Tourist's Journey: Calculating Total Travel Time

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Hey guys! Let's break down this travel time problem step by step. We've got a tourist who's been globetrotting using various modes of transport – a bus, a car, and good old-fashioned walking. The core question we're tackling today is: How much total time did this tourist spend on their entire journey? To figure this out, we need to add up the time spent on each leg of the trip. It sounds straightforward, but there's a little twist – we're dealing with mixed fractions and minutes, so we need to get everything into the same units before we can add them up. Now, before we dive into the numbers, why is this kind of problem important? Well, in real life, we're often juggling different units of time. Think about planning a trip, scheduling meetings, or even cooking a meal – you're constantly converting between hours, minutes, and sometimes even seconds. So, mastering these conversions and additions is a super practical skill. We'll start by converting the mixed fractions into improper fractions. This will make it easier to work with them mathematically. Then, we'll convert the 24 minutes into a fraction of an hour so that all our time measurements are in hours. Finally, we'll add up all the times to find the total journey time. Ready to put on your math hats and get started? Let's do this!

Breaking Down the Journey

Okay, let's dive into the specifics of the tourist's journey. This is where we take each leg of the trip and convert them into consistent units, which in this case, will be hours. Our tourist spent 1 rac{1}{6} hours on the bus. The first step is converting this mixed fraction into an improper fraction. Remember how to do that? We multiply the whole number (1) by the denominator (6) and then add the numerator (1). This gives us (1 * 6) + 1 = 7. We keep the same denominator, so 1 rac{1}{6} becomes 76\frac{7}{6} hours. Easy peasy, right? Next up, we have the car journey, which took 2 rac{2}{3} hours. Let's repeat the process. Multiply the whole number (2) by the denominator (3) and add the numerator (2): (2 * 3) + 2 = 8. So, 2 rac{2}{3} hours transforms into 83\frac{8}{3} hours. We're on a roll! Now, for the walking portion, the tourist hoofed it for 24 minutes. This is where we need to convert minutes into hours. There are 60 minutes in an hour, so 24 minutes is 2460\frac{24}{60} of an hour. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 12. So, 2460\frac{24}{60} becomes 25\frac{2}{5} hours. Fantastic! We've now successfully converted all the travel times into hours: 76\frac{7}{6} hours by bus, 83\frac{8}{3} hours by car, and 25\frac{2}{5} hours on foot. The next step? Adding these fractions together to find the total travel time. Before we do that, it’s important to understand why we do this conversion. Imagine trying to add 1 apple, 2 oranges, and 3 bananas directly. You can't, right? You need a common unit, like “pieces of fruit.” Similarly, we need a common unit (hours) to add these different time segments together. This meticulous approach ensures we get an accurate final answer. Let's move on to the next part where the real addition happens!

Adding Up the Travel Time

Alright, guys, we've got our travel times all neatly converted into hours: 76\frac{7}{6} hours by bus, 83\frac{8}{3} hours by car, and 25\frac{2}{5} hours walking. Now comes the fun part – adding these fractions together to find the total travel time. But remember, we can't just add fractions willy-nilly; they need a common denominator first. Finding the least common denominator (LCD) is the key here. The denominators we have are 6, 3, and 5. To find the LCD, we need to find the smallest number that all these denominators divide into evenly. One way to do this is to list the multiples of each number until we find a common one. Multiples of 6: 6, 12, 18, 24, 30, ... Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... Multiples of 5: 5, 10, 15, 20, 25, 30, ... Aha! We see that 30 is the smallest number that appears in all three lists. So, the LCD is 30. Now we need to convert each fraction into an equivalent fraction with a denominator of 30. For 76\frac{7}{6}, we multiply both the numerator and denominator by 5 (because 6 * 5 = 30): 7565=3530\frac{7 * 5}{6 * 5} = \frac{35}{30}. For 83\frac{8}{3}, we multiply both the numerator and denominator by 10 (because 3 * 10 = 30): 810310=8030\frac{8 * 10}{3 * 10} = \frac{80}{30}. For 25\frac{2}{5}, we multiply both the numerator and denominator by 6 (because 5 * 6 = 30): 2656=1230\frac{2 * 6}{5 * 6} = \frac{12}{30}. Great! Now we have all our fractions with the same denominator: 3530\frac{35}{30}, 8030\frac{80}{30}, and 1230\frac{12}{30}. We can finally add them together! To add fractions with the same denominator, we simply add the numerators and keep the denominator the same: 35+80+1230=12730\frac{35 + 80 + 12}{30} = \frac{127}{30}. So, the total travel time is 12730\frac{127}{30} hours. But wait, this is an improper fraction (the numerator is bigger than the denominator). We usually want to express our answer as a mixed number to make it easier to understand. Let's convert this improper fraction to a mixed number in the next section.

