Solving \(\frac{75}{2}-\frac{125}{3}+50-\left(\frac{12}{2}+\frac{8}{3}-20\right)\) A Step-by-Step Guide

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Hey guys! Ever stumbled upon a math problem that looks like a tangled mess of fractions and parentheses? Don't sweat it! We're diving into one such puzzle today, and we're going to break it down piece by piece. Our mission? To solve:

${ \frac{75}{2}-\frac{125}{3}+50-\left(\frac{12}{2}+\frac{8}{3}-20\right) }$

Sounds intimidating, right? But trust me, with a sprinkle of strategy and a dash of arithmetic, we'll conquer this equation together. So, grab your thinking caps, and let's get started!

Unraveling the Complexity: A Journey Through the Equation

In this mathematical expedition, we're tackling a problem that combines fractions, whole numbers, and those tricky parentheses. But fear not! We'll navigate this maze by following the golden rule of math: the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Think of PEMDAS as our trusty map, guiding us through the twists and turns of the equation.

Step 1: Conquering the Parentheses

Our first checkpoint is the parentheses: ${\left(\frac{12}{2}+\frac{8}{3}-20\right)}$. Inside this cozy enclosure, we have a mix of fractions and a whole number. To make things easier, let's find a common denominator for our fractions. Remember, a common denominator is like a universal language that allows us to add and subtract fractions seamlessly. In this case, the least common multiple of 2 and 3 is 6. So, we'll convert ${\frac{12}{2}}$ and ${\frac{8}{3}}$ to equivalent fractions with a denominator of 6. This means multiplying the numerator and denominator of ${\frac{12}{2}}$ by 3 and the numerator and denominator of ${\frac{8}{3}}$ by 2. This gives us:

${ \frac{12}{2} \times \frac{3}{3} = \frac{36}{6} }$

${ \frac{8}{3} \times \frac{2}{2} = \frac{16}{6} }$

Now our expression inside the parentheses looks like this: ${\frac{36}{6} + \frac{16}{6} - 20}$. We can now add the fractions: ${\frac{36}{6} + \frac{16}{6} = \frac{52}{6}}$. Next, we need to subtract 20 from this fraction. To do this, we'll convert 20 into a fraction with a denominator of 6. So, 20 becomes ${\frac{20 \times 6}{6} = \frac{120}{6}}$. Now we can subtract: ${\frac{52}{6} - \frac{120}{6} = \frac{-68}{6}}$. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2. This gives us ${\frac{-34}{3}}$. So, the expression inside the parentheses simplifies to ${\frac{-34}{3}}$.

Step 2: Taming the Fractions Outside

Now that we've conquered the parentheses, let's turn our attention to the fractions outside. We have ${\frac{75}{2}}$ and ${\frac{125}{3}}$. To combine these fractions, we'll need a common denominator, just like before. The least common multiple of 2 and 3 is still 6, so we'll convert both fractions to have a denominator of 6. This means multiplying the numerator and denominator of ${\frac{75}{2}}$ by 3 and the numerator and denominator of ${\frac{125}{3}}$ by 2. This gives us:

${ \frac{75}{2} \times \frac{3}{3} = \frac{225}{6} }$

${ \frac{125}{3} \times \frac{2}{2} = \frac{250}{6} }$

Now our equation looks like this: ${\frac{225}{6} - \frac{250}{6} + 50 - \left(\frac{-34}{3}\right)}$. We can now subtract the fractions: ${\frac{225}{6} - \frac{250}{6} = \frac{-25}{6}}$.

Step 3: Embracing the Double Negative

Ah, the double negative! It's like a mathematical plot twist. Remember, subtracting a negative is the same as adding a positive. So, ${-\left(\frac{-34}{3}\right)}$ becomes ${+\frac{34}{3}}$. Our equation now reads: ${\frac{-25}{6} + 50 + \frac{34}{3}}$.

Step 4: Unifying the Denominators

We're in the home stretch! To combine the remaining terms, we need a common denominator for all the fractions. The least common multiple of 6 and 3 is 6, so we'll convert ${\frac{34}{3}}$ to a fraction with a denominator of 6. This means multiplying the numerator and denominator of ${\frac{34}{3}}$ by 2. This gives us:

${ \frac{34}{3} \times \frac{2}{2} = \frac{68}{6} }$

We also need to convert the whole number 50 into a fraction with a denominator of 6. So, 50 becomes ${\frac{50 \times 6}{6} = \frac{300}{6}}$. Our equation now looks like this: ${\frac{-25}{6} + \frac{300}{6} + \frac{68}{6}}$.

