2007 AMC 10B Problem 2

The rule $a \star b = (a+b)b$ says: add the two numbers, then multiply by the second number. Watch which number is second each time.

1. For $3 \star 5$ the second number is $5$:
$$3 \star 5 = (3 + 5) \times 5 = 8 \times 5 = 40.$$
2. For $5 \star 3$ the second number is $3$:
$$5 \star 3 = (5 + 3) \times 3 = 8 \times 3 = 24.$$
3. Subtract:
$$40 - 24 = 16.$$

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

A made-up operation is only a formula with blanks; fill the blanks in the right order. A quick check: both results start with the same sum $8$, so the difference is $8$ times the difference of the multipliers, $8 \times (5 - 3) = 16$. That also tells you the sign is positive.

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 $\boxed{\textbf{(E)}\ 16}$.