Evaluate (3 * 2 * 2 * 4 - 1 * 5) / (0.02 * (13.5 / 4.5)) Using BODMAS

This article provides a step-by-step guide on how to evaluate the expression ${ \frac{3 \cdot 2 \times 2 \cdot 4 - 1 \cdot 5}{0.02(13.5 \div 4.5)} }$ using the BODMAS (PEMDAS) rule. The BODMAS rule is a mnemonic used to remember the order of operations to be followed while solving mathematical expressions. It stands for Brackets, Orders (powers and square roots, etc.), Division and Multiplication, Addition and Subtraction. In some regions, the acronym PEMDAS is used, which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Both acronyms represent the same order of operations.

Understanding the BODMAS rule is crucial for accurate calculations. The rule ensures that mathematical expressions are evaluated in a consistent and logical manner, preventing ambiguity and errors. By following the correct order of operations, we can arrive at the correct solution efficiently. In this comprehensive guide, we will break down the given expression and apply the BODMAS rule step by step. We will explain each operation in detail, ensuring that you understand the underlying concepts and can apply them to similar problems. Whether you are a student learning the basics of arithmetic or someone looking to refresh your math skills, this guide will provide you with the knowledge and confidence to tackle complex expressions with ease. Our goal is to make the process of evaluating mathematical expressions clear and straightforward. We will illustrate each step with examples and explanations, so you can see exactly how the BODMAS rule is applied in practice. By the end of this guide, you will be able to confidently evaluate expressions involving various operations, including multiplication, division, addition, and subtraction, as well as expressions that involve parentheses and exponents. Let's dive in and explore the world of mathematical expressions and the BODMAS rule!

Step-by-Step Solution

To accurately evaluate the given expression, we will follow the BODMAS (PEMDAS) order of operations. This involves breaking down the expression into smaller parts and solving them in the correct sequence:

${ \frac{3 \cdot 2 \times 2 \cdot 4 - 1 \cdot 5}{0.02(13.5 \div 4.5)} }$

1. Simplify the Numerator

In the numerator, we have a series of multiplications and a subtraction. According to BODMAS, we perform multiplications before subtraction. The numerator is ${ 3 \cdot 2 \times 2 \cdot 4 - 1 \cdot 5 }$.

First, we perform the multiplications from left to right:

• ${ 3 \cdot 2 = 6 }$
• ${ 6 \times 2 = 12 }$
• ${ 12 \cdot 4 = 48 }$

So, the first part of the numerator simplifies to 48. Next, we multiply 1 by 5:

• ${ 1 \cdot 5 = 5 }$

Now, we subtract the second result from the first:

• ${ 48 - 5 = 43 }$

The numerator simplifies to 43. The order in which we perform the operations is crucial here. Multiplying from left to right ensures that we maintain the correct sequence as per the BODMAS rule. The step-by-step approach helps in avoiding common mistakes and ensures accuracy. By breaking down the multiplication into smaller steps, we can easily manage the calculations and keep track of the intermediate results. This methodical approach is particularly useful when dealing with longer expressions or those involving multiple operations. Understanding the rationale behind each step is essential for grasping the overall concept. The numerator, initially a complex series of multiplications and a subtraction, is now reduced to a single number, 43. This simplification is a significant step forward in evaluating the entire expression. By carefully following the BODMAS rule, we have successfully navigated the first part of the problem, setting the stage for the next phase of the calculation. The key takeaway here is the importance of adhering to the correct order of operations to achieve the correct result. The numerator, initially a complex series of multiplications and a subtraction, is now reduced to a single number, 43. This simplification is a significant step forward in evaluating the entire expression. By carefully following the BODMAS rule, we have successfully navigated the first part of the problem, setting the stage for the next phase of the calculation. The key takeaway here is the importance of adhering to the correct order of operations to achieve the correct result.

2. Simplify the Denominator

The denominator is ${ 0.02(13.5 \div 4.5) }$.

According to BODMAS, we need to solve the expression inside the parentheses first. The operation inside the parentheses is division:

• ${ 13.5 \div 4.5 = 3 }$

Now, we multiply the result by 0.02:

• ${ 0.02 \cdot 3 = 0.06 }$

The denominator simplifies to 0.06. The division operation within the parentheses takes precedence over the multiplication outside the parentheses, as per the BODMAS rule. This order is essential to ensure we arrive at the correct answer. By performing the division first, we simplify the expression inside the parentheses, making it easier to handle the subsequent multiplication. The step-by-step simplification of the denominator not only ensures accuracy but also enhances understanding of the process. Each operation is performed in the correct sequence, reinforcing the importance of the BODMAS rule. The division, resulting in 3, is a crucial intermediate step that sets the stage for the final calculation in the denominator. Multiplying this result by 0.02 yields 0.06, which is the simplified form of the denominator. This meticulous approach to simplifying the denominator highlights the significance of careful calculation and adherence to the order of operations. The denominator, initially a complex expression involving division and multiplication, is now reduced to a single decimal number, 0.06. This simplification mirrors the process we followed for the numerator, illustrating the consistent application of the BODMAS rule across the entire expression. By breaking down the denominator into manageable steps, we have successfully navigated the second part of the problem, bringing us closer to the final solution.

