Mastering Order Of Operations A Step By Step Guide

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Hey guys! Ever find yourself staring at a math problem with a bunch of different operations and feeling totally lost? You're not alone! The key to cracking these problems is understanding the order of operations. It's like a secret code that tells you exactly which steps to take in what order. In this guide, we're going to break down some examples, making sure you've got the order of operations down pat. So, grab your pencils, and let's dive in!

1. 10 + 6 - [7 + (10 - 5)]

Let's tackle this one step by step. The most important thing to remember with these kinds of problems is to follow the order of operations, often remembered by the acronym PEMDAS (or BODMAS):

  • Parentheses (or Brackets)
  • Exponents (or Orders)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

So, for this problem, we start inside the innermost parentheses:

  1. (10 - 5) = 5. Now our expression looks like: 10 + 6 - [7 + 5]
  2. Next, we deal with the brackets: [7 + 5] = 12. Now we have: 10 + 6 - 12
  3. Now it’s just addition and subtraction, which we do from left to right:
    • 10 + 6 = 16
    • 16 - 12 = 4

So, the final answer is 4.

Remember, this order of operations is super important. If you don't follow it, you'll end up with the wrong answer. It’s like following a recipe – you need to add the ingredients in the right order for the dish to turn out perfectly. In math, if you mix up the steps, the “dish” (aka the answer) will taste a little off!

2. (8 + 18 + 9) _ (4 x 3 - 7) = 23

Okay, let's break down the second problem. Again, we're sticking with PEMDAS/BODMAS, so parentheses first!

  1. Let's handle the first set of parentheses: (8 + 18 + 9). This is straightforward addition:

    • 8 + 18 = 26
    • 26 + 9 = 35

  2. Now for the second set of parentheses: (4 x 3 - 7). Inside here, we have both multiplication and subtraction. Multiplication comes first:

    • 4 x 3 = 12
    • Now we subtract: 12 - 7 = 5

  3. So, now our problem looks like: 35 _ 5. The underscore symbol (_) usually means multiplication. So:

    • 35 x 5 = 175

Wait a minute! The original problem states that the answer should be 23. There seems to be a mistake or a misunderstanding in the original equation. If the result is intended to be 23, the operation between the parentheses should be subtraction or division, not multiplication. However, based on the stated operations, the correct answer is 175.

It's always a good idea to double-check your work, especially when the final result seems off. Math can be tricky, and sometimes a little mistake can lead to a big difference in the answer. Think of each step as a link in a chain – if one link is weak, the whole chain might break!

3. (15 - 6) x 12 + 3

Alright, let's keep the ball rolling with this one. Remember, PEMDAS/BODMAS is our best friend here. We're all about that order of operations!

  1. First up, the parentheses: (15 - 6) = 9. So, now we have: 9 x 12 + 3
  2. Next, we spot multiplication: 9 x 12 = 108. Our equation is now: 108 + 3
  3. Finally, we do the addition: 108 + 3 = 111

So, the answer to this one is 111. See how breaking it down step-by-step makes it so much easier? It’s like eating an elephant – you can’t do it in one bite! You need to take it one step at a time, and the same goes for math problems.

Don't rush through the steps. It's better to go slow and be accurate than to speed through and make a mistake. Think of it like building a house – you need a solid foundation before you can start adding the walls and roof. In math, each step is part of that foundation, and if one step is shaky, the whole thing could come tumbling down.

4. 24 + 12 _ 6 x 4 - 5 x 2 = 22

Here we go, another one! This looks a little more complex, but we're not scared, right? We have PEMDAS/BODMAS on our side!

  1. No parentheses here, so let's jump straight to multiplication and division. Remember, we do these from left to right. The underscore (_) again signifies multiplication.
    • Let’s start with division: 12 _ 6 = 2.
    • Now, our expression looks like: 24 + 2 x 4 - 5 x 2
  2. Now, let’s do the remaining multiplications from left to right.
    • First multiplication: 2 x 4 = 8.
    • Second multiplication: 5 x 2 = 10.
    • Now our expression looks like: 24 + 8 - 10
  3. Finally, let's tackle the addition and subtraction, again from left to right:
    • 24 + 8 = 32
    • 32 - 10 = 22

The answer is indeed 22, as stated in the original problem. We nailed it! This problem really highlights the importance of working from left to right when you have operations of the same priority (like multiplication and division, or addition and subtraction). If you did the subtraction before the addition, you'd end up with a different answer.

5. 18 _ 3 + 6 - 4 x 2 = 16

Let's keep practicing! This one has a mix of operations, so we'll be flexing our PEMDAS/BODMAS muscles.

