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
📘 View easy version →
Problem
Compute (8 × 4 + 2) and (8 + 4 × 2) separately, then subtract the second value from the first to get the final answer.

Pick an answer.

(A)
0
(B)
6
(C)
10
(D)
18
(E)
24

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

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.

1STEP 1

First part: multiply before adding, so (8 × 4 + 2) gives 34.

8 × 4 + 2 = 32 + 2 = 34
2STEP 2

Second part: compute 4 × 2 first, then add 8, giving 16.

8 + 4 × 2 = 8 + 8 = 16
3STEP 3

Combine the two subresults with the final subtraction: 34 - 16 = 18.

34 - 16 = 18
4STEP 4

The computed value lands exactly on choice (D); the others don't match, so eliminate them.

18 = (D)
Answer
18
The two parentheses look similar but their multiplications differ. 8 × 4 = 32 is much larger than 4 × 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.
💡Key takeaway

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

More like this

Same archetype — closest grade level first.