Evaluating algebraic expressions is a fundamental skill in mathematics. This article will guide you through the process of evaluating the expression 0.1m + 8 - 12n when m = 30 and n = 1/4. We will break down each step, explaining the order of operations and how to substitute the given values into the expression. By the end of this guide, you'll have a clear understanding of how to solve similar problems and enhance your algebraic manipulation skills.
Understanding the Expression
The expression 0.1m + 8 - 12n is an algebraic expression involving two variables, m and n. To evaluate this expression, we need to substitute the given values for m and n and then simplify the expression using the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Understanding the structure of the expression is crucial before we begin substituting values. The expression consists of three terms: 0.1m, 8, and -12n. The first term involves multiplication between a decimal (0.1) and the variable m. The second term is a constant, 8. The third term involves multiplication between a constant (-12) and the variable n. Recognizing these components helps us to approach the evaluation methodically and accurately. We must pay close attention to the order in which operations are performed to avoid common mistakes. Mastering the evaluation of such expressions builds a strong foundation for more complex algebraic concepts.
Step-by-Step Evaluation
To accurately evaluate the expression 0.1m + 8 - 12n for m = 30 and n = 1/4, we follow a step-by-step approach: First, substitute the given values of m and n into the expression. Replace m with 30 and n with 1/4. This yields the expression: 0.1(30) + 8 - 12(1/4). Next, perform the multiplication operations as per the order of operations (PEMDAS/BODMAS). Multiply 0.1 by 30, which equals 3. Then, multiply 12 by 1/4, which equals 3. Now our expression looks like this: 3 + 8 - 3. Finally, perform the addition and subtraction operations from left to right. Add 3 and 8 to get 11. Then, subtract 3 from 11, which equals 8. Therefore, the value of the expression 0.1m + 8 - 12n when m = 30 and n = 1/4 is 8. This methodical approach ensures accuracy and clarity in solving algebraic expressions. Each step is crucial, and understanding the order of operations is key to arriving at the correct answer. By following these steps diligently, one can confidently evaluate similar expressions in the future.
Detailed Substitution
The initial step in evaluating an algebraic expression is accurate substitution. For the expression 0.1m + 8 - 12n with m = 30 and n = 1/4, we replace each variable with its corresponding value. Substituting m with 30 transforms the first term, 0.1m, into 0.1 * 30. Similarly, substituting n with 1/4 changes the third term, -12n, into -12 * (1/4). The entire expression now reads 0.1 * 30 + 8 - 12 * (1/4). This substitution step is critical because any error here will propagate through the rest of the calculation. It’s important to double-check that each variable has been replaced correctly. After the substitution, the expression is ready for simplification according to the order of operations. The multiplication operations will be performed before addition and subtraction. Proper substitution not only sets the stage for correct evaluation but also highlights the importance of precision in algebraic manipulations. This meticulous approach ensures that the subsequent steps are based on accurate information, leading to a reliable final result. The clarity and accuracy of the substitution are foundational for solving algebraic problems effectively.
Performing Multiplication
After the substitution, the next crucial step in evaluating the expression 0.1(30) + 8 - 12(1/4) involves performing the multiplication operations. According to the order of operations (PEMDAS/BODMAS), multiplication takes precedence over addition and subtraction. First, we multiply 0.1 by 30. This calculation yields 3. Next, we multiply 12 by 1/4. This can be thought of as dividing 12 by 4, which also results in 3. It’s essential to manage the signs correctly; in this case, the term is -12 * (1/4), so the result is -3. After performing these multiplication operations, the expression simplifies to 3 + 8 - 3. This step significantly reduces the complexity of the expression, making it easier to perform the remaining addition and subtraction. Ensuring accurate multiplication is vital because it directly impacts the final result. Any error in this step will lead to an incorrect evaluation of the original expression. Therefore, careful attention to detail and a clear understanding of multiplication principles are necessary for successful algebraic problem-solving. The correct execution of multiplication sets the stage for the final steps of addition and subtraction.
Addition and Subtraction
With the multiplication steps completed, we now focus on the addition and subtraction operations in the simplified expression 3 + 8 - 3. Following the order of operations, addition and subtraction are performed from left to right. First, we add 3 and 8, which equals 11. This transforms the expression to 11 - 3. Next, we subtract 3 from 11, resulting in 8. Thus, the final value of the expression 0.1m + 8 - 12n when m = 30 and n = 1/4 is 8. This final step demonstrates how a series of operations, each performed in the correct order, leads to the solution. It’s important to note that performing addition and subtraction in the wrong order could lead to a different, incorrect result. For instance, subtracting 3 from 8 first and then adding 3 would yield 8, which, in this specific case, happens to be the correct answer but is a result of a flawed process. Consistent adherence to the order of operations ensures accuracy across various algebraic problems. The successful completion of addition and subtraction brings the evaluation to its conclusion, providing the final numerical answer.
Final Result and Conclusion
In conclusion, after meticulously following the steps of substitution, multiplication, addition, and subtraction, we have evaluated the expression 0.1m + 8 - 12n when m = 30 and n = 1/4. The final result is 8. This process highlights the importance of understanding and applying the order of operations (PEMDAS/BODMAS) to ensure accurate calculations. Each step, from the initial substitution to the final arithmetic, plays a crucial role in arriving at the correct answer. Mistakes at any stage can propagate through the calculation, leading to an incorrect result. The detailed breakdown provided here offers a clear methodology for solving similar algebraic problems. By mastering these fundamental skills, one can confidently tackle more complex mathematical challenges. Evaluating expressions is a foundational concept in algebra, and proficiency in this area is essential for further studies in mathematics. The result, 8, is not just a number; it represents the value of the expression under the given conditions, demonstrating the power and precision of algebraic manipulation.