Problem 56 - Algebra and Trigonometry

Question

Mixtures A chemist has three acid solutions at various concentrations. The first is 10\(\%\) acid, the second is \(20 \%,\) and the third is \(40 \% .\) How many milliters of each should she use to make 100 \(\mathrm{mL}\) of 18\(\%\) solution, if she has to use four times as much of the 10\(\%\) solution as the 40\(\%\) solution?

Step-by-Step Solution

Answer

Use 40 mL of 10% solution, 50 mL of 20% solution, and 10 mL of 40% solution.

Step 1: Understanding the Problem

We have three acid solutions: 10\(\%\), 20\(\%\), and 40\(\%\). We need to create 100 mL of an 18\(\%\) acid solution. Additionally, the amount of 10\(\%\) solution must be four times the amount of the 40\(\%\) solution.

Step 2: Define Variables

Let \( x \) be the volume in mL of the 10\(\%\) solution, \( y \) be the volume in mL of the 20\(\%\) solution, and \( z \) be the volume in mL of the 40\(\%\) solution.

Step 3: Set Up Equations

We have three equations based on the problem: 1. Total volume: \( x + y + z = 100 \) 2. Acid content equation: \( 0.1x + 0.2y + 0.4z = 0.18 \times 100 \) 3. Relationship between 10\(\%\) and 40\(\%\): \( x = 4z \).

Step 4: Solve the System of Equations

First, substitute \( x = 4z \) into the first equation to get \( 4z + y + z = 100 \), leading to \( y = 100 - 5z \).Then, substitute \( x = 4z \) and \( y = 100 - 5z \) in the second equation: \[ 0.1(4z) + 0.2(100 - 5z) + 0.4z = 18 \].This simplifies to \( 0.4z + 20 - z + 0.4z = 18 \).

Step 5: Simplifying the Acid Equation

Combine like terms in the acid content equation: \( 0.4z + 0.4z - z + 20 = 18 \) simplifies to \( 0.8z - z = -2 \) or \( -0.2z = -2 \). Solve for \( z \): \( z = 10 \).

Step 6: Find All Variables

With \( z = 10 \), calculate \( x = 4z = 40 \) and substitute \( z = 10 \) into the equation for \( y \): \( y = 100 - 5 \times 10 = 50 \).

Step 7: Verification

Verify the solution by checking:1. Total volume: \( 40 + 50 + 10 = 100 \) 2. Acid content: \( 0.1(40) + 0.2(50) + 0.4(10) = 4 + 10 + 4 = 18 \), matching the desired 18 mL.All conditions of the problem are satisfied.

Key Concepts

Acid Solutions

Acid solutions are mixtures where a specific percentage of acid is dissolved in water or another solvent. The concentration of the acid in these solutions is expressed as a percentage. For example, a 10\(\%\) acid solution means that 10\(\%\) of the solution's total volume is acid, and the remaining 90\(\%\) is the solvent. Knowing these percentages helps in creating mixture problems, where various acid concentrations are combined to achieve a particular result.

When working with acid solutions, it’s critical to understand that the percentage concentration directly impacts the properties of the resulting solution. To solve problems involving different acid concentrations, like the one presented, it’s often necessary to balance equations that account for these varying levels.

System of Equations

In mixture problems, we frequently use a system of equations to find unknown volumes of ingredients that satisfy multiple conditions. In this problem, three equations are crafted to solve for the volumes of the acid solutions used. Each equation represents a significant condition:

• The sum of all solution volumes must equal the total final volume.
• The sum of the acid content adjusted for each solution’s concentration should match the desired total acid content.
• Additional conditions or constraints, such as specific volume ratios between different solutions, add another equation.

By strategically choosing our variables (here, \( x, y, \text{ and } z\) to represent the respective volumes of the 10\(\%\), 20\(\%\), and 40\(\%\) solutions), these equations can be used sequentially to express all unknowns in terms of a single variable. This allows us to solve computationally complex problems that include multidimensional considerations of different concentrations and total volume constraints.

Concentration Calculations

Concentration calculations are essential for achieving the right balance in a mixture. In this particular problem, we need to create a final solution with an 18\(\%\) acid concentration from solutions of 10\(\%\), 20\(\%\), and 40\(\%\). We start by calculating the total amount of acid required, which is done by multiplying the target concentration by the total volume of the final mixture. Here, 18\(\%\) of 100 mL means 18 mL of acid is needed in the final solution.

To achieve this, the individual contributions of acid from each starting solution need to sum up to 18 mL. This calculation is represented in the system of equations as \(0.1x + 0.2y + 0.4z = 18\). Solving this equation alongside the other conditions ensures the mixture not only mixes to the correct total volume but also to the precise concentration needed. This ability to manipulate equations to balance both total and individual concentrations is a cornerstone of solving mixture problems efficiently.