Select The Proper Placement For Parentheses To Speed Up The Addition For The Expression $4+6+5$.A. $(5+6)+4$ B. $ 4 + ( 6 + 5 ) 4+(6+5) 4 + ( 6 + 5 ) [/tex] C. $(5+4)+6$ D. $(4+6)+5$

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Introduction

In mathematics, the order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. One of the most common operations is addition, and parentheses play a crucial role in determining the order in which numbers are added. In this article, we will explore the concept of optimizing parentheses for efficient addition and apply it to the expression $4+6+5$. We will examine four different options for placing parentheses and determine which one results in the fastest addition.

Understanding the Order of Operations

The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Applying the Order of Operations to the Expression

Now that we understand the order of operations, let's apply it to the expression $4+6+5$. We can see that there are no parentheses, exponents, multiplication, or division operations in the expression. Therefore, we can simply add the numbers from left to right.

However, let's consider the options for placing parentheses:

A. $(5+6)+4$ B. $4+(6+5)$ C. $(5+4)+6$ D. $(4+6)+5$

Analyzing Option A

Let's start by analyzing option A: $(5+6)+4$. To evaluate this expression, we need to follow the order of operations:

  1. Evaluate the expression inside the parentheses: $(5+6) = 11$
  2. Add 4 to the result: $11 + 4 = 15$

Analyzing Option B

Now let's analyze option B: $4+(6+5)$. To evaluate this expression, we need to follow the order of operations:

  1. Evaluate the expression inside the parentheses: $(6+5) = 11$
  2. Add 4 to the result: $4 + 11 = 15$

Analyzing Option C

Next, let's analyze option C: $(5+4)+6$. To evaluate this expression, we need to follow the order of operations:

  1. Evaluate the expression inside the parentheses: $(5+4) = 9$
  2. Add 6 to the result: $9 + 6 = 15$

Analyzing Option D

Finally, let's analyze option D: $(4+6)+5$. To evaluate this expression, we need to follow the order of operations:

  1. Evaluate the expression inside the parentheses: $(4+6) = 10$
  2. Add 5 to the result: $10 + 5 = 15$

Conclusion

In conclusion, all four options result in the same final answer: 15. However, the order in which we add the numbers can affect the speed at which we arrive at the answer. By analyzing the options, we can see that option B: $4+(6+5)$ is the most efficient way to add the numbers, as it requires only two steps: evaluating the expression inside the parentheses and adding 4 to the result.

Recommendation

Based on our analysis, we recommend using option B: $4+(6+5)$ when adding the numbers in the expression $4+6+5$. This option requires the fewest number of steps and is therefore the most efficient way to arrive at the answer.

Final Answer

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed when there are multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Why is it important to follow the order of operations?

A: Following the order of operations is crucial to ensure that mathematical expressions are evaluated correctly. If the order of operations is not followed, the result of the expression may be incorrect.

Q: How can I optimize parentheses for efficient addition?

A: To optimize parentheses for efficient addition, you can follow these steps:

  1. Identify the numbers that need to be added.
  2. Determine the most efficient order in which to add the numbers.
  3. Use parentheses to group the numbers in the most efficient order.

Q: What is the most efficient way to add the numbers in the expression 4+6+5?

A: The most efficient way to add the numbers in the expression 4+6+5 is to use the option: $4+(6+5)$. This option requires only two steps: evaluating the expression inside the parentheses and adding 4 to the result.

Q: Can I use parentheses to make the expression 4+6+5 easier to evaluate?

A: Yes, you can use parentheses to make the expression 4+6+5 easier to evaluate. For example, you can use the option: $(4+6)+5$. This option requires only two steps: evaluating the expression inside the parentheses and adding 5 to the result.

Q: What is the difference between the options (5+6)+4 and 4+(6+5)?

A: The options (5+6)+4 and 4+(6+5) are two different ways to evaluate the expression 4+6+5. The option (5+6)+4 requires three steps: evaluating the expression inside the parentheses, adding 4 to the result, and then adding 5 to the result. The option 4+(6+5) requires only two steps: evaluating the expression inside the parentheses and adding 4 to the result.

Q: Can I use parentheses to make the expression 4+6+5 easier to evaluate for a large number of additions?

A: Yes, you can use parentheses to make the expression 4+6+5 easier to evaluate for a large number of additions. For example, if you need to add 4+6+5+7+8, you can use the option: $(4+6+5)+(7+8)$. This option requires only two steps: evaluating the expression inside the parentheses and adding the result to the next set of numbers.

Q: What is the final answer to the expression 4+6+5?

A: The final answer to the expression 4+6+5 is 15.