Solving (-5/7 + 1/3) ÷ (12/21 + 1) A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like it belongs in a complicated puzzle? Today, we're going to break down one of those – specifically, how to solve the expression (-5/7 + 1/3) ÷ (12/21 + 1). Don't worry, it's not as scary as it looks! We'll go through each step super clearly, so you'll be solving these like a pro in no time. Let's dive in!

Understanding the Order of Operations

Before we even think about fractions, we need to remember our good old friend PEMDAS (or BODMAS, depending on where you went to school). This is our order of operations bible: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This acronym ensures we tackle the problem in the correct sequence, so our answer is spot on. In our case, we have parentheses and division, followed by addition and subtraction within the parentheses. Sticking to this order is paramount; otherwise, we might end up with a completely different (and incorrect) result. Think of PEMDAS/BODMAS as the golden rule of math – break it, and chaos ensues! We need to first simplify the expressions inside the parentheses before we can perform the division. This means finding common denominators for the fractions and then adding or subtracting them accordingly. Understanding this foundational principle sets us up for success in solving more complex mathematical problems down the line. So, let's keep PEMDAS/BODMAS at the forefront as we proceed, ensuring a smooth and accurate problem-solving journey. Ignoring the order of operations is like trying to build a house without a blueprint – it might stand for a while, but eventually, something's going to collapse. Let's avoid any mathematical collapses by being diligent about our order of operations!

Step 1: Solving the First Parenthesis (-5/7 + 1/3)

The first part of our equation we're going to tackle is (-5/7 + 1/3). To add these fractions, they need to have the same denominator. Think of it like trying to add apples and oranges – you need a common unit (like 'fruit') to make sense of the sum. So, we need to find the least common multiple (LCM) of 7 and 3. What's that, you ask? Well, the LCM is the smallest number that both 7 and 3 can divide into evenly. In this case, it's 21 (since 7 x 3 = 21). Now, we need to convert both fractions to have this denominator. To convert -5/7, we multiply both the numerator (the top number) and the denominator (the bottom number) by 3. This gives us -15/21. For 1/3, we multiply both the numerator and denominator by 7, resulting in 7/21. Now we can rewrite our expression inside the first parenthesis as (-15/21 + 7/21). See how much friendlier it looks already? When fractions share a common denominator, it's like they're speaking the same language. We can now easily add (or subtract) their numerators. Remember, we’re not changing the value of the fractions; we’re just expressing them in a different way. This is a crucial skill in fraction manipulation, and mastering it opens doors to solving all sorts of mathematical puzzles. Once the fractions have a common denominator, the rest of the process is a breeze. So, let’s recap: We identified the need for a common denominator, found the LCM, converted our fractions, and are now ready to combine them. It’s like we’re building a mathematical bridge, one step at a time.

Adding the Fractions

Okay, now that we have -15/21 and 7/21, we can finally add these fractions. When the denominators are the same, adding is super straightforward: we just add the numerators and keep the denominator the same. So, -15 + 7 = -8. That means -15/21 + 7/21 equals -8/21. And boom! We've conquered the first parenthesis. This might seem like a small victory, but each step we take brings us closer to the final solution. Think of it like climbing a staircase – every step counts! Understanding how to add fractions with common denominators is a fundamental skill in arithmetic. It's like the alphabet of mathematics – without it, we can't spell out more complex solutions. The key is to remember that the denominator acts as a unit, so when they’re the same, we’re simply counting how many of those units we have. It’s like saying we have -15 slices of a 21-slice pizza and then we add 7 slices – we’re left with -8 slices (or, more accurately, a deficit of 8 slices). So, let’s celebrate this milestone! We’ve successfully navigated the tricky terrain of adding fractions with different signs. This process reinforces the idea that math is a series of manageable steps, each building upon the last. Now, with this parenthesis solved, we can confidently move on to the next challenge, armed with our newfound fraction-adding prowess. Remember, every problem is just a series of smaller problems in disguise, and we’re becoming experts at uncovering them.

