SOLUTION: A taxidermist has a container with a solution of 10% tanning chemical and a container with a solution of 50% tanning chemical. If the taxidermist wants to mix the solutions to get

Let "t" be the volume of the 10% tanning chemical to mix (in gallons), and
let "f" be the volume of the 50% tanning chemical to mix (in gallons).

Then you have these two equations

   t +    f = 10,        (1)    for the total volume
0.1t + 0.5f = 0.25*10    (2)    for the pure chemical amount.

Multiply (2) by 10. You will get

   t +    f = 10,        (1')   ( still (1) )
   t +   5f = 2.5*10     (2') 

Now distract (1') from (2') (both sides). You will get

   5f - f = 25 - 10,

   4f = 15   --->  f = .

Thus you just found the volume of the 50% solution. It is  gallons. 

Then the volume of the 10% solution is t =  =  =  gallons.

This equation is

0.1t + 0.5*(10-t) = 0.25*10.

Simplify and solve for "t":  t = , the same answer.