Mastering Math Operations A Step-by-Step Guide

This comprehensive guide delves into the essential mathematical operations, providing step-by-step solutions and explanations for a variety of problems. Whether you're a student looking to reinforce your understanding or someone seeking to brush up on your math skills, this article will equip you with the knowledge and confidence to tackle a wide range of mathematical challenges. We will cover topics including evaluating expressions with integers, simplifying expressions, performing operations with fractions, converting between decimals and fractions, and simplifying decimal expressions.

Evaluating expressions with integers often involves a combination of operations such as multiplication, addition, and subtraction, along with the use of parentheses. The key to solving these problems lies in following the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means performing operations inside parentheses first, followed by exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). In this section, we will dissect the problem: (-5) × [(-6) + 2] - (-3) × 4. By meticulously applying the order of operations, we will arrive at the correct solution, unraveling each step to provide clarity and understanding. Let's embark on this mathematical journey together, breaking down complex expressions into manageable components and mastering the art of integer evaluation.

First, we need to solve the expression within the brackets: (-6) + 2 = -4. Now, we can rewrite the expression as (-5) × (-4) - (-3) × 4. Next, we perform the multiplications: (-5) × (-4) = 20 and (-3) × 4 = -12. The expression now becomes 20 - (-12). Subtracting a negative number is the same as adding its positive counterpart, so we have 20 + 12, which equals 32. Therefore, the final answer to the expression (-5) × [(-6) + 2] - (-3) × 4 is 32. This meticulous breakdown showcases the importance of adhering to the order of operations, ensuring accuracy and a deep comprehension of integer evaluation. Each step, from simplifying within parentheses to the final addition, is a testament to the systematic approach that mathematics demands. This skill is not just about arriving at the correct answer; it's about cultivating a logical and sequential mindset that is invaluable in various problem-solving scenarios.

When it comes to simplifying expressions, the goal is to reduce an expression to its most basic form while maintaining its original value. This often involves combining like terms, applying the distributive property, and adhering to the order of operations. Let’s tackle the expression: (-3) × 2 + (-4) × (-1) - 6. Our mission is to break it down step by step, revealing the elegance of mathematical simplification. By mastering this skill, we not only enhance our ability to solve equations but also gain a deeper appreciation for the structure and harmony within mathematical expressions. The process of simplification is akin to refining a raw material, transforming it into a polished gem of mathematical clarity. It's a journey of precision, where each step is carefully considered, and the destination is a concise and elegant expression that encapsulates the essence of the original form.

Following the order of operations, we first perform the multiplications: (-3) × 2 = -6 and (-4) × (-1) = 4. Now, the expression becomes -6 + 4 - 6. Next, we perform the additions and subtractions from left to right: -6 + 4 = -2, and then -2 - 6 = -8. Therefore, the simplified form of the expression (-3) × 2 + (-4) × (-1) - 6 is -8. This process illustrates the power of simplification, taking a seemingly complex expression and distilling it down to a single, concise value. This skill is fundamental in algebra and beyond, allowing us to manipulate and solve equations with greater ease and efficiency. The beauty of simplification lies in its ability to unveil the underlying simplicity within complex structures, a testament to the inherent order and logic that governs the mathematical universe.

Adding fractions requires a common denominator. To add 3/4 + 2/5, we first need to find the least common multiple (LCM) of 4 and 5, which is 20. Then, we convert each fraction to an equivalent fraction with a denominator of 20. For 3/4, we multiply both the numerator and the denominator by 5: (3 × 5) / (4 × 5) = 15/20. For 2/5, we multiply both the numerator and the denominator by 4: (2 × 4) / (5 × 4) = 8/20. Now, we can add the fractions: 15/20 + 8/20 = (15 + 8) / 20 = 23/20. The result is an improper fraction, which can be left as is or converted to a mixed number. To convert 23/20 to a mixed number, we divide 23 by 20, which gives us a quotient of 1 and a remainder of 3. Thus, 23/20 is equal to 1 3/20. This process highlights the fundamental principle of fraction addition: the necessity of a common denominator. It's akin to adding apples and oranges – you can't do it directly until you express them in the same unit, like