Solving Mathematical Operations 2 * 8 × 9 ÷ (48 ÷ 6) A Step-by-Step Guide
In this article, we will provide a step-by-step solution to the mathematical operation 2 · 8 × 9 ÷ (48 ÷ 6). This problem involves multiple arithmetic operations, including multiplication, division, and the use of parentheses. By following the correct order of operations, we can arrive at the correct answer. Understanding and applying the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial in mathematics. This ensures that we perform calculations in the correct sequence, leading to accurate results. Mastering these fundamental concepts is essential for anyone looking to improve their mathematical skills.
The order of operations, often remembered by the acronyms PEMDAS or BODMAS, is a set of rules that dictate the sequence in which mathematical operations should be performed. These rules ensure consistency and accuracy in mathematical calculations. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In some regions, the acronym BODMAS is used, which stands for Brackets, Orders (exponents), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Both acronyms represent the same order of operations and are crucial for solving complex mathematical expressions.
The first step in any mathematical problem is to address the Parentheses or Brackets. Operations inside parentheses are always performed first. This is because parentheses group numbers and operations together, indicating that they should be treated as a single unit. For example, in the expression 2 × (3 + 4), we must first add 3 and 4, resulting in 7, and then multiply by 2. Ignoring the parentheses would lead to an incorrect result. The same principle applies to more complex expressions involving multiple sets of parentheses or nested parentheses, where the innermost parentheses are dealt with first.
Next, we deal with Exponents or Orders. Exponents represent the power to which a number is raised (e.g., 2^3 means 2 raised to the power of 3, which equals 8). Calculations involving exponents are performed before multiplication, division, addition, or subtraction. For instance, in the expression 3 × 2^2, we first calculate 2^2, which is 4, and then multiply by 3. Exponents can significantly change the value of an expression, so it is vital to address them correctly in the order of operations.
After parentheses and exponents, we perform Multiplication and Division. These operations have equal priority and are performed from left to right. This means that if both multiplication and division are present in an expression, we solve them in the order they appear. For example, in the expression 10 ÷ 2 × 3, we first divide 10 by 2, which gives 5, and then multiply by 3, resulting in 15. If we were to multiply first, we would get a different result, highlighting the importance of the left-to-right rule.
Finally, we handle Addition and Subtraction, which also have equal priority and are performed from left to right. Similar to multiplication and division, the order in which these operations appear matters. For example, in the expression 8 + 5 - 2, we first add 8 and 5, which equals 13, and then subtract 2, giving us 11. Adhering to the left-to-right rule for addition and subtraction ensures accurate calculations.
By consistently applying the order of operations, whether using PEMDAS or BODMAS, we can avoid ambiguity and ensure that mathematical expressions are evaluated correctly. This foundational principle is essential for success in algebra, calculus, and other advanced mathematical topics. Understanding and practicing the order of operations not only improves accuracy but also builds a strong foundation for more complex problem-solving in mathematics and related fields.
To solve the operation 2 · 8 × 9 ÷ (48 ÷ 6), we need to follow the order of operations (PEMDAS/BODMAS). Here’s a detailed, step-by-step solution:
Step 1: Solve the Parentheses
The first step is to address the expression within the parentheses: (48 ÷ 6).
48 ÷ 6 = 8
So, the expression becomes:
2 · 8 × 9 ÷ 8
Step 2: Perform Multiplication and Division from Left to Right
Next, we perform multiplication and division from left to right. First, we have 2 · 8:
2 · 8 = 16
Now the expression is:
16 × 9 ÷ 8
Next, we multiply 16 by 9:
16 × 9 = 144
The expression is now:
144 ÷ 8
Finally, we divide 144 by 8:
144 ÷ 8 = 18
Final Answer
Therefore, the solution to the operation 2 · 8 × 9 ÷ (48 ÷ 6) is 18. This step-by-step breakdown ensures clarity and accuracy in solving the problem, making it easier to follow along and understand the process.
When solving mathematical operations, especially those involving multiple steps, it's common to make mistakes if the order of operations is not strictly followed. Recognizing these common errors and understanding how to avoid them can significantly improve accuracy and problem-solving skills. Here, we will discuss some frequent mistakes made in solving operations and strategies to prevent them.
1. Ignoring the Order of Operations
The most common mistake is not adhering to the order of operations (PEMDAS/BODMAS). This can lead to performing operations in the wrong sequence, resulting in incorrect answers. For example, consider the expression 3 + 4 × 2. If one adds 3 and 4 first, they get 7, then multiply by 2, resulting in 14. However, the correct order is to multiply 4 by 2 first, which gives 8, and then add 3, resulting in 11. Ignoring the order of operations can lead to significant errors, especially in more complex expressions.
