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My situation:

I have not yet taken a proof based math class but I am more interested in applied math, creating mathematical models. The issue is that my school does not offer an applied math degree (only "pure") but I can take applied math classes as electives. I've mapped out my four year schedule and found that I can double major in both math and engineering but I will have to take two semesters with 19 hours (6 courses) of purely math & engineering. I have a 4.0 right now and would like to keep my GPA up around a 3.7 at least, in order to be competitive for graduate school.

I'm guessing that most people will tell me to forget about the math major and just take relevant applied math courses that match my interests. This would make sense, except for that I'm looking at graduate programs in applied math as well as computational engineering. From what I've seen on most applied math websites is that they require the Math GRE subject test, and the only way that I would feel at all prepared for that is to double major in math, forcing me to take the pure classes.

Is it even physically possible to double major and do research 12 hours a week? (I am a very hard worker)

or

Should I forget about the double major and just take the more relevant applied math courses and try my hand at applied math programs?

Any and all advice is much appreciated