SOLUTION: How many gallons each of 10% chloride solution and 30% chloride solution should be mixed in order to obtain 10 gallons of 25% chloride solutions?

Let x = the amount of the 10%  chloride solution needed (in gallons).
The the amount of the 30%  chloride solution needed is (10-x) (in gallons).

The amount of the "pure" chloride in these solutions is 0.1*x + 0.3*(10-x).

It is exactly 25%, or 0.25 of 10 gallons, i.e. 0.25*10.

So, your "pure chloride amount" equation is 

0.1*x + 0.3*(10-x) = 0.25*10.

Simplify and solve it for x:

0.1x + 3 - 0.3x = 2.5,

-0.2x = 2.5 - 3,

-0.2x = -0.5,

x =  = 2.5.

Answer. 2.5 gallons of 10% chloride solution and 7.5 = 10-2.5 gallons of 30% chloride solutions are needed.