LP problem

A man deals with French currency (the franc) and American currency (the dollar). At $12$ midnight, he can buy francs by paying $.25$ dollars per franc and dollars by paying $3$ francs per dollar. Let $x_{1} = \text{number of dollars bought (by paying francs)}$ and $x_{2} = \text{number of francs bought (by paying dollars)}$ . Assume that both types of transactions take place simultaneously, and the only constraint it that at $12:01 \ \text{A.M.}$ the man must have a nonnegative number of francs and dollars.

(a) Formulate an LP that enables the man to maximize the number of dollars he has after all transactions are completed.

Let $X_{1} = x_{1}+ x_{1}'$ where $x_{1}'$ is the number of dollars the man has before the transaction. Similarly, let $X_{2} = x_{2} + x_{2}'$ where $x_{2}'$ is the number of francs the man has before the transaction. So is this the correct LP formulation:

maximize $X_{1}$ subject to constraints that $X_{1} \geq 0, \ X_{2} \geq 0$ . We can solve this right? This formulation allows for the man to not enter any transaction, right?

I'm not familiar with linear programming, but here is what I'd say: $X_1=x'_1+x_1-\frac{x_2}{3}$ (since $\frac{x_2}{3}$ is the number of dollars spent to buy francs) and $X_2=x'_2+x_2-\frac{x_1}{0.25}$ . Maximize $X_1$ subject to the constraints $X_1\geq0,\ X_2\geq 0$ .

(By the way, euros would be more up-to-date (Nod) than francs)

Laurent.

Thats all there is too it right? No ambiguities?