Problem 19 - Elementary Algebra

Question

A student doing a chemistry experiment has a beaker that contains \(84 \mathrm{ml}\) (milliliters) of an alcohol and water solution. Her lab directions tell her that there is 4.6 times as much water as alcohol in the solution. How many milliliters of alcohol are in the solution? How many milliliters of water?

Step-by-Step Solution

Answer

Answer: The approximate volume of alcohol in the solution is 15 ml, while the volume of water in the solution is 69 ml.

Step 1: Set up an equation representing the volumes of alcohol and water.

Let x represent the volume of alcohol in milliliters. Since there is 4.6 times as much water as alcohol, the volume of water can be represented as 4.6x. The total volume of the solution is 84 ml. Our equation will look like this: x + 4.6x = 84

Step 2: Combine like terms and solve the equation for x.

Combine the x terms: 5.6x = 84 Now, divide both sides by 5.6 to solve for x: x = 84 / 5.6 x ≈ 15

Step 3: Find the volume of water in the solution.

Now we will use our previously determined expression for the volume of water, which is 4.6x, and substitute the value of x that we found in the previous step: Water volume = 4.6 * 15 Water volume = 69

Step 4: Write down the final answer for the volumes of alcohol and water.

The volume of alcohol in the solution is approximately 15 ml. The volume of water in the solution is 69 ml.

Key Concepts

Equations

In algebra, equations are fundamental tools used to define relationships between different quantities. An equation represents a balance between expressions on either side of an equals sign. In this exercise, we set up an equation to model the relationship between alcohol and water in a chemical solution.
When crafting equations:

• Identify the variable: Here, we let \(x\) represent the volume of alcohol.
• Express related quantities: The water volume was expressed as \(4.6x\), based on the problem description that water is 4.6 times the alcohol.
• Combine into a complete equation: Our equation was \(x + 4.6x = 84\).

This equation highlights how variables can represent unknowns and how coefficients represent multiplication by constants. Understanding these concepts is key to solving algebraic equations in chemistry problems.

Ratios

Ratios are a way to define the relative sizes of two or more values. They describe the proportion of one quantity to another. In the exercise, the ratio of the amount of water to alcohol is given as 4.6 to 1.
To work with ratios:

• Understand the context: Ratios express relative quantities, such as 4.6 parts of water for every 1 part of alcohol.
• Translate ratios into algebra: Here, we represent the water as \(4.6x\) compared to the \(x\) of alcohol.
• Maintain the same units: Make sure all quantities are in the same unit (milliliters), so the ratio remains consistent.

A strong grasp of ratios allows you to understand how quantities compare to each other, crucial for interpreting scientific data and results.

Problem Solving

Problem solving in algebra involves using logical steps to determine unknown values in a scenario. For our chemistry experiment problem, it required a combination of understanding both equations and ratios.
Key elements of problem solving include:

• Defining the problem: Understand what you're being asked to find - in this case, the milliliters of alcohol and water.
• Developing a plan: Set up an equation that accurately represents the physical scenario.
• Carrying out the plan: Use mathematical operations to solve for the unknown variable \(x\).
• Reviewing the solution: Verify that the volumes calculated add up to the total solution (84 ml).

By systematically working through each stage, problem solving becomes a powerful, repeatable approach to tackle various types of algebraic problems.

Chemistry Experiment

Chemistry experiments often involve precise measurements and calculations to determine specific quantities. This example displays a scenario where chemical concentrations need to be derived from given parameters.
Important considerations for chemistry experiments:

• Accurate measurements: Experiments depend on precise initial data, such as the 84 ml total solution in this case.
• Understanding substance properties: Knowing that water volume is 4.6 times alcohol helps set the basis for calculations.
• Application of algebra: Using algebraic problem-solving methods to deduce the individual components of a mixture.
• Verification of results: Ensure the sum of the calculated volumes aligns with the total provided amount.

Careful attention to detail in chemistry helps ensure results are reliable and repeatable, which is critical in scientific experimentation.