¿Cuántos Pollitos Tenía El Estudiante Inicialmente?
Hey guys, ever wondered how to solve a tricky math problem that involves figuring out the initial number of items someone had? Let's dive into this fun word problem about a student selling chicks! We'll break it down step by step so it's super easy to understand. This kind of problem often pops up in math classes and can seem daunting at first, but don't worry, we've got your back.
Breaking Down the Pollitos Problem
So, the core of this problem revolves around understanding inequalities and how they change with each transaction. Let's first focus on the initial conditions presented in the problem. The student starts with an unknown number of chicks, let's call this x. He sells 35 chicks, meaning he now has x - 35 chicks left. The problem states that this amount is more than half of what he started with. This is our first crucial piece of information, and it helps us establish the foundation for solving the puzzle. To put it mathematically, we can represent this as: x - 35 > x/2. This inequality is the first stepping stone towards finding our answer. It tells us that the number of chicks remaining after the first sale is greater than half the original number, which gives us a lower bound for x.
Next, let's look at what happens after the student sells the chicks initially. He gets 3 chicks back, increasing his total to x - 35 + 3, which simplifies to x - 32. Then, he sells 18 more chicks, leaving him with x - 32 - 18, which further simplifies to x - 50 chicks. The problem tells us that this final amount is less than 22 chicks. This gives us another inequality: x - 50 < 22. This second inequality provides an upper bound for the number of chicks the student initially had. Combining both inequalities, we'll be able to narrow down the possible solutions and pinpoint the exact number. Understanding these steps and how they translate into mathematical expressions is vital for cracking this kind of problem. Let's keep going and solve it!
Remember, guys, math problems like these are puzzles. Each piece of information is a clue, and the key is to put them together in the right way. So, stay with me, and let's figure out how many chicks this student started with!
Setting Up the Inequalities
Alright, let's dive deeper into setting up those crucial inequalities! This is where we translate the word problem into mathematical language, making it easier to solve. Remember, guys, the key is to break down the problem into smaller, manageable chunks. Our main goal here is to accurately represent the given information using inequality symbols.
First, consider the phrase "le quedaron más de la mitad," which means "more than half were left." If the student initially had x chicks, then more than x/2 chicks remained after selling 35. This translates directly into our first inequality: x - 35 > x/2. This inequality is super important because it sets a lower limit on the number of chicks the student could have started with. It tells us that after the first sale, he had more than half of his original chicks. Understanding this relationship is crucial for moving forward. Think of it like this: If he started with a small number of chicks, selling 35 would leave him with a negative number, which doesn't make sense in this context. So, x has to be large enough that selling 35 still leaves him with more than half his initial amount.
Now, let's tackle the second part of the problem. After getting 3 chicks back and selling 18 more, the student has less than 22 chicks remaining. We can express this as x - 35 + 3 - 18 < 22. Simplifying this expression gives us x - 50 < 22. This inequality gives us an upper limit on the number of chicks. It tells us that even after these transactions, the final number of chicks is less than 22. This upper limit is just as important as the lower limit we found earlier because it helps us narrow down the possibilities.
Combining these two inequalities, x - 35 > x/2 and x - 50 < 22, gives us a range of possible values for x. We're essentially creating a window where the correct answer must lie. Remember, guys, setting up these inequalities correctly is half the battle! Once we have them, solving for x becomes much easier. So, let's move on to the next step: solving these inequalities and finding the solution!
Solving the Inequalities
Okay, guys, now for the exciting part – solving the inequalities! This is where we put our algebra skills to the test to find the possible range for the initial number of chicks. Don't worry, it's not as scary as it sounds. We'll take it one step at a time, making sure everything is crystal clear.
