10 Solved Examples: Operations With Grouping Symbols

Hey guys! Ever stumbled upon math problems with parentheses, brackets, and braces and felt a bit lost? Don't worry, you're not alone! Operations with grouping symbols can seem tricky at first, but once you understand the order of operations, they become much easier to handle. This guide will walk you through 10 solved examples of operations with grouping symbols, making sure you grasp the concepts thoroughly. We'll break down each step, so you can confidently tackle any similar problem in the future. Let's dive in and conquer these mathematical challenges together!

Understanding Grouping Symbols

Before we jump into the examples, let's quickly recap what grouping symbols are and why they're important. Grouping symbols, such as parentheses (), brackets [], and braces {}, are used in mathematical expressions to indicate the order in which operations should be performed. They tell us which parts of the expression to calculate first, ensuring we get the correct answer. Think of them as road signs in a mathematical journey, guiding you through the right path. The standard order of operations, often remembered by the acronym PEMDAS (or BODMAS), is crucial here:

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

Grouping symbols are essential because they can change the entire outcome of an expression. For example, 2 + 3 * 4 would be calculated as 2 + 12 = 14 if we follow the standard order of operations (multiplication before addition). However, if we add parentheses like this: (2 + 3) * 4, we first calculate the expression inside the parentheses, giving us 5 * 4 = 20. See how different the results are? This highlights the significance of understanding and correctly applying grouping symbols.

Now that we've got the basics covered, let's move on to the exciting part: solving some examples! Each example will illustrate different scenarios and techniques, helping you build a solid understanding of operations with grouping symbols. So, grab your pen and paper, and let's get started!

Example 1: Simple Parentheses

Let's begin with a straightforward example to illustrate the basic principle of using parentheses. Consider the expression:

(5 + 3) * 2

The first step, as always, is to look for any grouping symbols. In this case, we have parentheses around 5 + 3. According to the order of operations (PEMDAS/BODMAS), we need to perform the operation inside the parentheses first. So, we add 5 and 3:

5 + 3 = 8

Now, we replace the expression inside the parentheses with the result, giving us:

8 * 2

Next, we perform the multiplication:

8 * 2 = 16

Therefore, the final answer is 16. This example highlights the fundamental rule of addressing parentheses first. It's a simple illustration, but it sets the stage for more complex problems. Always remember, whatever is inside the parentheses takes precedence. This ensures that we follow the correct order and arrive at the accurate solution. Mastering this basic step is crucial for tackling more intricate expressions involving multiple grouping symbols and operations. Practice with similar examples will solidify your understanding and build your confidence in handling parentheses.

Example 2: Multiple Parentheses

Now, let's tackle an example with multiple sets of parentheses to understand how to handle them. Consider this expression:

(2 * 4) + (10 / 2)

In this case, we have two sets of parentheses: (2 * 4) and (10 / 2). According to the order of operations, we need to evaluate each set of parentheses independently before performing the addition. Let's start with the first set:

2 * 4 = 8

Now, let's evaluate the second set of parentheses:

10 / 2 = 5

We replace the expressions inside the parentheses with their respective results:

8 + 5

Finally, we perform the addition:

8 + 5 = 13

Thus, the final answer is 13. This example illustrates how to handle multiple parentheses in an expression. The key is to treat each set of parentheses as a separate mini-problem. Evaluate the operations within each set independently, and then combine the results according to the remaining operations. This approach ensures that you maintain the correct order of operations and avoid common mistakes. When you encounter expressions with multiple parentheses, take a moment to identify each set and solve them one by one. This systematic approach will make the problem more manageable and increase your accuracy.

Example 3: Parentheses and Subtraction

Let's look at an example that combines parentheses with subtraction. This will help us see how different operations interact with grouping symbols. Consider the expression:

15 - (6 + 2)

Here, we have parentheses around 6 + 2. As always, we start by evaluating the expression inside the parentheses:

6 + 2 = 8

Now we substitute the result back into the original expression:

15 - 8

Next, we perform the subtraction:

15 - 8 = 7

So, the final answer is 7. This example highlights the importance of addressing the parentheses before any other operation outside of them. Even though subtraction comes before addition in the acronym PEMDAS/BODMAS, the parentheses dictate that we add 6 and 2 first. This principle is crucial for correctly solving expressions. When you see parentheses, your immediate focus should be on what's inside them. Completing the operations within the parentheses simplifies the expression and sets the stage for the remaining calculations. Practicing with examples like this will help you develop a strong understanding of how parentheses interact with different operations.

