AMC 8 · 2023 · #1

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

order-of-operationsmulti-digit-arithmeticmental-arithmetic identify-subproblems ↑ Prerequisites: order-of-operationsmulti-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)$ ?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2023 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.

View mode:

Toolkit + CCSS Solution

Understand

Restated: Compute $(8 \times 4 + 2)$ and $(8 + 4 \times 2)$ separately, then subtract the second value from the first to get the final answer.

Givens: Expression: $(8 \times 4 + 2) - (8 + 4 \times 2)$; Numbers used: 8, 4, 2; Operations: multiplication, addition, subtraction

Unknowns: The value of the whole expression

Understand

Restated: Compute $(8 \times 4 + 2)$ and $(8 + 4 \times 2)$ separately, then subtract the second value from the first to get the final answer.

Givens: Expression: $(8 \times 4 + 2) - (8 + 4 \times 2)$; Numbers used: 8, 4, 2; Operations: multiplication, addition, subtraction

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

Doing the whole expression in one shot invites mistakes. The expression already splits into two parenthesized parts, so we evaluate each small piece independently and finish with one subtraction. Since the problem is multiple choice, we can also match the computed value to the answer choices as a final check.

Execute — Answer: D

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

Evaluate the first part $(8 \times 4 + 2)$.
By the order of operations, do the multiplication before the addition.

$$8 \times 4 + 2 = 32 + 2 = 34$$

💡 Inside parentheses we still follow the rule: multiply before add.

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

Evaluate the second part $(8 + 4 \times 2)$.
Again, compute $4 \times 2$ first, then add 8.

$$8 + 4 \times 2 = 8 + 8 = 16$$

💡 Order of operations applies inside parentheses too, even when the multiplication isn't written first.

#7 Identify Subproblems 3.NBT.A.2 Step 3

Combine the two subresults with the final subtraction.

$$34 - 16 = 18$$

💡 Subtracting two-digit numbers is basic Grade 3 fluency.

#3 Eliminate Possibilities 5.OA.A.1 Step 4

The value 18 matches choice (D) exactly.
Choices (A) 0, (B) 6, (C) 10, and (E) 24 don't match the computation, so we can eliminate them.

$$18 = \textbf{(D)}$$

💡 On a multiple-choice problem, always verify the computed value lands on one of the listed options.

[1] #7 5.OA.A.1 Evaluate the first part $(8 \times 4 + 2)$. By the order of operations, do the m

[2] #7 5.OA.A.1 Evaluate the second part $(8 + 4 \times 2)$. Again, compute $4 \times 2$ first,

[3] #7 3.NBT.A.2 Combine the two subresults with the final subtraction.

[4] #3 5.OA.A.1 The value 18 matches choice (D) exactly. Choices (A) 0, (B) 6, (C) 10, and (E) 2

Review

Reasonableness: The two parentheses look similar but their multiplications differ. $8 \times 4 = 32$ is much larger than $4 \times 2 = 8$, so the first parenthesis should be noticeably bigger than the second, and the difference should be sizable. Getting $34 - 16 = 18$ — a moderately large positive value — fits that expectation.

Alternative: Tool 6 (Guess & Check) could also work: estimate each parenthesis mentally and match to the choices. But because the expression is short and clearly splits in two, the subproblems approach is the cleanest.

CCSS standards used (min grade 5)

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions and evaluate (Applied the order of operations (multiplication before addition) inside each parenthesis to evaluate the numerical expression.)
3.NBT.A.2 Fluently add and subtract within 1000 (Used to carry out the final subtraction $34 - 16$ between the two evaluated parentheses.)

⭐ This AMC 8 problem only needs Grade 5 parentheses and order of operations you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 5)

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