Problem 31 - Algebra A Combined Function

Question

Dorren Schmidt is a chemist with Gemco Pharmaceutical. She needs to prepare 12 ounces of a \(9 \%\) hydrochloric acid solution. Find the amount of a \(4 \%\) solution and the amount of a \(12 \%\) solution she should mix to get this solution. $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Concentration } \\ \text { Rate } \end{array} & \begin{array}{c} \text { Liters of } \\ \text { Solution } \end{array} & \begin{array}{c} \text { Liters of } \\ \text { Pure Acid } \end{array} \\ \hline 0.04 & x & 0.04 x \\ \hline 0.12 & y & ? \\ \hline 0.09 & 12 & ? \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Mix 4.5 ounces of the 4% solution and 7.5 ounces of the 12% solution.

Step 1: Understanding the Problem

We need to prepare a 12-ounce solution with a concentration of 9% hydrochloric acid by mixing two solutions of different concentrations: 4% and 12%. The challenge is to find how much of each solution is needed to achieve this concentration.

Step 2: Setting up Variables

Let's designate the amount of 4% solution to be mixed as \( x \) ounces, and the amount of 12% solution as \( y \) ounces.

Step 3: Creating Equations for the Total Solution

We know the total volume of the solution required is 12 ounces, so we can set up the equation: \[ x + y = 12 \]

Step 4: Creating Equations for the Total Acid Content

The pure acid in the 4% solution is \( 0.04x \) ounces and in the 12% solution is \( 0.12y \) ounces. The desired concentration is 9%, which equals 9% of 12 ounces:\[ 0.04x + 0.12y = 0.09 imes 12 \] Simplifying the right side gives:\[ 0.04x + 0.12y = 1.08 \]

Step 5: Solving the System of Equations

We have the system:1. \( x + y = 12 \)2. \( 0.04x + 0.12y = 1.08 \)Substitute \( y = 12 - x \) from the first equation into the second:\[ 0.04x + 0.12(12 - x) = 1.08 \] Expanding and simplifying gives:\[ 0.04x + 1.44 - 0.12x = 1.08 \]\[ -0.08x + 1.44 = 1.08 \]Subtract 1.44 from both sides:\[ -0.08x = -0.36 \]Divide by -0.08:\[ x = 4.5 \]

Step 6: Finding y

Substitute \( x = 4.5 \) back into the first equation:\[ 4.5 + y = 12 \]Subtract 4.5 from both sides:\[ y = 7.5 \]

Step 7: Verify the Solution

Check the acid content:- For \( x = 4.5 \) ounces of the 4% solution: \( 0.04 imes 4.5 = 0.18 \) ounces of pure acid.- For \( y = 7.5 \) ounces of the 12% solution: \( 0.12 imes 7.5 = 0.9 \) ounces of pure acid.- The total acid should be \( 0.18 + 0.9 = 1.08 \) ounces of pure acid, which matches the requirement.The calculations are correct.

Key Concepts

System of Equations

Mixture problems often involve working with a system of equations, which is a set of two or more equations that share the same variables. In our problem, we are tasked with determining the amounts of two different solutions, each with a different concentration of acid, that need to be combined to achieve a specific concentration.

Two essential equations can help solve mixture problems:

• The first equation usually represents the total amount of the mixture. In our case, this is expressed as \( x + y = 12 \), where \( x \) is the amount of the 4% solution and \( y \) is the amount of the 12% solution. Together, they sum up to 12 ounces, the total volume of our desired solution.
• The second equation often represents the total amount of a certain ingredient, like pure acid in this case. This is represented by the equation \( 0.04x + 0.12y = 1.08 \), indicating the total pure acid we need in our final 9% solution.

By using systems of equations, we can determine the precise amounts of each ingredient to achieve our desired mixture.

Concentration Calculations

In mixture problems, concentration refers to the proportion of an ingredient (like acid) present in a solution. Here, concentration calculations are used to determine how much of each solution to mix to achieve a target concentration.

For Dorren's mixture problem, understanding concentration helps us determine:

• How much pure acid is in the initial solutions: The 4% solution offers \( 0.04x \) ounces of pure acid, and the 12% solution offers \( 0.12y \) ounces.
• How much pure acid is needed in the final solution to reach the desired 9% concentration: This is calculated as \( 9\% \times 12 \) ounces, which equals 1.08 ounces of pure acid.

By writing an equation for the concentrations of each component in the mixtures, we can ensure the final mixture meets the target specifications. This is crucial in many real-world applications and ensures precision in creating desired solutions.

Percentages in Solutions

Percentages play a vital role in mixture calculations, as they indicate the concentration of a component within a solution. In this problem, percentages help us understand how much of the original solutions contribute to the final mixture's total percentage.

Here's how percentages work in this context:

• The 4% hydrochloric acid solution means that every ounce of this solution contains 0.04 ounces of pure acid.
• Similarly, the 12% solution has 0.12 ounces of pure acid per ounce.
• The goal is to create a 9% hydrochloric acid solution from these mixtures, which requires that the total pure acid in 12 ounces be 1.08 ounces, that is, 9% of 12 ounces.

Understanding and working with percentages allows us to maintain consistency in chemical solutions or any mixture-related task. It's about balancing between different percentages to achieve the desired outcome accurately. This understanding is particularly useful in fields like chemistry, pharmaceuticals, and any science dealing with solutions and concentrations.