Total Miles Anthony Drives In A Week: A Fraction Problem

Hey guys! Let's dive into a fun math problem involving fractions and distances. We're going to figure out how many miles Anthony drives in a week, given his daily commute and trips. This is a great example of how we use math in our everyday lives, so let's break it down step by step.

Understanding the Daily Drive

First, let's focus on Anthony's daily driving routine. Anthony drives different segments each day: $\frac{2}{3}$ of a mile to work, then $\frac{1}{4}$ of a mile to his friend's house, and finally $\frac{5}{6}$ of a mile back home. To find the total distance he drives in a single day, we need to add these fractions together. This is where our fraction skills come into play! Remember, we need a common denominator to add fractions. The least common multiple of 3, 4, and 6 is 12. So, we'll convert each fraction to have a denominator of 12.

Let's convert each fraction:

• $\frac{2}{3}$ = $\frac{2 \times 4}{3 \times 4}$ = $\frac{8}{12}$
• $\frac{1}{4}$ = $\frac{1 \times 3}{4 \times 3}$ = $\frac{3}{12}$
• $\frac{5}{6}$ = $\frac{5 \times 2}{6 \times 2}$ = $\frac{10}{12}$

Now we can easily add them:

$\frac{8}{12}$ + $\frac{3}{12}$ + $\frac{10}{12}$ = $\frac{8 + 3 + 10}{12}$ = $\frac{21}{12}$

So, Anthony drives $\frac{21}{12}$ miles each day. This fraction can be simplified. Both 21 and 12 are divisible by 3, so let's simplify it:

$\frac{21}{12}$ = $\frac{21 \div 3}{12 \div 3}$ = $\frac{7}{4}$

We can also express this as a mixed number, which might give us a better sense of the distance. $\frac{7}{4}$ is equal to 1 whole (4/4) and 3/4 left over. So, it's 1$\frac{3}{4}$ miles. That means Anthony drives 1 and $\frac{3}{4}$ miles each day. Understanding these daily distances is crucial before we calculate the weekly total. It's like figuring out the basic building block before constructing the whole structure. Without this, the rest of the calculation wouldn't make sense. By ensuring we've correctly calculated the daily distance, we set ourselves up for an accurate final answer. Remember, guys, precision is key in math! Miscalculating the daily distance throws off the entire calculation. So, we took our time, found the common denominator, added the fractions correctly, simplified, and even converted to a mixed number for better understanding. This meticulous approach ensures we're on the right track. The cool thing about breaking down the problem like this is that it makes the whole thing feel less intimidating, right? Instead of a huge question about weekly mileage, we’ve turned it into a manageable daily calculation, which is then scaled up to the week. And that's a strategy you can use for a bunch of other problems too. So, with the daily distance nailed down, we're all set to tackle the next part: figuring out the total for the week. Let’s jump into it!

Calculating the Weekly Distance

Now that we know Anthony drives $\frac{7}{4}$ miles each day, we need to figure out how far he drives in a week. The problem states that Anthony drives this same path from Sunday through Saturday, which means he drives this distance 7 days a week. To find the total weekly distance, we simply multiply his daily distance by the number of days in a week. Multiplying fractions is generally straightforward; we multiply the numerators and the denominators. So, we'll multiply $\frac{7}{4}$ miles by 7 days. Think of 7 as the fraction $\frac{7}{1}$ to make the multiplication clearer.

Here's the calculation:

$\frac{7}{4}$ miles/day $\times$ 7 days = $\frac{7}{4}$ $\times$ $\frac{7}{1}$ = $\frac{7 \times 7}{4 \times 1}$ = $\frac{49}{4}$ miles

So, Anthony drives $\frac{49}{4}$ miles in a week. But again, this fraction might be easier to understand if we convert it to a mixed number. To do this, we divide 49 by 4.

49 divided by 4 is 12 with a remainder of 1. This means that $\frac{49}{4}$ is equal to 12 whole numbers and $\frac{1}{4}$ left over. So, the mixed number is 12$\frac{1}{4}$.

