Step-by-Step Guide To Solving 16 × (20 ÷ 4) + (57 – 12)

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Mathematical expressions can sometimes seem daunting, but breaking them down into smaller, manageable steps can make them much easier to solve. In this comprehensive guide, we'll walk you through the process of solving the expression 16 × (20 ÷ 4) + (57 – 12), explaining each step in detail. Whether you're a student learning the basics or just looking to brush up on your math skills, this guide will provide you with a clear and concise method for tackling similar problems. So, let's dive in and unravel this expression together!

Understanding the Order of Operations

Before we even think about crunching any numbers, it's super important to get our heads around the order of operations. Think of it as the golden rule of math – a set of instructions that tells us which parts of an expression to tackle first. You might have heard of the acronym PEMDAS, which is a handy way to remember the correct order:

  • Parentheses
  • Exponents
  • Multiplication and Division
  • Addition and Subtraction

PEMDAS essentially tells us to first deal with anything inside parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Ignoring this order can lead to some seriously wonky answers, so let's keep PEMDAS in mind as we solve our expression.

Breaking Down PEMDAS in Detail

Let's break down each part of PEMDAS a little further to ensure we're all on the same page. This understanding is crucial for accurately solving mathematical expressions, and it helps prevent common mistakes. So, grab your mental calculator, and let's dive in!

  1. Parentheses (P): When we see parentheses in an expression, they are like little VIP sections. We must deal with whatever is inside them first. This might involve performing addition, subtraction, multiplication, division, or even another set of operations. The key thing is to simplify the contents of the parentheses before moving on.
  2. Exponents (E): Exponents are the little numbers that hang out in the top right corner of a number, indicating how many times to multiply the base number by itself. For example, in 2^3 (2 cubed), the exponent is 3, meaning we multiply 2 by itself three times (2 × 2 × 2 = 8). If there are exponents in our expression, they come next in the order of operations.
  3. Multiplication and Division (MD): Multiplication and division are like close cousins; they have equal priority. When we encounter them in an expression, we work from left to right. So, if we have both multiplication and division, we perform whichever operation comes first as we read the expression from left to right.
  4. Addition and Subtraction (AS): Just like multiplication and division, addition and subtraction are also on the same level of priority. We handle them from left to right in the expression. This means that if addition comes before subtraction, we do the addition first, and vice versa.

By keeping PEMDAS in mind and understanding the nuances of each operation's priority, we can confidently approach any mathematical expression. Now, let's put this knowledge into action and solve our expression!

Step 1: Solving the Parentheses

Okay, guys, let's get down to business! Looking at our expression, 16 × (20 ÷ 4) + (57 – 12), the first thing that jumps out at us is the parentheses. According to PEMDAS, these are our first targets. We've got two sets of parentheses here: (20 ÷ 4) and (57 – 12).

Let's tackle the first one: (20 ÷ 4). This is a simple division problem. 20 divided by 4 is 5. So, we can replace (20 ÷ 4) with 5.

Now, let's move on to the second set of parentheses: (57 – 12). This is a subtraction problem. 57 minus 12 is 45. So, we can replace (57 – 12) with 45.

After completing this first step, our expression now looks like this: 16 × 5 + 45. We've successfully simplified the parentheses, and we're one step closer to the final answer!

Importance of Accurate Parentheses Solving

It's super important to get the parentheses right because messing up here can throw off the entire calculation. Imagine if we miscalculated (20 ÷ 4) or (57 – 12) – the subsequent steps would be based on incorrect numbers, leading to a wrong final answer. This is why PEMDAS puts parentheses at the top of the priority list; they're the foundation upon which the rest of the expression is built.

Moreover, parentheses can sometimes hide more complex operations within them. For instance, you might have nested parentheses like (2 + (3 × 4)), which requires solving the innermost parentheses first. This highlights the need for a systematic approach and careful attention to detail.

By diligently solving the parentheses first, we ensure that we're working with the correct intermediate values. This not only increases our chances of getting the right answer but also makes the rest of the calculation smoother and less prone to errors. So, always double-check your work within the parentheses before moving on – it's a small effort that can make a big difference!

