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I was asked to explain why relying on can not be used instead of depending on in the following sentence - but my explanation is, well, not so hot - I am relying on you to help me! Thanks

Example:
I would like a different ring tone on my mobile depending on who is calling me.
Comments  
depending on=according to.
relying on=depend on with full trust or confidence.

So, in your example, depending on is correct, relying on is wrong.

http://oxforddictionaries.com/definition/rely?q=relying+on
http://oxforddictionaries.com/definition/depend?q=depending+on+#depend__7

Prajwal
In the given sentence, 'depending on' functions as 'according to'.

I have set the ringtone according to the people who call me.

Somebody/something relies on somebody/something for some self-interest/purpose.
e.g.
We rely on trees for oxygen.(depend on)
We rely on you to give a clear explanation. (trust/ have faith)
Teachers: We supply a list of EFL job vacancies
thanks for the explanation
and thank you too