Example 4: Nested Parentheses

Now let's step it up a notch and explore nested parentheses, which means parentheses inside other parentheses. This might seem a bit daunting, but the principle is the same: work from the innermost grouping symbols outwards. Consider the expression:

2 * (8 - (3 + 1))

In this case, we have a set of parentheses (3 + 1) inside another set (8 - (3 + 1)). We start with the innermost parentheses:

3 + 1 = 4

Now we replace (3 + 1) with its result, giving us:

2 * (8 - 4)

Next, we evaluate the remaining parentheses:

8 - 4 = 4

Finally, we perform the multiplication:

2 * 4 = 8

Therefore, the final answer is 8. When dealing with nested parentheses, the key is to tackle them layer by layer. Start with the innermost set, solve it, and then work your way outwards. This systematic approach prevents confusion and ensures that you follow the correct order of operations. Nested parentheses might appear complex initially, but with practice, you'll become comfortable unraveling them. Each layer you solve simplifies the expression, bringing you closer to the final answer. Remember to take it one step at a time, and you'll master this concept in no time!

Example 5: Brackets and Parentheses

Let's introduce another type of grouping symbol: brackets []. Brackets are used similarly to parentheses, often to add another layer of grouping and make the expression easier to read. Consider the expression:

4 + [3 * (2 + 1)]

In this example, we have both brackets and parentheses. According to the order of operations, we start with the innermost grouping symbol, which is the parentheses:

2 + 1 = 3

Now we replace (2 + 1) with its result:

4 + [3 * 3]

Next, we evaluate the expression inside the brackets:

3 * 3 = 9

Finally, we perform the addition:

4 + 9 = 13

So, the final answer is 13. This example shows that brackets function just like parentheses in terms of the order of operations. They simply provide a visual cue to help organize complex expressions. When you encounter both brackets and parentheses, remember to start with the innermost grouping symbol and work your way outwards. This ensures that you follow the correct order and avoid errors. Brackets are particularly useful when you have multiple levels of grouping, as they can help distinguish between different parts of the expression. Practice with these types of examples will help you become comfortable with various grouping symbols.

Example 6: Braces, Brackets, and Parentheses

Now, let's add braces {} to the mix! Braces are another type of grouping symbol, often used to further clarify the order of operations in complex expressions. Consider this example:

10 - {2 + [3 * (4 - 1)]}

Here, we have braces, brackets, and parentheses. Remember, we start with the innermost grouping symbol, which is the parentheses:

4 - 1 = 3

Replace (4 - 1) with its result:

10 - {2 + [3 * 3]}

Next, we evaluate the expression inside the brackets:

3 * 3 = 9

Replace [3 * 3] with its result:

10 - {2 + 9}

Now, we evaluate the expression inside the braces:

2 + 9 = 11

Finally, we perform the subtraction:

10 - 11 = -1

So, the final answer is -1. This example demonstrates how to handle multiple levels of grouping symbols: braces, brackets, and parentheses. The key is to always start with the innermost set and work your way outwards, following the order of operations. Braces, like brackets, help to visually organize complex expressions, making them easier to understand and solve. Don't be intimidated by multiple grouping symbols; just remember the systematic approach, and you'll be able to tackle any problem.

Example 7: Multiplication and Division within Grouping Symbols

Let's examine an example that includes multiplication and division within grouping symbols. This will reinforce the importance of following the correct order of operations within the groups. Consider the expression:

(12 / 3) * (2 * 4)

We have two sets of parentheses here. Let's start with the first set:

12 / 3 = 4

Now, let's evaluate the second set of parentheses:

2 * 4 = 8

Replace the expressions inside the parentheses with their results:

4 * 8

Finally, we perform the multiplication:

4 * 8 = 32

Thus, the final answer is 32. This example emphasizes that we perform multiplication and division within grouping symbols from left to right, following the PEMDAS/BODMAS rule. Even though we have two separate sets of parentheses, we solve the operations within each set before combining the results. This ensures that we maintain the correct order and arrive at the accurate solution. When dealing with multiplication and division within grouping symbols, take a moment to identify each operation and perform them in the correct sequence. This systematic approach will minimize errors and build your confidence.

