MHB Solving for $f(600)$ Given $f(500)=3$

[Albert1]

given a function $f$, satisfying :
(1) $f(xy)=\dfrac {f(x)}{y}$ for all $x,y>0$
(2) $f(500)=3$
what is the value of $f(600)$

 

[topsquark]

given a function $f$, satisfying :
(1) $f(xy)=\dfrac {f(x)}{y}$ for all $x,y>0$
(2) $f(500)=3$
what is the value of $f(600)$

Is this a trick question or did I just stumble upon a quick way to do this? I usually go insane when you post one of these.

[sp]
[math]f(xy) = \frac{f(x)}{y}[/math]

Let x = 500 and xy = 600. Then y = 600/500 = 6/5.

Thus
[math]f(600) = \frac{f(500)}{\frac{6}{5}} = \frac{3}{\frac{6}{5}} = \frac{5}{2}[/math]
[/sp]

-Dan

 

[Albert1]

Is this a trick question or did I just stumble upon a quick way to do this? I usually go insane when you post one of these.

[sp]
[math]f(xy) = \frac{f(x)}{y}[/math]

Let x = 500 and xy = 600. Then y = 600/500 = 6/5.

Thus
[math]f(600) = \frac{f(500)}{\frac{6}{5}} = \frac{3}{\frac{6}{5}} = \frac{5}{2}[/math]
[/sp]

-Dan

please don't go insane,your answer is correct,thanks for participating.
By the way ,do you prefer more challenging problems (need a lot of tricks)?

 

[topsquark]

please don't go insane,your answer is correct,thanks for participating.
By the way ,do you prefer more challenging problems (need a lot of tricks)?

I don't usually lack self-confidence but whenever I can quickly answer a problem from you or the POTWs I get the feeling I've overlooked a critical point! Like this one...I wasn't expecting a one line solution so I was unsure about it. Personally I prefer the challenge, though I usually don't post my attempts. I like these. (Yes)

-Dan