Evaluating The Expression Three Less Than The Quotient Of Ten And A Number Increased By Six When N Equals 2

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Introduction

In this comprehensive article, we will delve into the process of evaluating the mathematical expression "three less than the quotient of ten and a number, increased by six" when the variable n is equal to 2. This problem combines several fundamental arithmetic operations, including division, subtraction, and addition, making it an excellent exercise for honing our algebraic skills. We will break down the expression step by step, ensuring clarity and understanding at each stage. Understanding the order of operations is crucial in solving this type of problem accurately. By the end of this article, you will not only be able to solve this specific problem but also gain a deeper understanding of how to approach similar algebraic expressions. Mastering these skills is essential for success in mathematics and various fields that require analytical thinking.

Understanding the Expression

The given expression, "three less than the quotient of ten and a number, increased by six," can be translated into a mathematical equation. Let's break it down piece by piece:

  • "The quotient of ten and a number" implies division. If we represent the number as n, this part of the expression can be written as 10 / n. Quotients are the result of division, and understanding this terminology is key to interpreting the expression correctly. This step involves translating words into mathematical symbols, a foundational skill in algebra. The ability to accurately represent verbal expressions mathematically is crucial for problem-solving.
  • "Three less than the quotient" means we subtract 3 from the result of the division. So, we have (10 / n) - 3. This subtraction follows the division operation according to the order of operations. Subtraction is the inverse operation of addition, and both are fundamental to arithmetic. Recognizing the correct order in which to perform these operations is essential for arriving at the correct answer.
  • "Increased by six" indicates that we add 6 to the previous result. Therefore, the complete expression is (10 / n) - 3 + 6. This final addition completes the mathematical representation of the verbal expression. Addition is a basic arithmetic operation, and its proper application is vital for solving mathematical problems. The entire expression now accurately reflects the original verbal statement, setting the stage for the next step: substitution.

Thus, the algebraic expression that represents the given statement is:

(10 / n) - 3 + 6

This representation is the foundation for our next step, which involves substituting the given value of n into the expression.

Substituting n = 2

Now that we have the algebraic expression (10 / n) - 3 + 6, we can substitute the given value of n, which is 2. This means we replace every instance of n in the expression with the number 2. Substitution is a fundamental algebraic technique that allows us to evaluate expressions for specific values of variables. By replacing variables with their numerical values, we transform algebraic expressions into arithmetic ones, which can then be simplified to obtain a numerical result.

Substituting n = 2 into the expression, we get:

(10 / 2) - 3 + 6

This new expression consists only of numbers and arithmetic operations. It is now ready for simplification using the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistency and accuracy in calculations. By adhering to these rules, we can avoid ambiguity and arrive at the correct solution.

Applying the Order of Operations

To evaluate the expression (10 / 2) - 3 + 6, we must follow the order of operations (PEMDAS/BODMAS). This ensures we arrive at the correct answer. Understanding and applying the order of operations is paramount in mathematics. It is the backbone of arithmetic and algebra, guiding us through complex calculations in a systematic way. Without it, mathematical expressions could be interpreted in multiple ways, leading to inconsistent results.

  1. Division: The first operation we perform is division. 10 divided by 2 equals 5. So, the expression becomes:

    5 - 3 + 6

    Division takes precedence over both subtraction and addition in the order of operations. Division is the inverse operation of multiplication, and both are performed before addition and subtraction. This hierarchy ensures that mathematical expressions are evaluated consistently and accurately.

  2. Subtraction and Addition: Now we have subtraction and addition. These operations are performed from left to right. Subtraction and addition are inverse operations and are performed at the same level of precedence in the order of operations. This means we work from left to right, performing whichever operation comes first.

    • First, we subtract 3 from 5, which equals 2:

      2 + 6

      Subtraction is a fundamental arithmetic operation that involves finding the difference between two numbers. It is the inverse of addition and is essential for solving various mathematical problems.

    • Then, we add 6 to 2, which equals 8:

      8

      Addition is a basic arithmetic operation that combines two or more numbers to find their sum. It is the foundation of many mathematical concepts and is used extensively in everyday life.

Therefore, the value of the expression (10 / 2) - 3 + 6 is 8.

The Final Answer

After substituting n = 2 into the expression "three less than the quotient of ten and a number, increased by six" and applying the order of operations, we have arrived at the solution. The solution is the final result obtained after performing all the necessary calculations. It represents the value of the expression for the given value of the variable. In this case, the solution is a numerical value, but in other contexts, it could be an algebraic expression or a set of values.

The value of the expression when n = 2 is 8.

Therefore, the correct answer is:

8

This result demonstrates the importance of accurately translating verbal expressions into mathematical equations and meticulously following the order of operations. Accuracy in mathematical calculations is paramount. Even a small error in applying the order of operations or in performing basic arithmetic can lead to a completely incorrect result. Therefore, it is crucial to double-check each step of the calculation to ensure that the final answer is accurate and reliable.

Conclusion

In this article, we have successfully evaluated the mathematical expression "three less than the quotient of ten and a number, increased by six" when n = 2. We began by translating the verbal expression into an algebraic equation, then substituted the given value of n, and finally applied the order of operations to simplify the expression and arrive at the solution. This process exemplifies the core principles of algebra, which involves translating real-world situations into mathematical models, manipulating these models to solve problems, and interpreting the results in the context of the original situation.

Key takeaways from this exercise include:

  • The importance of accurately translating verbal expressions into mathematical equations. Translation is a crucial step in mathematical problem-solving. It involves converting real-world scenarios or verbal descriptions into mathematical symbols and equations, which can then be manipulated and solved using algebraic techniques.
  • The necessity of following the order of operations (PEMDAS/BODMAS) to ensure accurate calculations. The order of operations is a fundamental rule in mathematics that dictates the sequence in which operations should be performed. Adhering to this rule ensures consistency and accuracy in mathematical calculations.
  • The significance of substitution in evaluating algebraic expressions for specific values of variables. Substitution is a powerful technique that allows us to determine the value of an expression for particular values of the variables. It is a cornerstone of algebra and is used extensively in various mathematical applications.

By mastering these skills, you will be well-equipped to tackle a wide range of algebraic problems. Mathematical proficiency is a valuable asset in various fields, including science, engineering, finance, and computer science. By developing a strong foundation in mathematics, you open up a world of opportunities and enhance your problem-solving abilities in both academic and professional settings.