Question

Acid Mixture A chemist needs 10 liters of a \(25 \%\) acid solution. The solution is to be mixed from three solutions whose acid concentrations are \(10 \%, 20 \%\), and \(50 \%\). How many liters of each solution should the chemist use to satisfy the following? (a) Use as little as possible of the \(50 \%\) solution. (b) Use as much as possible of the \(50 \%\) solution. (c) Use 2 liters of the \(50 \%\) solution.

Short answer

The exact amount of solution used in each case will depend on the solving process of the system of equations. It will vary according to the constraints and goal of each case.

Step-by-step solution

Create the equations

Let's denote three solutions (10%, 20% and 50%) as x, y, and z, respectively. The task is to obtain a 10-liter solution of 25%, which gives us the following equations: 1. The total volume: \(x+y+z=10\)2. The total percentage: \(0.1x+0.2y+0.5z=0.25*10=2.5\)

Case (a): Use as little 50% solution as possible

The=minimize z and resolve the system of equations. Assume \(z=0\), and solve the equations for x and y. If there's no solution in which all variables are greater than or equal to 0, increment z by 1 until you find a valid solution.

Case (b): Use as much 50% solution as possible

Next, we need to maximize z and resolve the system of equations. Start by assuming z is as high as possible (under the constraint of the total volume), and then decrement it until a solution is found where all variables are non-negative.

Case (c): Use 2 liters of the 50% solution

In this case, z is given as 2 liters. Just substitute \(z=2\) into the equations and solve the resulting system for x and y. Ensure that the resulting values for x and y are non-negative.

Key concepts

Linear Equations

Linear equations are a type of algebraic equation where each term is either a constant or the product of a constant and a single variable. In the original exercise, the chemist needed to create a mixture using linear equations. The linear equations used in this situation help determine the quantities of each solution (10%, 20%, 50%) needed to create exactly 10 liters of a 25% solution.

Linear equations here can be written as follows:

• Total volume equation: \( x + y + z = 10 \)
• Total concentration equation: \( 0.1x + 0.2y + 0.5z = 2.5 \)

By solving these equations, you determine how much of each solution you need. Each linear equation represents a constraint on the problem: one for total volume, another for the acid concentration. They work together, guiding you toward the correct answer through basic algebraic techniques, such as substitution or elimination. Linear equations like these are a powerful tool for solving real-world problems efficiently.

Solution Concentration

Solution concentration refers to the amount of solute (in this case, acid) present in a mixture or solution. It is usually expressed as a percentage or fraction of the total volume or mass. In the exercise, concentration determines the percentage of acid in each solution the chemist uses, like 10%, 20%, and 50%.

Understanding solution concentration helps in ensuring that the final mixture meets the specific requirements, such as a 25% acid solution. To target the desired concentration, you balance different concentrations of the solvent diligently. Here, the exercise uses three different concentrations to find an optimal combination.

By writing and solving the concentration equation \( 0.1x + 0.2y + 0.5z = 2.5 \), you adjust the quantities in the mixture so that they collectively average to the target percentage, ensuring the proper balance is struck without overusing or underusing any single solution.

System of Equations

A system of equations involves multiple equations that you solve together, reflecting relationships between different quantities. The exercise uses a system of equations to balance both volume and concentration constraints.

In problems involving multiple variables like this, solving one equation alone is not enough. Each equation addresses a different aspect of the mixture; together, they govern the solution. Thus, a system of equations provides a comprehensive framework for finding a solution that satisfies all given conditions.

With the equations \( x + y + z = 10 \) and \( 0.1x + 0.2y + 0.5z = 2.5 \), the chemist finds how much of each solution to use. By graphing, substitution or elimination method, the chemist ensures that each requirement is met simultaneously. Solving a system of equations like this is critical in real-life scenarios where multiple criteria must be addressed in a single calculation.