Calculating 2/7 + 3/14 - 1/21: A Step-by-Step Guide

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Hey guys! Math can sometimes feel like navigating a maze, but don't worry, we're here to break down this fraction problem step by step. This article will guide you through calculating the sum of 2/7 + 3/14 - 1/21. We'll cover finding a common denominator, performing the addition and subtraction, and simplifying the result. So, grab your pencils, and let's dive in!

Understanding the Problem

Before we jump into the calculation, let's quickly understand what we're dealing with. We have three fractions: 2/7, 3/14, and 1/21. Our goal is to add the first two and then subtract the third. To do this effectively, especially when the denominators (the bottom numbers) are different, we need to find a common denominator.

Why do we need a common denominator? Think of it like trying to add apples and oranges. You can't directly add them until you express them in a common unit, like “fruit.” Similarly, fractions need the same denominator to be added or subtracted directly. This common denominator allows us to express each fraction with the same “size of the pieces,” making the addition and subtraction straightforward.

Now, let's talk about why focusing on keywords like "calculating fractions," "common denominator," and "fraction addition and subtraction" is super important. When you search for help online, these are the kinds of terms you're likely to use. So, by including them here, we're making sure this guide is easy to find and super helpful for anyone struggling with this type of math problem. We want to ensure that when you're looking for a simple explanation of how to add and subtract fractions with different denominators, you land right here!

Also, remember that math isn't just about getting the right answer; it's about understanding the process. When you get the process down, you can tackle all sorts of similar problems. It's like learning a recipe – once you know the basics, you can whip up all kinds of variations! So, let's keep this in mind as we move through each step.

So, gear up! We’re about to dive deep into the world of fractions. We're going to find a common denominator, do some adding and subtracting, and simplify our answer. By the end of this, you’ll not only be able to solve this problem but also feel confident tackling similar fraction challenges. Let's make math less of a headache and more of a piece of cake (or, you know, a slice of pi!).

Finding the Least Common Denominator (LCD)

The most important part of this question is finding the Least Common Denominator (LCD). The LCD is the smallest number that each of our denominators (7, 14, and 21) can divide into evenly. This makes our calculations simpler and keeps the fractions manageable.

How do we find the LCD? There are a couple of ways to do this, but one common method is to list the multiples of each denominator and identify the smallest multiple they share.

  • Multiples of 7: 7, 14, 21, 28, 35, 42, ...
  • Multiples of 14: 14, 28, 42, 56, ...
  • Multiples of 21: 21, 42, 63, ...

Looking at these lists, we can see that the smallest multiple that 7, 14, and 21 share is 42. Therefore, the LCD is 42. Alternatively, you could use the prime factorization method, which involves breaking down each number into its prime factors. This method is especially helpful for larger numbers but works just as well here.

  • Prime factorization of 7: 7
  • Prime factorization of 14: 2 x 7
  • Prime factorization of 21: 3 x 7

To find the LCD, we take each prime factor that appears in any of the factorizations and use the highest power of that factor. In this case, we have 2, 3, and 7 as prime factors. The LCD is 2 x 3 x 7 = 42. Using keywords like “least common denominator,” “LCD calculation,” and “finding common multiples” can help clarify this process. Remember, understanding the LCD is crucial for accurately adding and subtracting fractions with different denominators.

Let’s pause for a moment and think about why mastering the LCD is a game-changer in the world of fractions. Once you nail this, adding and subtracting fractions becomes so much smoother. It’s like finding the right tool for the job – suddenly, everything clicks into place. This is why we’re giving it a good chunk of our attention here. We want you to walk away feeling like you’ve truly conquered this skill. We're not just solving one problem; we’re building a foundation for handling all sorts of fraction scenarios!

Converting Fractions to Equivalent Fractions

Now that we've found our LCD (which is 42), the next step is to convert each of our original fractions (2/7, 3/14, and 1/21) into equivalent fractions with a denominator of 42. An equivalent fraction represents the same value as the original fraction but has a different denominator. Think of it like slicing a pizza – whether you cut it into 8 slices or 16, it's still the same pizza, just in different sized pieces.

How do we convert these fractions? To convert a fraction, we need to multiply both the numerator (the top number) and the denominator (the bottom number) by the same value. This keeps the fraction equivalent because we're essentially multiplying by 1 (e.g., 2/2, 3/3, etc.).

  1. Converting 2/7: To get from 7 to 42, we need to multiply by 6 (since 7 x 6 = 42). So, we multiply both the numerator and the denominator of 2/7 by 6: (2 x 6) / (7 x 6) = 12/42.
  2. Converting 3/14: To get from 14 to 42, we need to multiply by 3 (since 14 x 3 = 42). So, we multiply both the numerator and the denominator of 3/14 by 3: (3 x 3) / (14 x 3) = 9/42.
  3. Converting 1/21: To get from 21 to 42, we need to multiply by 2 (since 21 x 2 = 42). So, we multiply both the numerator and the denominator of 1/21 by 2: (1 x 2) / (21 x 2) = 2/42.

