Wooden Chair Construction Time: Austin And Elliott

Let's dive into a classic problem involving work rates and collaboration. This scenario involves Austin and Elliott, two classmates in shop class, building a wooden chair. The core of the problem lies in understanding how their individual work rates combine when they work together. We'll break down the problem step by step, making it super easy to follow.

Understanding the Problem

So, here’s the deal: Austin takes a whole 8 hours longer than Elliott to build a wooden chair all by himself. When these two team up, they can knock out one chair in just 3 hours. The big question here is: How long would it take each of them to build a chair if they were working solo? This is a classic work-rate problem, and we're going to solve it using algebra. The key is to translate the words into mathematical expressions. We will define our variables and build our equations to represent each person's work rate and their combined efforts.

Defining Variables

Let's start by defining our variables. Let's denote the time it takes for Elliott to construct a chair as t (in hours). Since Austin takes 8 hours longer, the time it takes for Austin to construct a chair will be t + 8 (in hours). With these variables defined, we are ready to translate their individual work rates into fractions, which will help us formulate our equations. So far, so good, right?

Expressing Work Rates

Now, let’s think about their work rates. Work rate is the amount of work done per unit of time. If Elliott takes t hours to complete one chair, his work rate is 1/t chairs per hour. Similarly, Austin's work rate is 1/(t + 8) chairs per hour. When they work together, their work rates add up. Given that they complete one chair in 3 hours when working together, their combined work rate is 1/3 chairs per hour. Setting up these work rates is crucial, as it allows us to form an equation that represents their collaborative effort. This equation will be the key to solving for t and, subsequently, finding out how long each of them takes individually.

Forming the Equation

Okay, so when they work together, their combined work rate is the sum of their individual work rates. This gives us the equation:

1/t + 1/(t + 8) = 1/3

This equation represents that Elliott's work rate plus Austin's work rate equals their combined work rate. This is a rational equation that we need to solve for t. Solving this equation will give us the time it takes Elliott to build a chair alone, and from there, we can easily find the time it takes Austin.

Solving the Equation

Now, let's solve this equation step by step. This part might seem a bit tricky, but don't worry, we'll take it slow.

Clearing the Fractions

To get rid of the fractions, we need to find the least common denominator (LCD) of t, t + 8, and 3. The LCD is 3t(t + 8). We'll multiply both sides of the equation by this LCD:

3t(t + 8) * (1/t + 1/(t + 8)) = 3t(t + 8) * (1/3)

Distributing the LCD on the left side, we get:

3(t + 8) + 3t = t(t + 8)

Simplifying the Equation

Now, let's simplify this equation by expanding and combining like terms:

3t + 24 + 3t = t^2 + 8t

Combining the terms on the left side, we have:

6t + 24 = t^2 + 8t

Rearranging to Quadratic Form

To solve for t, we need to rearrange the equation into a standard quadratic form, which is at^2 + bt + c = 0. Subtract 6t and 24 from both sides to get:

0 = t^2 + 2t - 24

Factoring the Quadratic

Now, we need to factor the quadratic equation. We're looking for two numbers that multiply to -24 and add to 2. Those numbers are 6 and -4. So, we can factor the quadratic as:

0 = (t + 6)(t - 4)

Finding the Values of t

Setting each factor equal to zero gives us the possible values for t:

t + 6 = 0 or t - 4 = 0

Solving for t, we get:

t = -6 or t = 4

Choosing the Correct Value

Since time cannot be negative, we discard t = -6. Therefore, t = 4. This means it takes Elliott 4 hours to construct a wooden chair by himself.

Calculating Austin's Time

Now that we know Elliott's time, we can easily find Austin's time. Remember, Austin takes 8 hours longer than Elliott, so:

Austin's time = t + 8 = 4 + 8 = 12

So, it takes Austin 12 hours to construct a wooden chair by himself.

Final Answer

• Elliott takes 4 hours to build a chair.
• Austin takes 12 hours to build a chair.

Verification

Let's quickly verify our solution. Elliott's work rate is 1/4 chairs per hour, and Austin's work rate is 1/12 chairs per hour. Working together, their combined work rate is:

1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3

Since their combined work rate is 1/3 chairs per hour, it would indeed take them 3 hours to complete one chair together. This confirms that our solution is correct.

Conclusion

So, there you have it! By carefully defining variables, setting up the correct equations, and solving them step by step, we were able to determine how long it takes each classmate to construct a wooden chair individually. These work-rate problems might seem tricky at first, but with practice, you'll get the hang of them. Remember to break the problem down into smaller, manageable steps, and always verify your solution to make sure it makes sense. You got this, guys! Understanding the individual rates and how they combine is key to solving this type of problem. We translated the word problem into a mathematical equation, solved it, and verified our results. This step-by-step approach is invaluable for tackling similar problems in the future. Keep practicing, and you'll become a pro at solving these types of questions in no time. Remember, the key is to translate the problem into manageable mathematical expressions and then systematically solve for the unknowns. And always, always check your answer to ensure it makes sense in the context of the original problem. Good luck, and happy problem-solving! The ability to dissect word problems and convert them into solvable equations is a valuable skill, not just in mathematics but also in various real-life scenarios where you need to optimize processes or allocate resources efficiently. The more you practice, the more intuitive this process becomes. Don't be discouraged by the complexity; each problem you solve brings you one step closer to mastery. And remember, there are plenty of resources available online and in textbooks to help you further develop your skills in this area. Keep exploring, keep learning, and most importantly, keep enjoying the process of problem-solving!