On Saturday afternoon or in the Saturday afternoon?

Solution 1:

The choice of prepositions depends upon the temporal context in which you're speaking. "On ~ afternoon" implies that the afternoon is a single point in time; thus, that temporal context would take the entire afternoon as one of several different afternoons, or in other words, one would use "on" when speaking within the context of an entire week.

"In ~ afternoon" suggests that the afternoon is a temporal space in-and-of-itself, wherein anything that happens will happen amongst many other events. In other words, the temporal context for this usage would be if one were speaking of a single day -- whether past, present, or future -- and of a single afternoon, during which many things might happen.

Think of it like this:

"She called me Saturday morning, and said she'd give me her answer in the afternoon."

In the above example here, one can only use "in", and not "on". Similarly, one uses "in" in the following example, as well:

"She will call early Saturday morning to check in, and will give me her final answer in the afternoon."

In the following example, though, one must use "on":

"She called me yesterday afternoon, and said her mornings are too busy to talk. She's still not sure what her plans are for Sunday, so she'll only be able to give me her answer on Saturday afternoon."

As the above commentator suggests, one can never say "in the Saturday afternoon" -- but i think you already know that. In any event, from the above two examples i think it's clear that the choice of "in the afternoon" versus "on Saturday afternoon" depends on the temporal frame of reference, and the context in which you're speaking.

Solution 2:

The idiomatic way to say this in American English is "on Saturday afternoon".

Solution 3:

A version of this statement that's likely to work for everyone would use on Saturday. In the makes sense if we say in the afternoon, but *in the Saturday afternoon does not work.

She said she would give me her final answer on Saturday, in the afternoon.
She said she would give me her final answer in the afternoon on Saturday.
*She said she would give me her final answer in the Saturday afternoon.

Your original is also grammatical, but while it is something that occurs frequently in speech, I feel tempted to add in the afternoon (as in the first example above) if the context is formal writing.

She said she would give me her final answer on Saturday afternoon.

This question is common; in and on are difficult to use. For example, on the afternoon of that day may not make sense at first, but it will probably take on because it is like on that day:

"on the afternoon of that day" = "(on) that day's afternoon" = "(on) that day, in the afternoon" = "in the afternoon (on) that day".