3. Divide the Numerator by the Denominator

Now we have the simplified numerator and denominator. We divide the numerator by the denominator:

${ \frac{43}{0.06} }$

To perform this division, it’s helpful to remove the decimal from the denominator. We can do this by multiplying both the numerator and the denominator by 100:

${ \frac{43 \times 100}{0.06 \times 100} = \frac{4300}{6} }$

Now, we divide 4300 by 6:

${ \frac{4300}{6} = 716.666...\approx 716.67 }$

The result of the division is approximately 716.67. Converting the decimal in the denominator to a whole number simplifies the division process. Multiplying both the numerator and the denominator by the same factor (in this case, 100) does not change the value of the fraction but makes it easier to compute. The division of 4300 by 6 may seem daunting at first, but breaking it down into smaller steps makes it manageable. We can use long division or a calculator to find the quotient. The result, 716.666..., is a repeating decimal. Depending on the context, we may choose to round it to a certain number of decimal places. In this case, rounding to two decimal places gives us 716.67. This final step demonstrates the importance of accuracy in division, especially when dealing with decimals. The division operation is the culmination of all the previous simplifications, bringing us to the final answer. The quotient, approximately 716.67, is the value of the original expression. This result underscores the power of the BODMAS rule in systematically solving complex mathematical problems. The fraction, initially a daunting division problem involving decimals, is now elegantly solved, thanks to the step-by-step application of the BODMAS rule and strategic simplification techniques.

Final Answer

Therefore, the expression ${ \frac{3 \cdot 2 \times 2 \cdot 4 - 1 \cdot 5}{0.02(13.5 \div 4.5)} }$ evaluates to approximately 716.67.

Summary of Steps

1. Simplify the Numerator: Perform multiplications and then subtraction to get 43.
2. Simplify the Denominator: Perform division inside parentheses and then multiplication to get 0.06.
3. Divide Numerator by Denominator: Divide 43 by 0.06 to get approximately 716.67.

Conclusion

In conclusion, evaluating the expression using BODMAS involves simplifying the numerator and the denominator separately before performing the final division. The BODMAS rule ensures that we tackle the operations in the correct sequence, leading to an accurate result. This method is not only applicable to this specific problem but also to a wide range of mathematical expressions. By consistently applying the BODMAS rule, you can confidently solve complex problems involving multiple operations. Understanding the order of operations is a fundamental skill in mathematics, and mastering it will significantly enhance your problem-solving abilities. The expression, initially appearing complex, is now solved with clarity, thanks to the systematic application of the BODMAS rule. This step-by-step approach not only ensures accuracy but also fosters a deeper understanding of mathematical principles. The process of simplifying the numerator and denominator separately allows us to manage the complexity of the expression more effectively. The final division then brings us to the solution, which is approximately 716.67. This comprehensive guide has demonstrated the power of the BODMAS rule in breaking down complex mathematical problems into manageable steps. By adhering to the correct order of operations, we can navigate through intricate expressions with confidence and precision. The BODMAS rule serves as a cornerstone of mathematical problem-solving, providing a framework for approaching and solving a wide variety of equations and expressions. Mastery of this rule is essential for anyone looking to excel in mathematics and related fields. The journey from the initial complex expression to the final solution highlights the beauty and logic of mathematical operations. The meticulous application of the BODMAS rule, combined with careful calculation, results in a clear and concise answer. This underscores the importance of both theoretical understanding and practical application in mathematics. The ability to confidently evaluate mathematical expressions is a valuable skill, applicable not only in academic settings but also in real-world scenarios. Whether you are calculating finances, measuring ingredients for a recipe, or analyzing data, the BODMAS rule will serve you well. By understanding and applying this rule, you empower yourself to tackle mathematical challenges with ease and accuracy.

This detailed explanation provides a clear understanding of how to use BODMAS to solve mathematical expressions, ensuring accuracy and efficiency in calculations. Remember to always follow the order of operations to avoid errors and achieve the correct result.