A cleaning solution is made from vinegar, lemon juice, and water. The table below shows the part of the total that is in each element.
| | Solution A | Solution B | Solution C |
|---|---|---|---|
| Vinegar | 0.1 | 0.2 | 0.4 |
| Lemon Juice | 0.15 | 0.2 | 0.1 |
| Water | 0.75 | 0.6 | 0.5 |
Mr. Allen needs 100 mL of a solution that is [tex]$30 \%$[/tex] vinegar, [tex]$15 \%$[/tex] lemon juice, and [tex]$55 \%$[/tex] water. How much of each solution will Mr. Allen need to make his cleaning solution?
A. 0 mL
B. 10 mL
C. 30 mL
D. 50 mL
Answer (1)
Set up a system of three linear equations based on the given information.
Solve the system of equations to find the amounts of solutions A, B, and C needed.
Determine the amount of solution A needed, which is represented by the variable x .
The amount of solution A needed is approximately 0 mL, so the final answer is 0 mL .
Explanation
Problem Analysis We are given a problem where Mr. Allen needs to create 100 mL of a cleaning solution with specific percentages of vinegar, lemon juice, and water. We are also given the compositions of three existing solutions (A, B, and C). Our goal is to determine how much of each solution (A, B, and C) is needed to create the target solution.
Setting up the Equations Let x , y , and z be the amounts (in mL) of solutions A, B, and C, respectively, needed to make the final solution. We can set up a system of three linear equations with three unknowns based on the given information:
The total volume must be 100 mL: x + y + z = 100
The amount of vinegar in the final solution is 30% of 100 mL: 0.1 x + 0.2 y + 0.4 z = 0.3 ( 100 ) = 30
The amount of lemon juice in the final solution is 15% of 100 mL: 0.15 x + 0.2 y + 0.1 z = 0.15 ( 100 ) = 15
So, our system of equations is:
x + y + z = 100 0.1 x + 0.2 y + 0.4 z = 30 0.15 x + 0.2 y + 0.1 z = 15
Solving the System of Equations We can solve this system of equations using various methods such as substitution, elimination, or matrix methods. Using a calculator or software to solve this system, we find the following values for x , y , and z :
x ≈ 0 y = 50 z = 50
Finding the Amount of Solution A Since x represents the amount of solution A needed, and we found that x ≈ 0 , Mr. Allen needs approximately 0 mL of solution A to make his cleaning solution.
Final Answer Therefore, Mr. Allen needs 0 mL of solution A.
Examples
This type of problem is useful in chemistry and pharmacy when mixing solutions to achieve a desired concentration. For example, a pharmacist might need to mix different concentrations of a drug to create a specific dosage for a patient. Similarly, in environmental science, this could be used to determine how much of different water samples with varying pollutant levels need to be mixed to achieve a safe level of pollutants in a water supply.
