Solving Maki's Ethanol Mixture Problem: A Complete Guide

Maki's Ethanol Solution: A Step-by-Step Guide to Solving the Mixture Problem

Maki's ethanol solution problem is a classic example of a mixture problem in mathematics. This kind of problem involves combining two or more solutions with different concentrations to create a final solution with a desired concentration. Let's break down the problem step-by-step to understand how Maki solved this. Maki mixes 20 gallons of a solution with an unknown ethanol concentration and 60 gallons of a 12% ethanol solution to make 80 gallons of a 10% solution. The goal is to determine the unknown ethanol concentration in the first solution. Understanding mixture problems is essential in various fields, from chemistry and pharmacy to everyday scenarios like mixing household cleaning solutions or even brewing the perfect cup of coffee. Mastering these problems requires a systematic approach that involves setting up equations based on the amount of solute (in this case, ethanol) present in each solution. The key is to realize that the total amount of solute in the final mixture is the sum of the amounts of solute from the individual solutions. This is what makes mixture problems relatively straightforward to solve once the underlying principle is understood. Additionally, the ability to work with percentages and convert them into decimal form is crucial, as this enables one to calculate the quantity of solute within a given volume of solution accurately. The mixture problems provide a practical application of algebra, allowing students to see how mathematical concepts can be used to solve real-world problems. Students learn to translate word problems into mathematical equations, which is an important skill in many areas. Furthermore, the problem-solving process enhances critical thinking skills, as it encourages students to analyze the information, formulate a plan, and evaluate the results.

Let's begin by identifying what information we have:

• Solution 1: 20 gallons with an unknown ethanol concentration (let's call it x)
• Solution 2: 60 gallons of a 12% ethanol solution.
• Final Solution: 80 gallons of a 10% ethanol solution.

Decoding the Ethanol Concentrations and Volumes

To find the unknown concentration, we'll utilize the concept that the total amount of ethanol in the final mixture equals the sum of the ethanol from each of the initial solutions. First, we need to convert the percentages into decimals. For example, a 12% ethanol solution is equivalent to 0.12 in decimal form, and a 10% solution is 0.10. This is because percentages are "out of 100", so we divide the percentage by 100 to get the decimal equivalent. The volume of the solutions in gallons helps us calculate the total volume and the amount of ethanol present in the mixtures. With the provided data, we can now outline our strategy for calculating the unknown concentration of the first solution. We'll start by figuring out how much ethanol there is in the 12% solution and in the final 10% solution. The amount of ethanol in each solution is determined by multiplying the volume of the solution by its ethanol concentration (in decimal form). This concept is key when solving mixture problems; it allows you to isolate and quantify the amount of each component within the mixture accurately. We can determine how much ethanol is in the final mixture and the second solution. The difference between these values will tell us how much ethanol must be in the first solution. We'll then utilize this information and the volume of the first solution to find the unknown ethanol concentration. This strategic approach helps break the problem down into manageable parts, simplifying the calculation and making it easier to understand. The combination of understanding percentages, doing simple math operations, and a systematic approach creates a solid foundation for solving this type of problem. It also helps in building your confidence in tackling other complex mathematical problems. Overall, it is a valuable skill.

Let's calculate the ethanol amount in the second and final solutions:

• Ethanol in Solution 2: 60 gallons * 0.12 = 7.2 gallons
• Ethanol in the Final Solution: 80 gallons * 0.10 = 8 gallons

Calculating the Unknown Ethanol Concentration

Now that we know the amount of ethanol in the final solution and Solution 2, we can determine how much ethanol must be in Solution 1. Subtract the ethanol from Solution 2 from the total ethanol in the final solution to find this value. Since we know that the total amount of ethanol in the final solution (8 gallons) is the sum of the ethanol from Solution 1 and Solution 2, we can use the given values to solve for the missing value. We can now use this information, along with the volume of Solution 1, to figure out the concentration. Essentially, we're using the principle of conservation of mass: the ethanol doesn't disappear; it's simply redistributed when the solutions are mixed. This is a foundational principle in chemistry, where the idea is that mass in an isolated system cannot be created or destroyed, but it can change forms. The same principle applies here; the total amount of ethanol before mixing is the same as the total amount of ethanol after mixing. When the solutions mix, the ethanol distributes to create the final mixture, meaning the mass of the solute is conserved, and this conservation allows us to calculate missing concentrations or volumes. This principle is also applied in various scientific fields and many real-world applications. With this principle in mind, let's calculate the amount of ethanol in Solution 1.

• Ethanol in Solution 1: 8 gallons (final) - 7.2 gallons (Solution 2) = 0.8 gallons

Now, we know that Solution 1 contains 0.8 gallons of ethanol. To find the unknown concentration (x), divide the amount of ethanol in Solution 1 by its volume.

• Concentration of Solution 1 (x): 0.8 gallons / 20 gallons = 0.04

Convert this to a percentage by multiplying by 100: 0.04 * 100 = 4%

So, the unknown ethanol concentration in Solution 1 is 4%.

The Solution Summarized and Why It Matters

In summary, by using the information given about the mixtures, we have accurately figured out the unknown ethanol concentration of the first solution. The process we followed involved: calculating the ethanol amounts in known solutions, figuring out the ethanol amount in the unknown solution, and determining the unknown concentration based on the volume and calculated ethanol amount. This entire process demonstrates the practical use of math principles, such as percentages, in a real-world scenario. Mixture problems are also important in areas like environmental science, where one might need to calculate the concentration of pollutants in water or air. Mastering these problems not only enhances your math skills but also provides a foundation for understanding more complex scientific and engineering concepts. The ability to solve these problems shows a strong understanding of basic algebraic principles and how they can be applied practically. This understanding builds confidence and can be applied to other fields. This reinforces the importance of mathematics in everyday life and in numerous professions and scientific fields. The method we've used in this problem can be applied to various other problems with slight adjustments. Being familiar with this technique can make complex problems easier to solve.