Step-by-Step Solution (6/14 + 3/7) - (1/3 + 1/6) A Detailed Guide

Hey guys! Let's dive into solving this math problem together. We've got a classic fractions question here: (6/14 + 3/7) - (1/3 + 1/6). Don't worry, we'll break it down step-by-step so it's super easy to follow. Math can seem intimidating at first, but with a little patience and a clear process, anyone can tackle these problems. We’re going to take a journey through fractions, common denominators, and basic arithmetic. So, grab your pencils, and let’s get started!

Understanding the Problem

Before we jump right into the calculations, let's make sure we understand what the problem is asking. We have two sets of fractions inside parentheses, and we're going to add the fractions within each set first. Then, we'll subtract the result of the second set from the result of the first set. This is a classic order of operations problem, and understanding the order is crucial to getting the correct answer. The order of operations reminds us to handle parentheses first, which means dealing with (6/14 + 3/7) and (1/3 + 1/6) before we do the subtraction. If we didn’t follow this order, we’d end up with a totally different answer – and that’s not what we want! So, always remember to check for parentheses and tackle them first.

Why is this order so important? Think of it like following a recipe. If you add ingredients in the wrong order, you might end up with a cake that doesn’t taste quite right. Math is similar; the order in which we perform operations determines the final result. So, let's keep this in mind as we move forward. Now, let’s break down each part of the problem individually to make it even more manageable. We'll start with the first set of fractions and find a common denominator.

Step 1: Solving (6/14 + 3/7)

The first part of our problem is (6/14 + 3/7). To add these fractions, we need a common denominator. Remember, you can only add fractions directly if they have the same denominator. So, what's a common denominator for 14 and 7? Well, 14 is a multiple of 7 (7 x 2 = 14), so we can use 14 as our common denominator. This makes things super convenient because we don’t need to change the first fraction, 6/14. We only need to adjust the second fraction, 3/7. To convert 3/7 to a fraction with a denominator of 14, we need to multiply both the numerator and the denominator by the same number. In this case, we multiply by 2. So, 3/7 becomes (3 x 2) / (7 x 2), which equals 6/14. Now we have two fractions with the same denominator: 6/14 and 6/14. This makes adding them together much easier.

Now that we have a common denominator, we can add the numerators. So, we add 6 and 6, which gives us 12. The denominator stays the same, which is 14. So, 6/14 + 6/14 = 12/14. But we're not quite done yet! The fraction 12/14 can be simplified. Both 12 and 14 are divisible by 2. So, we divide both the numerator and the denominator by 2. 12 ÷ 2 = 6 and 14 ÷ 2 = 7. This simplifies our fraction to 6/7. So, the first part of our problem, (6/14 + 3/7), simplifies to 6/7. We’ve tackled the first set of parentheses – great job! Now, let’s move on to the second part of the problem.

Step 2: Solving (1/3 + 1/6)

Now let's tackle the second set of parentheses: (1/3 + 1/6). Just like before, we need to find a common denominator to add these fractions. What do you guys think? What’s a common denominator for 3 and 6? Well, 6 is a multiple of 3 (3 x 2 = 6), so we can use 6 as our common denominator. This means we don't have to change the second fraction, 1/6, because it already has the denominator we need. We only need to convert the first fraction, 1/3, to have a denominator of 6. To do this, we multiply both the numerator and the denominator of 1/3 by 2. So, 1/3 becomes (1 x 2) / (3 x 2), which equals 2/6. Now we have two fractions with the same denominator: 2/6 and 1/6.

With a common denominator in place, we can add the numerators. We add 2 and 1, which gives us 3. The denominator stays the same, so we have 3/6. So, 2/6 + 1/6 = 3/6. But hold on, we can simplify this fraction further! Both 3 and 6 are divisible by 3. We divide both the numerator and the denominator by 3. 3 ÷ 3 = 1 and 6 ÷ 3 = 2. This simplifies our fraction to 1/2. So, the second part of our problem, (1/3 + 1/6), simplifies to 1/2. Awesome! We've handled both sets of parentheses. Now we’re ready for the final step: subtraction.

Step 3: Subtracting the Results

Okay, guys, we're in the home stretch now! We've simplified (6/14 + 3/7) to 6/7 and (1/3 + 1/6) to 1/2. Our original problem was (6/14 + 3/7) - (1/3 + 1/6), so now we can rewrite it as 6/7 - 1/2. To subtract these fractions, we need – you guessed it – a common denominator. What's a common denominator for 7 and 2? Since 7 and 2 don't have any common factors other than 1, we can find the common denominator by multiplying them together. 7 x 2 = 14. So, our common denominator is 14. Now we need to convert both fractions to have a denominator of 14.

Let's start with 6/7. To get a denominator of 14, we multiply both the numerator and the denominator by 2. 6/7 becomes (6 x 2) / (7 x 2), which equals 12/14. Next, we need to convert 1/2 to have a denominator of 14. We multiply both the numerator and the denominator by 7. 1/2 becomes (1 x 7) / (2 x 7), which equals 7/14. Now we have two fractions with the same denominator: 12/14 and 7/14. We can now subtract these fractions: 12/14 - 7/14. We subtract the numerators: 12 - 7 = 5. The denominator stays the same, which is 14. So, 12/14 - 7/14 = 5/14. This is our final answer!

Final Answer

So, after breaking down the problem step-by-step, we've found that (6/14 + 3/7) - (1/3 + 1/6) = 5/14. That's it! We've successfully solved the problem. Remember, the key to tackling these types of problems is to take them one step at a time. By finding common denominators, simplifying fractions, and following the order of operations, you can solve even the trickiest problems. Don't be afraid to break things down and take your time. Math is like building a tower – each step is important, and if you build a solid foundation, you can reach great heights!

Conclusion

Great job, everyone! We’ve walked through the process of solving (6/14 + 3/7) - (1/3 + 1/6), and hopefully, you feel more confident about tackling similar problems in the future. Remember, the key takeaways here are: always follow the order of operations, find common denominators when adding or subtracting fractions, and simplify your fractions whenever possible. Practice makes perfect, so keep working on these types of problems, and you’ll become a fraction master in no time! If you have any questions or want to try another problem, let me know. Keep up the great work, and happy calculating!