Solving 42 ÷ [1 + (18 - 15) × 2] A Step-by-Step Guide

JU07/11/2025 08, 2025 by THE IDEN 54 views

$Mastering Order of Operations: A Deep Dive into $42 \div [1+(18-15) \times 2]$$

In the realm of mathematics, precision and order are paramount. When faced with complex expressions involving multiple operations, adhering to a specific order is crucial to arrive at the correct solution. This article delves into the intricacies of solving the mathematical expression $42 \div [1+(18-15) \times 2]$ , providing a comprehensive breakdown of each step and highlighting the fundamental principles that govern the order of operations.

Understanding the Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. This standardized approach ensures consistency and accuracy in mathematical calculations. Let's break down what each letter in PEMDAS signifies:

Parentheses: Operations within parentheses (or other grouping symbols like brackets) are performed first.
Exponents: Exponents (powers and roots) are evaluated next.
Multiplication and Division: Multiplication and division are performed from left to right.
Addition and Subtraction: Addition and subtraction are performed from left to right.

By following this order, we can systematically solve complex expressions and avoid ambiguity.

Solving the Expression: A Step-by-Step Approach

Now, let's apply the order of operations to solve the expression $42 \div [1+(18-15) \times 2]$ .

Step 1: Parentheses

The first step is to address the operations within the parentheses. In this case, we have $(18-15)$ .

$18 - 15 = 3$

So, the expression now becomes:

$42 \div [1 + 3 \times 2]$

Step 2: Brackets (Innermost Grouping)

Next, we focus on the operations within the brackets. Inside the brackets, we have both addition and multiplication. According to PEMDAS, multiplication takes precedence over addition. So, we perform the multiplication first:

$3 \times 2 = 6$

The expression now looks like this:

$42 \div [1 + 6]$

We continue working within the brackets by performing the addition:

$1 + 6 = 7$

Now, the expression simplifies to:

$42 \div 7$

Step 3: Division

Finally, we perform the division operation:

$42 \div 7 = 6$

Therefore, the solution to the expression $42 \div [1+(18-15) \times 2]$ is 6.

Importance of Following the Order of Operations

The order of operations is not just an arbitrary set of rules; it is a fundamental principle that ensures consistency and accuracy in mathematical calculations. Without a standardized order, the same expression could yield different results depending on the sequence in which operations are performed. This ambiguity would render mathematical communication and problem-solving highly unreliable.

To illustrate this point, let's consider what would happen if we disregarded the order of operations and performed the addition before the multiplication in our example. If we added 1 and 3 first, we would get 4, and then multiplying by 2 would give us 8. Dividing 42 by 8 would result in a completely different answer than the correct solution of 6. This simple example demonstrates the critical importance of adhering to the order of operations.

Real-World Applications

The order of operations is not confined to the realm of abstract mathematics; it has numerous real-world applications in various fields, including:

Computer Programming: Programming languages rely heavily on the order of operations to execute code accurately. Incorrectly ordered operations can lead to unexpected results and program errors.
Engineering: Engineers use mathematical expressions extensively in their designs and calculations. Adhering to the order of operations is crucial for ensuring the safety and functionality of structures and systems.
Finance: Financial calculations, such as compound interest and investment returns, involve complex expressions that require strict adherence to the order of operations.
Everyday Life: Even in everyday situations, we implicitly use the order of operations when performing calculations such as calculating the total cost of items with discounts or determining the time required to complete a series of tasks.

Practice Problems

To solidify your understanding of the order of operations, let's tackle a few more practice problems:

$10 + 2 \times (8 - 3)$
$(15 \div 3) + 4^2 - 1$
$36 \div [2 \times (5 + 1) - 4]$

Solutions:

By working through these problems, you can reinforce your skills and gain confidence in applying the order of operations.

Conclusion

Mastering the order of operations is a fundamental skill in mathematics. By understanding and applying the PEMDAS rules, we can solve complex expressions accurately and efficiently. This knowledge is not only essential for academic success but also has practical applications in various fields and everyday life. So, embrace the order, and watch your mathematical prowess soar!

Understanding the intricacies of mathematical expressions and the order of operations is crucial for success in mathematics and various fields that rely on precise calculations. This article will provide a comprehensive guide to solving the mathematical expression 42 ÷ [1 + (18 - 15) × 2], offering a step-by-step explanation that emphasizes the importance of following the correct order of operations.

