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2007 AMC 10B Problem 2

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2007 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Functions · difficulty 1 of 5 · about 1 min · position P1-10

Define the operation $\star$ by $a \star b = (a+b)b.$ What is $(3 \star 5) - (5 \star 3)?$

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

Answer

Key insight

Both products share the factor a + b = 8, so the difference is 8 times (5 - 3) = 16.

Solution

Apply the definition to each pair, keeping track of which number is the second argument:
$3 \star 5 = (3 + 5)\cdot 5 = 40, \qquad 5 \star 3 = (5 + 3)\cdot 3 = 24.$
Then $40 - 24 = 16$ .

The answer is $\boxed{\textbf{(E)}\ 16}$ .

Why this works

A made-up operation is just a formula; substitute carefully. Since $a + b$ is the same in both terms, the difference is $(a+b)(b - a) = 8 \cdot 2 = 16$ , which is a quick sanity check on the sign.

The trap

Assuming the operation is commutative and answering 0, or swapping which argument multiplies the sum.

Common mistakes

Assuming the operation is commutative and answering 0, or swapping which argument multiplies the sum.
Computing $5 \star 3 - 3 \star 5$ instead and getting $-16$ .

Techniques

Set up the equation/formula and compute; no special trick needed