  1. First, we handle multiplication and division from left to right. Remember, the underscore (_) is multiplication.
    • 18 _ 3 = 54
    • Our equation now looks like: 54 + 6 - 4 x 2
  2. Next up is multiplication: 4 x 2 = 8
    • Now we have: 54 + 6 - 8
  3. Finally, we do addition and subtraction from left to right:
    • 54 + 6 = 60
    • 60 - 8 = 52

But wait! The original problem says the answer is 16. This is a good reminder that sometimes there can be errors in the problem itself. Based on the operations given, the correct answer is 52, not 16.

This is a crucial lesson: don't just blindly accept the given answer. Always work through the problem yourself and see if it makes sense. Math isn't just about getting the right answer; it's about understanding the process. And sometimes, that process reveals a mistake in the original question!

6. 15 + 40 _ 5 x 3 = 39

Let's keep rolling! We're getting good at this. This one has a nice mix of operations, so let's put our PEMDAS/BODMAS knowledge to work.

  1. First, we tackle multiplication and division from left to right. Let's start with division, which the underscore (_) stands for:
    • 40 _ 5 = 8
    • Now our equation looks like: 15 + 8 x 3
  2. Next up is multiplication: 8 x 3 = 24
    • Now we have: 15 + 24
  3. Finally, we do the addition:
    • 15 + 24 = 39

And there you have it! The answer is 39, just like the original problem stated. High five! This problem is a great example of how PEMDAS/BODMAS helps us keep things organized. If we didn't follow the order of operations, we'd likely end up with a completely different (and incorrect) answer.

Math is like a puzzle, and each piece needs to fit in the right place. PEMDAS/BODMAS is the instruction manual that tells us where each piece goes. With practice, you'll be solving these puzzles like a pro!

7. 5 x 6 _ 10 + 18 = 21

Okay, let’s jump into this one. We’ve got multiplication, division (signified by the underscore _), addition – a classic PEMDAS/BODMAS scenario!

  1. First things first, multiplication and division, from left to right:
    • 5 x 6 = 30
    • Now our expression is: 30 _ 10 + 18
    • Next, division: 30 _ 10 = 3
    • Our equation now looks like: 3 + 18
  2. Finally, we have addition:
    • 3 + 18 = 21

Success! The answer is 21, which matches the original problem. This one was pretty straightforward once we followed the correct order. It’s like following a map – if you stick to the route, you’ll reach your destination. In math, PEMDAS/BODMAS is your map!

Remember, it’s not just about getting the answer right; it’s about understanding why you got the answer you did. Can you explain each step to someone else? If you can, you know you’ve really got it!

8. 4 x [(6 + 8) _ 2] - 12

Alright, this one looks a little more intimidating with the brackets, but we know what to do! PEMDAS/BODMAS is our superpower. Those brackets are just another form of parentheses, so we'll tackle those first.

  1. Let's start inside the brackets: (6 + 8) = 14
    • Now our expression looks like: 4 x [14 _ 2] - 12
  2. Still inside the brackets, we have division (underscore _): 14 _ 2 = 7
    • So now we have: 4 x 7 - 12
  3. Next up is multiplication: 4 x 7 = 28
    • Our equation is now: 28 - 12
  4. Finally, subtraction:
    • 28 - 12 = 16

So, the answer is 16. We conquered those brackets! This problem shows how important it is to work from the inside out when you have nested parentheses or brackets. It’s like peeling an onion – you need to remove the outer layers before you can get to the center.

9. [(5 + 7)]

Okay, this one seems super simple compared to the others, but let's run through it to be thorough!

  1. We have brackets, so we start there:
    • (5 + 7) = 12

And… that's it! The answer is 12. Sometimes the problems are easier than they look. But it's still important to go through the steps and make sure you haven't missed anything.

Mastering the Order of Operations: Key Takeaways

Guys, you've tackled some great examples today, and you're well on your way to mastering the order of operations! Here are a few key takeaways to remember:

  • PEMDAS/BODMAS is your friend: This acronym is your guide to the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Work from the inside out: When you have parentheses or brackets nested inside each other, start with the innermost ones and work your way outwards.
  • Left to right matters: When you have operations of the same priority (like multiplication and division, or addition and subtraction), do them from left to right.
  • Double-check your work: It's always a good idea to go back and review your steps, especially if the final answer seems off. Math can be tricky, and it's easy to make a small mistake.
  • Practice makes perfect: The more you practice, the more comfortable you'll become with the order of operations. It's like learning any new skill – the more you do it, the better you'll get!

So, keep practicing, keep asking questions, and you'll be a math whiz in no time! You got this!