Mastering Mathematical Simplification A Comprehensive Guide

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In the realm of mathematics, simplification is a fundamental skill that empowers us to unravel intricate expressions and arrive at concise, manageable solutions. This comprehensive guide delves into the art of mathematical simplification, providing you with the knowledge and techniques to confidently tackle complex problems. We will explore a range of simplification challenges, from arithmetic expressions involving decimals and fractions to ratio problems, equipping you with the tools necessary to excel in this essential mathematical domain.

Simplify: 18.3 ÷ [2.640 + {9.246 + (2.7 × 8.2 + 7.02)}]

In this section, we embark on a step-by-step journey to simplify the given expression: 18.3 ÷ [2.640 + {9.246 + (2.7 × 8.2 + 7.02)}]. This problem exemplifies the importance of adhering to the order of operations, commonly known as PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Let's break down the simplification process:

  1. Prioritize the Innermost Parentheses: Our first task is to tackle the expression within the innermost parentheses: (2.7 × 8.2 + 7.02). We begin by performing the multiplication: 2.7 × 8.2 = 22.14. Next, we add 7.02 to the result: 22.14 + 7.02 = 29.16. Therefore, the innermost parentheses simplify to 29.16.

  2. Address the Curly Braces: Now, we move to the expression within the curly braces: {9.246 + 29.16}. Adding these two numbers yields: 9.246 + 29.16 = 38.406. Consequently, the curly braces simplify to 38.406.

  3. Tackle the Square Brackets: Next, we focus on the expression within the square brackets: [2.640 + 38.406]. Adding these numbers, we get: 2.640 + 38.406 = 41.046. Thus, the square brackets simplify to 41.046.

  4. Perform the Division: Finally, we perform the division operation: 18.3 ÷ 41.046. This yields approximately 0.4458. Therefore, the simplified value of the entire expression is approximately 0.4458.

In summary, by meticulously following the order of operations and breaking down the expression into manageable steps, we successfully simplified the complex expression to its numerical equivalent. This systematic approach is crucial for tackling any mathematical simplification problem.

Simplify: (3/5) ÷ [(1/15) × {1 1/2 + (3 1/3 ÷ 2 1/2 × 3 9/16)}]

This section focuses on simplifying a complex expression involving fractions and mixed numbers: (3/5) ÷ [(1/15) × {1 1/2 + (3 1/3 ÷ 2 1/2 × 3 9/16)}]. This problem requires a strong understanding of fraction operations and the order of operations. Let's dissect the simplification process step-by-step:

  1. Convert Mixed Numbers to Improper Fractions: The first step is to convert all mixed numbers into improper fractions. We have 1 1/2 = 3/2, 3 1/3 = 10/3, and 3 9/16 = 57/16. The expression now becomes: (3/5) ÷ [(1/15) × {3/2 + (10/3 ÷ 5/2 × 57/16)}].

  2. Prioritize Innermost Parentheses (Division and Multiplication): Within the innermost parentheses, we have a combination of division and multiplication. Following the order of operations, we perform division first: 10/3 ÷ 5/2 = 10/3 × 2/5 = 4/3. Now, we perform the multiplication: 4/3 × 57/16 = 19/4. The expression within the innermost parentheses simplifies to 19/4.

  3. Continue with Curly Braces (Addition): Moving to the curly braces, we need to add 3/2 and 19/4. To do this, we find a common denominator, which is 4. We have 3/2 = 6/4. Therefore, 6/4 + 19/4 = 25/4. The expression within the curly braces simplifies to 25/4.

  4. Address Square Brackets (Multiplication): Next, we focus on the expression within the square brackets: (1/15) × (25/4). Multiplying these fractions, we get: (1/15) × (25/4) = 5/12. The square brackets simplify to 5/12.

  5. Perform the Final Division: Finally, we perform the division operation: (3/5) ÷ (5/12). Dividing fractions is the same as multiplying by the reciprocal, so we have: (3/5) × (12/5) = 36/25. Therefore, the simplified value of the entire expression is 36/25 or 1 11/25.

By systematically converting mixed numbers to improper fractions, adhering to the order of operations, and performing each step carefully, we successfully simplified this complex expression involving fractions and mixed numbers.

Write the following in simplest form of ratio: (i) 9 kg and 1000 g

In this section, we tackle the task of expressing the given quantities, 9 kg and 1000 g, in their simplest ratio form. Ratios are used to compare two or more quantities, and simplifying them involves reducing them to their lowest terms while maintaining the same proportion. The key to simplifying ratios with different units is to first convert them to the same unit.

  1. Convert to the Same Unit: The first step is to convert both quantities to the same unit. Since 1 kg = 1000 g, we can convert 9 kg to grams: 9 kg × 1000 g/kg = 9000 g. Now we have both quantities in grams: 9000 g and 1000 g.

  2. Express as a Ratio: Next, we express the quantities as a ratio: 9000 g : 1000 g.

  3. Simplify the Ratio: To simplify the ratio, we find the greatest common divisor (GCD) of the two numbers and divide both parts of the ratio by the GCD. The GCD of 9000 and 1000 is 1000. Dividing both parts of the ratio by 1000, we get: 9000 ÷ 1000 : 1000 ÷ 1000 = 9 : 1.

Therefore, the simplest form of the ratio 9 kg and 1000 g is 9:1. This means that for every 9 units of the first quantity (9 kg), there is 1 unit of the second quantity (1000 g). Simplifying ratios allows us to easily compare quantities and understand their relative proportions.

In conclusion, mastering mathematical simplification is essential for success in various mathematical domains. By understanding the order of operations, fraction manipulations, and ratio simplification techniques, you can confidently tackle complex problems and arrive at accurate solutions. Remember to break down problems into manageable steps, pay attention to detail, and practice regularly to hone your simplification skills.