Every so often college students will ask me for advice regarding going into applied math. I’ll tell them the first step in an application, and often the hardest step, is formulating a problem, not solving it. People don’t approach you with mathematical problems per se but problems that can be turned into mathematical problems. Nobody is going to hand you a precisely formulated math problem. That only happens on exams.
Saying “nobody” is a bit of hyperbole. It happens, but not often. Most problems come to you in the language of the client’s business, such as “We want to make sure this machine doesn’t run into that machine” or “We’re trying to predict kidney failure.” 
Recently Charles McCreary from CRM Engineering sent me the most specific problem statement I’ve ever seen:
I need to calculate the von Mises yield envelope in the axial force-internal/external pressure domain for Lame’s solution …
That really took me by surprise. Sometimes a lead will mention mathematical terminology like “particle filters” or “finite elements,” though even this level of detail is uncommon. I’ve never seen something so specific.
It’s still the case that a lot of work goes into formulating a problem. I’m sure Charles’ client didn’t approach him with this statement. I’m consulting to a consultant who has already done the hard work of reducing the original application down to a purely mathematical problem. (I’d call it “purely mathematical” rather than “pure mathematical” because it’s definitely applied math.) I look forward to digging into it as soon as I wrap up what I’m currently working on.
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 Nobody has come to me wanting to predict kidney failure, but I’ve worked on predicting several other maladies that I’ll not mention to maintain confidentiality.