Example 8: Addition and Subtraction within Grouping Symbols

Now, let's consider an example with addition and subtraction within grouping symbols. This will further solidify our understanding of how these operations interact with grouping. Consider the expression:

[15 - (5 + 2)] + 8

We have brackets and parentheses in this example. We start with the innermost grouping symbol, which is the parentheses:

5 + 2 = 7

Replace (5 + 2) with its result:

[15 - 7] + 8

Next, we evaluate the expression inside the brackets:

15 - 7 = 8

Replace [15 - 7] with its result:

8 + 8

Finally, we perform the addition:

8 + 8 = 16

So, the final answer is 16. This example highlights the importance of performing addition and subtraction within grouping symbols from left to right, just like we do with multiplication and division. The parentheses dictate that we add 5 and 2 first, and then we subtract the result from 15 within the brackets. This systematic approach ensures that we follow the correct order of operations and avoid common mistakes. When you encounter addition and subtraction within grouping symbols, remember to prioritize the operations within the innermost groups and work your way outwards.

Example 9: Combining All Operations

Let's tackle a more complex example that combines all the operations we've discussed so far: addition, subtraction, multiplication, and division, along with grouping symbols. This will give you a comprehensive understanding of how to apply the order of operations. Consider the expression:

2 * {10 - [6 / (1 + 2)]}

We have braces, brackets, and parentheses in this expression. We start with the innermost grouping symbol, the parentheses:

1 + 2 = 3

Replace (1 + 2) with its result:

2 * {10 - [6 / 3]}

Next, we evaluate the expression inside the brackets:

6 / 3 = 2

Replace [6 / 3] with its result:

2 * {10 - 2}

Now, we evaluate the expression inside the braces:

10 - 2 = 8

Finally, we perform the multiplication:

2 * 8 = 16

So, the final answer is 16. This example demonstrates how to handle a complex expression with multiple operations and grouping symbols. The key is to break it down step by step, always starting with the innermost grouping symbol and following the order of operations (PEMDAS/BODMAS). By systematically addressing each operation in the correct sequence, you can confidently solve even the most challenging problems. Practice with examples like this will help you develop a strong foundation in handling operations with grouping symbols.

Example 10: A Real-World Application

Let's look at a real-world example to see how operations with grouping symbols can be used in practical situations. Imagine you're planning a trip and need to calculate the total cost. You have the following expenses:

• Transportation: 2 * (50 + 25) dollars (2 tickets, each costing $50 for the main fare and $25 for extra fees)
• Accommodation: $100
• Food: 3 * 20 dollars (3 meals, each costing $20)

To find the total cost, we need to add up all these expenses. Let's write the expression:

2 * (50 + 25) + 100 + 3 * 20

First, we evaluate the parentheses:

50 + 25 = 75

Replace (50 + 25) with its result:

2 * 75 + 100 + 3 * 20

Next, we perform the multiplications from left to right:

2 * 75 = 150

3 * 20 = 60

Now the expression looks like this:

150 + 100 + 60

Finally, we perform the additions:

150 + 100 = 250

250 + 60 = 310

So, the total cost of the trip is $310. This example illustrates how operations with grouping symbols can be used to solve real-world problems. By understanding the order of operations, you can break down complex calculations into manageable steps and arrive at the correct answer. Real-world applications often involve multiple operations and grouping symbols, so mastering these concepts is essential for practical problem-solving.

Conclusion

Alright guys, we've covered a lot in this guide! We've explored 10 solved examples of operations with grouping symbols, starting from simple parentheses to complex expressions with braces, brackets, and nested parentheses. You've learned how to apply the order of operations (PEMDAS/BODMAS) systematically and how to tackle different scenarios. Remember, the key is to always start with the innermost grouping symbol and work your way outwards, following the correct order of operations. By understanding and practicing these concepts, you'll be well-equipped to handle any mathematical expression involving grouping symbols.

Don't be afraid to tackle complex problems; break them down into smaller, manageable steps, and you'll find that they become much easier to solve. Practice is crucial, so keep working on similar examples to solidify your understanding. The more you practice, the more confident you'll become in handling operations with grouping symbols. And remember, math is like a puzzle – the more you solve, the better you get at it! Keep up the great work, and happy calculating!