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Magazine Chapter 2: Problem 23
Solve. For each exercise, a table is given for you to complete and use to write an equation that models the situation. How much pure acid should be mixed with 2 gallons of a \(40 \%\) acid solution in order to get a \(70 \%\) acid solution?
Short Answer
Expert verified
You need 2 gallons of pure acid.
Step by step solution
01
Understand the Problem
We need to find the amount of pure acid to mix with a 40% acid solution to achieve a 70% acid solution. We have 2 gallons of a 40% acid solution. The goal is to create a new solution that is 70% acid.
02
Set Up the Variables
Let \( x \) be the gallons of pure acid to be added. The amount of acid in the 2 gallons of 40% solution is \( 0.4 \times 2 = 0.8 \) gallons. The pure acid is 100% acid, so it contributes \( x \) gallons of acid.
03
Write the Equation
The total amount of acid in the final mixture is \( 0.8 + x \), and the total volume of the solution is \( 2 + x \) gallons. We set up the equation for 70% acid concentration as follows: \[\frac{0.8 + x}{2 + x} = 0.7\]
04
Solve the Equation
Multiply both sides by \( 2 + x \) to clear the fraction: \( 0.8 + x = 0.7(2 + x) \). Expand the right side: \( 0.8 + x = 1.4 + 0.7x \).Rearrange the equation: \( 0.8 + x - 0.7x = 1.4 \).Simplify and solve: \( 0.3x = 0.6 \), leading to \( x = 2 \).
05
Verify the Solution
Substitute \( x = 2 \) back into the equation to ensure correctness: Total acid is \( 0.8 + 2 = 2.8 \) gallons in \( 4 \) gallons of solution. Calculate concentration: \( \frac{2.8}{4} = 0.7 \) or 70%. The solution is verified.
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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 word problem in algebra that involve combining different substances to achieve a desired concentration of one component. They are often encountered in chemistry and cooking scenarios but can be applied to various contexts in everyday life.
Understanding mixture problems involves:
Understanding mixture problems involves:
- Identifying the components to be mixed, such as liquids, solids, or gases.
- Determining the quantity or concentration of each component in the mixture.
- Setting up an equation that represents the overall concentration or proportion of the target component in the final mixture.
Percent Concentration
Percent concentration refers to the percentage of a particular substance within a solution. Understanding this is crucial in mixture problems as it helps define the initial and desired states of the solution.
In the given problem:
In the given problem:
- The initial solution is a 40% acid, meaning 40% of the solution's volume is acid.
- The pure acid is considered 100% concentration because it is entirely composed of the acid without any dilution.
- The desired final mixture must be 70% acid, indicating how the concentrations of the initial solutions must be adjusted through mixing.
Equation Solving
Solving equations in mixture problems helps determine unknown quantities, like the amount of a component to be added. Equation solving is a fundamental skill in algebra, requiring the application of mathematical operations to find solutions.
In this exercise, once the equation has been formulated:
In this exercise, once the equation has been formulated:
- We recognize that the expression \( \frac{0.8 + x}{2 + x} = 0.7 \) represents the mixture's final concentration as a function of the pure acid added.
- Multiplying both sides by \( 2 + x \) eliminates the fraction, simplifying the solving process.
- Rearranging terms and isolating the variable \( x \) leads us to solve for the necessary amount of pure acid, which, in this case, is 2 gallons.
Problem Setup
Setting up the problem correctly is key to finding the right solution, especially in word problems that revolve around mixing substances.
A clear setup involves:
A clear setup involves:
- Defining variables, such as letting \( x \) represent the unknown quantity鈥攊n this case, the gallons of pure acid.
- Turning given information into mathematical expressions, as seen with the conversion of the 40% solution to its acid content: \( 0.4 \times 2 = 0.8 \) gallons.
- Constructing a sensible equation like \( \frac{0.8 + x}{2 + x} = 0.7 \), which models the desired concentration of the final solution.
Solution Verification
Verifying a solution ensures that the answer obtained is correct and consistent with the problem's requirements. This crucial step confirms that interpretations and calculations have been properly executed.
For this problem:
For this problem:
- After solving for \( x = 2 \), we substitute back into the original concentration set-up to check soundness.
- We calculate the total acid by adding pure acid and existing acid: \( 0.8 + 2 = 2.8 \) gallons.
- The concentration is verified by checking \( \frac{2.8}{4} = 0.7 \), confirming a 70% acid solution.
