Solving Math Expressions: A Step-by-Step Guide

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Solving Math Expressions: A Step-by-Step Guide

Hey guys! Ever feel like you're staring at a bunch of numbers and symbols that just don't make sense? You're not alone! Math expressions can seem intimidating, but with a little breakdown and the right steps, you can solve them like a pro. This guide will walk you through evaluating some tricky expressions, so let's dive in!

Understanding Order of Operations

Before we jump into the expressions themselves, let's quickly recap the order of operations. This is the golden rule of math that tells us which operations to perform first. Remember the acronym PEMDAS:

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

Think of PEMDAS as your roadmap for solving any mathematical expression. It ensures everyone arrives at the same correct answer!

Why Order of Operations Matters

The order of operations isn't just some arbitrary rule – it's crucial for getting the correct answer. Imagine if we didn't have a standard order; we could interpret the same expression in multiple ways and end up with different results. This would be chaos! By following PEMDAS, we ensure clarity and consistency in mathematical calculations. For instance, consider the expression 2 + 3 * 4. If we perform the addition first, we get 5 * 4 = 20. But if we follow PEMDAS and do the multiplication first, we get 2 + 12 = 14. The second answer is the correct one, highlighting the importance of the order of operations.

Real-World Applications of PEMDAS

The order of operations isn't just confined to textbooks and classrooms. It's a fundamental concept that applies to various real-world situations, especially in fields like computer programming, engineering, and finance. In programming, for example, compilers and interpreters use PEMDAS to evaluate expressions in code. Engineers use it when designing structures and calculating loads, while financial analysts rely on it to determine investment returns and manage finances. Even in everyday life, PEMDAS comes into play when calculating discounts, figuring out cooking times, or budgeting expenses. Understanding and applying the order of operations ensures accurate results and prevents costly mistakes in these contexts. Therefore, mastering PEMDAS is not just about solving equations; it's about developing a critical skill that is applicable across a wide range of disciplines and daily activities.

Evaluating the Expressions

Now, let's tackle those expressions one by one, breaking them down step by step. We'll be sure to follow PEMDAS, so we don't miss a beat!

Expression a: (12 + 3) / 4 - 5% of 2

Okay, let's get started with the first expression. We'll go through each step carefully to make sure we understand what's happening. Remember, PEMDAS is our friend!

  1. Parentheses: First up, we have (12 + 3). Adding those together gives us 15. So, the expression now looks like this: 15 / 4 - 5% of 2.
  2. Percentage: Next, let's deal with 5% of 2. To find a percentage of a number, we convert the percentage to a decimal and multiply. So, 5% becomes 0.05. Now we multiply: 0.05 * 2 = 0.1. Our expression is now: 15 / 4 - 0.1.
  3. Division: Now we handle the division: 15 / 4 = 3.75. So, the expression becomes: 3.75 - 0.1.
  4. Subtraction: Finally, we subtract: 3.75 - 0.1 = 3.65.

So, the answer to expression a) is 3.65. See? Not so scary when we break it down!

Expression b: (32 + 15) - 12 / 6 + 2 * (5 - 7% - 3)

Expression b looks a bit more complex, but don't worry, we'll handle it the same way we did the first one – step by step, following PEMDAS. Let's do this!

  1. Inner Parentheses: We have two sets of parentheses here. Let's start with the inner one: (5 - 7% - 3). First, we need to calculate 7% of something. But wait! There's no number after the percentage sign. It seems like there might be a typo or missing information here. For the sake of demonstrating the process, let's assume the 7% was intended to be 7% of 5. So, 7% of 5 is 0.07 * 5 = 0.35. Now we have (5 - 0.35 - 3). Subtracting, we get 5 - 0.35 - 3 = 1.65. So, the expression now looks like this: (32 + 15) - 12 / 6 + 2 * 1.65.
  2. Outer Parentheses: Now let's handle the outer parentheses: (32 + 15). Adding these gives us 47. Our expression becomes: 47 - 12 / 6 + 2 * 1.65.
  3. Division: Next up is division: 12 / 6 = 2. Now we have: 47 - 2 + 2 * 1.65.
  4. Multiplication: Let's do the multiplication: 2 * 1.65 = 3.3. Our expression is now: 47 - 2 + 3.3.
  5. Subtraction and Addition (from left to right): Finally, we do subtraction and addition from left to right. 47 - 2 = 45, and then 45 + 3.3 = 48.3.

