the correct procedure

Match to a choice

Review

CCSS standards used (min grade 5)

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