Problem 13 - Elementary Algebra

Question

Set up an equation and solve each of the following problems. (Objectives 1-3) How many milliliters of pure acid must be added to 100 milliliters of a \(10 \%\) acid solution to obtain a \(20 \%\) solution?

Step-by-Step Solution

Answer

12.5 ml of pure acid must be added.

Step 1: Understanding the Problem

We have 100 ml of a 10% acid solution and we are adding pure acid (100% acid) to this solution. We want the result to be a 20% acid solution. Our goal is to find the amount of pure acid that needs to be added.

Step 2: Define Variables

Let the amount of pure acid to be added be represented by \( x \) milliliters. This means that the total volume of the solution after adding the acid will be \( 100 + x \) milliliters.

Step 3: Set Up the Equation

Initially, we have 10% acid in a 100 ml solution, which means there are \( 0.1 \times 100 = 10 \) ml of acid. After adding \( x \) ml of pure acid, we will have \( 10 + x \) ml of acid. We want this to equal 20% of the total volume, \( 100 + x \). Thus, the equation is:\[ 10 + x = 0.2(100 + x) \]

Step 4: Solve the Equation

Expand and simplify the equation:\[ 10 + x = 20 + 0.2x \]Subtract \(0.2x\) from both sides:\[ 10 + 0.8x = 20 \]Subtract 10 from both sides:\[ 0.8x = 10 \]Divide both sides by 0.8:\[ x = \frac{10}{0.8} = 12.5 \]

Step 5: Verify the Solution

After adding 12.5 ml of pure acid, the total amount of acid is \( 10 + 12.5 = 22.5 \) ml. The total volume of the solution is \( 100 + 12.5 = 112.5 \) ml. The percentage of acid in the solution is \( \frac{22.5}{112.5} \times 100\% = 20\% \), confirming our solution is correct.

Key Concepts

Percent Concentration

Percent concentration is a way to express the proportion of a substance in a mixture or solution. It is usually expressed in terms of percent by volume or mass, representing how much of a component is present relative to the total volume or mass of the mixture.
In the given problem, we start with a 10% acid solution. This indicates that 10% of the solution's volume is pure acid, while the remaining 90% is some other substance (often water).
This concept is pivotal when adjusting mixtures to reach a desired concentration. In practice, this can involve:

• Calculating the necessary amount of substance to adjust the concentration.
• Understanding how concentration changes with the addition or removal of components.
• Employing the concentration formula: \( \text{Concentration} = \frac{\text{Amount of Substance}}{\text{Total Volume}} \times 100\% \).

Recognizing how changes in the mixture's components affect the concentration is crucial in fields ranging from chemistry to pharmacology.

Equation Solving

Equation solving involves finding the value of unknown variables that satisfy given equations. It requires setting up equations based on the relationship described in a problem statement and then working systematically to solve them.
In this exercise, we define the unknown, \( x \), as the milliliters of pure acid to add. We construct an equation that represents the condition after mixing, \( 10 + x = 0.2(100 + x) \).
Here’s how to solve it step-by-step:

• First, identify known quantities and arrange them into an equation.
• Use algebra rules to simplify expressions (like distributing and combining like terms).
• Rearrange the equation to isolate the variable on one side.
• Solve the resulting expression for the unknown value.

This process not only finds solutions but also helps check the feasibility of the outcomes by validating them against original conditions.

Mixture Problems

Mixture problems require combining different substances to achieve a specific goal, like reaching a desired concentration. They often use systems of equations because mixtures bring together several variables to be solved simultaneously.
In this situation, we mixed pure acid with a pre-existing solution to increase the concentration. The key steps in handling mixture problems include:

• Identifying the components of the mixture and their respective concentrations.
• Determining the desired outcome (such as final concentration or volume).
• Setting up equations which model these relationships and constraints.
• Solving these equations, often manipulating variables to achieve the goal.

This type of problem is prevalent in industries like chemical manufacturing and pharmacology, where specific concentrations are crucial for effectiveness and safety. By mastering these problems, you'll not only boost your equation solving skills but also understand the dynamics of combining solutions.