Short Answer 
To solve the mixture problem, define variables for the amounts of the two boric acid solutions and create two equations based on total weight and concentration. After solving the equations, the pharmacist should mix 40 grams of the 12% solution and 40 grams of the 20% solution to achieve a desired 15% concentration.
Step 1: Set Up the Equations
To solve the mixture problem, start by defining the variables. Let x be the amount of the 12% boric acid solution and y be the amount of the 20% boric acid solution. Set up the first equation based on the total weight of the two solutions:
- Equation 1: x + y = 80
This equation indicates that the total amount of both solutions combined equals 80 grams.
Step 2: Create the Concentration Equation
Next, form the second equation using the concentrations of each solution. Calculate the total mass of boric acid from both solutions and set it equal to the mass needed in the 15% solution:
- The mass in the 12% solution: 0.12x
- The mass in the 20% solution: 0.20y
Set up the equation based on the total mass of boric acid needed:
- Equation 2: 0.12x + 0.20y = 12
This indicates the combined boric acid mass from both solutions equals 12 grams.
Step 3: Solve the Equations
Now, solve the two equations simultaneously to find the values of x and y. Substitute one equation into the other: for instance, you can express y from Equation 1 as y = 80 – x and replace it in Equation 2. After solving, you will find both:
- x = 40 grams (12% solution)
- y = 40 grams (20% solution)
These results mean the pharmacist should mix 40 grams of the 12% solution and 40 grams of the 20% solution to achieve the desired 15% solution.