Evaluate 2.7(10) − 41/110 Expression A Step-by-Step Solution
The given expression is 2.7(10) − 41/110. This expression involves basic arithmetic operations such as multiplication and subtraction. The main goal here is to correctly follow the order of operations, often remembered by the acronym PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). To accurately find the value, each part of the expression should be simplified step-by-step, ensuring no arithmetic errors are made. Proper understanding and execution of these steps are vital in arriving at the correct final answer. This question tests not only the arithmetic skills but also the ability to adhere to the fundamental rules of mathematical calculations.
Step-by-Step Breakdown
The first part of the expression to address is 2.7 multiplied by 10. When multiplying a decimal by 10, the decimal point moves one place to the right. Thus, 2.7 multiplied by 10 equals 27. This operation simplifies the first term significantly and sets the stage for the next operation, which involves dealing with the fraction. The next part of the expression is the fraction 41/110. To subtract this fraction from 27, it is beneficial to convert 27 into a fraction with the same denominator as 41/110. This preparatory step ensures accurate subtraction. To achieve this, 27 is multiplied by 110/110, converting it into a fraction that can be easily subtracted from 41/110. This conversion highlights the importance of understanding fractional arithmetic, a crucial component of mathematical operations.
Following the multiplication, the expression now looks like 27 − 41/110. Converting 27 to a fraction with a denominator of 110 involves multiplying 27 by 110. The product is 2970, making the expression 2970/110 − 41/110. This step transforms the original expression into one involving just fractions, which simplifies the subsequent subtraction operation. Understanding the conversion of whole numbers into fractions with common denominators is a fundamental concept in arithmetic. After this conversion, the expression is set for the final subtraction, which will yield the ultimate answer. The process underscores the significance of mastering fraction manipulation in solving mathematical problems.
Performing the Subtraction
With both terms now having a common denominator, subtraction can proceed. Subtracting 41 from 2970 results in 2929. Therefore, the expression simplifies to 2929/110. This fraction represents the final value of the expression. To provide a more understandable result, this fraction can be converted into a mixed number or a decimal. Converting it to a decimal involves dividing 2929 by 110. The result is approximately 26.627. This decimal form provides an alternative representation of the answer, which might be more intuitive for some users. The ability to convert between fractions and decimals is a crucial skill in mathematics, providing flexibility in representing numerical values. The final answer, whether expressed as a fraction or a decimal, completes the solution to the original problem.
Final Answer and Interpretation
The final value of the expression 2.7(10) − 41/110 is 2929/110, or approximately 26.627 when expressed as a decimal. This result is achieved by meticulously following the order of operations, converting numbers into appropriate forms, and performing the arithmetic accurately. The solution showcases the interconnectedness of various mathematical concepts, such as decimal multiplication, fraction manipulation, and the order of operations. It also underscores the necessity of precision in mathematical calculations, as each step builds upon the previous one. Understanding how to solve such expressions is essential in various fields, ranging from academic studies to practical applications in everyday life. The final answer not only solves the immediate problem but also reinforces a broader understanding of mathematical principles.
When solving mathematical expressions, errors can arise from several sources. In the context of the expression 2.7(10) − 41/110, a common mistake is not adhering to the correct order of operations. For instance, some might subtract 41/110 from 2.7 first and then multiply by 10, leading to an incorrect result. The order of operations dictates that multiplication should be done before subtraction in this case. Another potential error is in the conversion of decimal and fractional forms. If the conversion isn't done correctly, it can lead to significant discrepancies in the final answer. For example, miscalculating 2.7 multiplied by 10 or improperly converting 41/110 to a decimal can affect the outcome. Understanding the principles of decimals and fractions is crucial for avoiding such mistakes. Arithmetic errors during multiplication, subtraction, or division are also common pitfalls. Careless calculations can easily lead to an incorrect answer, highlighting the importance of double-checking each step. Effective error analysis and attention to detail can significantly improve accuracy in mathematical problem-solving.
Strategies for Avoiding Mistakes
To minimize errors in mathematical calculations, implementing specific strategies is beneficial. One effective method is to write down each step clearly and methodically. This approach aids in tracking the calculations and makes it easier to identify any mistakes. It's also helpful to double-check each operation as it is performed, ensuring that no arithmetic errors are made. Another useful strategy is to estimate the answer before performing the detailed calculations. This estimation serves as a benchmark to compare the final answer against, helping to catch any major discrepancies. Understanding the fundamental mathematical principles is also crucial. A solid grasp of concepts like the order of operations, decimal and fraction manipulation, and basic arithmetic operations is essential for accurate problem-solving. Additionally, practicing similar problems can significantly improve proficiency and reduce the likelihood of errors. Regular practice reinforces the understanding of concepts and builds confidence in handling mathematical expressions. By employing these strategies, one can enhance their ability to solve mathematical problems accurately and efficiently.
Alternative Solution Methods
While the step-by-step method described earlier is effective, there are alternative approaches to solving the expression 2.7(10) − 41/110. One such method involves converting the entire expression into decimal form before performing any operations. The fraction 41/110 can be converted to a decimal by dividing 41 by 110, which yields approximately 0.3727. The expression then becomes 2.7(10) − 0.3727. Next, multiplying 2.7 by 10 gives 27, so the expression simplifies to 27 − 0.3727. Subtracting 0.3727 from 27 results in approximately 26.6273, which is very close to the fraction equivalent of 2929/110. This alternative method demonstrates the flexibility in mathematical problem-solving, where different approaches can lead to the same correct answer. It also highlights the usefulness of decimal representation in simplifying calculations. Another approach could involve using a calculator to directly compute the expression, but understanding the manual steps is crucial for developing mathematical intuition and problem-solving skills. The ability to choose the most efficient method based on the problem’s structure is a valuable asset in mathematics.
Real-World Applications
Understanding how to evaluate mathematical expressions like 2.7(10) − 41/110 has numerous real-world applications. In finance, for example, such calculations might be used to determine profits after deducting expenses, where one term represents income and the other represents costs. In science and engineering, these types of expressions can appear in various formulas, such as calculating forces, velocities, or electrical circuits. In everyday life, calculating discounts or figuring out the final cost of items often involves similar arithmetic operations. For instance, determining the sale price of an item after a certain percentage discount might involve subtracting a fraction of the original price from the initial amount. Cooking and baking also make use of such math skills, especially when scaling recipes up or down. Adjusting ingredient quantities often requires multiplying or dividing fractions and decimals. The ability to accurately evaluate mathematical expressions is thus a fundamental skill that extends beyond the classroom and into various aspects of daily living and professional endeavors. Mastering these skills enhances problem-solving capabilities and allows for more informed decision-making in numerous contexts.
Conclusion
In conclusion, finding the value of the expression 2.7(10) − 41/110 involves a combination of understanding the order of operations, converting between decimals and fractions, and performing basic arithmetic calculations accurately. The process begins with multiplying 2.7 by 10, resulting in 27. Then, the fraction 41/110 is considered, and the subtraction is performed. The final result, 2929/110, or approximately 26.627 in decimal form, is achieved by carefully executing each step. Common mistakes can be avoided by adhering to the order of operations, double-checking calculations, and employing strategies like writing down each step clearly. Alternative methods, such as converting the entire expression into decimal form, can also be used to solve the problem. The skills used in this calculation have broad applications in various fields, including finance, science, and everyday life. Ultimately, the ability to accurately evaluate mathematical expressions is a valuable skill that enhances problem-solving abilities and supports informed decision-making. Mastering these concepts is crucial for both academic and practical success, underscoring the importance of continuous learning and practice in mathematics.
