Problem 10 · 2016 AMC 8
Easy
Algebra & Patterns
custom-operationsubstitution
Suppose that a ∗ b means 3a − b. What is the value of x if
2 ∗ (5 ∗ x) = 1 ?
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Answer: D — x = 10.
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Hint 1 of 2
The ∗ symbol is just a made-up RECIPE: "triple the first thing, then subtract the second." It's not scary new math — treat it like a function and obey the recipe.
Still stuck? Show hint 2 →
Hint 2 of 2
Just like ordinary nested parentheses, do the INSIDE ∗ first: 5 ∗ x = 3(5) − x = 15 − x. Then feed that result into the outer ∗.
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Approach: apply the recipe inside-out
- Inner first: 5 ∗ x = 3·5 − x = 15 − x.
- Outer: 2 ∗ (15 − x) = 3·2 − (15 − x) = 6 − 15 + x = x − 9.
- Set equal to 1: x − 9 = 1, so x = 10.
- Sanity check: plug back — 5 ∗ 10 = 5, then 2 ∗ 5 = 6 − 5 = 1. ✓
- Why this transfers: any invented operator (∗, ⊕, ⊗) is just a substitution rule; replace the symbol with its definition and work the innermost piece outward exactly like nested parentheses.
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