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2001 AMC 12 Problem 9

Problem 9 of 25EasierAlgebra

Let ff be a function satisfying f(xy)=f(x)yf(xy) = \dfrac{f(x)}{y} for all positive real numbers xx and y.y. If f(500)=3,f(500) = 3, what is the value of f(600)?f(600)?

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Solution

Choose x=500x = 500 and y=65y = \dfrac{6}{5} so that xy=600.xy = 600. Then f(600)=f(500)65=365=52. f(600) = \dfrac{f(500)}{\frac{6}{5}} = \dfrac{3}{\frac{6}{5}} = \dfrac{5}{2}. Thus, the correct answer is C.
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Tagged: functional equation · substitution

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