Hey there, math enthusiasts! Today, we're diving into a problem that might seem a bit daunting at first glance: simplifying the expression 8.3 - 4 1/5 (6.1 + 1/2). But don't worry, we'll break it down step by step, focusing on the crucial concept of the order of operations. Understanding this order is like having a secret key to unlocking any mathematical puzzle. So, let's get started and discover which operations we need to tackle first, second, and last to arrive at the correct answer. We'll make sure to explain everything in a clear and friendly way, so even if you're new to this, you'll be simplifying expressions like a pro in no time!
The Order of Operations: A Mathematical Roadmap
Before we jump into our specific problem, let's quickly review the order of operations, often remembered by the acronym PEMDAS or BODMAS. Think of it as a roadmap that guides us through any mathematical expression, ensuring we perform the operations in the correct sequence.
- Parentheses (or Brackets): This is our starting point. We always simplify expressions inside parentheses or brackets first. It's like dealing with the inner workings of a machine before tackling the outer parts.
- Exponents (or Orders): Next up are exponents, those little numbers that tell us how many times to multiply a number by itself. Think of it as raising a number to a certain power.
- Multiplication and Division: These operations have equal priority, so we perform them from left to right, just like reading a sentence. It's like choosing which path to take first when they are side by side.
- Addition and Subtraction: Similar to multiplication and division, addition and subtraction also have equal priority and are performed from left to right. This is the final step in our journey, bringing everything together to reach the solution.
Now that we have our roadmap in mind, let's apply it to our expression and see how it helps us navigate to the correct answer. Remember, following the order of operations is key to avoiding errors and ensuring we simplify expressions accurately. Think of it as the golden rule of mathematics!
First Stop: Parentheses – The Inner Sanctum of the Expression
Alright, let's get our hands dirty with the expression 8.3 - 4 1/5 (6.1 + 1/2). According to PEMDAS/BODMAS, the first thing we need to address is the parentheses. Think of it as entering the inner sanctum of our mathematical temple. Inside the parentheses, we have (6.1 + 1/2). This is where our addition skills come into play.
Before we can add these two numbers, we need to make sure they're in the same format. We have a decimal (6.1) and a fraction (1/2). To make things easier, let's convert the fraction 1/2 into its decimal equivalent, which is 0.5. Now we have (6.1 + 0.5). This conversion is crucial because it allows us to perform the addition seamlessly. It's like speaking the same language so we can have a smooth conversation.
Now, the addition is straightforward: 6.1 + 0.5 = 6.6. So, we've successfully simplified the expression inside the parentheses to 6.6. It's like solving a mini-puzzle within the bigger puzzle. This might seem like a small step, but it's a crucial one in simplifying the entire expression. We've conquered the first hurdle, and we're well on our way to the final answer!
Second in Line: Multiplication – Connecting the Pieces
Great job, guys! We've successfully tackled the parentheses in our expression 8.3 - 4 1/5 (6.1 + 1/2), simplifying it to 8.3 - 4 1/5 (6.6). Now, according to the order of operations, the next operation we need to perform is multiplication. Notice that the term 4 1/5 (6.6) implies multiplication. It's like an invisible multiplication sign is sitting right there, waiting for us to unleash its power.
Before we can multiply, we need to convert the mixed number 4 1/5 into an improper fraction. Remember how to do that? We multiply the whole number (4) by the denominator (5) and add the numerator (1), then keep the same denominator. So, 4 1/5 becomes (4 * 5 + 1) / 5 = 21/5. Converting to an improper fraction is like transforming a clunky shape into a streamlined one that fits perfectly into our equation.
Now we have 8.3 - (21/5) (6.6). Let's make things even simpler by converting 6.6 into a fraction as well. 6.6 is the same as 66/10. So, our expression now looks like 8.3 - (21/5) (66/10). Now we are ready to multiply the fractions. When multiplying fractions, we multiply the numerators (top numbers) and the denominators (bottom numbers). So, (21/5) * (66/10) = (21 * 66) / (5 * 10) = 1386/50. This multiplication is like connecting two important pieces of the puzzle, bringing us closer to the final solution.
We can simplify the fraction 1386/50 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 693/25. Now, let's convert this improper fraction back into a decimal to make it easier to work with in our expression. 693/25 equals 27.72. So, our expression now looks like 8.3 - 27.72. We've successfully performed the multiplication, and we're one step closer to simplifying the entire expression! It might seem like a lot of steps, but each one is crucial in ensuring we arrive at the correct answer.
The Final Step: Subtraction – Unveiling the Solution
Alright, mathletes, we've reached the final stretch! Our expression 8.3 - 4 1/5 (6.1 + 1/2) has been simplified to 8.3 - 27.72. According to the order of operations, the last operation we need to perform is subtraction. This is the moment where we bring everything together and reveal the final answer. It's like the grand finale of a fireworks display!
Subtracting 27.72 from 8.3 might seem a bit tricky at first, because we're subtracting a larger number from a smaller one. This means our answer will be a negative number. Think of it as owing more money than you have – you'll end up with a debt. To perform the subtraction, we can think of it as finding the difference between 27.72 and 8.3, and then adding a negative sign.
So, let's subtract 8.3 from 27.72: 27.72 - 8.3 = 19.42. Now, remember that we're subtracting a larger number from a smaller one, so our final answer will be negative. Therefore, 8.3 - 27.72 = -19.42. And there we have it! We've successfully simplified the expression and arrived at the solution: -19.42. It's like reaching the summit of a challenging mountain, the view is definitely worth the climb!
In Conclusion: Mastering the Order of Operations
So, to recap, when simplifying the expression 8.3 - 4 1/5 (6.1 + 1/2), the operations were performed in the following order:
- First: The addition within the parentheses: (6.1 + 1/2)
- Second: The multiplication: 4 1/5 (6.6)
- Last: The subtraction: 8.3 - 27.72
By following the order of operations, we were able to break down a seemingly complex problem into manageable steps. Remember, PEMDAS/BODMAS is your friend in the world of mathematics. It's the secret code that unlocks the solution to any expression. Keep practicing, and you'll become a master of simplifying expressions in no time! You've got this, guys!
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