Define a@b = ab - b^2 and a#b = a + b - ab^2. What is (6@2)/(6#2)?

2007 AMC 10A Problem 2

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Define

a@b = ab - b^2

and

a\#b = a + b - ab^2.

What is

\dfrac{6@2}{6\#2}?

-\dfrac{1}{2}

-\dfrac{1}{4}

\dfrac{1}{8}

\dfrac{1}{4}

\dfrac{1}{2}

Substitute

a = 6

and

b = 2

into each definition

The numerator is

6 \cdot 2 - 2^2

and the denominator is

6 + 2 - 6 \cdot 2^2

The numerator is

6@2 = 6 \cdot 2 - 2^2 = 12 - 4 = 8.

The denominator is

6\#2 = 6 + 2 - 6 \cdot 2^2

= 8 - 24

= -16.

The quotient is

\dfrac{8}{-16} = -\dfrac{1}{2}.

Thus, the correct answer is A.