If Students Sit 3 Per Bench, We Are Short 3 Benches. If They Sit 5 Per Bench, 2 Benches Are Left Empty, And One Bench Only Has 4 Students. How Many Benches Are There? If The Number Of Benches Is *x*, How To Find X?

Have you ever encountered a seemingly simple problem that unravels into a captivating puzzle? Let's dive into a classic mathematical brain-teaser that involves students, benches, and a dash of logical deduction. This problem, often encountered in algebra or word problem exercises, challenges us to think critically and apply our problem-solving skills. It's not just about finding the answer; it's about the journey of unraveling the problem and understanding the underlying relationships.

Deconstructing the Bench Problem

This mathematical problem presents a scenario involving students and benches, posing a classic algebraic puzzle. At its core, the problem asks us to determine the total number of benches in a classroom or setting, given two different seating arrangements and their respective constraints. To fully grasp the challenge, let's break down the problem into its key components. The problem states: If students sit 3 per bench, we are short 3 benches. If they sit 5 per bench, 2 benches are left empty, and one bench only has 4 students. How many benches are there? The beauty of this problem lies in its simplicity and the way it subtly weaves together mathematical concepts. We need to translate these words into mathematical expressions, using variables to represent unknowns, and then solve for those variables. This is a fundamental skill in algebra, and this problem provides an excellent opportunity to practice it. First, we have a situation where the students are sitting 3 per bench, and we are short 3 benches. This implies that there aren't enough benches to accommodate all the students in groups of 3. Second, we have a situation where the students are sitting 5 per bench, leaving 2 benches completely empty and one bench with only 4 students. This adds another layer of complexity to the problem. The goal is to find the number of benches, which we can represent with a variable, say x. We need to set up equations that relate the number of benches to the number of students in both seating scenarios. This is where the algebraic magic happens, as we transform the word problem into a set of solvable equations.

Setting Up the Equations: The Algebra of Seating

To solve this classic bench puzzle, we must translate the word problem into the language of algebra. This involves identifying the unknowns, assigning variables, and constructing equations that represent the given relationships. This crucial step transforms the problem from a narrative into a solvable mathematical model. Let's begin by defining our variable. Let x represent the total number of benches. This is the quantity we ultimately want to find. Next, we need to express the number of students in terms of x for each seating scenario. This is where careful reading and logical deduction come into play. In the first scenario, students sit 3 per bench, and we are short 3 benches. This means that if we had 3 more benches, we could seat all the students. So, the number of students can be expressed as 3(x + 3). We add 3 to the number of benches because we are short 3 benches in the original scenario. Now, let's consider the second scenario. Students sit 5 per bench, 2 benches are left empty, and one bench has only 4 students. This is a bit more complex, but we can break it down. We have x benches in total, 2 of which are empty, and 1 has 4 students. This leaves x - 3 benches with 5 students each. So, the number of students can be expressed as 5(x - 3) + 4. We subtract 3 from the number of benches because 2 are empty, and 1 has 4 students instead of 5. We then multiply the remaining benches by 5 (students per bench) and add the 4 students on the partially filled bench. Now we have two expressions representing the same quantity – the total number of students. We can set these expressions equal to each other to form our equation: 3(x + 3) = 5(x - 3) + 4. This is the heart of the solution. We've successfully transformed the word problem into an algebraic equation. The next step is to solve this equation for x, which will give us the number of benches.