So, the answer to expression b) is 48.3 (assuming 7% was meant to be 7% of 5). Remember, always double-check your work and the original problem for any potential typos or missing information!

Expression c: (231 / 10) % 10 + 10 * (231 / 100) % 10

Alright, let's tackle expression c! This one introduces the modulo operator (%), which gives us the remainder of a division. Don't let it intimidate you; we'll break it down just like the others.

  1. Parentheses (Division): We have two sets of parentheses, each containing a division. Let's start with the first one: (231 / 10). This equals 23.1. The expression now looks like this: 23.1 % 10 + 10 * (231 / 100) % 10.
  2. Parentheses (Division): Now the second set: (231 / 100). This equals 2.31. The expression is now: 23.1 % 10 + 10 * 2.31 % 10.
  3. Modulo (First Instance): Let's handle the first modulo operation: 23.1 % 10. This means we want the remainder when 23.1 is divided by 10. Since 10 goes into 23.1 twice with a remainder, we calculate 23.1 - (2 * 10) = 3.1. So, 23.1 % 10 = 3.1. Our expression is now: 3.1 + 10 * 2.31 % 10.
  4. Modulo (Second Instance): Next modulo operation: 2.31 % 10. The remainder when 2.31 is divided by 10 is simply 2.31 because 10 doesn't go into 2.31 at all. So, 2.31 % 10 = 2.31. The expression becomes: 3.1 + 10 * 2.31.
  5. Multiplication: Now for the multiplication: 10 * 2.31 = 23.1. The expression is now: 3.1 + 23.1.
  6. Addition: Finally, we add: 3.1 + 23.1 = 26.2.

So, the answer to expression c) is 26.2. See how the modulo operator works? It's all about finding that remainder!

Expression d: (54 % 70) + 70 + 54 / 10 - (231 / 100) % 10

Okay, guys, let's wrap things up with expression d! This one combines a few things we've already seen, so we're in good shape. Remember, PEMDAS all the way!

  1. Parentheses (Modulo): Let's start with the parentheses. Inside, we have a modulo operation: (54 % 70). The remainder when 54 is divided by 70 is just 54 because 70 doesn't go into 54 at all. So, (54 % 70) = 54. The expression now looks like this: 54 + 70 + 54 / 10 - (231 / 100) % 10.
  2. Parentheses (Division): We have another set of parentheses with a division inside: (231 / 100). This equals 2.31. The expression is now: 54 + 70 + 54 / 10 - 2.31 % 10.
  3. Division: Now let's do the division: 54 / 10 = 5.4. The expression becomes: 54 + 70 + 5.4 - 2.31 % 10.
  4. Modulo: Next up is the modulo operation: 2.31 % 10. The remainder when 2.31 is divided by 10 is simply 2.31. So, 2.31 % 10 = 2.31. The expression is now: 54 + 70 + 5.4 - 2.31.
  5. Addition and Subtraction (from left to right): Finally, we perform the addition and subtraction from left to right. 54 + 70 = 124. Then, 124 + 5.4 = 129.4. And last, 129.4 - 2.31 = 127.09.

So, the answer to expression d) is 127.09. We made it! We tackled a complex expression by breaking it down into manageable steps.

Key Takeaways

  • PEMDAS is Your Best Friend: Always follow the order of operations (PEMDAS) to ensure accurate results.
  • Break It Down: Complex expressions become much easier when you break them down into smaller steps.
  • Double-Check: Always double-check your work and the original problem for any typos or missing information.
  • Practice Makes Perfect: The more you practice, the more comfortable you'll become with evaluating expressions.

Conclusion

So there you have it! We've walked through evaluating some math expressions step by step. Remember, the key is to understand the order of operations and break down complex problems into smaller, more manageable steps. Keep practicing, and you'll be solving even the trickiest expressions in no time! You got this!