Competition · AMC preparation · step 4 of 4

AMC 8 · 1999 · #1

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

order-of-operationsmental-arithmeticsystematic-enumeration guess-and-checksystematic-enumeration ↑ Prerequisites: multi-digit-arithmeticorder-of-operations

📏 Short solution 💡 2 insights

Problem

In the equation (6 ? 3) + 4 - (2 - 1) = 5, which of ÷, ×, +, -, or "None of these" should replace the question mark to make the statement true?

Pick an answer.

(A)

$\div$

(B)

$\times$

(C)

(D)

(E)

None of these

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

The equation already tells us the final value is 5. Tool #11 (Work Backwards) lets us undo the known pieces — first simplify (2-1), then peel off the +4 - 1 — until only the mystery expression (6 ? 3) is left equal to a number. Once we know what (6 ? 3) must equal, Tool #6 (Guess and Check) tests each of the four operators on the small numbers 6 and 3 — a one-line check per option.

1STEP 1

Simplify inside the parentheses

Simplify the second parenthesis first: (2 - 1) = 1, so the equation becomes (6 ? 3) + 4 - 1 = 5.

(6 ? 3) + 4 - 1 = 5

2STEP 2

Combine the known numbers

Combine the known numbers: 4 - 1 = 3, so the equation now reads (6 ? 3) + 3 = 5.

(6 ? 3) + 3 = 5

3STEP 3

Subtract 3 from both sides

Subtract 3 from both sides to isolate the mystery expression: (6 ? 3) = 2.

(6 ? 3) = 5 - 3 = 2

4STEP 4

Test each operator

Test each operator on 6 and 3 for the value 2: only 6 ÷ 3 = 2 works; the others give 18, 9, and 3.

6 ÷ 3 = 2 ✓, 6 × 3 = 18, 6 + 3 = 9, 6 - 3 = 3 → (A)

Answer

Plug ÷ back into the original equation: (6 ÷ 3) + 4 - (2 - 1) = 2 + 4 - 1 = 5. The left side equals the right side, so (A) is confirmed. The other three operators all fail the same check: (6 × 3) + 4 - 1 = 21, (6 + 3) + 4 - 1 = 12, (6 - 3) + 4 - 1 = 6. None equals 5, so (E) "None of these" is also ruled out.

💡Key takeaway

Clean up the parts you already know, see what the mystery piece has to equal, then try each operator. AMC 8 #1 is a Grade 6 balance-the-equation problem with a Grade 3 fact check at the end.

Simplify inside the parentheses
Combine the known numbers
Subtract 3 from both sides
Test each operator

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Simplify inside the parentheses

Combine the known numbers

Subtract 3 from both sides

Test each operator

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