Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #10
Grade 6 algebraPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
We are told the final value (the whole expression equals 1) and we have to recover the starting unknown x at the deepest layer — this is the textbook trigger for Tool #11 (Work Backwards). Tool #7 (Identify Subproblems) helps name the inner piece y = 5 * x so we only deal with one layer at a time: first peel the outer 2 * y = 1, then the inner 5 * x = y. We keep Tool #3 (Eliminate) on the bench to verify against the five answer choices once we have a candidate.
Name the inner result
Name the inner operation y = 5 * x, so the outer equation becomes the single-layer 2 * y = 1.
Giving the messy sub-expression its own name is the Tool #7 "identify subproblems" move — now there is only one * to unfold at a time.
6.EE.A.2Identify SubproblemsUndo the outer operation
Undo the outer layer: 2 * y = 1 means 6 - y = 1, so y = 5.
Knowing the end value (1) and undoing the last step is Tool #11 (Work Backwards) — it converts the outer layer into a simple one-step equation.
Because the rule a * b = 3a - b computes 2 * (5 * x) as 6 - (5 * x), the equation 2 * (5 * x) = 1 forces the inner value 5 * x to equal 5.
▸ Why?
Turning 2 * (5 * x) into 6 - (5 * x) uses the 3a part of the rule with a = 2, which is 3 · 2 = 6.
▸ Why?
3 · 2 means three equal groups of 2, which total 6.
▸ Why?
The equation 6 - (5 * x) = 1 says that 6 with the inner value taken away leaves 1, so the inner value is exactly what you add to 1 to climb back to 6; subtraction and addition undo each other, giving 6 - 1 = 5.
Undo the inner operation
Undo the inner layer: 5 * x = 5 means 15 - x = 5, so x = 10.
Same Tool #11 move applied to the inner layer: the last operation on x was "subtract from 15", so we reverse it.
6.EE.B.7Work BackwardsMatch against the choices
x = 10 matches choice (D) — lock it in.
Confirming the computed value appears among (A)-(E) is the Tool #3 "eliminate possibilities" check on a multiple-choice problem.
6.EE.B.5Eliminate PossibilitiesThis AMC 8 problem only needs Grade 6 one-step equations and the "undo the last step" idea you already know!
- Name the inner result
- Undo the outer operation
- Undo the inner operation
- Match against the choices
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