Problem 105 - College Algebra Essentials

Question

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide (100\% peroxide). If \(x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C\), of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}.$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?

Step-by-Step Solution

Answer

In order to obtain a mixture that is \(28 \%\) peroxide, approximately \(125\) liters of pure peroxide should be added to the initial \(10 \%\) solution.

Step 1: Understanding the given concentration formula

The concentration of peroxide in the mixture is given by the formula \(C=\frac{x+0.1(500)}{x+500}\). Here, \(x\) signifies the amount of pure peroxide is added to the mixture and \(C\) stands for the concentration of the resulted mixture.

Step 2: Substituting the desired concentration into the formula

The goal here is to increase the concentration of the mixture to \(28 \%\) peroxide. Therefore, we insert \(0.28\) in place of \(C\) in the formula, getting: \(0.28=\frac{x+0.1(500)}{x+500}\).

Step 3: Solving the equation for x

Now we have an equation, \(0.28=\frac{x+0.1(500)}{x+500}\), which we can solve for \(x\). To do that, first multiply both sides of the equation by \(x+500\) to get: \(0.28(x+500)=x+0.1(500)\). Then simplify it to get \(0.28x+140=x+50\). Solve for \(x\) by subtracting \(0.28x\) from both sides, which gives \(0.72x=90\). Then, divide both sides by \(0.72\) to find the value for \(x\), which is approximately \(125\) liters.

Key Concepts

Concentration Formula

Understanding how to work with concentration formulas is crucial when dealing with mixtures and solutions. A concentration formula helps us express the amount of a solute, like peroxide, in a given volume of solution. The specific formula provided here: \[C = \frac{x + 0.1(500)}{x + 500}\] is designed to calculate the concentration of peroxide in a mixture.

• In this formula, the numerator \(x + 0.1(500)\) indicates the total amount of peroxide in the solution.
• The term \(0.1(500)\) represents the initial amount of peroxide (10% of 500 liters).
• The denominator \(x + 500\) is the total volume of the solution after adding \(x\) liters.

The formula provides a straightforward way to calculate the concentration of the resulting solution when different amounts of pure substance are added.

Solving Equations

Solving equations is like unraveling a puzzle to find the unknown value, in this case, \(x\). When given an equation, \[0.28 = \frac{x + 0.1(500)}{x + 500}\], our goal is to make one side of the equation equal to the other by isolating \(x\). To start, multiply both sides by the denominator \(x + 500\) to eliminate the fraction:\[0.28(x + 500) = x + 0.1(500)\]. This step simplifies your equation. Next, distribute the \(0.28\):\[0.28x + 140 = x + 50\]. Subtract \(0.28x\) from both sides:\[140 = 0.72x + 50\]. After subtracting \(50\) from both sides, you simplify to:\[90 = 0.72x\]. Finally, solve for \(x\) by dividing both sides by \(0.72\):\[x = \frac{90}{0.72} \approx 125\]. It can feel complex, but breaking it down step by step makes it manageable.

Peroxide Solutions

Peroxide solutions are common in various applications, especially in cleaning and bleaching. Hydrogen peroxide (H\(_2\)O\(_2\)) is often found in concentrations varying from 3% to 90% or more.When talking about increasing the peroxide concentration, we mean adjusting the amount of pure peroxide mixed with a solution to achieve the desired strength. In industrial or household use, this can be crucial for efficacy.

• A 10% solution means there are 10 parts peroxide per 100 parts solution.
• A 28% solution, as desired in the exercise, has 28 parts peroxide per 100 parts solution.

Adjusting concentration isn't only about getting the percentage right; it’s about ensuring the end solution is effective and safe for its intended use, whether for cleaning, as an antiseptic, or other applications. Understanding how to mathematically adjust concentration helps ensure precise applications of chemistry in practical scenarios.