Competition · AMC preparation · step 4 of 4
AMC 8 · 2001 · #12
Grade 6 arithmeticalgebraPick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The expression has a clean inside-then-outside shape thanks to the parentheses, so Tool #7 (Identify Subproblems) is the natural fit: first compute the inner 6 ⊗ 4, then plug that result into the outer ⊗ 3. Each ⊗ is just a tiny recipe — read the letters a and b, line them up with the actual numbers on either side, and substitute. Tool #3 (Eliminate Possibilities) is the AMC multiple-choice safety net: after computing, check that the value matches one of the five offered choices and that the trap distractors can be ruled out.
Work the inner bracket first
Read the parentheses first: split (6 ⊗ 4) ⊗ 3 into an inner job and an outer job.
Reading the parentheses first is the Grade 5 "use parentheses in numerical expressions" rule, and it gives us our two subproblems.
5.OA.A.1Identify SubproblemsApply the rule to 6 and 4
Apply the ⊗ rule to the inner pair with a = 6, b = 4: 6 ⊗ 4 = .
Plugging numbers into letters a and b of a defined formula is exactly Grade 6 "evaluate expressions where letters stand for numbers".
The inner operation evaluates as 6 ⊗ 4 = (6+4)/(6-4) = 5.
▸ Why?
6 ⊗ 4 and (6+4)/(6-4) name the same value, because ⊗ is defined to be (a+b)/(a-b) and we have written 6 where a stands and 4 where b stands, so wherever 6 ⊗ 4 appears we may put (6+4)/(6-4) instead.
▸ Why?
(6+4)/(6-4) = 10/2, because the top joins 6 and 4 into a single whole and the bottom takes 4 away from 6.
▸ Why?
6+4 = 10: a part of size 6 and a part of size 4, with nothing missing or counted twice, combine into one whole of 10.
▸ Why?
6-4 = 2: taking 4 from 6 asks what must be added back to 4 to rebuild 6, and that missing amount is 2.
▸ Why?
10/2 = 5: sharing 10 into equal groups of 2 asks what number multiplied by 2 gives 10, since dividing reverses multiplying.
Simplify the fraction
Do the arithmetic: = = 5, so the inner ⊗ equals 5.
The numerator and denominator are simple whole-number sums and differences, and 10 ÷ 2 = 5 is Grade 6 whole-number division.
6.NS.B.2Identify SubproblemsFeed the result back in
Substitute back — now 5 ⊗ 3 = = = 4.
We reuse the same Grade 6 substitution move on the outer ⊗, and the arithmetic finishes the work.
6.EE.A.2Identify SubproblemsMatch against the choices
Only choice (A) equals 4; (E) 72 is the ⊗-as-multiply trap 6·4·3, and (C) 15 is just adding everything.
Lining the result up against the five choices is the AMC multiple-choice habit and catches the "⊗ is just times" trap labelled (E).
5.OA.A.1Eliminate PossibilitiesThis AMC 8 problem only needs Grade 6 expression evaluation — plug numbers into the letters of a formula, then do it once more — that you already know!
- Work the inner bracket first
- Apply the rule to 6 and 4
- Simplify the fraction
- Feed the result back in
- Match against the choices
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