Step 2: Multiplication

Alright, we've conquered the parentheses, and our expression is looking much simpler: 16 × 5 + 45. According to PEMDAS, the next operation we need to tackle is multiplication. We've got one multiplication operation here: 16 × 5.

Let's get this done. 16 multiplied by 5 is 80. So, we can replace 16 × 5 with 80. Our expression now transforms into 80 + 45. We're making progress, guys!

The Role of Multiplication in Mathematical Expressions

Multiplication is a fundamental operation in mathematics, and its proper execution is crucial for solving expressions accurately. It represents repeated addition and is often used to scale quantities or combine groups of items. In the context of PEMDAS, multiplication holds a higher priority than addition and subtraction, meaning we must perform all multiplication operations before moving on to these lower-priority operations.

In more complex expressions, multiplication can interact with other operations, such as exponents and division. For example, you might encounter expressions like 2 × (3^2 + 4), where you need to handle the exponentiation within the parentheses before multiplying. This highlights the importance of adhering to the order of operations and carefully considering the relationships between different operations.

By accurately performing multiplication, we ensure that we're combining quantities correctly and setting the stage for the final steps of the calculation. It's a building block that helps us construct the solution piece by piece. So, let's move on to the final step and complete our journey!

Step 3: Addition

We're almost there, guys! We've simplified our expression to 80 + 45. Now, all that's left to do is addition. This is the final step, and it's going to give us our answer.

So, let's add 80 and 45 together. 80 plus 45 equals 125. Therefore, the solution to our expression is 125!

The Significance of Addition as the Final Step

Addition is often the final operation in simplifying mathematical expressions because it combines the results of all previous operations into a single, cohesive answer. It represents the culmination of all the calculations we've performed, bringing together the individual components to form the complete solution.

In the context of PEMDAS, addition and subtraction are the lowest-priority operations, meaning they are performed last after parentheses, exponents, multiplication, and division have been addressed. This order ensures that we're combining the quantities in the correct sequence and arriving at the accurate final result.

Throughout the process of solving a mathematical expression, addition serves as the ultimate aggregator, pulling together the various intermediate values into a comprehensive answer. It's the final piece of the puzzle, and once we've performed the addition, we can confidently state that we've solved the expression. So, congratulations, we've reached the finish line!

Final Answer

Woo-hoo! We did it! By following the order of operations and breaking down the problem step by step, we've successfully solved the expression 16 × (20 ÷ 4) + (57 – 12). The final answer is 125.

Recap of the Solution Process

Let's quickly recap the steps we took to arrive at our solution. This will help reinforce the process and make it easier to remember for future problems:

  1. Parentheses: We started by solving the operations inside the parentheses: (20 ÷ 4) = 5 and (57 – 12) = 45. This simplified our expression to 16 × 5 + 45.
  2. Multiplication: Next, we performed the multiplication: 16 × 5 = 80. This further simplified our expression to 80 + 45.
  3. Addition: Finally, we added the remaining numbers: 80 + 45 = 125. This gave us our final answer.

By systematically working through each step according to PEMDAS, we were able to tackle a seemingly complex expression with confidence and accuracy.

Tips for Solving Mathematical Expressions

To help you master the art of solving mathematical expressions, here are a few additional tips to keep in mind:

  • Write it down: When solving expressions, especially longer ones, it's always a good idea to write down each step. This helps you keep track of your work and makes it easier to spot any mistakes.
  • Double-check: After each step, take a moment to double-check your calculations. This can save you from making simple errors that can throw off the entire solution.
  • Practice makes perfect: The more you practice solving mathematical expressions, the more comfortable and confident you'll become. So, don't be afraid to tackle new problems and challenge yourself.
  • Use resources: If you're struggling with a particular concept or type of expression, don't hesitate to use resources like textbooks, online tutorials, or even ask a friend or teacher for help.

With these tips and the step-by-step guide we've covered, you'll be well-equipped to solve all sorts of mathematical expressions. So, keep practicing, and happy calculating!