Simplify Expressions And Solve Math Problems

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#title: Simplify Expressions and Solve Math Problems A Comprehensive Guide

Introduction

This article provides a detailed walkthrough on simplifying mathematical expressions and solving practical problems. We will cover simplifying expressions involving order of operations, including brackets, exponents, multiplication, division, addition, and subtraction (BODMAS/PEMDAS). Furthermore, we will address converting units of measurement and applying these concepts to real-world scenarios. Let's dive into the world of mathematical simplification and problem-solving!

7. Simplify Numerical Expressions

In this section, we will focus on simplifying numerical expressions using the order of operations. This involves understanding the hierarchy of operations, commonly remembered by the acronyms BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Mastering this concept is crucial for accurate mathematical calculations.

a) 78 - [5 + 3 of (25 - 2 × 10)]

To simplify this expression, we must meticulously follow the order of operations. The key here is to breakdown the expression into manageable parts, tackling the innermost brackets first and working our way outwards. Understanding the role of each operator is crucial for correct simplification. Let's break down the problem step by step:

  1. Innermost Parentheses: Begin by simplifying the expression within the innermost parentheses: (25 - 2 × 10).
    • Multiplication: 2 × 10 = 20. Multiplication takes precedence over subtraction.
    • Subtraction: 25 - 20 = 5. Now the innermost parenthesis is simplified.
  2. 'Of' Operation: Next, we address the 'of' operation, which in this context means multiplication. 3 of 5 means 3 multiplied by 5.
    • Multiplication: 3 of 5 = 3 × 5 = 15. It's important to recognize 'of' as multiplication in this context.
  3. Brackets: Now we focus on the expression within the square brackets: [5 + 15].
    • Addition: 5 + 15 = 20. The content within the brackets is now a single value.
  4. Final Subtraction: Finally, we perform the subtraction operation outside the brackets.
    • Subtraction: 78 - 20 = 58. This final step gives us the simplified answer.

Therefore, 78 - [5 + 3 of (25 - 2 × 10)] simplifies to 58. This meticulous step-by-step approach, focusing on the order of operations, guarantees the correct solution. Remember, patience and accuracy are your allies in mathematical simplification.

b) [29 - (-2) (6 - (7 - 3))] ÷ [3 × (5 + (-3) × (-2))]

This expression appears more complex, but the same principles of BODMAS/PEMDAS apply. We must systematically break down the expression, dealing with brackets and operations in the correct order. Careful attention to signs, especially negative signs, is paramount to avoid errors. Let's simplify this step by step:

  1. Innermost Parentheses: Start with the innermost parentheses: (7 - 3).
    • Subtraction: 7 - 3 = 4. This simplifies the innermost part.
  2. Next Layer of Parentheses: Now consider the next set of parentheses: (6 - 4).
    • Subtraction: 6 - 4 = 2. We continue working outwards.
  3. Multiplication within Brackets (First Set): Within the first set of square brackets, we have (-2)(2).
    • Multiplication: (-2)(2) = -4. Remember the rules of multiplying negative numbers.
  4. Subtraction within Brackets (First Set): Now we have [29 - (-4)]. Subtracting a negative number is the same as adding its positive counterpart.
    • Subtraction: 29 - (-4) = 29 + 4 = 33. This simplifies the first set of brackets.
  5. Innermost Parentheses (Second Set): In the second set of square brackets, begin with (-3) × (-2).
    • Multiplication: (-3) × (-2) = 6. A negative times a negative is a positive.
  6. Addition within Parentheses (Second Set): Next, we have (5 + 6).
    • Addition: 5 + 6 = 11. The parentheses are simplified.
  7. Multiplication within Brackets (Second Set): Now we have 3 × 11.
    • Multiplication: 3 × 11 = 33. This simplifies the second set of brackets.
  8. Final Division: Finally, we perform the division operation.
    • Division: 33 ÷ 33 = 1. The final step.

Therefore, [29 - (-2) (6 - (7 - 3))] ÷ [3 × (5 + (-3) × (-2))] simplifies to 1. This detailed breakdown emphasizes the importance of systematically applying the order of operations and carefully handling signs. Practice and attention to detail are essential for mastering complex expression simplification.

8. Unit Conversion and Problem Solving

This section focuses on unit conversion, a fundamental skill in various fields, including mathematics, science, and engineering. We'll explore how to convert between different units of measurement, specifically focusing on length. Understanding conversion factors and applying them correctly is key to solving problems involving different units. Let's consider the given converting rule and apply it to the problem.

The Height Problem

Siddhartha's height is given as 4 ft 6 inches. We need to convert this height into inches and then potentially use this information to solve further problems. The ability to convert between feet and inches is essential for this problem.

a) Convert Siddhartha's height into inches.

To convert Siddhartha's height into inches, we need to know the conversion factor between feet and inches.

  • Conversion Factor: 1 foot (ft) = 12 inches (in).

Siddhartha's height is 4 ft 6 inches. We need to convert the 4 feet into inches and then add the remaining 6 inches. Breaking down the problem into smaller steps makes the conversion easier to understand.

  1. Convert feet to inches: Multiply the number of feet by the conversion factor (12 inches/foot).
    • 4 ft × 12 inches/ft = 48 inches. This gives us the equivalent of 4 feet in inches.
  2. Add the remaining inches: Add the 6 inches to the result from step 1.
    • 48 inches + 6 inches = 54 inches. This accounts for the additional inches in Siddhartha's height.

Therefore, Siddhartha's height is 54 inches. This straightforward conversion highlights the importance of knowing and applying the correct conversion factors.

b) Convert the... (The original question is incomplete)

Since the original question (8b) is incomplete (