Critiquing Saltwater Solution Problems A Math Analysis

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Hey everyone! Today, we're diving deep into a classic math problem – one that involves mixing solutions to achieve a desired concentration. This is a super practical skill, not just for the classroom, but also for real-world situations like chemistry experiments, cooking, or even creating the perfect saltwater aquarium. We're going to take a look at a student's approach to this problem, critique it, and see how we can refine the process for a better solution. Let's get started!

The Saltwater Solution Challenge

So, the scenario is this: a student has two saltwater solutions – 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, mimicking the salinity of the local beach water. The student has defined '$x$' as the amount of the 2% solution needed. This is a great starting point, but let's break down the problem further and see how we can ensure the student's approach is spot-on.

Understanding the Problem

Before we jump into any equations, it's crucial to grasp the core concepts. We're dealing with concentrations, which are essentially ratios – the amount of salt compared to the total amount of solution. We need to combine two solutions with different salt concentrations to achieve a specific target concentration. This is a classic mixture problem, and there's a systematic way to solve it.

First, let's think about what we know. We know the desired final volume (1 liter) and the desired final concentration (3.5%). We also know the concentrations of the two solutions we're mixing (2% and 7%). What we don't know is how much of each solution to use. This is where our variable '$x$' comes in. By defining '$x$' as the amount of the 2% solution, we can express the amount of the 7% solution in terms of '$x$'. Since the total volume needs to be 1 liter, the amount of the 7% solution will be (1 - $x$) liters. This is a crucial step in setting up the problem correctly.

Now, let's consider the amount of salt in each solution. The amount of salt in the 2% solution is 0.02$x$ (2% of $x$ liters). Similarly, the amount of salt in the 7% solution is 0.07(1 - $x$) (7% of (1 - $x$) liters). The total amount of salt in the final mixture should be 3.5% of 1 liter, which is 0.035 liters. This gives us the key equation to solve for $x$. By setting up the problem in this methodical way, we ensure that we're accounting for all the necessary factors.

Setting Up the Equation

Alright, now that we have a clear understanding of the problem, let's translate that into a mathematical equation. This is where the student's approach needs careful consideration. The fundamental principle here is that the total amount of salt in the final mixture is equal to the sum of the salt from the individual solutions. So, we need to express the amount of salt contributed by each solution and then add them up.

As we discussed earlier, the amount of salt in the 2% solution is 0.02$x$, and the amount of salt in the 7% solution is 0.07(1 - $x$). The total amount of salt in the 1 liter of 3.5% solution is 0.035 liters. Therefore, the equation should look something like this:

0.02x + 0.07(1 - x) = 0.035

This equation represents the heart of the problem. It states that the salt from the 2% solution (0.02$x$) plus the salt from the 7% solution (0.07(1 - $x$)) equals the total salt in the desired 3.5% solution (0.035). It’s crucial to ensure this equation is set up correctly, as any error here will propagate through the rest of the solution. We need to make sure that each term accurately reflects the amount of salt contributed by each solution and that the equation as a whole represents the conservation of salt.

Solving for x

Once the equation is set up correctly, the next step is to solve for '$x$'. This involves using algebraic techniques to isolate '$x$' on one side of the equation. Let's walk through the steps:

  1. Distribute: Expand the term 0.07(1 - $x$) to get 0.07 - 0.07$x$. The equation now becomes:
    0.  02x + 0.07 - 0.07x = 0.035
    
  2. Combine Like Terms: Combine the terms with '$x$' (0.02$x$ and -0.07$x$) to get -0.05$x$. The equation is now:
    0.  07 - 0.05x = 0.035
    
  3. Isolate the x Term: Subtract 0.07 from both sides of the equation to isolate the term with '$x$':
    -0.  05x = 0.035 - 0.07
    -0.05x = -0.035
    
  4. Solve for x: Divide both sides by -0.05 to solve for '$x$':
    x = -0.035 / -0.05
    x = 0.7
    

So, we've found that $x$ = 0.7. But what does this mean in the context of the problem? Remember, '$x$' represents the amount of the 2% solution needed. Therefore, we need 0.7 liters of the 2% solution.

Finding the Amount of the 7% Solution

Now that we know how much of the 2% solution we need, let's figure out how much of the 7% solution is required. We defined the amount of the 7% solution as (1 - $x$) liters. Since we found that $x$ = 0.7, we can substitute this value into the expression:

Amount of 7% solution = 1 - $x$ = 1 - 0.7 = 0.3 liters

So, we need 0.3 liters of the 7% solution. This is a critical step in completing the solution. We’ve not only found the value of '$x$', but we’ve also interpreted it in the context of the original problem. It’s important to always go back and make sure our answer makes sense in the real-world scenario.

Verifying the Solution

Before we declare victory, it's always a good idea to verify our solution. We can do this by plugging the values we found back into the original equation and making sure they satisfy the conditions of the problem. We know we need 0.7 liters of the 2% solution and 0.3 liters of the 7% solution. Let's check if this combination results in 1 liter of a 3.5% solution.

First, let's check the total volume: 0.7 liters + 0.3 liters = 1 liter. This checks out!

Now, let's check the total amount of salt: (0.02 * 0.7) + (0.07 * 0.3) = 0.014 + 0.021 = 0.035 liters of salt.

This is exactly 3.5% of the total volume (1 liter), as 0.035 / 1 = 0.035 or 3.5%. So, our solution is verified! We've successfully determined the amounts of each solution needed to create the desired mixture.

Common Mistakes and How to Avoid Them

Mixing problems can be tricky, and there are a few common pitfalls that students often encounter. Let's highlight some of these and discuss how to avoid them. By being aware of these common errors, we can approach these problems with greater confidence and accuracy.

  • Incorrectly Setting Up the Equation: This is perhaps the most common mistake. It often stems from a misunderstanding of the underlying principle – that the total amount of solute (in this case, salt) is conserved when the solutions are mixed. Students might forget to account for the different concentrations or misinterpret how the volumes combine. To avoid this, always start by clearly defining your variables and writing out the equation in words before translating it into mathematical symbols. For example, write