AMC 10 · 2002 · #2
Grade 5 arithmeticPick an answer.
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.
Recover the original number
Undo her steps backwards: multiply to reverse the division, 43 × 3 = 129, then add 9 to get 138.
Running a chain of operations backwards with their opposites lands you exactly where you started.
Running the wrong chain backwards with each operation's opposite recovers the number she started with.
▸ Why?
Subtracting undoes adding and dividing undoes multiplying, so each step can be reversed exactly.
▸ Why?
The chain is its steps applied one after another, so undoing them in reverse order returns the whole chain to its start.
Apply the correct procedure
Now run 138 through the intended recipe: 138 - 3 = 135, then 135 ÷ 9 = 15.
Once the starting number is known, the correct result is just a matter of doing the right two operations in order.
5.NBT.B.6Identify SubproblemsMatch to a choice
The result 15 is choice (A); 138 and 43 sit in the list to catch stopping early.
The question asks for the corrected answer, not the original number or the wrong answer.
4.OA.A.3Identify SubproblemsWhen 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.
- Recover the original number
- Apply the correct procedure
- Match to a choice