Mixing Salt Solutions: A Math Problem

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Mixing Salt Solutions: A Math Problem

Hey guys! Ever wonder how to mix different concentrations of saltwater to get the perfect blend? Today, we're diving into a fun math problem that's all about figuring out just that. Imagine a student, maybe you, who's got two types of saltwater: one with a 2% salt concentration and another with a 7% salt concentration. The goal? To create 1 liter of a 3.5% saltwater solution, just like the kind you might find at your favorite beach. It's like being a mad scientist, but with salt and water!

To get started, the student defines x as the amount (in liters) of the 2% solution they'll be using. This is our starting point, and it's super important to understand. We're essentially saying, "Hey, we're going to use x liters of this stuff." The beauty of this problem is that it uses a simple equation to solve it. We need to figure out how much of each solution is needed to get our desired 3.5% mix. This type of problem is incredibly useful for all sorts of real-world scenarios, from chemistry to even cooking. So, let's break it down and see how it works.

Setting Up the Equation

Okay, so we've got our 2% solution and our 7% solution. We know we want a final solution that is 1 liter total. We also know that x represents the amount of the 2% solution. Now, the magic happens as we set up our equation. The student writes: 0.02x + 0.07*(1-x) = 0.035. This equation is the heart of the problem, and it's essential to understand what each part represents. Let's break it down, shall we?

  • 0.02x: This part represents the amount of salt in the 2% solution. Since x is the amount of the 2% solution, we multiply it by 0.02 (which is the same as 2%) to find out how much salt is in that portion of the mix. Simple, right?
  • 0.07(1-x)*: This part is all about the 7% solution. We know that the total volume of the final solution is 1 liter. If we're using x liters of the 2% solution, then the rest (1-x) liters must be the 7% solution. We multiply that amount by 0.07 (or 7%) to find the amount of salt in the 7% solution.
  • 0.035: This is the total amount of salt we want in the final solution. Since we're making 1 liter of a 3.5% solution, we multiply 1 liter by 0.035 (or 3.5%) to get the total amount of salt. This is the end goal: the amount of salt we need in our perfect beach-like solution.

Now, the equation ties everything together. It says that the amount of salt from the 2% solution (0.02x) plus the amount of salt from the 7% solution (0.07(1-x)) equals the total amount of salt we want in our final solution (0.035). It's like a recipe where we're calculating exactly how much of each ingredient we need to achieve the perfect flavor. And now, all that's left is to solve it!

Solving the Equation and Finding x

Alright, buckle up, because we're about to solve this equation! This is where the real math magic happens, and it's super important to get this part right. We're going to use algebra to find the value of x, which represents the amount of the 2% solution we need. Let's go step by step, shall we?

  1. Simplify the equation: Start by simplifying the equation: 0.02x + 0.07*(1-x) = 0.035.
  2. Distribute the 0.07: Multiply 0.07 by both terms inside the parentheses: 0.02x + 0.07 - 0.07x = 0.035.
  3. Combine like terms: Combine the x terms: (0.02 - 0.07)x + 0.07 = 0.035. This simplifies to -0.05x + 0.07 = 0.035.
  4. Isolate the x term: Subtract 0.07 from both sides: -0.05x = 0.035 - 0.07. Which gives us -0.05x = -0.035.
  5. Solve for x: Divide both sides by -0.05: x = -0.035 / -0.05. Therefore, x = 0.7.

The Solution

So, what does this mean? It means the student needs 0.7 liters of the 2% solution. Remember, x represented the amount of the 2% solution. Now, to find out how much of the 7% solution is needed, we remember that the total volume is 1 liter. We know that we need 0.7 liters of the 2% solution, therefore, we need 1 - 0.7 = 0.3 liters of the 7% solution. And there you have it! The student needs 0.7 liters of the 2% solution and 0.3 liters of the 7% solution to create 1 liter of a 3.5% saltwater solution. It's amazing how a little bit of algebra can solve real-world problems, isn't it? This process can be applied to many different scenarios, like mixing chemicals, creating alloys, or even adjusting the strength of a cleaning solution. Understanding how to set up and solve these types of equations can be a valuable skill in many fields. It all boils down to understanding the proportion of each component needed to reach your desired result. Keep practicing, and you'll be a math whiz in no time!

Verification and Conclusion

Let's do a quick check to ensure our solution is accurate. We found that the student needs 0.7 liters of the 2% solution and 0.3 liters of the 7% solution. To verify, we'll plug these values back into our original equation to make sure it holds true.

  • Calculate the salt from the 2% solution: 0.02 * 0.7 = 0.014 liters of salt.
  • Calculate the salt from the 7% solution: 0.07 * 0.3 = 0.021 liters of salt.
  • Calculate the total salt in the mixture: 0.014 + 0.021 = 0.035 liters of salt.

Now, remember, we wanted a 1-liter solution with a 3.5% salt concentration, which means 0.035 liters of salt in total. Our calculation matches this exactly. So, our answer is correct! The student needs 0.7 liters of the 2% solution and 0.3 liters of the 7% solution. This is a perfect example of how math can be used to solve practical problems. The ability to calculate the amounts of different solutions needed to achieve a specific concentration is a valuable skill in many areas, from science and engineering to everyday life. So, the next time you need to mix solutions, remember this problem, and you'll be well on your way to success.

And that's a wrap, guys! Hopefully, this helps you understand how to solve this type of problem. Remember, practice makes perfect, so don't be afraid to try more problems on your own. Keep experimenting, keep learning, and keep having fun with math! You got this! Remember, understanding the concept is key. With each problem, you'll become more confident in your ability to solve complex equations. So, the next time you encounter a problem like this, you'll know exactly what to do. Now go forth and conquer the world of mixing salt solutions!