application problem

Question

You have two solutions one contains 40% acid, the other 10% acid. You require 10 gallons of a 20% solution. How much of the 10% should you use? I have to write an equation and solve.

Answer 1

This problem is a classic substitution problem.  We have two variables, gallons of 40% solution (x) and gallons of 20% solution (y).   In order to solve, we will need two equations containing the variables x & y.

 

The first equation is going to be just the sum of the two variables.  From the problem, we know we are mixing unknown amounts of x & y to get 10 gallons of final solution.  So,

                                                     

                                                           Equation 1:           x + y = 10gal

 

The second equation has to do with the concentration.  We want to end up with a 20% solution by mixing x & y, each of which has a specific concentration.  We can set up an equation to reflect this by multiplying the variables by the concentration. (remember to turn % into decimals)

 

                                                         Equation 2:       (0.4*x) + (0.1*y) = (0.2*10)

 

Now if we solve equation 1 for x, we can plug that into equation 2 and solve for y.

 

                                                                                 x + y = 10

                                                                                  -y          -y                              (subtract y from both sides)

                                                                                  x = 10 - y                               (equation 3)  

Substitute equation 3 into equation 2:

                                                                     (0.4*(10 - y)) + (0.1*y) = (0.2*10)

                                                                     (0.4*10) - (0.4*y) + (0.1*y) = (0.2*10)  (Distribute)

                                                                       4 - 0.4y + 0.1y = 2                             (Simplify)

                                                                        4 - 0.3y = 2                                       (Simplify)

                                                                            -4          -4                                     (subtract 4 from both sides)

                                                                         -0.3y = -2 

                                                                            /-0.3     /-0.3                                 (divide by -0.3 on both sides)

 

                                                                            y = 6.666gal