AMC 10 · 2017 · #1
Grade 3 arithmeticPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We are told the END of the chain (the switched number lands in 71–75) and asked for the START (Mary's number), so Tool #11 (Work Backwards) fits exactly: un-switch the digits, undo the +11, then undo the × 3. Tool #6 (Guess and Check) closes the problem by running the chain forward on the survivor to confirm it lands in the target range.
Switch the digits back
Switching digits undoes itself, so flip 71–75 back — those flips are the possible values of 3 × (number) + 11.
A two-digit number is just a tens digit and a ones digit, so switching them is a move you can simply do in reverse.
1.NBT.B.2Work BackwardsUndo the two operations
Undo each candidate — subtract 11, then divide by 3 — and only 47 gives a whole two-digit start: 47 − 11 = 36, 36 ÷ 3 = 12.
Undoing "times 3, then plus 11" means peeling the operations off in reverse order: subtract first, then divide.
Undoing a chain of operations means peeling them off in reverse order, opposite by opposite.
▸ Why?
Each operation has an opposite that exactly cancels it, so the original number returns.
▸ Why?
A two-digit number is just a tens digit and a ones digit, so swapping them is its own reverse.
Check by running forward
Run 12 forward — 12 × 3 = 36, + 11 = 47, switch to 74 — 74 lands in 71–75, confirming (B).
Running the only survivor through the original steps is the fastest way to be sure nothing went wrong.
3.NBT.A.2Guess And CheckWhen a problem hands you the ending and asks for the start, undo each step in reverse order — subtract before you divide.
- Switch the digits back
- Undo the two operations
- Check by running forward