Simplifying Complex Mathematical Expressions A Step By Step Guide

In this article, we will delve into the simplification of several mathematical expressions. These expressions involve a mix of arithmetic operations, including division, subtraction, multiplication, fractions, and decimals. Simplifying these expressions requires a solid understanding of the order of operations (PEMDAS/BODMAS) and proficiency in handling fractions and decimals. Our goal is to break down each expression step by step, ensuring clarity and accuracy in the process. Mastering the simplification of mathematical expressions is crucial for success in various fields, including engineering, finance, and computer science. This article aims to provide a comprehensive guide to tackle such problems, making it easier for students and professionals alike. Let's embark on this journey of simplification and enhance our mathematical prowess.

(i) 15.3 ÷ 3 - 1/4 of (19.6 ÷ 6.8) + 0.5 × 7.5

To simplify this expression, we must adhere to the order of operations, which dictates that we perform division and multiplication before addition and subtraction. Additionally, the term “of” implies multiplication. Let's begin by addressing the division and multiplication within the parentheses and other parts of the expression. The first operation we encounter is 15.3 ÷ 3, which yields 5.1. Next, we tackle the division within the parentheses: 19.6 ÷ 6.8. This calculation results in approximately 2.88. We then need to find 1/4 of this result, which translates to (1/4) × 2.88, giving us 0.72. Moving on, we have 0.5 × 7.5, which equals 3.75. Now, we can rewrite the expression with these intermediate results: 5.1 - 0.72 + 3.75. The next step is to perform the subtraction and addition from left to right. First, we subtract 0.72 from 5.1, which gives us 4.38. Finally, we add 3.75 to 4.38, resulting in 8.13. Therefore, the simplified value of the expression 15.3 ÷ 3 - 1/4 of (19.6 ÷ 6.8) + 0.5 × 7.5 is 8.13. This step-by-step approach ensures accuracy and clarity in our calculations, highlighting the importance of following the correct order of operations. Understanding each step not only helps in arriving at the correct answer but also reinforces the fundamental principles of arithmetic.

(ii) [4 1/2 - 2 2/3] ÷ 7/12 + 5 1/7 of 3 5/6

This expression involves mixed fractions and multiple operations, making it crucial to follow the order of operations meticulously. Our initial step is to convert all mixed fractions into improper fractions. The mixed fraction 4 1/2 can be converted to an improper fraction by multiplying the whole number (4) by the denominator (2) and adding the numerator (1), which gives us 9. We then place this result over the original denominator, resulting in 9/2. Similarly, 2 2/3 becomes (2 * 3 + 2) / 3 = 8/3. The mixed fraction 5 1/7 converts to (5 * 7 + 1) / 7 = 36/7, and 3 5/6 converts to (3 * 6 + 5) / 6 = 23/6. Now, the expression can be rewritten as [9/2 - 8/3] ÷ 7/12 + 36/7 of 23/6. Next, we address the subtraction within the brackets. To subtract 8/3 from 9/2, we need a common denominator, which is 6. We convert 9/2 to 27/6 and 8/3 to 16/6. Subtracting 16/6 from 27/6 gives us 11/6. The expression now becomes (11/6) ÷ 7/12 + 36/7 of 23/6. The term “of” implies multiplication, so we multiply 36/7 by 23/6. This gives us (36 * 23) / (7 * 6) = 828/42, which simplifies to 138/7. Now the expression is (11/6) ÷ 7/12 + 138/7. To divide 11/6 by 7/12, we multiply 11/6 by the reciprocal of 7/12, which is 12/7. This gives us (11 * 12) / (6 * 7) = 132/42, which simplifies to 22/7. Finally, we add 22/7 to 138/7, resulting in (22 + 138) / 7 = 160/7. Converting this improper fraction to a mixed fraction, we get 22 6/7. Therefore, the simplified value of the given expression is 22 6/7. The process of converting mixed fractions to improper fractions and back is fundamental in simplifying such expressions. This methodical approach minimizes errors and provides a clear path to the solution.

(iii) (1/2 + 1/3) ÷ (1/4 - 1/6) - [8 - {5 1/3 (3 - 2 1/2)}]

This expression requires a careful application of the order of operations, particularly when dealing with nested brackets and fractions. We begin by simplifying the innermost parentheses and brackets first. Let's start with the parentheses (1/2 + 1/3) and (1/4 - 1/6). To add 1/2 and 1/3, we find a common denominator, which is 6. Converting the fractions, we get 3/6 + 2/6 = 5/6. Similarly, for (1/4 - 1/6), the common denominator is 12. Converting the fractions, we have 3/12 - 2/12 = 1/12. Now, let's move to the innermost curly brackets. We have 5 1/3, which can be converted to an improper fraction as (5 * 3 + 1) / 3 = 16/3. Next, we address the parentheses (3 - 2 1/2). Converting 2 1/2 to an improper fraction, we get 5/2. So, 3 can be written as 6/2, and the subtraction becomes 6/2 - 5/2 = 1/2. Now, we multiply 16/3 by 1/2, which gives us (16 * 1) / (3 * 2) = 16/6, simplifying to 8/3. The expression inside the square brackets is now 8 - 8/3. To subtract 8/3 from 8, we write 8 as 24/3. The subtraction becomes 24/3 - 8/3 = 16/3. Returning to the main expression, we have (5/6) ÷ (1/12) - 16/3. To divide 5/6 by 1/12, we multiply 5/6 by the reciprocal of 1/12, which is 12/1. This gives us (5 * 12) / (6 * 1) = 60/6, which simplifies to 10. Finally, we subtract 16/3 from 10. Writing 10 as 30/3, the subtraction becomes 30/3 - 16/3 = 14/3. Converting this improper fraction to a mixed fraction, we get 4 2/3. Therefore, the simplified value of the expression is 4 2/3. Nested brackets and fractions can make expressions appear complex, but breaking them down systematically ensures an accurate solution. This methodical approach is essential for managing complex mathematical problems.

(iv) 2.3 - [1.89 - (3.6 - (3.7 - 0.8 - 0.03))]

This expression involves nested brackets and decimal numbers, requiring careful attention to the order of operations and decimal arithmetic. Our strategy is to simplify the expression from the innermost brackets outwards. We begin with the innermost parentheses: (3.7 - 0.8 - 0.03). First, subtract 0.8 from 3.7, which gives us 2.9. Then, subtract 0.03 from 2.9, resulting in 2.87. Now, the expression inside the outer parentheses is (3.6 - 2.87). Subtracting 2.87 from 3.6, we get 0.73. Next, we move to the square brackets. The expression inside the square brackets is [1.89 - 0.73]. Subtracting 0.73 from 1.89, we get 1.16. Finally, we have the main expression: 2.3 - 1.16. Subtracting 1.16 from 2.3, we get 1.14. Therefore, the simplified value of the expression is 1.14. Dealing with decimals requires precision, and working through the nested brackets step by step helps avoid errors. This methodical approach highlights the importance of accuracy in mathematical calculations and simplifies the overall process.

(v) 4.5 - 1/3 of [7.6 - 3.5] + 2.3 × 4.05

To simplify this expression, we need to follow the order of operations, which includes addressing the brackets first, then multiplication and division, and finally addition and subtraction. Let's start with the expression inside the brackets: [7.6 - 3.5]. Subtracting 3.5 from 7.6 gives us 4.1. Now, the expression becomes 4.5 - 1/3 of 4.1 + 2.3 × 4.05. The term