AMC 8 · 2016 · #10

Grade 6 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

function-evaluationlinear-equations-one-varformula-substitution identify-subproblemsconvert-to-algebra ↑ Prerequisites: linear-equations-one-varformula-substitution

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Problem

A new operation is defined by a * b = 3a - b. Given that 2 * (5 * x) = 1, find x.

Pick an answer.

(A)

$frac{1}{10}$

(B)

(C)

$frac{10}{3}$

(D)

(E)

AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

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.

1STEP 1

Name the inner operation y = 5 * x, so the outer equation becomes the single-layer 2 * y = 1.

Let y = 5 * x. Then 2 * y = 1.

2STEP 2

Undo the outer layer: 2 * y = 1 means 6 - y = 1, so y = 5.

2 * y = 1 → 6 - y = 1 → y = 5.

3STEP 3

Undo the inner layer: 5 * x = 5 means 15 - x = 5, so x = 10.

5 * x = 5 → 15 - x = 5 → x = 10.

4STEP 4

x = 10 matches choice (D) — lock it in.

x = 10 → (D).

Answer

Plug x = 10 back into the original expression. Inner: 5 * 10 = 3(5) - 10 = 15 - 10 = 5. Outer: 2 * 5 = 3(2) - 5 = 6 - 5 = 1. That matches the given 1 exactly, so x = 10 is correct.

💡Key takeaway

This AMC 8 problem only needs Grade 6 one-step equations and the "undo the last step" idea you already know!

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