Calculating The Third Day's Distance A Tourist Trip Problem

Hey guys! Today, let's dive into a fun math problem that involves calculating distances traveled over three days. This is a classic example of a multi-step problem where we need to break it down to solve it effectively. We'll walk through it step by step, making sure everyone understands the process. So, grab your thinking caps, and let's get started!

Understanding the Problem

The problem states that tourists traveled a total of 48 kilometers in three days. On the first day, they covered $\frac{1}{4}$ of the entire distance. On the second day, they traveled $\frac{5}{9}$ of the remaining distance after the first day. The question we need to answer is: How many kilometers did they travel on the third day?

To solve this, we need to follow a few key steps:

1. Calculate the distance traveled on the first day.
2. Determine the remaining distance after the first day.
3. Calculate the distance traveled on the second day.
4. Calculate the distance traveled on the third day by subtracting the distances of the first two days from the total distance.

Let's break down each of these steps.

Step 1 Calculating the Distance Traveled on the First Day

First things first, we need to figure out how many kilometers the tourists covered on the first day. The problem tells us they traveled $\frac{1}{4}$ of the total distance, which is 48 kilometers. To find $\frac{1}{4}$ of 48, we simply multiply:

$$\frac{1}{4} \times 48 = 12$$

So, on the first day, the tourists traveled 12 kilometers. This is a crucial first step because it sets the stage for calculating the remaining distances. Remember, this is just the beginning, guys! We've got more to figure out.

Why This Step is Important

Calculating the distance of the first day is important because it helps us determine the new total distance that needs to be covered in the remaining two days. Without knowing this, we can't accurately calculate the distance covered on the second day, which depends on the remaining distance. It's like building a house; you need a solid foundation before you can put up the walls!

Real-World Application

Understanding how to calculate fractions of a whole is super useful in everyday life. Imagine you're planning a road trip, and you want to cover a certain fraction of the distance on the first day. Knowing how to do this calculation helps you plan your trip effectively. Or, think about splitting a pizza with friends – you're using the same math principles to figure out how many slices each person gets.

Step 2 Determining the Remaining Distance After the First Day

Now that we know the tourists traveled 12 kilometers on the first day, we need to figure out how much distance was left to cover. This is a straightforward subtraction problem. We started with a total of 48 kilometers, and they covered 12 kilometers. So, the remaining distance is:

$$48 - 12 = 36$$

Therefore, after the first day, there were 36 kilometers left to travel. This remaining distance is what we'll use to calculate how far they went on the second day. Think of it as the new 'whole' we're working with. It’s like when you’ve eaten part of a cake, and you’re figuring out how much is left – same concept!

Why This Subtraction Matters

Finding the remaining distance is a critical step because the problem tells us the distance covered on the second day is a fraction of this remaining distance, not the original total distance. If we didn't subtract, we'd be working with the wrong number, and our final answer would be way off. This is why it’s so important to pay close attention to the details in word problems.

Practical Uses of Calculating Remainders

This type of calculation isn’t just for math problems; it’s something we do all the time in real life. For example, if you’re saving up for a new gadget, and you’ve already saved a certain amount, you need to calculate the remaining amount to know how much more you need to save. Or, if you’re baking and you’ve used some of an ingredient, you need to know how much is left. These are everyday scenarios where understanding subtraction and remainders comes in handy.

Step 3 Calculating the Distance Traveled on the Second Day

Okay, guys, let's move on to the second day! The problem states that on the second day, the tourists traveled $\frac{5}{9}$ of the remaining distance. We know from the previous step that the remaining distance was 36 kilometers. So, to find $\frac{5}{9}$ of 36, we multiply:

$$\frac{5}{9} \times 36$$

To do this, we can first divide 36 by 9, which gives us 4. Then, we multiply 5 by 4:

$$5 \times 4 = 20$$

So, on the second day, the tourists traveled 20 kilometers. We're making progress, guys! We’ve now figured out the distances for the first two days, and we’re one step closer to finding the distance for the third day.

The Magic of Fractions

This step highlights how useful fractions are in breaking down quantities into parts. Instead of thinking about the whole 36 kilometers, we’re focusing on a specific fraction of it. This is a common technique in many areas of math and science. Understanding fractions helps us deal with proportions and ratios, which are essential in many real-world situations.

Real-Life Scenarios with Fractions

Think about situations where you might use fractions in your daily life. If you’re following a recipe and need to use $\frac{1}{2}$ cup of flour, you’re working with fractions. Or, if you’re figuring out a discount of 25% on an item, you’re essentially calculating a fraction of the original price. Knowing how to work with fractions makes these tasks much easier.

Step 4 Calculating the Distance Traveled on the Third Day

Alright, we've reached the final stretch! We know the total distance (48 kilometers), the distance traveled on the first day (12 kilometers), and the distance traveled on the second day (20 kilometers). To find the distance traveled on the third day, we need to subtract the distances of the first two days from the total distance.

So, we have:

$$48 - (12 + 20)$$

First, let's add the distances of the first two days:

$$12 + 20 = 32$$

Now, we subtract this sum from the total distance:

$$48 - 32 = 16$$

Therefore, on the third day, the tourists traveled 16 kilometers. And there you have it, guys! We’ve solved the problem.

Putting It All Together

This final step is all about putting together the information we’ve gathered in the previous steps. It shows how each individual calculation contributes to the final answer. This is a great example of how problem-solving often involves breaking down a big problem into smaller, more manageable parts.

The Importance of Checking Your Work

Whenever you solve a multi-step problem like this, it’s a good idea to check your work. You can do this by adding up the distances for all three days and making sure they equal the total distance. In this case:

$$12 + 20 + 16 = 48$$

This confirms that our answer is correct. Checking your work helps you avoid simple mistakes and builds confidence in your solution.

Conclusion

So, guys, we’ve successfully calculated the distance the tourists traveled on the third day. By breaking the problem down into smaller steps, we made it much easier to solve. We calculated the distance for the first day, found the remaining distance, calculated the distance for the second day, and finally, figured out the distance for the third day. Remember, the key to solving these types of problems is to take it one step at a time and carefully follow the information given. You've got this!

This problem demonstrates not only how to work with fractions and subtraction but also how to approach complex problems in general. By breaking them down into smaller, more manageable steps, we can tackle even the trickiest challenges. Keep practicing, and you'll become math whizzes in no time!