A Chemist Needs 15 Liters Of A 60% Alcohol Solution. The Chemist Has A Solution That Is 50% Alcohol. How Many Liters Of The 50% Alcohol Solution And Pure Alcohol Solution Should The Chemist Mix Together To Make 15 Liters Of A 60% Alcohol Solution?

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The Problem

A chemist needs to create 15 liters of a 60% alcohol solution. However, the chemist only has a solution that is 50% alcohol and pure alcohol. The question is, how many liters of each solution should the chemist mix together to achieve the desired concentration?

Understanding the Problem

To solve this problem, we need to understand the concept of concentration and how it relates to the amount of alcohol in a solution. Concentration is typically expressed as a percentage, which represents the amount of solute (in this case, alcohol) present in a solution. In this problem, we have two solutions: a 50% alcohol solution and pure alcohol (which is 100% alcohol).

Setting Up the Equation

Let's assume the chemist needs to mix x liters of the 50% alcohol solution with y liters of pure alcohol to create 15 liters of a 60% alcohol solution. We can set up an equation based on the amount of alcohol in each solution. The total amount of alcohol in the final solution should be 60% of 15 liters, which is 9 liters.

The Equation

We can set up the following equation:

0.5x + 1y = 9

This equation represents the amount of alcohol in the 50% solution (0.5x) plus the amount of alcohol in the pure solution (1y) equals the total amount of alcohol in the final solution (9 liters).

The Volume Equation

We also know that the total volume of the final solution should be 15 liters. We can set up another equation based on the volume of each solution:

x + y = 15

Solving the Equations

Now we have two equations and two variables. We can solve for x and y by substituting the expression for y from the second equation into the first equation.

Substituting y

From the second equation, we can express y as:

y = 15 - x

Substituting y into the First Equation

Substituting y into the first equation, we get:

0.5x + 1(15 - x) = 9

Simplifying the Equation

Simplifying the equation, we get:

0.5x + 15 - x = 9

Combining Like Terms

Combining like terms, we get:

-0.5x + 15 = 9

Subtracting 15 from Both Sides

Subtracting 15 from both sides, we get:

-0.5x = -6

Dividing Both Sides by -0.5

Dividing both sides by -0.5, we get:

x = 12

Finding y

Now that we have found x, we can find y by substituting x into the second equation:

y = 15 - x y = 15 - 12 y = 3

Conclusion

The chemist should mix 12 liters of the 50% alcohol solution with 3 liters of pure alcohol to create 15 liters of a 60% alcohol solution.

Real-World Applications

This problem may seem simple, but it has real-world applications in various fields, such as chemistry, pharmacy, and manufacturing. In these fields, it's essential to achieve the correct concentration of a solution to ensure the quality and safety of the final product.

Tips and Tricks

When solving this type of problem, it's essential to:

  • Understand the concept of concentration and how it relates to the amount of solute in a solution.
  • Set up the correct equation based on the amount of solute in each solution.
  • Solve the equation using algebraic methods.
  • Check the solution by substituting the values back into the original equation.

Common Mistakes

When solving this type of problem, common mistakes include:

  • Failing to set up the correct equation.
  • Not using the correct units (e.g., liters, percentage).
  • Not checking the solution by substituting the values back into the original equation.

Conclusion

In conclusion, solving this problem requires a clear understanding of concentration and how it relates to the amount of solute in a solution. By setting up the correct equation and solving it using algebraic methods, we can find the correct mixture of solutions to achieve the desired concentration. This problem has real-world applications in various fields, and it's essential to understand the concept of concentration to ensure the quality and safety of the final product.

Frequently Asked Questions

Q: What is the main goal of the chemist in this problem?

A: The main goal of the chemist is to create 15 liters of a 60% alcohol solution by mixing a 50% alcohol solution and pure alcohol.

Q: Why is it essential to achieve the correct concentration of a solution?

A: Achieving the correct concentration of a solution is essential to ensure the quality and safety of the final product. In various fields, such as chemistry, pharmacy, and manufacturing, the correct concentration of a solution can affect the efficacy and safety of the final product.

Q: What are some common mistakes that people make when solving this type of problem?

A: Common mistakes include failing to set up the correct equation, not using the correct units (e.g., liters, percentage), and not checking the solution by substituting the values back into the original equation.

Q: How can I ensure that I am using the correct units when solving this type of problem?

A: To ensure that you are using the correct units, make sure to read the problem carefully and understand what units are required. In this problem, the units are liters and percentage.

Q: What is the importance of checking the solution by substituting the values back into the original equation?

A: Checking the solution by substituting the values back into the original equation is essential to ensure that the solution is correct. This step helps to verify that the solution satisfies the original equation and that the values are reasonable.

Q: Can I use other methods to solve this type of problem?

A: Yes, you can use other methods to solve this type of problem, such as using a graphing calculator or a computer program. However, the method used in this article is a straightforward and easy-to-understand approach.

Q: How can I apply this problem to real-world scenarios?

A: This problem can be applied to real-world scenarios in various fields, such as chemistry, pharmacy, and manufacturing. For example, in a pharmacy, a pharmacist may need to mix different solutions to create a medication with the correct concentration.

Q: What are some tips for solving this type of problem?

A: Some tips for solving this type of problem include:

  • Understanding the concept of concentration and how it relates to the amount of solute in a solution.
  • Setting up the correct equation based on the amount of solute in each solution.
  • Solving the equation using algebraic methods.
  • Checking the solution by substituting the values back into the original equation.

Q: Can I use this problem as a teaching tool?

A: Yes, you can use this problem as a teaching tool to help students understand the concept of concentration and how it relates to the amount of solute in a solution. This problem can be used to teach students about algebraic methods and how to solve equations.

Q: How can I modify this problem to make it more challenging?

A: You can modify this problem by changing the concentration of the solutions, the volume of the solutions, or the number of solutions. For example, you can change the concentration of the 50% solution to 40% or 60%.

Q: Can I use this problem to create a word problem?

A: Yes, you can use this problem to create a word problem. For example, you can create a word problem that involves a chemist who needs to mix different solutions to create a medication with the correct concentration.

Conclusion

In conclusion, this Q&A article provides answers to frequently asked questions about the problem of mixing solutions to achieve the perfect concentration. The article covers topics such as the main goal of the chemist, common mistakes, and tips for solving the problem. The article also provides information on how to apply the problem to real-world scenarios and how to modify the problem to make it more challenging.