Solving Combined Operations (15-3)-4÷2+5x2+(4-2)+7 A Step-by-Step Guide

by Axel Sørensen 72 views

Hey there, math enthusiasts! Ever stumbled upon a math problem that looks like a jumbled mess of numbers and operations? You know, the kind that makes you go, "Where do I even start?" Well, you're not alone! Combined operations can seem daunting at first, but trust me, once you understand the rules, they become a fun puzzle to solve. Let's dive into this particular problem: (15-3)-4÷2+5x2+(4-2)+7 and break it down step by step.

Understanding the Order of Operations: PEMDAS/BODMAS

Before we even think about crunching those numbers, we need to talk about the golden rule of combined operations: the order of operations. This is like the secret code that unlocks the solution. There are two acronyms you might have heard: PEMDAS and BODMAS. They both mean the same thing, just with slightly different words:

  • PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • BODMAS: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

See? Same order, different words. The important thing is to remember the hierarchy. Operations within parentheses or brackets come first, then exponents or orders, then multiplication and division (we handle these from left to right), and finally, addition and subtraction (also from left to right). This order ensures that we all arrive at the same answer, no matter who's solving the problem. Without it, math would be a chaotic free-for-all!

Think of it like building a house. You wouldn't start painting the walls before you've built the frame, right? The order of operations is the blueprint for our mathematical house. It tells us exactly what to do first, second, and so on, until our equation is beautifully constructed and we have our final answer. It might seem a little rigid, but it's this structure that brings clarity and consistency to math. So, let's embrace the order and watch the magic unfold as we tackle our problem!

Cracking the Code: Applying PEMDAS/BODMAS to Our Problem

Okay, guys, now that we've got the order of operations firmly in our minds, let's get back to our original problem: (15-3)-4÷2+5x2+(4-2)+7. It's time to put PEMDAS/BODMAS into action and watch how it transforms this seemingly complex equation into something manageable. Remember, we're not just blindly calculating; we're following a strategic plan. Think of it like a dance – each operation has its place and time.

First up, we spot those parentheses – or brackets, if you're a BODMAS person. These are our top priority. We have two sets of parentheses in this equation: (15-3) and (4-2). Let's tackle them one at a time.

  • (15-3) is a straightforward subtraction problem. 15 minus 3 equals 12. So, we can replace (15-3) with 12 in our equation.
  • Next, we have (4-2), another simple subtraction. 4 minus 2 equals 2. We'll replace (4-2) with 2 as well.

Our equation now looks like this: 12-4÷2+5x2+2+7. See how much simpler it's becoming already? We've conquered the parentheses, and we're ready to move on to the next step in our PEMDAS/BODMAS adventure.

Division and Multiplication: The Left-to-Right Rule

Alright, folks, we've successfully navigated the parentheses, and now we're at the M and D of PEMDAS/BODMAS – Multiplication and Division. This is where things get a little more nuanced. We don't just automatically do multiplication before division (or vice versa). The rule here is to work from left to right. Think of it like reading a sentence; you process the words in the order they appear.

Looking at our equation: 12-4÷2+5x2+2+7, we see a division operation 4÷2 and a multiplication operation 5x2. Since division appears first as we read from left to right, we'll tackle that one first.

  • 4÷2 is a simple division. 4 divided by 2 equals 2. So, we replace 4÷2 with 2 in our equation.

Now our equation looks like this: 12-2+5x2+2+7. Notice how we're carefully substituting each solved operation back into the equation. This helps us keep track of where we are and prevents errors. It's like marking your progress on a map – you always know how far you've come and what's still ahead.

Next up is the multiplication: 5x2. This is another straightforward calculation.

  • 5x2 equals 10. We replace 5x2 with 10 in our equation.

Our equation is now: 12-2+10+2+7. We've conquered the division and multiplication, and we're left with a string of additions and subtractions. We're in the home stretch now!

Addition and Subtraction: The Final Countdown

Okay, mathletes, we've reached the final stage of our combined operations adventure! We've tackled the parentheses, conquered the division and multiplication, and now we're left with a string of additions and subtractions. Just like with multiplication and division, we don't have a strict order between addition and subtraction. The rule is to work from left to right. This is super important to remember, guys, because changing the order here can change your final answer!

Looking at our equation: 12-2+10+2+7, we start with the leftmost operation, which is a subtraction: 12-2.

  • 12-2 equals 10. We replace 12-2 with 10 in our equation.

Now our equation looks like this: 10+10+2+7. See how we're just chipping away at the problem, one step at a time? It's like building a tower of blocks – each calculation is a block that brings us closer to the top.

Next, we have 10+10, which is a simple addition.

  • 10+10 equals 20. We replace 10+10 with 20 in our equation.

Our equation is now: 20+2+7. We're getting closer and closer to the finish line!

Now we have 20+2, another addition.

  • 20+2 equals 22. We replace 20+2 with 22 in our equation.

Our equation is now: 22+7. Just one more step to go!

Finally, we have 22+7, our last addition.

  • 22+7 equals 29.

And there you have it! We've successfully navigated the twists and turns of our combined operations problem. The final answer is 29. We did it! Give yourselves a pat on the back, guys. You've shown those numbers who's boss!

The Grand Finale: Our Solution

So, after all that calculating, strategizing, and PEMDAS/BODMAS-ing, we've arrived at our final answer. Drumroll, please... The solution to the combined operations problem (15-3)-4÷2+5x2+(4-2)+7 is 29. 🎉

But more than just getting the right answer, we've learned a valuable lesson: breaking down complex problems into smaller, manageable steps makes them much less intimidating. It's like eating an elephant – you don't try to swallow it whole! You take it one bite at a time. Similarly, with combined operations, we tackle each operation in the correct order, and before you know it, we've conquered the entire problem.

