Salt Concentration Challenge: A Step-by-Step Solution

by TextBrain Team 54 views

Hey everyone! Let's dive into a classic math problem involving salt concentrations. We've got two containers, A and B, each holding a saltwater solution with different salt percentages. We'll mix them around, and the question asks us to figure out the final salt concentration in container A. This kind of problem pops up in all sorts of places – from basic chemistry to understanding how mixtures work in real life. I'll walk you through each step, so it's super clear. Let's get started!

The Initial Setup: Understanding the Problem

Alright, so here's the situation. In container A, we have 72 liters of saltwater, and 30% of it is salt. In container B, there are 36 liters of saltwater, and 10% of it is salt. The first move involves taking half of the solution from container A and pouring it into container B. After that, we take half of the new mixture from container B and pour it back into container A. The aim of the question is to figure out the final salt concentration in container A. This looks a bit complex, but we'll break it down into smaller, easier-to-manage steps. The key here is to keep track of how much salt we have in each container at each stage.

Let's start with container A. Initially, we have 72 liters of solution, and 30% of it is salt. To find out the amount of salt, we multiply the total volume by the percentage of salt: 72 liters * 0.30 = 21.6 liters of salt. The rest of the solution in A is water. Now, for container B, we have 36 liters, with a 10% salt concentration. So, the amount of salt is 36 liters * 0.10 = 3.6 liters of salt. This will be our base calculation. We will now focus on the process that is described in the question and the changes made to each container at each step. To clarify things as much as possible, let's make a table to keep track of these changes. This method will make it easy for us to avoid mistakes, especially when dealing with multiple transfers. We will also make sure to provide detailed explanations for each step.

Step 1: Transfer from A to B

First, we're taking half of the solution from container A and moving it to container B. Half of 72 liters is 36 liters. This transfer affects both containers. In container A, we're removing 36 liters of the solution. Since the initial salt concentration in A is 30%, we're also removing 30% of these 36 liters of solution. The amount of salt transferred from A to B is 36 liters * 0.30 = 10.8 liters of salt. After this transfer, container A will have 72 - 36 = 36 liters of solution left. And, the amount of salt left in A is 21.6 - 10.8 = 10.8 liters. Now, let's calculate what happens in container B. We're adding 36 liters of solution to B. This means that the total volume in B increases. B originally has 36 liters. So the new total volume in B will be 36 + 36 = 72 liters. Also, we have to add the salt transferred from A to B. B had 3.6 liters of salt originally and receives 10.8 liters from A. So, the total amount of salt in B is 3.6 + 10.8 = 14.4 liters.

Step 2: Transfer from B to A

In this step, we'll take half of the new solution in container B and pour it back into container A. Remember that the total volume in B is now 72 liters. So, half of that is 36 liters. This is what we'll move back to A. The salt concentration in B at this point is 14.4 liters of salt in 72 liters of solution. The salt concentration in B is (14.4 / 72) * 100 = 20%. So, the amount of salt we're transferring from B to A is 36 liters * 0.20 = 7.2 liters. As container A receives 36 liters from B, its total volume increases. A had 36 liters before. So, the new total volume in A is 36 + 36 = 72 liters. The total amount of salt in A will also increase. It had 10.8 liters of salt and gets 7.2 liters more. So, the total amount of salt in A becomes 10.8 + 7.2 = 18 liters. This is the last step of the process, so now we can proceed to calculate the final salt concentration in A.

Calculating the Final Salt Concentration in A

Now that we've gone through all the steps, we can finally calculate the final salt concentration in container A. We've worked out that container A has a total of 72 liters of solution, and 18 liters of it is salt. To find the salt concentration, we divide the amount of salt by the total volume and multiply by 100%: (18 liters / 72 liters) * 100% = 25%. So, the final salt concentration in container A is 25%. This is the answer to our question. It's worth noting that this final concentration is a result of the mixing and re-mixing of solutions with different salt concentrations. Each transfer changed the overall composition of the solutions. To recap, we started with A having 30% salt and B having 10% salt, and through the transfers, container A ended with a 25% salt concentration. These problems demonstrate how important it is to break down complex problems into manageable steps. By meticulously tracking the volume and salt content, we can accurately solve them. Let's summarize the key points.

Summary

  1. Initial Setup: Container A (72L, 30% salt), Container B (36L, 10% salt).
  2. A to B Transfer: Half of A's solution (36L) moved to B.
  3. B to A Transfer: Half of B's new solution (36L) moved to A.
  4. Final Calculation: Final concentration in A is 25%.

Simplifying Complex Math Problems

Solving math problems like this one isn't just about finding the right answer; it's also about developing critical thinking and problem-solving skills. Breaking down a complex problem into smaller, more manageable steps makes it much easier to understand and solve. Visualizing each step, like we did with the containers and the salt, helps in keeping track of the details. And hey, don't be afraid to make mistakes. Mistakes are a great way to learn and improve your understanding. If you are working on similar problems, always double-check your calculations and make sure you understand the units and percentages. With practice, these types of problems will become a lot easier. Keep practicing, and don't be discouraged if it takes a few tries to get it right. The most important part is the learning process! So, next time you encounter a salt concentration problem, or any other complex math question, remember to break it down, take it step by step, and keep practicing. You've got this!