The Significance of Order of Operations

At the heart of mathematical computation lies the order of operations, a set of rules that dictate the sequence in which mathematical operations should be executed. This established order guarantees uniformity in calculations, preventing ambiguity and guaranteeing reliable results. The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), serves as a useful mnemonic device for remembering the correct order.

Parentheses: Operations enclosed within parentheses, brackets, or other grouping symbols are always performed first. This allows us to isolate and simplify specific parts of an expression before addressing the larger context.
Exponents: Exponents, which represent repeated multiplication, are evaluated after parentheses. This includes powers and roots.
Multiplication and Division: Multiplication and division hold equal precedence and are performed from left to right. This means that if both operations are present, we execute them in the order they appear in the expression.
Addition and Subtraction: Addition and subtraction also have equal precedence and are performed from left to right, similar to multiplication and division.

The order of operations ensures that every mathematical expression has only one correct answer, regardless of who is solving it. This consistency is vital for clear communication and accurate problem-solving in mathematics and its applications. Without this standardized approach, the same expression could be interpreted in multiple ways, leading to confusion and errors.

Solving the Expression Step-by-Step

Now, let’s apply the PEMDAS principles to solve the expression 42 ÷ [1 + (18 - 15) × 2]. By meticulously following each step, we will arrive at the correct solution.

Step 1: Parentheses (Innermost Grouping)

According to PEMDAS, the first step is to address the operation within the parentheses: (18 - 15).

Performing the subtraction, we get:

18 - 15 = 3

Now, the expression is simplified to:

42 ÷ [1 + 3 × 2]

Step 2: Brackets (Outermost Grouping)

Next, we address the operations within the brackets. Inside the brackets, we have both addition and multiplication. Following the order of operations, multiplication takes precedence over addition. Therefore, we perform the multiplication first:

3 × 2 = 6

The expression now becomes:

42 ÷ [1 + 6]

We continue working within the brackets by performing the addition:

1 + 6 = 7

Now, the expression simplifies to:

42 ÷ 7

Step 3: Division

Finally, we perform the division operation:

42 ÷ 7 = 6

Therefore, the solution to the expression 42 ÷ [1 + (18 - 15) × 2] is 6.

Why the Order of Operations Matters

The order of operations is not an arbitrary rule; it is a fundamental principle that ensures mathematical consistency and accuracy. Imagine if we ignored the order and performed the addition before the multiplication in our example. We would get a completely different result!

For instance, if we added 1 and 3 first, we would get 4. Multiplying that by 2 would give us 8. Then, dividing 42 by 8 would yield 5.25, which is incorrect. This stark difference highlights the critical role of PEMDAS in achieving the correct answer.

The order of operations is the bedrock of mathematical calculations, preventing ambiguity and ensuring that everyone arrives at the same solution when working with the same expression. This consistency is vital in fields like computer programming, engineering, and finance, where accurate calculations are paramount.

Real-World Applications of Order of Operations

Beyond the classroom, the order of operations plays a vital role in numerous real-world applications. Here are a few examples:

Computer Programming: In programming, the order of operations dictates how code is executed. Incorrectly ordered operations can lead to bugs and unexpected program behavior. Understanding PEMDAS is essential for writing accurate and efficient code.
Engineering: Engineers rely heavily on mathematical calculations in their designs and analyses. From calculating stress on a bridge to designing electrical circuits, the order of operations is critical for ensuring the safety and functionality of engineering projects.
Finance: Financial calculations, such as compound interest, loan payments, and investment returns, involve complex formulas that require strict adherence to the order of operations. Accurate financial calculations are crucial for sound decision-making.
Everyday Life: Even in everyday situations, we implicitly use the order of operations. For example, when calculating the total cost of items with discounts or figuring out the time needed to complete a series of tasks, we follow the principles of PEMDAS.

Practice Makes Perfect

To reinforce your understanding of the order of operations, try solving the following practice problems:

2 × (5 + 3) – 10 ÷ 2
18 ÷ (6 – 3) + 4²
(15 + 9) ÷ (2 × 3) – 1

Solutions:

By practicing these types of problems, you'll build confidence in your ability to apply the order of operations and solve more complex mathematical expressions.

Conclusion

Mastering the order of operations is a fundamental skill that unlocks the door to mathematical proficiency. By following the PEMDAS guidelines, you can confidently tackle complex expressions and arrive at accurate solutions. This knowledge is not only essential for academic success but also has far-reaching applications in various fields and everyday life. So, embrace the power of PEMDAS, and watch your mathematical abilities flourish!