Step 5: The Grand Finale: Combining the Terms

Finally, we can combine all the fractions! ${\frac{-25}{6} + \frac{300}{6} + \frac{68}{6} = \frac{-25 + 300 + 68}{6} = \frac{343}{6}}$. And there you have it! The solution to our mathematical puzzle is ${\frac{343}{6}}$.

The Final Verdict: ${\frac{343}{6}}$ and Why It Matters

So, after our mathematical adventure, we've arrived at the answer: ${\frac{343}{6}}$. But what does this number really mean? Well, it's the simplified form of our original equation, a single fraction that represents the culmination of all our calculations. We could also express this as a mixed number (57 ${\frac{1}{6}}$) or a decimal (approximately 57.17), depending on the context. The beauty of math is that it gives us different ways to represent the same value, each with its own advantages.

Why Understanding Order of Operations is Crucial

Throughout this journey, we've emphasized the importance of the order of operations (PEMDAS). This isn't just some arbitrary rule; it's the foundation upon which all mathematical calculations are built. Imagine if we decided to add before dealing with the parentheses – we'd end up with a completely different answer! PEMDAS ensures that we follow a consistent path, leading us to the correct solution every time. It's like a universal agreement among mathematicians, ensuring that everyone speaks the same language.

Fractions: More Than Just Slices of Pie

Fractions often get a bad rap, but they're actually incredibly versatile tools. They allow us to represent parts of a whole, express ratios, and even describe probabilities. In our equation, fractions were the building blocks of the problem, and mastering them is essential for tackling more complex mathematical concepts. Understanding fractions helps us in everyday situations, from splitting a bill with friends to adjusting a recipe.

The Power of Simplification

Simplifying fractions, like we did in Step 1, might seem like a minor detail, but it can make a big difference. A simplified fraction is easier to understand and work with, reducing the chances of errors in later calculations. It's like tidying up your workspace before starting a project – it sets you up for success.

Math as a Journey, Not Just a Destination

Solving this equation wasn't just about getting the right answer; it was about the process. We broke down a complex problem into manageable steps, applied fundamental mathematical principles, and celebrated each milestone along the way. This journey is what makes math so rewarding. It's about developing problem-solving skills, building confidence, and appreciating the elegance of mathematical logic.

So, the next time you encounter a daunting math problem, remember our adventure today. Embrace the challenge, break it down, and enjoy the ride! And remember, ${\frac{343}{6}}$ is more than just a number; it's a testament to your mathematical prowess.

Practice Makes Perfect: Sharpening Your Skills

Now that we've conquered this equation together, the best way to solidify your understanding is through practice. Try tackling similar problems on your own, and don't be afraid to make mistakes – they're valuable learning opportunities! Here are a few ideas to get you started:

1. Create your own equations: Mix and match fractions, whole numbers, and parentheses to create your own mathematical puzzles. This is a great way to challenge yourself and explore different combinations.
2. Find real-world applications: Look for situations in your daily life where you can use the concepts we've discussed. For example, you could calculate discounts while shopping or measure ingredients while cooking.
3. Seek out resources: There are tons of fantastic resources available online and in libraries, from interactive websites to comprehensive textbooks. Explore different options and find what works best for you.

Remember, math is a skill that improves with practice. The more you engage with it, the more confident and capable you'll become. So, keep exploring, keep experimenting, and keep having fun with math!

In Conclusion: Embracing the Mathematical Adventure

We've journeyed through the intricate world of fractions, parentheses, and the order of operations, emerging victorious with the solution ${\frac{343}{6}}$. But more importantly, we've reinforced the power of breaking down complex problems into manageable steps, the importance of understanding fundamental mathematical principles, and the joy of mathematical discovery. Math isn't just about numbers and equations; it's about critical thinking, problem-solving, and the ability to see patterns and connections in the world around us.

So, keep exploring the fascinating realm of mathematics, embrace the challenges, and celebrate the triumphs. And remember, every equation solved is a step further on your mathematical adventure!