Problem 5 - Prealgebra

Question

Solve each of the following word problems by translating the statement into a proportion. Be sure to show the proportion used in each case. [Examples \(1-4]\) Mixture A solution contains 8 pints of antifreeze and 5 pints of water. How many pints of water must be added to 24 pints of antifreeze to get a solution with the same concentration?

Step-by-Step Solution

Answer

Add 15 pints of water to the antifreeze.

Step 1: Understand the Problem

We need to find out how many pints of water need to be added to 24 pints of antifreeze to make a solution with the same concentration as Mixture A.

Step 2: Identify the Concentration of Mixture A

Mixture A has 8 pints of antifreeze and 5 pints of water, totaling 13 pints of solution. The concentration of antifreeze in Mixture A is \( \frac{8}{13} \).

Step 3: Set Up the Proportion for the New Mixture

Let \( x \) be the number of pints of water added to 24 pints of antifreeze. The total amount of the new solution will be \( 24 + x \) pints. The concentration of antifreeze should be \( \frac{24}{24 + x} \) to match Mixture A.

Step 4: Formulate the Proportion Equation

Set the concentration of Mixture A equal to the concentration of the new mixture: \[ \frac{8}{13} = \frac{24}{24 + x} \]

Step 5: Solve the Proportion Equation

Cross-multiply to solve for \( x \):\[ 8(24 + x) = 13 \cdot 24 \]This simplifies to:\[ 192 + 8x = 312 \]Subtract 192 from both sides:\[ 8x = 120 \]Divide both sides by 8:\[ x = 15 \]

Step 6: Conclusion

We found that 15 pints of water need to be added to 24 pints of antifreeze to achieve the same concentration as Mixture A.

Key Concepts

Concentration Calculation

Calculating concentration involves finding the proportion of one substance within a mixture in comparison to the entire mixture. This can be expressed as a ratio or fraction. For example, to determine the concentration of antifreeze in a solution, you would divide the amount of antifreeze by the total volume of the solution.
In Mixture A, which contains 8 pints of antifreeze and 5 pints of water, the concentration of antifreeze is determined by forming the fraction \( \frac{8}{13} \). This means that the antifreeze makes up \( \frac{8}{13} \) of the total mixture. Calculating concentration helps in understanding the strength or potency of a mixture, which is crucial in many practical applications such as chemical solutions, medications, and beverages.
By maintaining consistent concentration in different mixtures, one ensures uniformity, which is critical for effectiveness, safety, and quality control in practical scenarios.

Word Problems in Pre-Algebra

Word problems are a fundamental part of pre-algebra and lay the foundation for critical thinking in algebra. They require turning a verbal statement into a mathematical equation or proportion to find a solution. Often, these problems involve real-life scenarios and help students develop problem-solving skills.
To effectively tackle word problems, it’s essential to understand the problem fully. This includes identifying the known and unknown quantities and deciding how they relate to each other mathematically. In our example, you need to find out how many pints of water must be added to achieve the desired concentration.

• Identify the problem: What are you solving for?
• Translate words into numbers: Convert the scenario into a mathematical equation.
• Set up a relationship: Create equations based on these relationships.
• Solve the equation: Use algebraic techniques to find the unknown.

By following these steps, word problems become less daunting, leading to a systematic approach to problem-solving.

Mixture Problems

Mixture problems are a specific type of word problem in algebra where different substances are combined to form a new mixture. These problems often involve calculating concentrations, percentages, or amounts needed to achieve a specific characteristic in a mixture.
In our exercise, we used a concentration-based mixture problem to determine how much water was required to be added to antifreeze to maintain a similar concentration as another mixture. This required setting up a proportion to find the relationship between the different components.
The essential part of solving mixture problems is establishing the right equations that represent the proportions of the components.

• Understand each substance's role: For instance, knowing that antifreeze needs to be a specific part of the solution.
• Formulate the equation: Write equations that compare the sample concentration with the desired concentration.
• Solve for the unknown: Use algebra to find how much of each component is needed.

Being adept at solving mixture problems empowers students to tackle complex real-world issues involving ratios and concentrations, enhancing their mathematical versatility.