Exploring the Mathematical Expression: 1/2 ÷ 1/2 × 1/2 1/2 - 1/2

Today, let's delve into a specific mathematical expression:

1/2 ÷ 1/2 × 1/2 1/2 - 1/2

Understanding how to solve such expressions correctly is crucial for anyone looking to master the basics of mathematics, especially in fields such as computer science and engineering where precision is paramount.

Introduction to the Problem

We start with the expression:

Y 1/2 ÷ 1/2 × 1/2 1/2 - 1/2

To solve this expression, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) or BODMAS (Brackets, Orders (i.e., powers and square roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Step-by-Step Solution

Let's break it down step by step:

First, we focus on division and multiplication, working from left to right. Next, we address any subtraction and addition, working from left to right.

Step 1: Resolving Division and Multiplication

Y 1/2 ÷ 1/2 × 1/2 1/2 - 1/2

1/2 ÷ 1/2 is equivalent to multiplying by the reciprocal, which results in 1. Thus, the expression simplifies to:

Y 1 × 1/2 × 1/2 1/2 - 1/2

Step 2: Addressing the Expression Further

The expression can be further broken down as:

Y 1/2 1/2 - 1/2

At this point, the expression simplifies to:

Y (1/2 × 1/2) 1/2 - 1/2

Which simplifies to:

Y 1/4 × 1/2 - 1/2

Step 3: Solving the Final Expression

Calculating the multiplication and then subtraction, we get:

Y 1/8 - 1/2

Converting 1/2 to a fraction with the same denominator as 1/8, we get 4/8.

So, the expression now is:

Y 1/8 - 4/8 -3/8

This yields the decimal fraction:

Y -0.375

Alternative Perspectives

Let's discuss the alternative perspectives and results provided in the question:

1. First Alternative:

1/2 ÷ 1/2 × 1/2 1/2 - 1/2 1 × 1/2 1/2 - 1/2

Followed by:

1/2 1/2 - 1/2 11/2 - 1/2 2 - 1/2 1/2

This result is incorrect if we follow the correct order of operations.

2. Second Alternative:

1/2 1/2 1/2 - 1/2 x 1/2 1 1/2 - 1/2 x 1/2 3/2 - 1/4 6/4 - 1/4 5/4 1.25

This result is also incorrect and does not adhere to the correct order of operations.

Conclusion

To solve the given expression, one must strictly follow the order of operations. The correct answer, as derived, is:

Y -3/8 -0.375

It’s important to always double-check your work and follow the correct order of operations (BODMAS/PEMDAS) to ensure accuracy in your calculations.

Related Mathematical Concepts

1. Order of Operations: Understanding how to solve expressions correctly is crucial, especially when dealing with more complex mathematical problems.

2. Arithmetic Precision: Mathematics relies on precise calculations, and mastering the basic operations ensures that you can tackle more advanced problems.

3. BODMAS/PEMDAS: Knowing the mnemonic for the order of operations and applying it correctly guarantees accurate results in mathematical expressions.