AMC 8 · 2023 · #1
Easy mode Grade 5Problem
We have two little number puzzles to work out, then we subtract one from the other.
The first puzzle is .
The second puzzle is .
What do you get when you take the second answer away from the first?
Pick an answer.
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
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.
💡 Inside parentheses we still follow the rule: multiply before add.
5.OA.A.1 Step 2 - Evaluate the second part $(8 + 4 \times 2)$.
- Again, compute $4 \times 2$ first, then add 8.
💡 Order of operations applies inside parentheses too, even when the multiplication isn't written first.
3.NBT.A.2 Step 3 Combine the two subresults with the final subtraction.
💡 Subtracting two-digit numbers is basic Grade 3 fluency.
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.
💡 On a multiple-choice problem, always verify the computed value lands on one of the listed options.
5.OA.A.1 Evaluate the first part $(8 \times 4 + 2)$. By the order of operations, do the m 5.OA.A.1 Evaluate the second part $(8 + 4 \times 2)$. Again, compute $4 \times 2$ first, 3.NBT.A.2 Combine the two subresults with the final subtraction. 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.1Use 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.2Fluently 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!
⭐ This AMC 8 problem only needs Grade 5 parentheses and order of operations you already know!
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