Let's start with the first inequality: x - 35 > x/2. Our goal here is to isolate x on one side of the inequality. To do this, we can start by subtracting x/2 from both sides. This gives us x - x/2 - 35 > 0. Simplifying the left side, we get x/2 - 35 > 0. Now, we add 35 to both sides, which leaves us with x/2 > 35. To finally isolate x, we multiply both sides by 2. This gives us x > 70. So, this inequality tells us that the student initially had more than 70 chicks. This is a crucial piece of information as it eliminates any options less than or equal to 70.
Now, let's tackle the second inequality: x - 50 < 22. This one is a bit simpler to solve. We just need to add 50 to both sides of the inequality. This gives us x < 72. This inequality tells us that the student initially had less than 72 chicks. This upper bound is equally important as our lower bound because it narrows down our possible solutions even further.
Combining both inequalities, we have x > 70 and x < 72. This means that the initial number of chicks, x, must be greater than 70 but less than 72. Since we're dealing with a whole number of chicks, the only possible solution is x = 71. However, this option is not available in the given choices. Looking back at the options provided (a) 69, (b) 70, it seems there might be a slight discrepancy. Given our mathematical constraints, let's re-evaluate the provided options in the next section.
Remember guys, math is all about precision, but sometimes there might be slight differences between the theoretical solution and the options provided. It's important to stay flexible and think critically.
Evaluating the Options
Alright, guys, let's put on our detective hats and evaluate the given options in light of the inequalities we've solved. Remember, we found that x > 70 and x < 72. This narrows our choices down significantly. Let’s see which option fits best.
We have two options to consider: (a) 69 and (b) 70. Option (a), 69, is less than 70, so it doesn't satisfy our first inequality, x > 70. Therefore, we can eliminate option (a) right away. Option (b), 70, initially seems problematic because x must be greater than 70 according to our inequality. However, let’s plug 70 back into the original problem to see if it works. If the student started with 70 chicks:
- After selling 35, he has 70 - 35 = 35 chicks. This is exactly half of the initial amount, but the problem states he has more than half, so initially, option (b) seems incorrect.
- If we proceed with 70, after getting 3 chicks back, he has 70 - 35 + 3 = 38 chicks.
- After selling 18 more, he has 38 - 18 = 20 chicks, which is less than 22, so this condition checks out.
However, we need to re-emphasize that the condition x > 70 means that starting with exactly 70 doesn’t perfectly align with the initial condition of having more than half remaining after selling 35. There seems to be a subtle conflict between the exactness of the mathematical solution and the phrasing of the problem.
Given the available options and the strictness of the "more than half" condition, none of the options perfectly fits. The closest solution we derived mathematically was 71, which isn’t an option. In a real-world test scenario, it might be worth noting this discrepancy to the instructor or test administrator. However, in the context of selecting the best available option, 70 might be considered the closest, albeit imperfect, answer.
Final Answer and Key Takeaways
So, guys, after carefully analyzing the problem and evaluating the options, the closest answer we can reasonably select is (b) 70. It’s essential to acknowledge that this answer doesn't perfectly align with the strict mathematical inequality x > 70, but it’s the most logical choice given the options provided and the constraints of the problem.
Key Takeaways from this problem:
- Breaking Down Word Problems: The most important step is to break down the word problem into smaller, manageable parts. Identify the key information and translate it into mathematical expressions.
- Setting Up Inequalities: Accurately setting up inequalities is crucial for solving problems involving ranges or conditions like “more than” or “less than.”
- Solving Inequalities: Use algebraic techniques to isolate the variable and find the possible range of solutions.
- Evaluating Options: Always evaluate the given options in the context of the problem and the inequalities you've derived. Sometimes, the perfect answer might not be available, and you need to choose the best possible option.
- Critical Thinking: Math problems can sometimes have nuances. It’s important to think critically and consider the context of the problem, especially when the mathematical solution doesn't perfectly match the available choices.
Remember, guys, solving problems like these is a journey. It’s not just about getting the right answer; it’s about understanding the process and developing your problem-solving skills. Keep practicing, and you'll become math whizzes in no time!