AMC 10 · 2007 · #2

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
function-evaluationformula-substitutionorder-of-operations formula-substitution ↑ Prerequisites: order-of-operations
📏 Short solution 💡 1 insight
Problem
For any two numbers a and b, a new operation ★ is defined by the rule a ★ b = (a+b)b: add the two numbers, then multiply that sum by the second number b. Using this rule, find the value of (3 ★ 5) - (5 ★ 3).

Pick an answer.

(A)
-16
(B)
-8
(C)
0
(D)
8
(E)
16

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The expression is a difference of two separate ★ computations, so Tool #7 (Identify Subproblems) says: evaluate 3 ★ 5 on its own, evaluate 5 ★ 3 on its own, then subtract. Tool #4 (Introduce a Variable) keeps the rule a ★ b = (a+b)b in view so the first and second inputs land in the right slots — the trap here is assuming ★ behaves like ordinary multiplication and commutes. Tool #3 (Eliminate Possibilities) uses that trap: choice (C) 0 is exactly what you get if you wrongly believe 3 ★ 5 = 5 ★ 3.

1STEP 1

Read the rule and its two slots

The rule takes two inputs: add them to get a+b, then multiply that sum by the second input b alone.

a ★ b = (a+b) · b
2STEP 2

Evaluate the first term

Put a=3, b=5: the inner sum is 3+5=8 and the outside multiplier is 5, so 3 ★ 5 = 8·5 = 40.

3 ★ 5 = (3+5) · 5 = 8 · 5 = 40
3STEP 3

Evaluate the second term

Now a=5, b=3: the inner sum is still 8, but the multiplier drops to 3, so 5 ★ 3 = 8·3 = 24.

5 ★ 3 = (5+3) · 3 = 8 · 3 = 24
4STEP 4

Subtract the two results

Subtract: 40 - 24 = 16, choice (E). It would be 0 only if the terms matched, but the outside factor changed.

(3 ★ 5) - (5 ★ 3) = 40 - 24 = 16 → (E)
Answer
16
Both terms share the same sum 3+5=5+3=8, so the difference is 8 · 5 - 8 · 3 = 8(5-3) = 8 · 2 = 16 — matching the step-by-step result. Because the first term uses the larger outside factor (5 vs 3), the difference must be positive, which immediately kills the negative choices (A) -16 and (B) -8. A difference of 16 is small and sensible for numbers this size, and the shared sum of 8 makes the clean factor-of-8 answer believable.
💡Key takeaway

A made-up symbol is just a recipe: read which slot each number goes into, because swapping the inputs can change the result.

  • Read the rule and its two slots
  • Evaluate the first term
  • Evaluate the second term
  • Subtract the two results

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