Therefore, Anthony drives 12$\frac{1}{4}$ miles in a week. This gives us a much clearer picture of the total distance. Guys, visualizing this distance can be helpful. Imagine driving 12 miles and then another quarter of a mile – that's Anthony's weekly travel! We've successfully calculated the total distance, but let's think about why this process is so important. Knowing how to work with fractions is essential in many real-world situations, not just math problems. Whether you're calculating cooking measurements, figuring out how much material you need for a project, or even planning a road trip, fractions are everywhere. Mastering these skills builds confidence and gives you the tools to solve all sorts of problems. Plus, it’s pretty satisfying to break down a problem and arrive at the correct answer, right? Each step we took – from finding the daily distance to multiplying by the number of days – was like piecing together a puzzle. And now, the puzzle is complete! We know exactly how far Anthony drives each week. But hold on, let's recap the entire process to make sure we’ve got it all locked in.

Recapping the Solution

Okay, let's do a quick recap of how we solved this problem. This is always a good practice to ensure we understand each step and why we did it. First, we identified that Anthony's daily drive consisted of three segments: $\frac{2}{3}$ of a mile to work, $\frac{1}{4}$ of a mile to his friend's house, and $\frac{5}{6}$ of a mile home. The crucial first step was to add these fractions together to find his total daily distance. To add fractions, we needed a common denominator, which we found to be 12. We converted each fraction to an equivalent fraction with a denominator of 12, making the addition straightforward.

$\frac{2}{3}$ became $\frac{8}{12}$, $\frac{1}{4}$ became $\frac{3}{12}$, and $\frac{5}{6}$ became $\frac{10}{12}$. Adding these gave us $\frac{21}{12}$ miles per day. We then simplified this fraction to $\frac{7}{4}$ and converted it to a mixed number, 1$\frac{3}{4}$ miles, for a clearer understanding of the daily distance. Next, we tackled the weekly distance. Since Anthony drives this route every day for a week (7 days), we multiplied his daily distance, $\frac{7}{4}$ miles, by 7. This resulted in $\frac{49}{4}$ miles. Finally, we converted this improper fraction to a mixed number to make it easier to grasp. $\frac{49}{4}$ miles is equal to 12$\frac{1}{4}$ miles. So, Anthony drives a total of 12$\frac{1}{4}$ miles each week.

Guys, notice how converting between improper fractions and mixed numbers helped us better understand the distances? Sometimes, a mixed number just makes more sense in our heads than an improper fraction. This is a great skill to have when solving real-world problems. Breaking down complex problems into smaller, manageable steps is a fantastic strategy. We took the big question of weekly mileage and turned it into smaller questions about daily mileage and the number of days in a week. This made the whole process much less daunting. And that's a key takeaway here: don't be afraid to break things down! Thinking about the steps we took, we can appreciate how each one built on the previous one. Adding fractions, simplifying fractions, multiplying fractions, and converting between fraction types – all these skills came together to help us solve this problem. That's the beauty of math – it's like a toolbox filled with useful tools that you can use in so many different situations. Now, let's wrap up with some final thoughts on the importance of these skills.

Final Thoughts on Fraction Skills

So, guys, we've successfully calculated the total distance Anthony drives in a week, which is 12$\frac{1}{4}$ miles. But beyond getting the right answer, it’s important to appreciate the process and the skills we used along the way. Working with fractions is a fundamental skill in mathematics, and it pops up in so many areas of life. Whether it's cooking, measuring, budgeting, or even planning a trip, fractions are everywhere! The better you understand them, the more confident you'll feel in tackling all sorts of challenges.

This problem also highlights the importance of breaking down complex tasks into smaller, more manageable steps. When we first looked at the question, figuring out the total weekly distance might have seemed a bit daunting. But by focusing on the daily distance first and then scaling up to the week, we made the problem much easier to solve. This is a valuable problem-solving strategy that you can apply to many other situations, both in math and in life. And remember, guys, practice makes perfect! The more you work with fractions, the more comfortable you'll become with them. Don't be afraid to make mistakes – they're a natural part of the learning process. The key is to learn from those mistakes and keep practicing. There are tons of resources available online and in textbooks to help you hone your fraction skills. Explore different methods, try different problems, and most importantly, have fun with it! Math can be challenging, but it can also be incredibly rewarding. The satisfaction of cracking a tough problem and arriving at the correct answer is a fantastic feeling. Plus, the skills you develop in math can benefit you in so many other areas of your life. So, keep practicing, keep exploring, and keep challenging yourselves. And who knows, maybe the next time you're planning a road trip, you'll be able to calculate the distances with fractions like a pro! We hope this breakdown helped you understand how to tackle fraction problems and appreciate their real-world applications. Keep up the great work, guys!