If n m=n^3m^2, what is (2 4)/(4 2)?

2016 AMC 10B Problem 2

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n\heartsuit m=n^3m^2,

what is

\frac{2\heartsuit 4}{4\heartsuit 2}?

\ \dfrac{1}{4}

\ \dfrac{1}{2}

\ 1

\ 2

\ 4

Compute the operation ratio symbolically first

\frac{n\heartsuit m}{m\heartsuit n}=\frac nm

For any nonzero

a,b,

we have

\dfrac{a\heartsuit b}{b\heartsuit a} = \dfrac{a^3b^2}{b^3 a^2} = \dfrac ab .

Using

a=2,b=4,

we get that

\frac{ 2}{4} = \frac 12.

Thus, the correct answer is B .