Solving -75 Divided By (56 Divided By (-5 + 12)) A Mathematical Discussion

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Hey guys! Today, we're diving deep into a fascinating mathematical problem that might seem a bit daunting at first glance: solving -75 divided by (56 divided by (-5 + 12)). Don't worry, we'll break it down step-by-step to make sure everyone understands the process. Math can be fun, especially when we tackle complex problems together! Let's get started and unlock the secrets behind this equation.

Understanding the Order of Operations

Before we even think about punching numbers into a calculator, it's super important to understand the order of operations. Think of it as the golden rule of math – if you don't follow it, you'll end up with the wrong answer. We often use the acronym PEMDAS (or BODMAS in some countries) to remember this order. PEMDAS stands for:

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

This tells us exactly what to tackle first. In our problem, -75 ÷ (56 ÷ (-5 + 12)), we have parentheses, so that's where we'll begin. Remember, we always work from the innermost parentheses outwards.

Tackling the Innermost Parentheses: (-5 + 12)

The first thing we need to do is solve the expression inside the innermost parentheses: (-5 + 12). This is a simple addition problem. We're adding a negative number (-5) to a positive number (12). Think of it like starting at -5 on a number line and moving 12 steps to the right. Where do you end up? You land on 7! So, (-5 + 12) = 7. This simplifies our original equation quite a bit. Now we have: -75 ÷ (56 ÷ 7). See? We're making progress already!

Moving to the Outer Parentheses: (56 ÷ 7)

Now that we've conquered the innermost parentheses, let's move to the outer ones. We need to solve (56 ÷ 7). This is a straightforward division problem. How many times does 7 go into 56? If you know your times tables, you'll know that 56 ÷ 7 = 8. Great! Our equation is becoming even simpler. We now have: -75 ÷ 8. We're almost there, guys!

Final Division: -75 ÷ 8

Okay, the final step! We need to divide -75 by 8. This is where things get a little interesting because 8 doesn't go into 75 evenly. But don't worry, we can handle this. When we divide 75 by 8, we get 9 with a remainder. So, 8 goes into 75 nine times (8 x 9 = 72), leaving us with a remainder of 3 (75 - 72 = 3).

Since we're dealing with a negative number (-75) divided by a positive number (8), our answer will be negative. We can express the answer in a few ways. We could say it's -9 with a remainder of 3, but it's more common to express it as a decimal or a mixed number.

Expressing the Answer as a Decimal

To express the answer as a decimal, we need to continue the division. We have a remainder of 3, so we can add a decimal point and a zero to 75, making it 75.0. Now we're essentially dividing 30 (the 3 with the added zero) by 8. 8 goes into 30 three times (8 x 3 = 24), leaving a remainder of 6 (30 - 24 = 6). We add another zero, making it 60. 8 goes into 60 seven times (8 x 7 = 56), leaving a remainder of 4 (60 - 56 = 4). Add another zero, making it 40. 8 goes into 40 five times (8 x 5 = 40), with no remainder.

So, when we divide 75 by 8, we get 9.375. Remember, our original problem was -75 divided by 8, so our final answer is -9.375. Woohoo! We did it!

Expressing the Answer as a Mixed Number

Alternatively, we can express the answer as a mixed number. We know that 8 goes into 75 nine times with a remainder of 3. So, we can write the answer as -9 and 3/8. This means -9 whole numbers and 3 eighths. Both -9.375 and -9 3/8 are correct ways to represent the answer.

Key Takeaways

  • The order of operations (PEMDAS/BODMAS) is crucial for solving mathematical problems correctly. Always tackle parentheses first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Ignoring this order will almost certainly lead to the wrong answer.
  • Breaking down complex problems into smaller, manageable steps makes them much less intimidating. We started with a seemingly complicated expression and simplified it step-by-step until we reached the solution. This approach works wonders in math and in life!
  • Understanding how to express answers in different forms (decimals, mixed numbers) is a valuable skill. It allows you to choose the representation that's most appropriate for the situation.
  • Dividing negative numbers by positive numbers always results in a negative answer. Keep track of the signs throughout the problem to avoid making careless errors.

Common Mistakes to Avoid

Let's chat about some common pitfalls folks often stumble into when tackling problems like this. Being aware of these potential slip-ups can save you a lot of headaches and ensure you nail the correct answer.

Ignoring the Order of Operations

We've hammered this point home, but it's worth repeating: the order of operations is king! A frequent mistake is to simply work through the equation from left to right, disregarding PEMDAS/BODMAS. For example, someone might try to divide -75 by 56 first, which would lead to a completely different (and incorrect) answer. Always, always, always prioritize parentheses, exponents, multiplication and division (from left to right), and then addition and subtraction (from left to right). Make it your mantra!

Sign Errors

Working with negative numbers can be a bit like navigating a minefield if you're not careful. A common mistake is to mishandle the signs during division or multiplication. Remember, a negative divided by a positive is a negative, and vice versa. Double-check your signs at every step to ensure accuracy. It’s super easy to drop a negative sign, especially when you're working quickly, so take your time and be meticulous.

Misunderstanding Remainders

When a division doesn't result in a whole number, we end up with a remainder. It's crucial to understand what to do with that remainder. As we saw earlier, we can express the answer as a decimal or as a mixed number. A common mistake is to simply ignore the remainder or to miscalculate it. Practice converting remainders to decimals and mixed numbers to build your confidence.

Calculator Dependency

Calculators are fantastic tools, but they shouldn't be a crutch. Relying too heavily on a calculator without understanding the underlying mathematical principles can hinder your problem-solving skills. It's essential to be able to perform these calculations manually, at least to some extent. This not only deepens your understanding but also helps you catch errors that might occur if you input something incorrectly into the calculator. Think of your brain as the primary calculator and the electronic one as a helpful assistant.

Skipping Steps

In the rush to solve a problem, it's tempting to skip steps and try to do things in your head. While mental math is a valuable skill, skipping steps in complex problems can increase the likelihood of making errors. Write out each step clearly and methodically. This not only helps you keep track of your work but also makes it easier to spot mistakes if you make them. Think of it like showing your work – it's not just for your teacher; it's for yourself too!

Practice Makes Perfect

The best way to avoid these mistakes and become a math whiz is to practice! The more problems you solve, the more comfortable you'll become with the concepts and the less likely you'll be to make errors. Seek out similar problems and work through them step-by-step. Don't be afraid to make mistakes – they're part of the learning process. Just learn from them and keep going. Math is like a muscle; the more you exercise it, the stronger it gets.

So, there you have it! We've not only solved the problem -75 divided by (56 divided by (-5 + 12)) but also explored the underlying principles and common pitfalls. Remember, math is a journey, not a destination. Enjoy the ride, embrace the challenges, and keep on learning!