Adding Fractions & Distance Problems: Step-by-Step Solutions

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Hey guys! Let's dive into some math problems involving fractions and distances. We'll tackle these step-by-step, so it's super easy to follow along. We've got two main problems here: one about adding a fraction to the difference of two other fractions, and another about calculating distances run by Akmal. Let’s break it down!

Problem 1: Adding Fractions

So, the first part of our math adventure asks us to calculate the result of adding 3 5/12 to the difference between 2 5/6 and 1 1/4. Sounds like a mouthful, right? But don't worry, we'll take it one step at a time. Our main keywords here are adding fractions, difference, and mixed numbers, so let's keep those in mind as we go through the solution.

Step 1: Calculate the Difference

First, we need to find the difference between 2 5/6 and 1 1/4. To do this, we need to subtract the second fraction from the first. But before we can subtract, we need to make sure the fractions have a common denominator. Remember denominators? They're the bottom numbers in a fraction. In this case, our denominators are 6 and 4. The least common multiple (LCM) of 6 and 4 is 12. So, we'll convert both fractions to have a denominator of 12.

  • Converting 2 5/6: To convert 5/6 to a fraction with a denominator of 12, we multiply both the numerator (5) and the denominator (6) by 2. This gives us 10/12. So, 2 5/6 becomes 2 10/12.
  • Converting 1 1/4: To convert 1/4 to a fraction with a denominator of 12, we multiply both the numerator (1) and the denominator (4) by 3. This gives us 3/12. So, 1 1/4 becomes 1 3/12.

Now we can subtract! We have 2 10/12 - 1 3/12. Subtract the whole numbers: 2 - 1 = 1. Then subtract the fractions: 10/12 - 3/12 = 7/12. So, the difference between 2 5/6 and 1 1/4 is 1 7/12. Understanding these conversions is key to mastering fraction operations, guys!

Step 2: Add the Result to 3 5/12

Next, we need to add this difference (1 7/12) to 3 5/12. Again, we're dealing with mixed numbers, so let's break it down. Add the whole numbers: 3 + 1 = 4. Then add the fractions: 5/12 + 7/12 = 12/12. But wait! 12/12 is just equal to 1. So, we add that 1 to our whole number sum: 4 + 1 = 5. See how breaking it into smaller steps makes it much easier?

So, the final answer to the first problem is 5. We've successfully added 3 5/12 to the difference between 2 5/6 and 1 1/4. Great job! Now, let's move on to the next challenge.

Problem 2: Akmal's Run

Alright, this problem is about Akmal and his running adventures! Akmal ran 2 1/5 km from point A to point B, and then another 3 7/8 km from point B to point C. We have two questions to answer here:

a) What's the total distance Akmal ran? b) How much shorter is the distance between points A and B compared to the distance between points B and C?

The keywords here are total distance and distance comparison, so let's keep those in mind. It's all about adding and comparing distances, which is something we can totally handle!

Part a: Total Distance

To find the total distance Akmal ran, we need to add the distance from A to B and the distance from B to C. That means we're adding 2 1/5 km and 3 7/8 km. Just like before, we need a common denominator to add these fractions. Our denominators are 5 and 8. The least common multiple of 5 and 8 is 40. So, we'll convert both fractions to have a denominator of 40.

  • Converting 2 1/5: To convert 1/5 to a fraction with a denominator of 40, we multiply both the numerator (1) and the denominator (5) by 8. This gives us 8/40. So, 2 1/5 becomes 2 8/40.
  • Converting 3 7/8: To convert 7/8 to a fraction with a denominator of 40, we multiply both the numerator (7) and the denominator (8) by 5. This gives us 35/40. So, 3 7/8 becomes 3 35/40.

Now we can add! We have 2 8/40 + 3 35/40. Add the whole numbers: 2 + 3 = 5. Then add the fractions: 8/40 + 35/40 = 43/40. Uh oh, our fraction is greater than 1! This is called an improper fraction. We can convert 43/40 to a mixed number. 43/40 is equal to 1 and 3/40. So, we add that 1 to our whole number sum: 5 + 1 = 6. And we're left with the fraction 3/40. So, the total distance Akmal ran is 6 3/40 km. See, even when things look tricky, we can break them down!

Part b: Distance Comparison

Now, we need to figure out how much shorter the distance between points A and B is compared to the distance between points B and C. This means we need to subtract the distance from A to B (2 1/5 km) from the distance from B to C (3 7/8 km). We already converted these fractions to have a common denominator of 40 in the previous step, so we know that 2 1/5 is 2 8/40 and 3 7/8 is 3 35/40.

So, we're subtracting 2 8/40 from 3 35/40. Subtract the whole numbers: 3 - 2 = 1. Then subtract the fractions: 35/40 - 8/40 = 27/40. So, the distance between points A and B is 1 27/40 km shorter than the distance between points B and C. Fantastic work! We've tackled both parts of this problem.

Conclusion

So, guys, we’ve solved two pretty cool math problems today! We added fractions, calculated the difference between fractions, and figured out distances. Remember, the key to solving these problems is to break them down into smaller, manageable steps. Keep practicing, and you'll become a math whiz in no time! If you ever feel stuck, just remember the keywords and take it one step at a time. You got this!