Problem 50 - Precalculus: Mathematical for Calculus

Question

A biologist has two brine solutions, one containing \(5 \%\) salt and another containing \(20 \%\) salt. How many milliliters of each solution should he mix to obtain \(1 \mathrm{L}\) of a solution that contains \(14 \%\) salt?

Step-by-Step Solution

Answer

Mix 400 mL of the 5% solution with 600 mL of the 20% solution.

Step 1: Understand the Problem

We need two solutions: a 5% salt solution and a 20% salt solution. We aim to mix these solutions to produce a 1 L (1000 mL) solution containing 14% salt.

Step 2: Define Variables

Let's denote the volume of the 5% solution as \( x \) mL and the volume of the 20% solution as \( y \) mL.

Step 3: Set Up Equations

We have two equations: \( x + y = 1000 \) (the total volume equation) and \( 0.05x + 0.20y = 0.14 \times 1000 \) (the salt concentration equation).

Step 4: Solve the First Equation

From the equation \( x + y = 1000 \), express \( y \) in terms of \( x \): \( y = 1000 - x \).

Step 5: Substitute and Solve the Second Equation

Substitute \( y = 1000 - x \) into the second equation: \( 0.05x + 0.20(1000 - x) = 140 \). Simplify to find \( x \): \( 0.05x + 200 - 0.20x = 140 \).

Step 6: Simplify and Solve for x

Combine the terms: \( -0.15x + 200 = 140 \). Subtract 200 from both sides: \( -0.15x = -60 \). Divide both sides by -0.15 to find \( x = 400 \) mL.

Step 7: Solve for y

Using \( y = 1000 - x \), replace \( x \) with 400: \( y = 1000 - 400 \), so \( y = 600 \) mL.

Step 8: Verify Solution

Check that the salt mass from both solutions equals the salt needed in the final solution. \( 0.05 \times 400 + 0.20 \times 600 = 20 + 120 = 140 \), which satisfies the concentration condition.

Key Concepts

System of Equations

One common approach to solving mixture problems is using a system of equations. This method allows us to tackle multiple equations simultaneously to find the values of unknown variables. In the exercise, we have two solutions with different salt percentages. The aim is to mix them to achieve a specific concentration in the final solution. First, we define variables to represent the amounts of each solution. Typically, these are denoted by \(x\) for the first solution and \(y\) for the second. The system of equations then comprises:

• The total volume equation: \(x + y = 1000\) mL, because we need a final volume of 1 L.
• The concentration equation: \(0.05x + 0.20y = 0.14 \times 1000\), representing the salt content.

Using these equations, we aim to find the specific amounts of each type of solution. For this, we express one variable in terms of the other, then substitute and solve. This step-by-step process exemplifies how systems of equations can provide multiple conditions, making it easier to pinpoint an exact solution.

Percentage Concentration

Understanding percentage concentration is vital when dealing with solutions involving mixtures. It tells us how much of a component is present in a specific volume of a solution. For example, a 5% salt solution means that 5% of the solution's total volume is salt. In our exercise, we work with solutions of 5% and 20% concentrations, mixing to obtain one that is 14% concentrated.
The concentration of our final solution tells us the ratio of salt to the total solution volume. To calculate the salt content based on given concentrations, we use the formula:

• Amount of salt in a solution = percentage concentration \(\times\) total volume.

Thus, for a target concentration of 14%, we need:

• \(0.14 \times 1000\) mL of salt = 140 mL in the final mixture.

This focuses our calculations, as we check that the combined salt masses from both solutions equal the target concentration. Mastering percentage concentration calculations aids in understanding and solving mixture problems effectively.

Problem Solving

Problem solving in the context of mixture problems involves dissecting a real-world scenario and translating it into mathematical terms. Here's a simple strategy:

• Identify the problem's goals, such as the desired concentration and total volume in this exercise.
• Define variables for unknown quantities. In this exercise, they represent the volumes of each salt solution.
• Set up equations to capture the relationships involved, like total volume or concentrations.
• Solve the equations step by step, ensuring each action moves you closer to the desired outcome.

In this exercise, our main task is to find the volumes of two different solutions to meet a specific salt concentration. It's achieved by creating a system of equations. Problem solving also involves checking that the solution meets the initial conditions of the problem. Here, verifying that the total salt content corresponds to the final solution's specified concentration ensures accuracy. Emphasizing a clear, strategic approach helps in tackling mixture problems with greater ease.