Math Problem: Road Repaving Time Calculation

Hey guys, let's dive into a classic math problem that's super relevant for understanding rates and work. We've got a road crew that's on a mission to repave a stretch of road, and we need to figure out just how long that mission will take. This isn't just about crunching numbers; it's about understanding how to break down a task when you know the rate at which you can complete a piece of it. So, stick around, and we'll unravel this problem step-by-step, making sure we end up with the simplest, most accurate answer possible. Whether you're a student tackling homework or just someone who enjoys a good brain teaser, this problem will give you a solid workout in fractions and division.

Understanding the Problem: Rates and Work

Alright, let's break down what we're dealing with here. We have a road crew, and their superpower is repaving roads. We're given their speed, or more accurately, their rate of work: they can repave $\frac{1}{48}$ miles of road each hour. That's a pretty specific rate, right? Now, they have a total job to do: repave a road that's $\frac{3}{8}$ miles long. The big question is, how long will it take them to complete this entire job? This is a classic rate problem, often found in mathematics, where you need to figure out the total time required given a rate and a total amount of work. Think of it like baking cookies – if you can bake 12 cookies in 30 minutes, how long will it take you to bake 36 cookies? It's the same principle, just with road miles instead of cookies.

To solve this, we need to use division. We have the total amount of work (the length of the road) and the amount of work done per unit of time (the miles repaved per hour). When you divide the total work by the rate of work, you get the total time. In mathematical terms, it looks like this:

Total Time = Total Work / Rate of Work

In our case, the Total Work is the length of the road, which is $\frac{3}{8}$ miles. The Rate of Work is how fast the crew is working, which is $\frac{1}{48}$ miles per hour. So, the formula becomes:

Total Time = $\frac{3}{8}$ miles / $\frac{1}{48}$ miles per hour

This is where our fraction skills really come into play. Dividing by a fraction is the same as multiplying by its reciprocal. So, we'll flip the second fraction ($\frac{1}{48}$) and multiply. This step is crucial, and getting it right ensures our final answer is accurate. It's like reversing the operation to find the original quantity. Don't get bogged down if fractions seem tricky; we'll walk through it, and you'll see it's totally manageable. The key is to remember the rules of fraction operations, and you'll be golden. This problem is a fantastic way to reinforce those foundational math concepts, guys, so let's really focus on nailing this calculation.

Calculating the Time: Division of Fractions

Now for the fun part – the actual calculation! We established that to find the total time, we need to divide the total road length by the crew's repaving rate. So, we have:

Time = $\frac{3}{8} \div \frac{1}{48}$

Remember what we said about dividing fractions? We keep the first fraction the same, change the division sign to a multiplication sign, and flip the second fraction (find its reciprocal). So, $\frac{1}{48}$ becomes $\frac{48}{1}$. Our equation now looks like this:

Time = $\frac{3}{8} \times \frac{48}{1}$

When multiplying fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we get:

Time = $\frac{3 \times 48}{8 \times 1}$

Time = $\frac{144}{8}$

Woah, 144 over 8! That looks like a big number, but we're not done yet. The problem asks for the answer in its simplest form. This means we need to simplify this fraction. We can do this by finding the greatest common divisor (GCD) of 144 and 8, or by simply performing the division. Let's simplify by finding common factors first. Both 144 and 8 are divisible by 8.

Let's divide 144 by 8:

$144 \div 8 = 18$

And divide 8 by 8:

$8 \div 8 = 1$

So, the fraction simplifies to $\frac{18}{1}$, which is just 18.

Alternatively, we could have simplified before multiplying. This is often a smarter move with larger numbers. Look at our multiplication step again:

Time = $\frac{3}{8} \times \frac{48}{1}$

See how 8 and 48 share a common factor? Both are divisible by 8. So, we can divide 8 by 8 (which gives us 1) and divide 48 by 8 (which gives us 6).

Time = $\frac{3}{1} \times \frac{6}{1}$

Now, multiply the numerators and the denominators:

Time = $\frac{3 \times 6}{1 \times 1}$

Time = $\frac{18}{1}$

Which, again, simplifies to 18.

This calculation shows that it will take the road crew 18 hours to repave the $\frac{3}{8}$ mile road. It's pretty cool how breaking down the problem and using the rules of fraction division leads us directly to the answer. Remember, simplifying before multiplying can save you a lot of headache with bigger numbers, guys!

Final Answer and Implications

So, after all that fraction fun, we've arrived at our answer: it will take the road crew 18 hours to repave the $\frac{3}{8}$ mile road. This is our final answer, and it's already in its simplest form because 18 is a whole number. It's important to ensure that we've addressed all parts of the original question, and we have. We were asked how long it would take and to write the answer in simplest form.

What does this 18 hours mean in the real world? Well, it means that if the crew works consistently at their rate of $\frac{1}{48}$ miles per hour, they'll need a full 18-hour shift (or perhaps spread over multiple days) to complete the job. This kind of calculation is vital for project planning. Road construction companies need to estimate how long a project will take to schedule labor, order materials, and inform the public about disruptions. If this were a major highway, 18 hours of repaving might be broken down into overnight shifts or weekend work to minimize traffic impact. It also highlights the scale of road repair; even a seemingly short section of road ($\frac{3}{8}$ of a mile) requires a significant amount of time and effort when you consider the rate of work.

This problem elegantly demonstrates the power of fractions and division in solving practical, real-world scenarios. It's a reminder that the math skills we learn in school aren't just abstract concepts; they have direct applications in how things get done in the world around us. Whether it's construction, logistics, or even cooking, understanding rates and proportions helps us make sense of the time and resources required for any task. So, next time you see roadwork, you can mentally picture the calculations happening behind the scenes to get that road fixed up! Keep practicing these kinds of problems, guys, because the more you do them, the more intuitive they become. And who knows, you might even find yourself enjoying the challenge of solving them!

Remember, the core of this problem was understanding the relationship between total work, rate, and time. By dividing the total length of the road by the rate at which the crew repaves, we directly calculated the time needed. The steps involved converting the division of fractions into multiplication by the reciprocal and then simplifying the resulting fraction. This process ensures we get the most accurate and concise answer. It's a solid mathematical technique that applies to countless other problems. So, pat yourselves on the back for tackling this one, and keep those math brains sharp!