How to Avoid: Always remember and apply the order of operations. Write down PEMDAS/BODMAS as a reminder and systematically work through each part of the expression. Start with parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right). This structured approach helps ensure that operations are performed in the correct sequence.
2. Incorrectly Handling Parentheses
Parentheses indicate the operations that should be performed first. A common mistake is to overlook parentheses or to perform operations inside them incorrectly. For instance, in the expression 2 × (5 - 3), the subtraction inside the parentheses must be done before multiplication. If someone mistakenly multiplies 2 by 5 first and then subtracts 3, they will arrive at the wrong answer. Similarly, with nested parentheses, it’s crucial to work from the innermost parentheses outwards.
How to Avoid: Always start by solving the innermost parentheses first and work your way outwards. Ensure each operation inside the parentheses is performed correctly before moving on to the next step. If the expression contains nested parentheses, take it one layer at a time to avoid confusion and errors.
3. Errors in Multiplication and Division
Multiplication and division should be performed from left to right, but it's easy to make mistakes if the order is not followed correctly. For example, in the expression 12 ÷ 3 × 2, division should be done before multiplication. If someone multiplies 3 by 2 first and then divides 12 by the result, they will get an incorrect answer. Similarly, arithmetic errors in the multiplication or division process can lead to wrong results.
How to Avoid: Perform multiplication and division from left to right as they appear in the expression. Double-check each calculation to ensure accuracy. If necessary, use a calculator or write out the steps to avoid simple arithmetic errors. Practicing these operations regularly can also improve speed and accuracy.
4. Errors in Addition and Subtraction
Like multiplication and division, addition and subtraction should also be performed from left to right. A common mistake is to add or subtract numbers in the wrong order, which can lead to incorrect results. For example, in the expression 7 + 5 - 3, addition should be done before subtraction. If someone subtracts 3 from 5 first and then adds 7, they will get an incorrect answer.
How to Avoid: Perform addition and subtraction from left to right. Double-check each calculation to ensure accuracy. Pay attention to the signs of the numbers and ensure they are correctly applied. If necessary, rewrite the expression to group positive and negative numbers separately before performing the operations.
5. Neglecting Negative Signs
Negative signs can be a source of confusion and errors, especially when combined with multiple operations. For example, in the expression 5 - (-3), the double negative should be simplified to a positive before proceeding with the subtraction. Neglecting to do so can lead to an incorrect result. Similarly, when multiplying or dividing negative numbers, it's essential to remember the rules for signs (e.g., a negative number multiplied by a negative number results in a positive number).
How to Avoid: Pay close attention to negative signs and ensure they are correctly applied throughout the calculation. Simplify double negatives to positive signs before performing other operations. Remember the rules for multiplying and dividing negative numbers to avoid sign errors. Practicing with expressions involving negative numbers can help build confidence and accuracy.
6. Careless Arithmetic Mistakes
Even when the correct order of operations is followed, simple arithmetic mistakes can lead to incorrect answers. These mistakes can range from misreading numbers to making errors in basic calculations like addition, subtraction, multiplication, and division. For example, accidentally writing 7 × 8 as 54 instead of 56 is a common arithmetic error.
How to Avoid: Double-check each calculation step to ensure accuracy. Write neatly and clearly to avoid misreading numbers. If necessary, use a calculator for complex calculations or break down the problem into smaller, more manageable steps. Regular practice can also help improve arithmetic skills and reduce the likelihood of careless mistakes.
By identifying these common mistakes and implementing strategies to avoid them, individuals can significantly improve their accuracy in solving mathematical operations. Consistent practice, attention to detail, and a systematic approach are key to mastering the order of operations and achieving correct results.
In conclusion, solving mathematical operations accurately requires a clear understanding and consistent application of the order of operations (PEMDAS/BODMAS). In this article, we addressed the operation 2 · 8 × 9 ÷ (48 ÷ 6) step by step, demonstrating how to prioritize parentheses, multiplication, and division to arrive at the correct answer, which is 18. We also discussed common mistakes, such as ignoring the order of operations, mishandling parentheses, and making arithmetic errors, and provided strategies to avoid them. By following the correct sequence and paying close attention to detail, one can confidently tackle complex mathematical problems. Regular practice and a systematic approach are essential for mastering these skills and achieving accuracy in mathematical calculations. This knowledge not only aids in academic pursuits but also in various real-life scenarios where mathematical problem-solving is crucial.
