AMC 8 · 2016 · #10

Grade 6 algebra

Archetype Arithmetic Expression Decomposition

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

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Problem

Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if
$2 * (5 * x)=1$
$\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$

Pick an answer.

(A)

$frac{1}{10}$

(B)

(C)

$frac{10}{3}$

(D)

(E)

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Toolkit + CCSS Solution

Understand

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

Givens: Custom operation: $a * b = 3a - b$; Equation: $2 * (5 * x) = 1$; Answer choices: (A) $\tfrac{1}{10}$, (B) $2$, (C) $\tfrac{10}{3}$, (D) $10$, (E) $14$

Unknowns: The value of $x$ that makes the equation true

Understand

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

Givens: Custom operation: $a * b = 3a - b$; Equation: $2 * (5 * x) = 1$; Answer choices: (A) $\tfrac{1}{10}$, (B) $2$, (C) $\tfrac{10}{3}$, (D) $10$, (E) $14$

Plan

Primary tool: #11 Work Backwards

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

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.

Execute — Answer: D

#7 Identify Subproblems 6.EE.A.2 Step 1

Name the inner operation.
Let $y = 5 * x$.
The outer equation becomes $2 * y = 1$, a single-layer equation we can attack first.

$$\text{Let } y = 5 * x. \;\text{Then}\; 2 * y = 1.$$

💡 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.

#11 Work Backwards 6.EE.B.7 Step 2

Work backwards through the outer $*$.
By the definition $a * b = 3a - b$, the equation $2 * y = 1$ means $3(2) - y = 1$, i.e., $6 - y = 1$.
The last operation done to $y$ was "subtract from $6$", so undo it: $y = 6 - 1 = 5$.

$$2 * y = 1 \;\Rightarrow\; 6 - y = 1 \;\Rightarrow\; y = 5.$$

💡 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.

#11 Work Backwards 6.EE.B.7 Step 3

Work backwards through the inner $*$.
Now $5 * x = y = 5$.
By definition, $5 * x = 3(5) - x = 15 - x$, so $15 - x = 5$.
Undo the subtraction: $x = 15 - 5 = 10$.

$$5 * x = 5 \;\Rightarrow\; 15 - x = 5 \;\Rightarrow\; x = 10.$$

💡 Same Tool #11 move applied to the inner layer: the last operation on $x$ was "subtract from $15$", so we reverse it.

#3 Eliminate Possibilities 6.EE.B.5 Step 4

Match $x = 10$ to the answer choices and lock it in.

$$x = 10 \;\Rightarrow\; \textbf{(D)}.$$

💡 Confirming the computed value appears among (A)-(E) is the Tool #3 "eliminate possibilities" check on a multiple-choice problem.

[1] #7 6.EE.A.2 Name the inner operation. Let $y = 5 * x$. The outer equation becomes $2 * y = 1

[2] #11 6.EE.B.7 Work backwards through the outer $*$. By the definition $a * b = 3a - b$, the eq

[3] #11 6.EE.B.7 Work backwards through the inner $*$. Now $5 * x = y = 5$. By definition, $5 * x

[4] #3 6.EE.B.5 Match $x = 10$ to the answer choices and lock it in.

Review

Reasonableness: 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.

Alternative: Tool #6 (Guess and Check) directly on the five choices: for each candidate $x$, compute $5 * x = 15 - x$, then $2 * (15 - x) = 6 - (15 - x) = x - 9$, and set equal to $1$. Only $x = 10$ gives $10 - 9 = 1$. The other choices give $\tfrac{1}{10} - 9$, $2 - 9 = -7$, $\tfrac{10}{3} - 9$, $14 - 9 = 5$ — none equal $1$.

CCSS standards used (min grade 6)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Reading the custom operator definition $a * b = 3a - b$ and substituting numbers (and the helper variable $y$) into it.)
6.EE.B.7 Solve real-world and mathematical problems by writing and solving one-step equations of the form $x + p = q$ and $px = q$ (Undoing $6 - y = 1$ to get $y = 5$, then undoing $15 - x = 5$ to get $x = 10$ — each is a one-step linear equation.)
6.EE.B.5 Understand solving an equation as a process of answering a question: which values from a specified set make the equation true (Verifying $x = 10$ is the choice (D) that satisfies $2 * (5 * x) = 1$, ruling out the other listed values.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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