Step-by-Step Guide To Solving (-3/4 X 16/81 + 0.3) X (-3/4)

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Hey guys! Today, we're going to break down a math problem that might look a little intimidating at first glance, but trust me, it's totally manageable once we tackle it step by step. We're diving into the expression (-3/4 x 16/81 + 0.3) x (-3/4). We’ll go through each operation, making sure you understand the logic behind every move. So, grab your calculators (or your trusty mental math skills) and let’s get started!

Understanding the Order of Operations

Before we jump into the nitty-gritty, let's quickly recap the order of operations. Remember PEMDAS/BODMAS? It’s an acronym that tells us the sequence in which we should perform calculations:

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

Keeping this order in mind will prevent us from making common mistakes and ensure we arrive at the correct answer. Think of it as the golden rule of mathematical expressions!

Breaking Down the Expression: A Detailed Walkthrough

Okay, let's apply PEMDAS/BODMAS to our expression: (-3/4 x 16/81 + 0.3) x (-3/4). The first thing we need to address is the parentheses. Inside the parentheses, we have both multiplication and addition. According to our order of operations, multiplication comes first.

Step 1: Multiplication Inside the Parentheses

So, we start by multiplying -3/4 by 16/81. When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, we have:

(-3/4) x (16/81) = (-3 x 16) / (4 x 81)

Now, let’s do the math: -3 multiplied by 16 is -48, and 4 multiplied by 81 is 324. This gives us:

-48/324

But hold on! We can simplify this fraction. Both -48 and 324 are divisible by 12 (and even higher numbers, but let’s take it one step at a time). Dividing both by 12, we get:

(-48 ÷ 12) / (324 ÷ 12) = -4/27

So, the result of our first multiplication is -4/27. Great job! We’re one step closer.

Step 2: Addition Inside the Parentheses

Next up, we need to add 0.3 to -4/27. Now, here’s a little trick: to add or subtract fractions and decimals, it’s easiest to convert everything to either fractions or decimals. In this case, let’s convert 0.3 to a fraction. 0.3 is the same as 3/10.

Now we have:

-4/27 + 3/10

To add these fractions, we need a common denominator. The least common multiple (LCM) of 27 and 10 is 270. So, we need to convert both fractions to have a denominator of 270.

To convert -4/27, we multiply both the numerator and the denominator by 10:

(-4 x 10) / (27 x 10) = -40/270

To convert 3/10, we multiply both the numerator and the denominator by 27:

(3 x 27) / (10 x 27) = 81/270

Now we can add them:

-40/270 + 81/270 = (-40 + 81) / 270 = 41/270

So, the result inside the parentheses is 41/270. Awesome! We’re halfway there.

Step 3: Final Multiplication

Now we have: (41/270) x (-3/4). This is the final multiplication step. Again, we multiply the numerators and the denominators:

(41/270) x (-3/4) = (41 x -3) / (270 x 4)

Multiplying the numbers, we get:

-123/1080

Can we simplify this fraction? Yes, we can! Both 123 and 1080 are divisible by 3:

(-123 ÷ 3) / (1080 ÷ 3) = -41/360

And there we have it! The final answer is -41/360. You did it!

Common Mistakes to Avoid

Math can be tricky, and it’s easy to slip up if you're not careful. Here are a few common mistakes people make when solving expressions like this:

  • Forgetting the Order of Operations: This is the biggest pitfall. Always remember PEMDAS/BODMAS. Multiply and divide before you add and subtract.
  • Sign Errors: Keep track of those negative signs! A single missed negative can throw off the entire calculation.
  • Fraction Simplification: Always simplify your fractions to their lowest terms. It makes the final answer cleaner and easier to work with.
  • Incorrectly Finding Common Denominators: When adding or subtracting fractions, make sure you find the least common denominator correctly. A wrong denominator will lead to an incorrect result.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in solving mathematical expressions.

Practice Makes Perfect: Try These Examples

Okay, now that we’ve walked through this example together, it’s time for you to put your skills to the test! Here are a couple of similar expressions for you to try. Remember to take it slow, follow the order of operations, and double-check your work. The more you practice, the better you'll get!

  1. (1/2 x 4/5 - 0.2) x (2/3)
  2. (-5/6 + 2/9) x (3/4)

Work through these problems step by step, and if you get stuck, revisit our example and see if you can apply the same principles. Math is like a puzzle; each step is a piece that fits together to reveal the solution. Keep practicing, and you’ll become a math whiz in no time!

Real-World Applications of Order of Operations

You might be thinking, “Okay, this is great, but when am I ever going to use this in real life?” Well, the order of operations isn't just some abstract math concept. It has practical applications in many areas, from everyday finances to advanced engineering.

Financial Planning

Let's say you're calculating your monthly expenses. You might need to add up your bills, subtract your expenses from your income, and then figure out how much you can save. The order in which you perform these operations matters! You need to calculate your total expenses before you can subtract them from your income. That's order of operations in action.

Cooking and Baking

Recipes often involve multiple steps and measurements. If a recipe says, “Mix 2 cups of flour with 1/2 cup of sugar, then add 1/4 cup of butter,” you need to follow those steps in the correct order to get the desired result. Otherwise, you might end up with a culinary disaster!

Computer Programming

In programming, the order of operations is crucial for writing correct code. Computers follow instructions in a specific sequence, and if you mess up the order, your program won't work as expected. Programmers use the order of operations to ensure their code runs smoothly and produces accurate results.

Engineering and Construction

Engineers and construction workers use math every day to design buildings, bridges, and other structures. They need to calculate loads, stresses, and dimensions, and they must follow the order of operations to ensure their calculations are accurate and the structures are safe.

These are just a few examples, but they illustrate how the order of operations is a fundamental concept that underlies many aspects of our lives. So, mastering this skill isn't just about acing math tests; it's about developing a valuable problem-solving tool that you can use in countless situations.

Conclusion: You've Got This!

So, there you have it! We've successfully solved the expression (-3/4 x 16/81 + 0.3) x (-3/4), and you've learned a ton along the way. Remember, the key to mastering math is to break down complex problems into smaller, manageable steps. Keep practicing, stay patient, and don't be afraid to ask for help when you need it.

Math might seem daunting at times, but with the right approach and a little bit of effort, you can conquer any mathematical challenge. You've got this, guys! Keep up the great work, and I’ll see you in the next math adventure!