AMC 8 · 2016 · #10
Easy mode Grade 6Problem
Suppose we make up a new operation, written with a star ∗. The rule is: a∗b means "take three times a, then subtract b." So a∗b=3a−b.
For example, 4∗7=3×4−7=5.
Now we want a number x that makes this true:
2∗(5∗x)=1.
Start with the inside part (5∗x) first, then handle the outside.
What is x?
Pick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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.
💡 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.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: $y = 6 - 1 = 5$.
💡 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.
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: $x = 15 - 5 = 10$.
💡 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.5 Step 4 Match $x = 10$ to the answer choices and 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.A.2 Name the inner operation. Let $y = 5 * x$. The outer equation becomes $2 * y = 1 6.EE.B.7 Work backwards through the outer $*$. By the definition $a * b = 3a - b$, the eq 6.EE.B.7 Work backwards through the inner $*$. Now $5 * x = y = 5$. By definition, $5 * x 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.2Write, 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.7Solve 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.5Understand 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!
⭐ 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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