100 Divided By 4 2 3 Pemdas

100 Divided by 4, 2, and 3: Demystifying PEMDAS/BODMAS

The seemingly simple question, "What is 100 divided by 4, 2, and 3?" often leads to confusion. This isn't a trick question; it highlights the crucial importance of understanding the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) depending on your region. This article will delve into the nuances of solving this problem, clarifying the correct method and explaining why other approaches are incorrect. We'll explore the fundamental principles behind PEMDAS/BODMAS and provide practical examples to solidify your understanding.

Understanding PEMDAS/BODMAS: The Order of Operations

PEMDAS/BODMAS is a mnemonic device used to remember the standard order of operations in mathematics. It ensures consistency and avoids ambiguity when evaluating mathematical expressions. Let's break down each part:

P/B - Parentheses/Brackets:

This is the first step. Any calculations within parentheses or brackets must be performed before any other operations. This includes nested parentheses, where you work from the innermost set outwards.

E/O - Exponents/Orders:

Next, we handle exponents (powers or indices). This includes operations like squaring, cubing, or raising to any other power.

MD - Multiplication and Division:

Multiplication and division are performed together, from left to right. It's a common misconception that multiplication always comes before division. The order depends on their position in the expression.

AS - Addition and Subtraction:

Finally, addition and subtraction are performed together, also from left to right. Similar to multiplication and division, the order isn't strictly defined; it follows the sequence of the operations as they appear in the equation.

Solving 100 Divided by 4, 2, and 3

Now, let's tackle the original problem: 100 divided by 4, 2, and 3. The crucial part is recognizing that this is a series of divisions, and we must follow the order from left to right. This means we don't divide 100 by the product of 4, 2, and 3 (4 x 2 x 3 = 24). Instead, we perform the divisions sequentially:

1. 100 ÷ 4 = 25
2. 25 ÷ 2 = 12.5
3. 12.5 ÷ 3 = 4.166666... (approximately)

Therefore, the correct answer to 100 divided by 4, 2, and 3 is approximately 4.17.

Common Mistakes and Misinterpretations

Many individuals make the mistake of calculating the product of the divisors (4 x 2 x 3) before performing the division. This leads to an incorrect answer. Let's illustrate the incorrect approach:

Incorrect Method:

1. 4 x 2 x 3 = 24
2. 100 ÷ 24 = 4.166666... (approximately)

While this result is numerically close, the process is mathematically flawed. It violates the established order of operations.

The Importance of Clarity in Mathematical Notation

The ambiguity in the original problem highlights the importance of clear mathematical notation. Using parentheses or fractions can eliminate any doubt about the intended order of operations. For example, if the intention was to divide 100 by the product of 4, 2, and 3, the expression should be written as:

100 ÷ (4 x 2 x 3)

This makes the order of operations unambiguous. Similarly, using a fraction format achieves the same clarity:

100/(423)

Advanced Applications of PEMDAS/BODMAS

The principles of PEMDAS/BODMAS extend far beyond simple arithmetic problems. They are fundamental to more complex mathematical concepts, including:

Algebra:

Solving algebraic equations often involves manipulating expressions that require applying PEMDAS/BODMAS to simplify and isolate variables.

Calculus:

Derivatives and integrals, the cornerstones of calculus, involve complex expressions that necessitate adherence to the order of operations for accurate calculations.

Programming:

In programming, the order of operations is critical. Programming languages strictly enforce the order of operations to ensure that computations are performed as intended. Ignoring this can lead to significant errors in program execution.

Practical Examples to Enhance Understanding

Let’s look at some more examples to reinforce the application of PEMDAS/BODMAS:

Example 1:

(10 + 5) x 2 - 4

1. Parentheses first: 10 + 5 = 15
2. Multiplication: 15 x 2 = 30
3. Subtraction: 30 - 4 = 26

Therefore, the answer is 26.

Example 2:

2² + 3 x 4 - 6 ÷ 2

1. Exponents: 2² = 4
2. Multiplication: 3 x 4 = 12
3. Division: 6 ÷ 2 = 3
4. Addition: 4 + 12 = 16
5. Subtraction: 16 - 3 = 13

Therefore, the answer is 13.

Example 3:

15 ÷ (3 + 2) x 4 - 1

1. Parentheses first: 3 + 2 = 5
2. Division: 15 ÷ 5 = 3
3. Multiplication: 3 x 4 = 12
4. Subtraction: 12 - 1 = 11

Therefore, the answer is 11.

Conclusion: Mastering the Order of Operations

Understanding and applying PEMDAS/BODMAS is crucial for accurate mathematical calculations. It's not merely a set of rules to memorize; it's a fundamental principle that ensures consistency and precision in mathematics and related fields. By carefully following the order of operations, we can avoid common mistakes and arrive at the correct answers, even in seemingly simple problems like 100 divided by 4, 2, and 3. Remember to always prioritize parentheses or brackets, then exponents, followed by multiplication and division from left to right, and finally, addition and subtraction from left to right. Practice these examples and others to solidify your understanding and improve your mathematical skills. Through consistent practice and a clear understanding of the principles, you can confidently tackle more complex mathematical problems and excel in your studies or professional endeavors. Mastering PEMDAS/BODMAS unlocks a deeper understanding of mathematics and allows for more efficient and accurate problem-solving.