Tourist's Journey: Calculating Total Distance Walked

Hey guys! Let's dive into a fun little math problem about a tourist's journey. Our main goal here is to figure out the total distance this tourist covered over two hours. We know they walked a certain distance in the first hour and a slightly different distance in the second, so let's break it down step by step.

Breaking Down the Problem

So, here's the deal: In the first hour, our tourist trekked $3\frac{1}{2}$ kilometers. Not bad, right? But the second hour was a bit different. They walked $\frac{3}{4}$ kilometers less than they did in the first hour. This means we need to do a little subtraction to find out exactly how far they went in that second hour. Once we know the distance for both hours, we can simply add them together to get the total distance. Easy peasy!

Converting Mixed Numbers

First things first, let's make that mixed number, $3\frac{1}{2}$, a bit easier to work with. We can convert it into an improper fraction. To do this, we multiply the whole number (3) by the denominator (2) and then add the numerator (1). This gives us $(3 \times 2) + 1 = 7$. So, $3\frac{1}{2}$ is the same as $\frac{7}{2}$.

Calculating the Distance of the Second Hour

Alright, now we need to figure out how far the tourist walked in the second hour. We know they walked $\frac{3}{4}$ km less than the first hour, so we need to subtract $\frac{3}{4}$ from $\frac{7}{2}$. But before we can subtract fractions, they need to have the same denominator. The least common denominator for 2 and 4 is 4. So, let's convert $\frac{7}{2}$ to have a denominator of 4. To do this, we multiply both the numerator and the denominator by 2: $\frac{7 \times 2}{2 \times 2} = \frac{14}{4}$.

Now we can subtract! $\frac{14}{4} - \frac{3}{4} = \frac{11}{4}$. So, the tourist walked $\frac{11}{4}$ kilometers in the second hour.

Converting Back to a Mixed Number

$\\frac{11}{4}$ is an improper fraction, which isn't super helpful for understanding the distance. Let's convert it back to a mixed number. We divide 11 by 4. 4 goes into 11 two times (2 x 4 = 8) with a remainder of 3. So, $\frac{11}{4}$ is the same as $2\frac{3}{4}$. This means the tourist walked $2\frac{3}{4}$ kilometers in the second hour.

Finding the Total Distance

Okay, we're in the home stretch! We know the tourist walked $3\frac{1}{2}$ km in the first hour and $2\frac{3}{4}$ km in the second hour. Now we just need to add these two distances together. Let's use the improper fractions we already calculated: $\frac{7}{2}$ and $\frac{11}{4}$. Again, we need a common denominator, which we already know is 4. So, $\frac{7}{2}$ becomes $\frac{14}{4}$.

Now we add: $\frac{14}{4} + \frac{11}{4} = \frac{25}{4}$.

Converting to a Mixed Number Again

Let's convert that improper fraction, $\frac{25}{4}$, back into a mixed number. 4 goes into 25 six times (6 x 4 = 24) with a remainder of 1. So, $\frac{25}{4}$ is the same as $6\frac{1}{4}$.

The Final Answer

Drumroll, please! The tourist walked a total of $6\frac{1}{4}$ kilometers in two hours. So, the correct answer is:

• $6\frac{1}{4}$ km

Key Concepts Used

To solve this problem, we used a few important math concepts. Let's recap:

• Mixed Numbers: Understanding how to work with mixed numbers (like $3\frac{1}{2}$) is super important. We converted them into improper fractions to make the calculations easier.
• Improper Fractions: Improper fractions (where the numerator is larger than the denominator, like $\frac{7}{2}$) are useful for calculations, but mixed numbers are often easier to understand in terms of real-world quantities.
• Finding a Common Denominator: You absolutely need a common denominator when adding or subtracting fractions. We found the least common denominator to make our calculations as simple as possible.
• Adding and Subtracting Fractions: Once we had a common denominator, adding and subtracting fractions was a breeze!

Understanding these concepts will help you tackle similar problems in the future. Keep practicing, and you'll become a fraction master in no time!

Why This Matters

Okay, so you might be thinking, "Why do I need to know this stuff?" Well, understanding fractions and how to work with them is essential for all sorts of real-life situations. Here are just a few examples:

• Cooking: Recipes often use fractions (like $\frac{1}{2}$ cup of flour or $\frac{1}{4}$ teaspoon of salt). If you can't work with fractions, you might end up with a culinary disaster!
• Home Improvement: Measuring lengths, areas, and volumes often involves fractions. Whether you're hanging a picture or building a deck, you'll need to know how to work with them.
• Shopping: Sales and discounts are often expressed as fractions or percentages (which are just fractions in disguise!). Knowing how to calculate these discounts can save you money.
• Travel: Calculating distances, travel times, and fuel consumption often involves fractions. Understanding these calculations can help you plan your trips more effectively.

So, as you can see, fractions are all around us. The better you understand them, the better equipped you'll be to navigate the world!

Practice Problems

Want to test your skills? Here are a few practice problems similar to the one we just solved:

1. A baker used $2\frac{1}{3}$ cups of flour for one cake and $\frac{5}{6}$ cups less for a second cake. How much flour did the baker use in total?
2. A runner ran $5\frac{1}{4}$ miles on Monday and $\frac{2}{3}$ miles less on Tuesday. How far did the runner run over the two days?
3. A student spent $1\frac{1}{2}$ hours studying math and $\frac{3}{4}$ hours less studying science. How much time did the student spend studying in total?

Try solving these problems on your own, and check your answers with a friend or teacher. The more you practice, the more confident you'll become in your fraction skills!

Conclusion

So, there you have it! We successfully calculated the total distance the tourist walked by breaking the problem down into smaller, manageable steps. Remember, the key to solving these types of problems is to understand the underlying concepts and to take your time. Don't be afraid to make mistakes – they're a natural part of the learning process. Keep practicing, and you'll be a math whiz in no time! And remember math is everywhere.

This detailed explanation should help anyone understand the problem and the math concepts involved. Keep practicing and you'll get there! Don't be afraid to ask for help and good luck!