AMC 10 · 2017 · #1

Easy mode Grade 3

Problem

Mary picked a two-digit number. She multiplied it by $3$ , then added $11$ . She switched the two digits of her answer and got a number from $71$ to $75$ . What number did Mary start with?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2017 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: Mary picked a positive two-digit number, multiplied it by $3$, and added $11$. She then switched the two digits of that result and got a number from $71$ to $75$, inclusive. Find Mary's original number.

Givens: Mary's number is a positive two-digit number; She computes $3 \times (\text{number}) + 11$; Switching the digits of that result gives a number between $71$ and $75$ inclusive; Answer choices: (A) $11$, (B) $12$, (C) $13$, (D) $14$, (E) $15$

Unknowns: Mary's original two-digit number

Understand

Plan

Primary tool: #11 Work Backwards

Secondary: #6 Guess and Check

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 $\times 3$. Tool #6 (Guess and Check) closes the problem by running the chain forward on the survivor to confirm it lands in the target range.

Execute — Answer: B

#11 Work Backwards 1.NBT.B.2 Step 1

Switching the digits is its own undo: switch them back.
So the result of $3 \times (\text{number}) + 11$, before Mary flipped it, was a number whose digits flip to land in $71$–$75$.
Flipping each of $71, 72, 73, 74, 75$ gives the possible values of $3 \times (\text{number}) + 11$.

$$71{\to}17,\;72{\to}27,\;73{\to}37,\;74{\to}47,\;75{\to}57$$

💡 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.

#11 Work Backwards 3.OA.C.7 Step 2

Now undo the two operations on each candidate: subtract $11$, then divide by $3$.
A valid start must come out as a whole two-digit number.
Only $47$ works: $47 - 11 = 36$ and $36 \div 3 = 12$.
The others fail — $17{\to}6{\to}2$ is one digit, and $27, 37, 57$ minus $11$ give $16, 26, 46$, none divisible by $3$.

$$47 - 11 = 36,\quad 36 \div 3 = 12$$

💡 Undoing "times $3$, then plus $11$" means peeling the operations off in reverse order: subtract first, then divide.

#6 Guess and Check 3.NBT.A.2 Step 3

Check $12$ by running the chain forward: triple it, add $11$, then switch the digits.
The result is $74$, which sits inside $71$–$75$, so $12$ is Mary's number and the answer is (B).

$$12 \times 3 = 36,\; 36 + 11 = 47,\; 47 \to 74 \;\Rightarrow\; \textbf{(B)}$$

💡 Running the only survivor through the original steps is the fastest way to be sure nothing went wrong.

[1] #11 1.NBT.B.2 Switching the digits is its own undo: switch them back. So the result of $3 \tim

[2] #11 3.OA.C.7 Now undo the two operations on each candidate: subtract $11$, then divide by $3$

[3] #6 3.NBT.A.2 Check $12$ by running the chain forward: triple it, add $11$, then switch the di

Review

Reasonableness: The forward check is airtight: $12 \to 36 \to 47 \to 74$, and $71 \le 74 \le 75$. Every other choice fails the divisibility-by-$3$ test in the backward step, so $12$ is the unique answer, matching (B).

Alternative: Tool #6 (Guess and Check) on its own: just test the five choices. $11 \to 44 \to 44$, $12 \to 47 \to 74$, $13 \to 50 \to 05$, $14 \to 53 \to 35$, $15 \to 56 \to 65$. Only $12$ produces a switched number in $71$–$75$, confirming (B).

CCSS standards used (min grade 3)

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones (Switching the digits of a two-digit number, and switching them back, by treating it as a tens digit and a ones digit.)
3.OA.C.7 Fluently multiply and divide within 100 (Undoing the $\times 3$ by dividing by $3$ (and checking divisibility), and tripling a candidate in the forward check.)
3.NBT.A.2 Fluently add and subtract within 1000 (Undoing the $+11$ by subtracting $11$, and adding $11$ when running the chain forward to verify.)

⭐ When a problem hands you the ending and asks for the start, undo each step in reverse order — subtract before you divide.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 3)

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