AMC 8 · 2017 · #1

Easy mode Grade 5

Problem

Look at the five expressions below. Each one mixes the digits $2$ , $0$ , $1$ , $7$ with some addition signs and multiplication signs.

Work out the value of each expression one at a time. Remember the rule: multiplication is done before addition.

Which expression gives the biggest value?

$\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 + 7\qquad\textbf{(D) }2+0+1 \times 7\qquad\textbf{(E) }2 \times 0 \times 1 \times 7$

Pick an answer.

(A)

2+0+1+7

(B)

2 imes 0 +1+7

(C)

2+0 imes 1 + 7

(D)

2+0+1 imes 7

(E)

2 imes 0 imes 1 imes 7

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Toolkit + CCSS Solution

Understand

Restated: Five short arithmetic expressions are listed as answer choices (A)-(E). Each one uses the digits $2, 0, 1, 7$ glued together with $+$ and $\times$ signs in different spots. The job is to evaluate every expression using the standard order of operations and pick the one with the largest value.

Givens: (A) $2+0+1+7$; (B) $2 \times 0 + 1 + 7$; (C) $2 + 0 \times 1 + 7$; (D) $2 + 0 + 1 \times 7$; (E) $2 \times 0 \times 1 \times 7$

Unknowns: Which of the five expressions evaluates to the largest number

Understand

Givens: (A) $2+0+1+7$; (B) $2 \times 0 + 1 + 7$; (C) $2 + 0 \times 1 + 7$; (D) $2 + 0 + 1 \times 7$; (E) $2 \times 0 \times 1 \times 7$

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #7 Identify Subproblems

The five answer choices ARE the five candidates to test, which is the textbook setup for Tool #3 (Eliminate Possibilities) — compute each candidate's value, then keep only the winner. Tool #7 (Identify Subproblems) is the natural helper: each expression is its own self-contained mini-arithmetic problem, so we solve five tiny subproblems and compare. Algebra (Tool #13) is overkill here; the work is pure order-of-operations bookkeeping.

Execute — Answer: A

#7 Identify Subproblems 2.NBT.B.5 Step 1

Evaluate (A) $2+0+1+7$.
Only addition, so the order does not matter.
Add the four numbers.

$$2+0+1+7 = 10$$

💡 Adding four small whole numbers within $20$ is a Grade 2 fluency skill.

#7 Identify Subproblems 5.OA.A.1 Step 2

Evaluate (B) $2 \times 0 + 1 + 7$.
Order of operations says do the multiplication first: $2 \times 0 = 0$.
Then add the rest.

$$2 \times 0 + 1 + 7 = 0 + 1 + 7 = 8$$

💡 Knowing to do $\times$ before $+$ in a mixed expression is the Grade 5 order-of-operations idea.

#7 Identify Subproblems 5.OA.A.1 Step 3

Evaluate (C) $2 + 0 \times 1 + 7$.
Multiplication first: $0 \times 1 = 0$.
Then add.

$$2 + 0 \times 1 + 7 = 2 + 0 + 7 = 9$$

💡 The hidden $0 \times 1$ in the middle only "counts" after order of operations is applied.

#7 Identify Subproblems 5.OA.A.1 Step 4

Evaluate (D) $2 + 0 + 1 \times 7$.
Multiplication first: $1 \times 7 = 7$.
Then add.
Note: without order of operations a student might wrongly compute left-to-right and get $(2+0+1) \times 7 = 21$ — that wrong path is exactly what this choice is testing.

$$2 + 0 + 1 \times 7 = 2 + 0 + 7 = 9$$

💡 Order of operations protects you from the $21$ trap and gives the correct value $9$.

#7 Identify Subproblems 3.OA.B.5 Step 5

Evaluate (E) $2 \times 0 \times 1 \times 7$.
Any product that contains $0$ is $0$, so this expression is $0$ no matter how the multiplications are grouped.

$$2 \times 0 \times 1 \times 7 = 0$$

💡 The zero-product property — "any number times $0$ is $0$" — is a Grade 3 property of multiplication.

#3 Eliminate Possibilities 4.NBT.A.2 Step 6

Compare the five values and eliminate everything except the largest.
Values are (A) $10$, (B) $8$, (C) $9$, (D) $9$, (E) $0$.
The largest is $10$, so (A) wins.

$$\max(10, 8, 9, 9, 0) = 10 \;\Rightarrow\; \textbf{(A)}$$

💡 Comparing five whole numbers within $20$ to pick the largest is a Grade 4 multi-digit comparison skill.

[1] #7 2.NBT.B.5 Evaluate (A) $2+0+1+7$. Only addition, so the order does not matter. Add the fou

[2] #7 5.OA.A.1 Evaluate (B) $2 \times 0 + 1 + 7$. Order of operations says do the multiplicatio

[3] #7 5.OA.A.1 Evaluate (C) $2 + 0 \times 1 + 7$. Multiplication first: $0 \times 1 = 0$. Then

[4] #7 5.OA.A.1 Evaluate (D) $2 + 0 + 1 \times 7$. Multiplication first: $1 \times 7 = 7$. Then

[5] #7 3.OA.B.5 Evaluate (E) $2 \times 0 \times 1 \times 7$. Any product that contains $0$ is $0

[6] #3 4.NBT.A.2 Compare the five values and eliminate everything except the largest. Values are

Review

Reasonableness: Intuitively, (A) keeps every digit alive by only adding, while every other choice either multiplies by $0$ (which zeroes out a term) or replaces an addition with a multiplication-by-$1$ (which doesn't help). So pure addition should be the biggest, and indeed $10$ beats $8, 9, 9, 0$. The answer (A) matches that intuition.

Alternative: Tool #5 (Look for a Pattern) gives a quick shortcut: any expression that includes $\times 0$ at all (choices B and E) collapses that whole product to $0$, losing value compared to addition. Multiplying by $1$ (choices C and D) is at best neutral but still kills the $+$ contribution of that term. So the all-addition choice (A) is structurally guaranteed to be the largest — no need to compute any choice in detail.

CCSS standards used (min grade 5)

2.NBT.B.5 Fluently add and subtract within 100 (Computing the all-addition value of choice (A): $2+0+1+7 = 10$.)
3.OA.B.5 Apply properties of operations as strategies to multiply and divide (Using the zero-product property to see that $2 \times 0 \times 1 \times 7 = 0$ for choice (E).)
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols (Comparing the five computed values $\{10, 8, 9, 9, 0\}$ to identify the largest.)
5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate (Applying the order of operations (multiplication before addition) to choices (B), (C), and (D), which is what makes this problem genuinely a Grade 5 problem rather than a Grade 2 addition exercise.)

⭐ This AMC 8 problem only needs Grade 5 order of operations (multiplication before addition) that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 5)

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