Step 2: Solving the Second Parenthesis (12/21 + 1)

Alright, let's move on to the second parenthesis: (12/21 + 1). Now, we're adding a fraction to a whole number. To make this easier, we can think of the whole number '1' as a fraction. How? Well, any number divided by itself equals 1. So, we can rewrite '1' as 21/21. This is because our other fraction has a denominator of 21, and we want a common denominator so we can add them together. It's like translating different languages into a common one so everyone can understand. Rewriting whole numbers as fractions is a handy trick in math, especially when dealing with addition or subtraction. It allows us to maintain consistency in our operations and avoids confusion. By expressing 1 as 21/21, we're not changing its value; we're just changing its form. This is a common strategy in mathematics – manipulating numbers without altering their inherent value to make calculations smoother. Think of it as putting on a different outfit; the person inside remains the same. Now our expression inside the second parenthesis looks much more manageable: (12/21 + 21/21). This transformation is key to unlocking the solution. We’ve turned a potentially confusing problem into a straightforward one. Remember, math is often about finding the simplest way to express a problem. This step highlights the importance of flexibility and creative thinking in problem-solving. We're not just following rules; we're understanding the underlying principles and applying them in innovative ways. So, with our whole number now masquerading as a fraction, we’re all set to conquer this second parenthesis.

Adding the Fraction and the Whole Number

Now that we've rewritten 1 as 21/21, our expression is 12/21 + 21/21. Since the denominators are the same, we can simply add the numerators. So, 12 + 21 = 33. This means 12/21 + 21/21 equals 33/21. We've successfully solved the second parenthesis! High five! This step is a perfect illustration of how simplifying a problem can make it much easier to solve. By converting the whole number into a fraction with a common denominator, we transformed the addition into a simple numerator addition problem. It’s like turning a winding road into a straight path. The result, 33/21, represents the combined value of the fraction and the whole number. However, it’s also an improper fraction, meaning the numerator is larger than the denominator. This isn’t necessarily a problem, but it’s often good practice to simplify or convert it to a mixed number later on, depending on the context of the problem. For now, we’ve accurately calculated the value within the second parenthesis, and that’s a major accomplishment. Think of this as reaching the second base in a baseball game – we’re halfway home! The skill we’ve honed here – adding fractions with common denominators – is a cornerstone of fraction arithmetic. It’s a skill that will serve us well in many mathematical scenarios. So, let’s bask in the glow of our success and prepare to tackle the final act: the division.

Step 3: Dividing the Results

Okay, we're on the home stretch! We've simplified the expression inside both parentheses, and now we have (-8/21) ÷ (33/21). Remember our PEMDAS/BODMAS? We've handled the parentheses, and now it's time for the division. Dividing fractions can seem a bit tricky at first, but there's a neat trick to it: dividing by a fraction is the same as multiplying by its reciprocal. What's a reciprocal? It's simply flipping the fraction – swapping the numerator and the denominator. So, the reciprocal of 33/21 is 21/33. This “flip and multiply” rule is a cornerstone of fraction division. It transforms a division problem into a multiplication problem, which is often easier to handle. Think of it like using a mathematical shortcut! Understanding why this works requires delving into the properties of fractions and division, but for now, let’s embrace the simplicity of the rule. It’s like having a secret code that unlocks the solution. By changing the division to multiplication, we've made the problem much more approachable. Now, instead of dividing -8/21 by 33/21, we’re going to multiply -8/21 by 21/33. This transformation is a game-changer! We’re taking a potentially daunting task and turning it into something much more manageable. So, let’s recap: We’ve identified the division operation, recalled the reciprocal rule, and are now ready to multiply. We’re in the final lap of this mathematical race, and the finish line is in sight!