Solving for x: Unveiling the Number of Benches

With our algebraic equation firmly in place – 3(x + 3) = 5(x - 3) + 4 – we now embark on the process of solving for x. This is where our algebraic manipulation skills come into play, as we carefully isolate x to reveal the solution. To solve for x, we begin by expanding both sides of the equation, carefully distributing the constants. This means multiplying 3 by both terms inside the first set of parentheses and 5 by both terms inside the second set. Expanding the left side, we get 3x + 9. Expanding the right side, we get 5x - 15 + 4. Now our equation looks like this: 3x + 9 = 5x - 11. The next step is to gather the x terms on one side of the equation and the constant terms on the other. We can do this by subtracting 3x from both sides and adding 11 to both sides. Subtracting 3x from both sides, we get 9 = 2x - 11. Adding 11 to both sides, we get 20 = 2x. Now we have a simple equation with x on one side. To isolate x, we divide both sides of the equation by 2. Dividing both sides by 2, we get 10 = x. Therefore, x = 10. This is our solution! We've successfully solved for x, which represents the number of benches. But before we celebrate, it's crucial to verify our solution.

Verifying the Solution: Ensuring Accuracy

Having arrived at a potential solution, x = 10 benches, it's crucial to verify its accuracy. This step ensures that our algebraic manipulations were correct and that our answer logically fits the original problem statement. Verification is a cornerstone of mathematical problem-solving, preventing errors and building confidence in our results. To verify our solution, we substitute x = 10 back into the original equation and check if both sides are equal. Our equation was 3(x + 3) = 5(x - 3) + 4. Substituting x = 10, we get 3(10 + 3) = 5(10 - 3) + 4. Now we simplify both sides independently. On the left side, we have 3(13) = 39. On the right side, we have 5(7) + 4 = 35 + 4 = 39. Since both sides of the equation equal 39 when x = 10, our solution is verified. But we're not quite done yet! We also need to check if our solution makes sense in the context of the original word problem. Does having 10 benches satisfy both seating scenarios? In the first scenario, if students sit 3 per bench, we are short 3 benches. With 10 benches, we would need 10 + 3 = 13 benches to seat all students at 3 per bench. This means there are 13 * 3 = 39 students. In the second scenario, if students sit 5 per bench, 2 benches are left empty, and one bench has only 4 students. With 10 benches, this means 2 benches are empty, 7 benches have 5 students, and 1 bench has 4 students. This gives us a total of 7 * 5 + 4 = 35 + 4 = 39 students. Since the number of students is the same in both scenarios (39 students), our solution of 10 benches is consistent with the problem's conditions. We can confidently conclude that there are 10 benches.

The Power of Problem-Solving: More Than Just an Answer

We've successfully navigated the bench puzzle, employing algebraic techniques to arrive at the solution: 10 benches. However, the true value of this exercise extends far beyond the numerical answer. It's about the journey of problem-solving itself, the process of transforming a word problem into a mathematical representation, and the satisfaction of unraveling a complex scenario. This type of problem hones critical thinking skills, teaching us to break down complex situations into smaller, manageable parts. We learned to identify key information, define variables, and construct equations that capture the relationships between them. These are skills that are transferable to a wide range of real-world situations. The process of solving the equation itself reinforced our algebraic manipulation skills. We practiced distributing, combining like terms, and isolating variables – fundamental techniques in mathematics. And perhaps most importantly, we learned the importance of verification. Checking our solution not only ensures accuracy but also deepens our understanding of the problem and our confidence in our results. This bench problem, seemingly simple on the surface, provides a valuable lesson in the power of problem-solving. It demonstrates how mathematical tools can be used to model and analyze real-world scenarios, fostering a deeper appreciation for the beauty and utility of mathematics. So, the next time you encounter a challenging problem, remember the lessons learned from the bench puzzle: break it down, translate it into a mathematical model, solve it systematically, and always verify your solution. The journey of problem-solving is as rewarding as the destination.

In conclusion, the bench problem exemplifies the beauty and power of mathematical problem-solving. By carefully analyzing the given information, translating it into algebraic equations, and systematically solving for the unknown, we were able to determine the number of benches. The process not only honed our mathematical skills but also reinforced the importance of critical thinking, logical deduction, and verification. This type of problem serves as a valuable reminder that mathematics is not just about numbers and formulas; it's a powerful tool for understanding and solving real-world challenges.