Problem 50 - Algebra and Trigonometry

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 L of a solution that contains 14% salt?

Step-by-Step Solution

Answer

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

Step 1: Define the Variables

Let \( x \) be the volume in milliliters of the 5% solution, and \( y \) be the volume in milliliters of the 20% solution. We need to find \( x \) and \( y \) such that their sum is equal to 1000 mL (1 L) and the final solution is 14% salt.

Step 2: Write the System of Equations

The first equation comes from the total volume: \( x + y = 1000 \). The second equation comes from the requirement of the final concentration: \( 0.05x + 0.20y = 0.14 \times 1000 \).

Step 3: Simplify the Concentration Equation

Substitute \( 0.14 \times 1000 = 140 \) into the equation from Step 2 to get \( 0.05x + 0.20y = 140 \).

Step 4: Solve the System of Equations for One Variable

Solve the first equation for \( y \): \( y = 1000 - x \). Substitute this into the second equation: \( 0.05x + 0.20(1000 - x) = 140 \).

Step 5: Solve for \( x \)

Distribute and combine like terms in the equation: \( 0.05x + 200 - 0.20x = 140 \), which simplifies to \( -0.15x = -60 \). Divide both sides by -0.15 to find \( x = 400 \).

Step 6: Calculate \( y \)

Substitute \( x = 400 \) back into the equation \( y = 1000 - x \) to find \( y = 600 \).

Step 7: Verify the Solution

Ensure \( 0.05 \times 400 + 0.20 \times 600 = 140 \) to confirm accuracy. Calculation: \( 20 + 120 = 140 \). The solution satisfies the concentration requirement.

Key Concepts

Percent Concentration

Understanding percent concentration is crucial when dealing with solutions. It describes the amount of solute present in a solution as a percentage of the total solution volume or mass. For our problem, we are dealing with two solutions of different salt concentrations. The first one has 5% salt and the second one 20%.
This means that in every 100 mL of the solution:

• The first solution contains 5 mL of salt.
• The second solution contains 20 mL of salt.

Percent concentration helps us understand how concentrated or diluted a solution is and is fundamental in mixing problems.
When combining measurements with different concentrations, percent concentration calculations guide us in achieving the desired mixture by balancing proportions of each component.

Mixture Problems

Mixture problems often involve finding the unknown quantities of two or more substances that combine to form a desired solution with a specific concentration. In our exercise, we are tasked with combining two salt solutions to achieve a specific percent concentration in the final mixture.

• The goal is to determine how much of each solution to mix to reach the desired concentration.
• We need to ensure the total volume is exactly one liter, which influences the equation setup.

Using mixture problems in real-life chemistry or biology helps in creating solutions with precise properties.
These problems can involve solids, liquids, or even gases, but the basic principles of balancing proportions remain the same. Understanding mixture problems develops your skills in logical thinking and mathematical reasoning.

Linear Equations

Linear equations provide a structured method to solve mixture problems. They involve creating equations that describe relationships between various quantities and constraints.
For this exercise, we set up a system of linear equations:

• One equation represents the total volume: \(x + y = 1000\).
• The other represents the concentration balance: \(0.05x + 0.20y = 140\).

These equations allow us to solve for the volumes \(x\) and \(y\), which represent the amounts of 5% and 20% solutions, respectively.
Solving these involves algebraic techniques such as substitution and simplification to find precise values. Linear equations are a powerful tool in solving real-world problems where different conditions must be balanced to achieve an optimal result.