MMP.10:5.3 - Keep quantities and domain restrictions together

A preparation requires one litre containing 35 percent solute by volume, using solutions A and B at 20 and 80 percent. Assume solute is conserved and component volumes add in this preparation. Those subject assumptions supply the relations. Let x and y be the respective volumes in litres; choose nonnegative real domains because the amounts can initially be divided freely.

The joint conditions are x+y=1 and 0.2*x+0.8*y=0.35. Substitution gives x=0.75, y=0.25. Both the total and solute requirements hold for those amounts. Separate bounds 0<=x<=1 and 0<=y<=1 would lose their required total.

Now only whole half-litre doses may be used. Change the representation to x=m/2, y=n/2, with nonnegative integers m and n. The volume equation becomes m+n=2. Its possibilities (m,n)=(2,0),(1,1),(0,2) give solute amounts 0.2, 0.5 and 0.8 litre. None supplies 0.35 litre. Rounding the former solution changes the preparation; it does not satisfy its original condition.

If the required concentration changes to 50 percent, one dose of each solution works. The subject relations and unit remain, while the requirement and feasible assignment change. If mixing changes volume or solute, obtain the replacement subject relation before revising its mathematical expression.