Multiplying by the Reciprocal

So, we're now faced with the problem (-8/21) x (21/33). Multiplying fractions is actually quite straightforward: you multiply the numerators together and the denominators together. So, -8 x 21 = -168, and 21 x 33 = 693. This gives us the fraction -168/693. We’ve done the multiplication, but our work isn’t quite done yet. This fraction looks a little… intimidating, doesn't it? Large numbers in fractions often suggest that we can simplify. This is where our fraction-simplifying skills come into play. Think of it as polishing a gem to reveal its true brilliance. Large fractions can hide simpler, more elegant forms. Simplifying fractions means finding a common factor (a number that divides evenly into both the numerator and the denominator) and dividing both by it. This reduces the fraction to its lowest terms. Simplifying fractions is not just about making the numbers smaller; it's about expressing the fraction in its most basic form. It’s like speaking the language of math fluently. A simplified fraction is easier to understand, compare, and work with in future calculations. So, let’s not be intimidated by -168/693. We have the tools to tame this beast! We’ll embark on a quest to find the greatest common factor (GCF) and divide both the numerator and denominator by it. This process will lead us to the simplest, most beautiful form of our fraction. We’re transforming from mathematicians to mathematical artists, sculpting the perfect solution.

Step 4: Simplifying the Result

Now, let's simplify -168/693. To do this, we need to find the greatest common factor (GCF) of 168 and 693. Finding the GCF can sometimes feel like a puzzle, but there are a few ways to tackle it. One way is to list the factors of each number and find the largest one they have in common. Another way is to use the Euclidean algorithm, which is a more systematic approach. For our purposes, let's try listing factors. The factors of 168 are 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168. The factors of 693 are 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, and 693. Looking at these lists, we can see that the greatest common factor is 21. This means that 21 is the largest number that divides evenly into both 168 and 693. Finding the GCF is like finding the perfect key to unlock a mathematical simplification. It’s a crucial step in expressing fractions in their simplest form. There are different techniques for finding the GCF, and each has its own advantages. Choosing the right method depends on the numbers involved and your personal preference. The important thing is to find the largest factor that the numbers share, as this will lead to the most simplified fraction in a single step. It’s like choosing the most efficient route on a map – it gets you to your destination faster. So, with our GCF of 21 in hand, we’re ready to divide both the numerator and denominator of our fraction. We’re about to perform the final act of simplification, transforming our complex fraction into a sleek, elegant result.

Dividing by the GCF

So, we're going to divide both the numerator and the denominator of -168/693 by 21. -168 ÷ 21 = -8, and 693 ÷ 21 = 33. This simplifies our fraction to -8/33. And there you have it! We've reached the end of our mathematical journey. This is the simplified form of our answer, and it's much easier to work with than -168/693. Congratulations, you've successfully navigated a multi-step problem involving fractions, parentheses, and division! Simplifying the fraction is like putting the finishing touches on a masterpiece. It’s the final step that transforms a good answer into a great one. By dividing by the GCF, we ensure that the fraction is in its lowest terms, meaning there are no more common factors between the numerator and the denominator. This simplified form is not only easier to work with, but it also represents the fraction in its most fundamental form. It’s like stripping away the unnecessary layers to reveal the core essence of the number. The process of simplification reinforces the importance of precision and attention to detail in mathematics. Each step, from finding the GCF to performing the division, requires careful execution. It’s like following a recipe – each ingredient must be measured accurately to achieve the desired result. So, let’s celebrate our victory! We’ve not only solved the problem but also honed our skills in fraction manipulation and simplification. This experience will serve us well in future mathematical adventures. We’ve proven that even complex problems can be conquered with a step-by-step approach and a dash of mathematical know-how.

Final Answer

Therefore, (-5/7 + 1/3) ÷ (12/21 + 1) = -8/33. Great job, everyone! You've tackled a challenging problem and come out on top. Remember, math is like any other skill – the more you practice, the better you get. So, keep those problem-solving muscles flexed, and you'll be a math whiz in no time!