Natsano has at most $55,000 to invest in the common stocks of two companies. He estimates that an investment in Company A will yield a return of 10%, whereas an investment in Company B, which he feels is a riskier investment, will yield a return of 20%. If he decides that his investment in the stocks of Company A is to exceed his investment in the stocks of Company B by at least $20,000, determine how much he should invest in the stocks of each company to maximize the return on his investment.

Natsano can satisfy his constraints by investing the first $20,000 in Company A, then splitting the remaining $35,000 evenly between the companies. For best return, he needs to invest as much as possible in Company B, but each such dollar (after the first 20k) must be matched by a dollar invested in Company A. That is, his investments should be ...

Company A: $37,500
Company B: $17,500

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The attached graph shows the feasible region of investments (doubly shaded). The vertex that maximizes the objective function (return on investment) is the one highlighted. (It puts the objective function line as far as possible from the origin.)

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Sometimes graphing the constraints is more work than necessary if there is some simple logic that quickly identifies the solution.