Solving The Wine Mixture Problem: Find 'x' Liters

Hey guys! Let's dive into a classic mixture problem. We're going to break down this word problem step-by-step so you can totally nail it. The original problem goes like this: "Juan has x liters of wine at S/.5 per liter. He adds 40 liters at S/.4 per liter. Then, from the resulting mixture, he takes out 20 liters and replaces them with 20 liters of wine at S/.3.6 per liter. The final mixture costs S/.4.4 per liter. Find x." This type of problem is super common in math exams, and it's all about understanding how to mix different things with different prices to get a desired average price. Let's get started!

Breaking Down the Problem: Understanding the Setup

Okay, before we get all crazy with equations, let's make sure we totally get what's going on. Juan's got some wine, and the amount he has is our unknown, x. This x liters of wine has a specific value: S/.5 per liter. Then, Juan does a little wine upgrade! He adds 40 more liters, but this time, it's cheaper wine, costing S/.4 per liter. This changes the overall average price of the wine. Think of it like this: if you mix expensive coffee with cheap coffee, the final price of the coffee per cup would not be the same as the price of just the expensive coffee. Now the tricky part: Juan removes some of the mixture and replaces it with even cheaper wine (S/.3.6 per liter). This final step also changes the average price of the wine, and we know that the final average price of the entire mixture is S/.4.4 per liter. This is the key information we'll use to solve for x. It's all about keeping track of the total value and the total volume of the wine at each stage. Remember, the total value of the mixture is equal to the sum of the value of each component. So, we'll need to figure out the value of Juan's initial wine, the value of the added wine, and the value after the replacement.

The Importance of Mixtures

Mixture problems, like the one we're tackling, show up all the time in real life, even if we don't always think of them that way. They're all about combining different things to get something new with specific properties. For example, think about making a recipe. You might mix expensive ingredients with cheaper ones to control the cost and taste of the final dish. Or, in science, you might mix different chemicals to create a specific solution with certain characteristics. This kind of problem helps you think logically and understand how changing proportions affects the final result. Understanding this concept goes beyond just solving a math problem. It helps you analyze real-world situations, make smart decisions, and understand the trade-offs involved in mixing different components to achieve a desired outcome. So, let's keep going and see how we can solve this problem step-by-step!

Step-by-Step Solution: Finding 'x'

Alright, let's get down to the nitty-gritty and work our way through the solution. We will make an equation that allows us to find x. This equation represents the idea that the total value of the wine at the end of the process is equal to the total value of the wine at the start. Here's how we'll do it:

Step 1: Initial Mixture Value

First, figure out the initial value of Juan's wine. He has x liters at S/.5 per liter. Therefore, the total value is 5x (S/.).

Step 2: Adding the Second Wine

Next, Juan adds 40 liters of wine at S/.4 per liter. The value of this wine is 40 * 4 = S/.160. So, the new total volume of wine is now x + 40 liters, and the total value is now 5x + 160.

Step 3: Removing and Replacing

This step is a bit tricky, so pay close attention! Juan removes 20 liters from the mixture, but he doesn't remove a specific type of wine. He removes a mixture of both types. To find out the value of the 20 liters he removes, we need the price per liter of the current mixture. At this point, the mixture has a total value of (5x + 160) and a volume of (x+40) liters, so the price of the current mixture per liter is (5x + 160) / (x + 40). When Juan removes 20 liters, he removes 20 * [(5x + 160) / (x + 40)] value, this should be taken into account when calculating the total value of the new mixture. After the replacement, Juan adds 20 liters of wine at S/.3.6 per liter. This adds a value of 20 * 3.6 = S/.72.

Step 4: Final Equation

We know that the final average price per liter is S/.4.4. The total volume of wine after removing and adding is still (x + 40) liters, because Juan just swapped 20 liters for 20 liters. This lets us set up an equation. The value of the wine after removing the 20 liters and adding 20 liters of wine, is (5x + 160) - 20 * [(5x + 160) / (x + 40)] + 72. So we need the following equation:

(5x + 160) - 20 * [(5x + 160) / (x + 40)] + 72 = 4.4(x + 40)

Step 5: Solving for x

Let's go through the steps of solving the equation:

1. Expand the right side of the equation: 4.4(x + 40) = 4.4x + 176.
2. Simplify the equation and isolate x.
3. Calculate x.

Step 6: Finding The Value of x

Let's solve for x. The equation is:

(5x + 160) - 20 * [(5x + 160) / (x + 40)] + 72 = 4.4(x + 40)

(5x + 160) - (100x + 3200) / (x + 40) + 72 = 4.4x + 176

(5x + 160) - (100x + 3200) / (x + 40) + 72 - 4.4x - 176 = 0

Multiplying both sides by (x+40):

(5x + 160)(x+40) - (100x + 3200) + 72(x+40) - 4.4x(x+40) - 176(x+40) = 0

Expanding each term:

5x^2 + 200x + 160x + 6400 - 100x - 3200 + 72x + 2880 - 4.4x^2 - 176x - 7040 = 0

(5x^2 - 4.4x^2) + (200x + 160x - 100x + 72x - 176x) + (6400 - 3200 + 2880 - 7040) = 0

0.6x^2 + 156x - 960 = 0

(0.6x^2) + 156x - 960 = 0

x^2 + 260x - 1600 = 0

Using the quadratic formula to solve for x

x = [-b ± √(b^2 - 4ac)] / 2a

where a = 1, b = 260, and c = -1600:

x = [-260 ± √(260^2 - 4(1)(-1600))] / 2(1)

x = [-260 ± √(67600 + 6400)] / 2

x = [-260 ± √74000] / 2

x = [-260 ± 272.03] / 2

We have two possible solutions for x:(-260 + 272.03) / 2 = 6.015 and (-260 - 272.03) / 2 which gives a negative value, so, that cannot be correct. Therefore x = 6.015, which is closest to the option A. This means Juan initially had approximately 6 liters of wine.

Key Takeaways and Tips

Alright, guys, what did we learn? First, mixture problems are all about keeping track of the total value and volume. Second, the tricky part is when you remove a portion of the mixture. Remember to calculate the value of what you're removing based on the current average price. Then, just replace the removed wine with the new wine.

Tips for Tackling Similar Problems

• Draw a Diagram: Visual aids can be super helpful. Draw a diagram or a table to organize the information. This will help keep track of the value and the volume at each step of the problem.
• Define Variables: Clearly define your variables (like x in this case). It’ll make it easier to set up the equations. Label everything clearly, so you don't confuse yourself later.
• Double-Check Your Work: After you think you've solved it, take a moment to reread the problem and make sure your answer makes sense in the context of the question. Does the answer fit with the original situation? Also, do the math again to make sure everything adds up correctly.
• Practice, Practice, Practice: The best way to get good at mixture problems (and any math problem) is to practice. Work through different examples. The more you practice, the easier it will become to identify the problem type and solve it quickly.

So, there you have it! We've solved the wine mixture problem. Keep up the good work and you will be amazing!