Let f be a function satisfying f(xy) = (f(x))/(y) for all positive real numbers x and y. If f(500) = 3, what is the value of f(600)?

2001 AMC 12 Problem 9

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Let

f

be a function satisfying

f(xy) = \dfrac{f(x)}{y}

for all positive real numbers

x

and

y.

f(500) = 3,

what is the value of

f(600)?

1

2

\dfrac{5}{2}

3

\dfrac{18}{5}

Write

600

500

times a convenient factor

f(600) = f\!\left(500 \cdot \dfrac{6}{5}\right) = \dfrac{f(500)}{\frac{6}{5}}

Choose

x = 500

and

y = \dfrac{6}{5}

so that

xy = 600.

Then

f(600) = \dfrac{f(500)}{\frac{6}{5}} = \dfrac{3}{\frac{6}{5}} = \dfrac{5}{2}.

Thus, the correct answer is C.