Solving -75 Divided By 56 Divided By -5 Plus 12 A Step-by-Step Guide

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Hey guys! Let's tackle a fascinating math problem today that involves division, negative numbers, and order of operations. We're going to break down the expression -75 ÷ (56 ÷ (-5 + 12)) step by step, so you can not only understand the solution but also grasp the underlying concepts. Math can be like a puzzle, and our job is to find the right pieces and put them together correctly. So, let's dive in!

Understanding the Order of Operations

Before we even think about negative numbers or division, we need to talk about the order of operations. This is like the golden rule of math – a set of guidelines that tell us in what order to perform calculations. If we don't follow these rules, we might end up with the wrong answer, and nobody wants that! The most common way to remember the order of operations is by using the acronym PEMDAS, which stands for:

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

So, whenever we see a mathematical expression, we first look for parentheses. Operations inside parentheses are always done first. Then, we look for exponents. After that, we perform multiplication and division from left to right, and finally, addition and subtraction from left to right. Think of it as a hierarchy – some operations are more important than others and need to be done before the less important ones.

Now, let's apply this to our problem: -75 ÷ (56 ÷ (-5 + 12)). The first thing we see is a set of parentheses. Inside the parentheses, we have another set of parentheses! This means we need to start with the innermost parentheses first. This nested structure might seem a little intimidating at first, but don't worry, we'll take it one step at a time.

Step-by-Step Breakdown of the Expression

Let's break down this expression piece by piece. Remember, the key is to follow the order of operations and take our time. No need to rush – accuracy is more important than speed in math!

1. Solving the Innermost Parentheses: (-5 + 12)

Our first task is to tackle the innermost parentheses: (-5 + 12). This is a simple addition problem involving a negative number. Think of it like starting at -5 on a number line and moving 12 spaces to the right. What do we get? We end up at 7. So, (-5 + 12) = 7. This is our first step, and we've already made progress!

Now we can rewrite our original expression as -75 ÷ (56 ÷ 7). See how we've simplified it already? By focusing on the parentheses first, we've made the expression much easier to manage.

2. Solving the Outer Parentheses: (56 ÷ 7)

Next up, we need to deal with the remaining parentheses: (56 ÷ 7). This is a straightforward division problem. How many times does 7 go into 56? If you know your multiplication tables, you'll know that 7 multiplied by 8 equals 56. So, (56 ÷ 7) = 8. Another step down, and the problem is becoming clearer.

Our expression now looks like this: -75 ÷ 8. We're almost there! We've simplified the expression inside the parentheses, and now we're left with a single division operation.

3. Performing the Final Division: -75 ÷ 8

Finally, we need to perform the division: -75 ÷ 8. This is where things get a little trickier because 75 is not perfectly divisible by 8. We're going to end up with a decimal or a fraction. But don't worry, we can handle it!

Let's think about the division first. 8 goes into 75 nine times (8 x 9 = 72). That leaves us with a remainder of 3. So, we can write the answer as 9 with a remainder of 3, or 9 and 3/8. But remember, we have a negative sign in front of the 75. A negative number divided by a positive number is always negative. So, our answer is -9 and 3/8.

If we want to express this as a decimal, we can divide 3 by 8, which gives us 0.375. So, the final answer in decimal form is -9.375. We've done it! We've successfully navigated the order of operations, handled the negative numbers, and arrived at the solution.

Key Takeaways and Practice Tips

Okay, guys, that was a pretty intense journey through the world of division and negative numbers! But hopefully, you've learned a lot along the way. Let's recap the key takeaways:

  • Order of Operations (PEMDAS): Always follow the order of operations to ensure you get the correct answer. Parentheses first, then exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).
  • Negative Numbers: Remember the rules for working with negative numbers. A negative number divided by a positive number is negative, and vice versa.
  • Fractions and Decimals: Don't be afraid of fractions and decimals! They're just different ways of representing numbers. Practice converting between fractions and decimals to become more comfortable with them.

Now, the best way to master these concepts is to practice! Try solving similar problems on your own. You can even make up your own expressions with parentheses, negative numbers, and different operations. The more you practice, the more confident you'll become in your math skills.

Here are a few tips to help you with your practice:

  • Start Simple: Begin with simpler expressions and gradually increase the complexity.
  • Show Your Work: Write down each step of your solution. This will help you track your progress and identify any errors.
  • Check Your Answers: Use a calculator or online tool to check your answers. This will help you learn from your mistakes.
  • Don't Give Up: Math can be challenging, but don't get discouraged! Keep practicing, and you'll eventually get it.

Conclusion: Math is a Journey, Not a Destination

So, guys, we've reached the end of our exploration of -75 ÷ (56 ÷ (-5 + 12)). We've seen how the order of operations guides us through complex expressions, and we've learned how to handle negative numbers and fractions. But the most important thing to remember is that math is not just about getting the right answer. It's about the process of problem-solving, the journey of discovery, and the satisfaction of understanding how things work.

Keep exploring, keep practicing, and keep asking questions. Math is a vast and fascinating world, and there's always something new to learn. And remember, if you ever get stuck, don't hesitate to ask for help. There are plenty of resources available, including teachers, tutors, online forums, and friends who are good at math.

So, go forth and conquer those math problems! You've got this!