Converting to a Mixed Number and Finding the Answer

Okay, we've landed on a total travel time of 12730\frac{127}{30} hours, which is an improper fraction. To truly understand how long the tourist spent traveling, we need to convert this into a mixed number – a whole number and a fraction. So, how do we convert an improper fraction to a mixed number? It's all about division! We divide the numerator (127) by the denominator (30). 127 divided by 30 is 4 with a remainder of 7. The whole number part of our mixed number is the quotient (4). The remainder (7) becomes the numerator of the fractional part, and we keep the same denominator (30). So, 12730\frac{127}{30} hours is equal to 4 rac{7}{30} hours. Awesome! We've now got our answer in a more digestible format. The tourist spent a total of 4 rac{7}{30} hours on their journey. But let's pause for a second and think about what this mixed number means in real-world terms. The '4' represents 4 whole hours, and the 730\frac{7}{30} represents a fraction of an hour. To get a better sense of this fraction, we could convert it into minutes. Remember, there are 60 minutes in an hour, so 730\frac{7}{30} of an hour is 73060=14\frac{7}{30} * 60 = 14 minutes. So, the tourist traveled for 4 hours and 14 minutes. Now, let's circle back to the original question. We were given a few answer choices, and we need to identify the correct one. Looking back, the options were: A) 4 rac{7}{30} B) 5 rac{1}{30} C) 3 rac{29}{30} D) 4 rac{11}{30} The correct answer is A) 4 rac{7}{30}! We've successfully calculated the total travel time and matched it to the correct option. Pat yourselves on the back, guys! We navigated through fractions, conversions, and mixed numbers like pros. But before we wrap up, let’s think about what we’ve learned and how we can apply it to other situations.

Key Takeaways and Real-World Applications

Alright, guys, we've successfully navigated this travel time problem, but the journey doesn't end here! Let's recap the key takeaways and see how these skills apply to the real world. First and foremost, we tackled the art of converting between mixed numbers and improper fractions. This is a fundamental skill in mathematics, especially when dealing with fractions in various operations. Remember, mixed numbers are great for understanding the quantity in a real-world context (like 4 and a bit hours), while improper fractions are often easier to work with in calculations. We also mastered the process of finding the least common denominator (LCD) and using it to add fractions. The LCD is crucial because it allows us to add fractions with different denominators by expressing them with a common base. Think of it as finding a common language for the fractions so they can “talk” to each other and be combined. Beyond the specific math skills, we practiced problem-solving strategies. We broke down a complex problem into smaller, manageable steps. We converted units, added fractions, and then interpreted our answer in a meaningful way. This step-by-step approach is invaluable in any problem-solving situation, whether it's in math, science, or everyday life. So, how can we apply these skills in the real world? Think about cooking. Recipes often use fractions (like 12\frac{1}{2} cup or 34\frac{3}{4} teaspoon), and you might need to adjust quantities based on how many people you're serving. Understanding fractions and conversions is essential for accurate cooking and baking. Another example is managing your time. If you have several tasks to complete with different time estimates, you need to add those times together to see if you can fit everything into your day. This involves converting between minutes and hours and potentially dealing with fractions of time. Financial planning also involves fractions and percentages. Calculating interest rates, discounts, or dividing expenses among roommates all require a solid understanding of these concepts. In essence, the skills we've practiced today are not just about solving math problems; they're about developing critical thinking and problem-solving abilities that are applicable in countless situations. So, keep practicing, keep exploring, and keep applying these skills to make your life a little bit easier and a whole lot more interesting!