Time To Install Floor: Jasper, Yolanda, And Teamwork

Hey guys! Let's dive into a classic problem involving work rates and teamwork. This is a scenario where we figure out how long it takes different people to complete a task, both individually and together. It's like a puzzle, and we're going to solve it step by step. So, let’s break down this question:

If Jasper takes 7 hours to install a floor by himself, and when he works with Yolanda, it takes them only 3 hours, the big question is: How long would it take Yolanda to install the floor if she were working alone? This is a common type of problem you might see in math classes, especially when you're dealing with rates and time. Don't worry, it's not as tricky as it might seem at first glance. We'll use a bit of algebra and some logical thinking to crack this one. Let's get started and see how we can find the answer together!

Understanding Work Rate

To solve this problem effectively, we first need to grasp the concept of work rate. Think of work rate as the amount of a job someone can complete in a single unit of time, like an hour. In our case, it's how much of the floor Jasper or Yolanda can install in one hour. Understanding this is key to figuring out how long it takes them to work together or individually.

Let’s start with Jasper. If he can install an entire floor in 7 hours, we can say his work rate is 1/7 of the floor per hour. This means that in each hour he works, he completes one-seventh of the total job. Now, let's think about why this is important. When we know the fraction of work someone does in an hour, we can compare it to someone else's work rate and also figure out how much they accomplish when working together. This approach helps us translate the time it takes to complete a job into a manageable fraction, making the math a whole lot easier.

Similarly, we’ll apply the same logic to Yolanda. The goal here is to find her individual work rate so we can calculate how long it would take her to install the floor on her own. Once we’ve got both Jasper's and Yolanda's individual work rates, we can combine them to understand their combined work rate. This is super useful for determining how quickly they can finish the job when they team up. So, stick with me as we unravel the mystery of work rates and apply them to solve our floor-installing puzzle!

Setting Up the Equation

Now, let's get to the heart of the problem and set up an equation that will help us find the answer. This is where the magic happens, and we start turning our understanding of work rates into a concrete mathematical expression. Remember, the key is to translate the given information into a format we can work with. Let's break it down step by step.

We already know Jasper’s work rate: he completes 1/7 of the floor per hour. Let's denote Yolanda’s work rate as 1/x, where x is the number of hours it would take her to install the floor alone. This is what we’re trying to find. When they work together, they complete the floor in 3 hours. This means their combined work rate is 1/3 of the floor per hour. So, how do we put this all together in an equation? Well, the combined work rate is simply the sum of their individual work rates. That gives us the equation:

(1/7) + (1/x) = 1/3

This equation is the crux of our solution. It represents the relationship between Jasper's work rate, Yolanda's work rate, and their combined work rate. Now, it might look a bit intimidating with those fractions, but don't worry! We’re going to tackle it together. The next step is to solve this equation for x, which will give us the time it takes Yolanda to install the floor by herself. So, let's roll up our sleeves and get ready to do some algebra!

Solving for Yolanda's Time

Alright, let's dive into solving the equation we set up. This is where we put on our algebra hats and manipulate the numbers to find the value of x, which represents the time it takes Yolanda to install the floor alone. Remember, our equation is:

(1/7) + (1/x) = 1/3

The first thing we want to do is isolate the term with x, which is 1/x. To do this, we'll subtract 1/7 from both sides of the equation. This keeps the equation balanced and moves us closer to our goal. So, we get:

1/x = 1/3 - 1/7

Now, we need to subtract the fractions on the right side. To do this, we need a common denominator. The least common multiple of 3 and 7 is 21, so we'll convert both fractions to have this denominator. This gives us:

1/x = (7/21) - (3/21)

Subtracting the fractions, we get:

1/x = 4/21

We're almost there! Now, we need to solve for x. Since we have 1/x, we can take the reciprocal of both sides to find x. This means flipping the fractions:

x = 21/4

So, x equals 21/4, which is an improper fraction. To make it easier to understand, let's convert it to a mixed number. 21 divided by 4 is 5 with a remainder of 1, so we get:

x = 5 1/4

This tells us that it would take Yolanda 5 and 1/4 hours to install the floor by herself. But let's break that down even further. 1/4 of an hour is 15 minutes, so Yolanda would take 5 hours and 15 minutes to complete the job alone. Yay, we solved it! But, before we celebrate, let's recap our steps to make sure we fully understand the process. This way, we can tackle similar problems with confidence.

Checking the Answer

Before we wrap things up, it's always a good idea to check our answer. This is like the final polish on our work, ensuring we haven't made any mistakes along the way. It's a crucial step in problem-solving because it gives us confidence in our solution. So, let's verify if our answer—5 and 1/4 hours for Yolanda to install the floor alone—makes sense in the context of the problem.

We found that Yolanda’s work rate is 1/(5 1/4), which is the same as 1/(21/4), or 4/21 of the floor per hour. Now, let's add her work rate to Jasper’s work rate (1/7) and see if it equals their combined work rate (1/3). We have:

(1/7) + (4/21) = ?

To add these fractions, we need a common denominator, which is 21. Converting 1/7, we get 3/21. So, the equation becomes:

(3/21) + (4/21) = 7/21

Simplifying 7/21, we get 1/3. And guess what? That’s exactly their combined work rate! This confirms that our answer is correct. Yolanda’s individual time to install the floor is indeed 5 hours and 15 minutes. Checking our work not only validates our answer but also reinforces our understanding of the problem-solving process. It’s a habit that will serve you well in math and beyond. Now that we've nailed this problem, let's summarize the key takeaways so you can tackle similar challenges with ease.

Key Takeaways

Okay, guys, let's recap the key takeaways from this problem. Understanding these concepts will help you tackle similar questions with confidence. We've covered a lot, from work rates to solving equations, so let's distill the main points.

First and foremost, we learned about work rate. Remember, work rate is the amount of work completed per unit of time. In our case, it was the fraction of the floor Jasper or Yolanda could install in an hour. Understanding work rate is crucial because it allows us to compare different people's productivity and calculate how much they can achieve together.

Next, we tackled the importance of setting up the equation correctly. We translated the word problem into a mathematical expression that represented the relationship between Jasper's work rate, Yolanda's work rate, and their combined work rate. This step is vital because the equation is the roadmap to our solution. A well-set-up equation makes the rest of the problem-solving process much smoother.

Then, we dived into the algebra and solved for the unknown. We used techniques like finding common denominators, isolating variables, and taking reciprocals to find the value of x, which represented the time it takes Yolanda to install the floor alone. This part highlights the importance of algebraic skills in solving real-world problems.

Finally, we emphasized the significance of checking our answer. We verified that our solution made sense in the context of the problem, ensuring we hadn't made any computational errors. This step is often overlooked but is crucial for building confidence in our solutions.

In summary, remember to break down the problem, understand the concept of work rate, set up the equation carefully, solve for the unknown using your algebraic skills, and always, always check your answer. With these takeaways in mind, you're well-equipped to handle similar work-rate problems. Keep practicing, and you'll become a pro in no time!

So, that's how we solved this problem. I hope you found it helpful and that you're now more confident in tackling similar questions. Remember, practice makes perfect, so keep at it! Until next time, happy problem-solving!