Now, we have three equivalent fractions: 12/42, 9/42, and 2/42. These fractions have the same denominator, which means we're ready to perform the addition and subtraction. Using keywords such as “converting fractions,” “equivalent fractions,” and “same denominator” helps reinforce this concept.

Think of converting fractions as giving them a makeover so they can play together nicely. It's like making sure everyone speaks the same language before trying to have a conversation. Once they're all speaking the same language (or, in this case, have the same denominator), the math gets way easier. Plus, this skill isn't just for this specific problem. It's a fundamental part of working with fractions, and you'll use it again and again. So, if you're feeling a bit unsure, take a deep breath and maybe go through this section one more time. It's worth getting this step down solid!

Performing the Addition and Subtraction

Now that we have our fractions with a common denominator (42), we can finally perform the addition and subtraction. Our problem now looks like this: 12/42 + 9/42 - 2/42. Remember, when fractions have the same denominator, we simply add or subtract the numerators (the top numbers) and keep the denominator the same. It’s like we’re just counting the pieces, and the size of the pieces stays the same.

Let's break it down:

  1. Addition: First, we add 12/42 and 9/42. To do this, we add the numerators (12 + 9) and keep the denominator (42): 12 + 9 = 21. So, 12/42 + 9/42 = 21/42.
  2. Subtraction: Next, we subtract 2/42 from the result we just obtained (21/42). We subtract the numerators (21 - 2) and keep the denominator (42): 21 - 2 = 19. So, 21/42 - 2/42 = 19/42.

Therefore, 2/7 + 3/14 - 1/21 = 19/42. Keywords like “adding fractions,” “subtracting fractions,” and “common denominator operations” are essential for understanding this step.

At this point, we’re really in the home stretch! Adding and subtracting fractions might seem like a big deal when you first encounter it, but look at you now – you’re doing it! This is where all that work we put in earlier, finding the LCD and converting fractions, really pays off. It's like building a solid foundation for a house; once you have that in place, everything else goes up much more smoothly. And it's not just about getting the right answer; it’s about building your math skills and feeling confident in your abilities. So, take a moment to appreciate how far you've come in this problem. We're almost there, and the final answer is just around the corner!

Simplifying the Result (If Possible)

The final step is to simplify our answer, 19/42, if possible. Simplifying a fraction means reducing it to its lowest terms. We do this by dividing both the numerator and the denominator by their greatest common factor (GCF). The GCF is the largest number that divides evenly into both numbers.

In this case, let's look at 19 and 42. Is there a number that divides evenly into both? 19 is a prime number, which means its only factors are 1 and itself. Since 19 does not divide evenly into 42, the greatest common factor of 19 and 42 is 1. This means that 19/42 is already in its simplest form. Sometimes, you'll encounter fractions that can be simplified, but in this instance, we’re good to go!

So, our final answer is 19/42. We can't simplify it any further, so we've reached the end of our calculation. Understanding “simplifying fractions,” “reducing fractions,” and “greatest common factor” is crucial for this step.

And there you have it! We've gone from the starting problem all the way to the simplified answer. How awesome is that? Simplifying is like putting the final polish on a masterpiece. It ensures your answer is in its most elegant form. Plus, it shows a deeper understanding of fractions. Always check if your answer can be simplified – it’s a great habit to get into. But hey, even if it can’t be simplified, like in this case, you still get to bask in the glow of solving the problem. You’ve tackled a math challenge head-on, and you’ve come out on top. So, give yourself a pat on the back – you’ve earned it!

Conclusion

Alright guys, we've successfully calculated 2/7 + 3/14 - 1/21, and our final answer is 19/42. We walked through each step, from finding the least common denominator to converting fractions and performing the addition and subtraction. We even checked to see if we could simplify our answer (and we found that it was already in its simplest form!). Remember the key steps:

  1. Find the Least Common Denominator (LCD).
  2. Convert fractions to equivalent fractions with the LCD.
  3. Perform the addition and subtraction.
  4. Simplify the result, if possible.

By understanding these steps, you can tackle similar fraction problems with confidence. This skill is crucial not just in math class but also in everyday situations, like cooking, measuring, and problem-solving. So, keep practicing, and you'll become a fraction master in no time!

We hope this guide has made working with fractions a little less daunting and a lot more understandable. Math might have its tricky moments, but with a step-by-step approach and a bit of practice, you can conquer it. Keep up the great work, and remember, every problem you solve makes you a little bit stronger in the world of math! Keep practicing, keep exploring, and most importantly, keep believing in yourself. You've got this!