A 12% solution and a 24% solution are to be mixed to create 80 mL of a 16% solution. How much of each should be used?
If you have x mL of 12%, then you also have 80-x of 24%.
.12x + .24(80-x) = .16(80)
ok thanks. i can never start these problems so I can go from this.
To solve this problem, we can use the method of mixture or concentration. Let's break it down step by step.
Step 1: Assign variables.
Let's assign variables to the unknown quantities. Let x represent the volume (in mL) of the 12% solution and y represent the volume (in mL) of the 24% solution.
Step 2: Write the equation for the total volume.
The total volume of the solution is given as 80 mL. We can write this as an equation:
x + y = 80.
Step 3: Write the equation for the concentration.
The concentration of the solution is given as 16%. The concentration can be calculated by taking the weighted average of the two solutions based on their volumes. The equation can be written as follows:
(0.12x + 0.24y) / 80 = 0.16.
Step 4: Solve the system of equations.
Now, we have a system of two equations with two variables. We can solve it to find the values of x and y.
Using the equation x + y = 80, we can rewrite it as x = 80 - y and substitute into the concentration equation:
(0.12(80 - y) + 0.24y) / 80 = 0.16.
Simplifying the equation:
(9.6 - 0.12y + 0.24y) / 80 = 0.16,
(9.6 + 0.12y) / 80 = 0.16,
9.6 + 0.12y = 0.16 * 80,
9.6 + 0.12y = 12.8,
0.12y = 12.8 - 9.6,
0.12y = 3.2.
Dividing both sides by 0.12:
y = 3.2 / 0.12,
y ≈ 26.67.
Now, substitute the value of y back into x = 80 - y:
x = 80 - 26.67,
x ≈ 53.33.
Step 5: Check the answer.
To confirm if the solution is correct, check the total volume and concentration values.
Total volume:
x + y = 53.33 + 26.67 ≈ 80.
Concentration:
(0.12(53.33) + 0.24(26.67)) / 80 = (6.40 + 6.40) / 80 ≈ 12.80 / 80 ≈ 0.16.
Both the total volume and the concentration match the given conditions, so the solution is correct.
Therefore, approximately 53.33 mL of the 12% solution should be used, and approximately 26.67 mL of the 24% solution should be used to create 80 mL of a 16% solution.