Problem 58 Solve each of Problems by settin... [FREE SOLUTION]

Chapter 2: Problem 58

Solve each of Problems by setting up and solving an appropriate algebraic equation. How many gallons of a \(12 \%\)-salt solution must be mixed with 6 gallons of a \(20 \%\)-salt solution to obtain a \(15 \%\)-salt solution?

Short Answer

Expert verified

10 gallons of the 12% solution are needed.

Step by step solution

Understand the Problem

We need to find out how many gallons of a 12% salt solution need to be mixed with 6 gallons of a 20% salt solution to create a solution that is 15% salt.

Define the Variables

Let \( x \) represent the number of gallons of the 12% salt solution needed.

Set Up the Equation

The total amount of salt in the mixture is equal before and after mixing. The equation is based on this principle: \(0.12x + 0.20 \times 6 = 0.15(x + 6)\).

Simplify and Solve the Equation

Expand and simplify the equation:1. \(0.12x + 1.2 = 0.15x + 0.9\)2. Subtract \(0.12x\) from both sides: \(1.2 = 0.03x + 0.9\)3. Subtract 0.9 from both sides: \(0.3 = 0.03x\)4. Divide both sides by 0.03: \(x = 10\).

Verify the Solution

Substitute \(x = 10\) back into the context of the problem. Mixing 10 gallons of 12% solution with 6 gallons of 20% solution gives:- Total salt from 12%: \(0.12 \times 10 = 1.2\) gallons- Total salt from 20%: \(0.20 \times 6 = 1.2\) gallons- Combined, the solution is 1.2 + 1.2 = 2.4 gallons of salt, in \(10 + 6 = 16\) gallons total.- The resulting concentration is \(\frac{2.4}{16} = 0.15\) or 15%.The solution checks out.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mixture Problems

Mixture problems are a common type of algebraic puzzle where you combine different substances to reach a desired property or concentration. In many cases, like our problem, we're blending solutions with differing percentages of a substance to obtain a new solution with a specific percentage concentration. Understanding the structure of mixture problems involves considering different components and their proportions.

Identify what is being mixed鈥攈ere, solutions with salt concentrations.
Define what the desired property of the final mixture should be鈥攁 15% salt solution.
Understand the relationships between the components and their contributions to the mixture.

By organizing these elements and relationships into an algebraic equation, you can effectively tackle mixture problems. Ensuring every variable and quantity is clearly defined lays the groundwork for solving these problems step by step.

Percentage Concentration

Percentage concentration is an important concept when dealing with solutions, as it quantifies the amount of a solute in a given amount of solution. In the case of salt solutions, you're often given a percentage which tells you what fraction of the solution is made up of salt. This is crucial in problems where we need to mix solutions to achieve a particular concentration.

An understanding of what each percentage represents is necessary. For example, a 12% salt solution contains 12 grams of salt per 100 grams of solution.
In our problem, we find how the individual concentrations contribute to the final mix by converting percentages into equations, like converting 12% into 0.12 in the equation.
Calculating both before the mixture and after the expected outcome gives clarity on the constancy of the total solute in the process, emphasizing conservation principles.

Getting to your correct solution means ensuring all these concentration values correctly sync into the algebraic equation you're working out.

Problem-Solving in Algebra

Approaching problem-solving in algebra, especially for mixture equations, requires a structured approach. This involves more than just applying formulas; it requires logically arranging the given information and variables.

Start by clearly defining your variables, like using \(x\) for the unknown quantity.
Set up equations based on the relationships between the variables, often representing contributions from each part of the mixture.
Simplify and solve the equation methodically. First, expand the terms if needed, then move terms involving the variable you're solving for one side and constants to the other.
Verification is key; after calculative processes, substitute back into the original framework to confirm accuracy. This ensures that the constructed solution adheres to the original problem statement.

Such detailed and stepwise methods in algebra underscore the importance of strategic thinking and meticulous calculations, helping students approach complex mixture problems with increased confidence.

91影视

Solve each of Problems by setting up and solving an appropriate algebraic equation. How many gallons of a \(12 \%\)-salt solution must be mixed with 6 gallons of a \(20 \%\)-salt solution to obtain a \(15 \%\)-salt solution?

Short Answer

Step by step solution

Understand the Problem

Define the Variables

Set Up the Equation

Simplify and Solve the Equation

Verify the Solution

Key Concepts

Mixture Problems

Percentage Concentration

Problem-Solving in Algebra

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