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Mixture problems often involve finding amounts of ingredients to form a desired final product. To achieve this, we can use systems of equations as a mathematical tool. In a system of equations, you have two or more equations working together. The goal is to find values for the variables that make all the equations true simultaneously.

### Why Use Systems of Equations?

• **Balance**: They help keep quantities in balance by taking multiple relationships into account at once.
• **Precision**: They ensure accurate and consistent solutions.

Applying this to our problem, we used two equations: 1. Total volume equation: \( x + y = 100 \) gallons2. Chlorine percentage equation: \( 0.03x + 0.15y = 6 \) gallons of pure chlorine to meet the required 6% mixture.Each equation captures a different aspect of the problem. Solving these equations involves substitution or elimination methods to find the right amounts of the 3% and 15% solutions that mix into exactly 100 gallons with precisely 6% chlorine. It’s like solving a puzzle with numbers!

Working with percentages is crucial in mixture problems, especially for determining concentrations and proportions. A percentage represents a part of a whole and it is used to express how much of a certain component is present compared to the total quantity.

### How Percentages Work

• **Representation**: A percentage like 3% or 15% is a way to express parts per hundred. It tells you how many parts of something are in each 100 parts of the whole mixture.
• **Conversion**: To work with percentages in equations, convert them to decimals by dividing by 100. For example, 3% becomes 0.03 and 15% becomes 0.15.

In this problem, understanding and applying percentages is essential to calculating how the different chlorine solutions come together. Each percentage helps determine how much actual chlorine exists in each part of the solutions. This helps us create a new solution with a specific required concentration by mixing different parts correctly.

Solution concentration refers to how much solute is dissolved in a solvent. In a mixture problem, it's concerned with determining how strong or dilute a solution is by the proportions of solute and solvent.

### Calculating Concentration

• **Solution Terms**: The solution here is a bleach-water mixture, where chlorine is the common solute.
• **Concentration Determination**: The concentration of 6% means that out of 100 gallons, 6 gallons must be pure chlorine.

To achieve this, we mixed solutions with known concentrations (3% and 15%) using specific amounts calculated to result in a new solution that has the desired 6% concentration. By carefully selecting how much of each solution to use, based on the initial concentrations, we can create a mixture that precisely fits the required concentration goal. Solving such problems needs strategic use of equations and percentage understanding.