This approach isn't just useful in math, guys. It's a life skill! Whether you're planning a big project at work, organizing a party, or even just trying to declutter your house, breaking things down into smaller steps makes the task seem less overwhelming and more achievable. So, remember the lessons we've learned today, both mathematical and life-related, and go out there and conquer your own challenges!

Practice Makes Perfect: Level Up Your Skills

Okay, you've successfully solved one combined operations problem with me, which is awesome! But let's be real, math skills are like muscles – you need to keep exercising them to stay strong. You wouldn't expect to run a marathon after just one training session, right? Same goes for math. The more you practice, the more comfortable and confident you'll become with these types of problems. It's all about building that mental muscle memory.

So, what's the best way to practice? Well, the good news is that combined operations problems are everywhere! You can find them in textbooks, online worksheets, or even create your own. The key is to start with simpler problems and gradually work your way up to the more challenging ones. It's like learning to play an instrument – you start with basic chords and scales before you try to play a complicated song.

Here are a few ideas for practicing your combined operations skills:

  1. Find online resources: There are tons of websites and apps that offer practice problems with step-by-step solutions. This is a great way to check your work and see where you might be making mistakes.
  2. Use textbooks or workbooks: If you have a math textbook or workbook, look for the section on order of operations. These often have a variety of practice problems with increasing difficulty.
  3. Create your own problems: This is a fun way to challenge yourself and really test your understanding. Try mixing up different operations and numbers to create unique problems.
  4. Work with a friend or study group: Math is often more fun when you're learning with others. You can quiz each other, discuss different approaches, and help each other when you get stuck.
  5. Don't be afraid to make mistakes: Mistakes are a natural part of the learning process. The important thing is to learn from them. When you make a mistake, try to figure out why you made it and how you can avoid making the same mistake in the future.

Remember, guys, practice doesn't just make perfect; it makes permanent. The more you practice, the more these concepts will become ingrained in your mind, and the easier it will be to tackle even the most complex math problems. So, grab your pencil, dust off your calculator, and get practicing! You've got this!

Mastering Combined Operations: Tips and Tricks

Alright, you're practicing, you're feeling more confident, but let's take it to the next level! Mastering combined operations isn't just about knowing the order; it's about developing strategies and tricks that make the process smoother and more efficient. It's like learning the shortcuts on your favorite video game – they help you level up faster and achieve your goals more easily.

Here are some tips and tricks that can help you become a combined operations pro:

  1. Write it out step by step: This might seem obvious, but it's crucial! Don't try to do everything in your head. Write out each step of the process, showing how you're applying PEMDAS/BODMAS. This helps you stay organized, prevents errors, and makes it easier to track your progress.
  2. Double-check your work: It's easy to make a small mistake, especially with so many operations involved. Take a few extra seconds to double-check each step, making sure you haven't made any calculation errors. It's like proofreading a paper – a fresh look can catch those sneaky typos.
  3. Use parentheses strategically: Sometimes, adding extra parentheses can help clarify the order of operations, especially in more complex problems. If you're feeling unsure about which operation to perform first, add parentheses to group the numbers and operations that should be done together.
  4. Look for patterns and shortcuts: As you solve more problems, you'll start to notice patterns and shortcuts that can save you time and effort. For example, you might notice that multiplying by 10 is as easy as adding a zero to the end of the number.
  5. Estimate your answer: Before you even start solving the problem, try to estimate what the answer should be. This can help you catch any major errors along the way. For example, if you're solving a problem that involves adding several numbers, and your final answer is smaller than one of the numbers you added, you know you've made a mistake somewhere.
  6. Break it down into smaller chunks: If the problem looks really intimidating, try breaking it down into smaller, more manageable chunks. Focus on solving one part of the problem at a time, and then put the pieces together at the end. This is similar to the strategy we used earlier – tackling the parentheses first, then the multiplication and division, and so on.
  7. Use a calculator wisely: Calculators can be helpful tools, but they shouldn't replace your understanding of the order of operations. Use a calculator to perform calculations, but make sure you're still applying PEMDAS/BODMAS correctly.

By using these tips and tricks, you'll not only solve combined operations problems more efficiently but also develop a deeper understanding of mathematical principles. It's like becoming a math ninja – you'll have the skills and strategies to conquer any problem that comes your way! So, keep practicing, keep learning, and keep those math muscles strong!

Conclusion: Embracing the Power of Order

Well, math adventurers, we've reached the end of our combined operations quest! We started with a seemingly tangled problem, (15-3)-4÷2+5x2+(4-2)+7, and step by step, we unraveled its mysteries using the power of the order of operations. We learned about PEMDAS/BODMAS, the golden rule that guides us through the maze of calculations. We tackled parentheses, conquered multiplication and division, and sailed smoothly through addition and subtraction. And in the end, we arrived at our triumphant answer: 29.

But more than just finding a number, we've gained a valuable skill – the ability to approach complex problems with a clear strategy. We've learned that breaking things down into smaller steps, following a logical order, and double-checking our work can lead us to success in math and in life.

Remember, guys, combined operations might seem intimidating at first, but they're really just puzzles waiting to be solved. With a little practice and the right tools, you can conquer any equation that comes your way. So, embrace the challenge, enjoy the process, and celebrate your mathematical victories!