Understanding the Expression 3 ÷ 4 × 34: A Comprehensive Guide
Understanding the Basics of Order of Operations
When dealing with mathematical expressions, it is crucial to follow a specific sequence of steps to ensure accurate results. This sequence is known as the order of operations. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) serves as a helpful mnemonic to remember the correct order. In this article, we will explore how to solve the expression 3 ÷ 4 × 34 by strictly following the order of operations.
Evaluating the Expression: A Step-by-Step Guide
The expression in question is 3 ÷ 4 × 3 4. Let's break down the process, adhering to the principles of PEMDAS, to find the correct solution.
Step 1: Division
According to PEMDAS, we start by addressing the division operation first.
3 ÷ 4 0.75
Step 2: Multiplication
Next, we multiply the result from the division by 3.
0.75 × 3 2.25
Step 3: Addition
Finally, we add 4 to the result of the multiplication.
2.25 4 6.25
Therefore, the value of 3 ÷ 4 × 3 4 is 6.25.
Another Approach to Evaluating the Expression
Let's consider another interpretation of the expression, ensuring that we prioritize multiplication before performing addition.
3 ÷ 4 × 34 3/4 × 3 4 division before multiplication
First, we perform the division:
3 ÷ 4 0.75
Next, we multiply 0.75 by 3:
0.75 × 3 2.25
Then, we add 4 to the result of the multiplication:
2.25 4 6.25
Hence, the value of 3 ÷ 4 × 3 4 is 6.25, which aligns with the first method of solution.
Ensuring Clarity in Mathematical Expressions
It is essential to pay careful attention to the order of operations and the clarity of the mathematical expression. In our example, the expression 3 ÷ 4 × 3 4 could be ambiguous if the intended operations are not clearly defined.
To avoid such ambiguities, it is beneficial to use parentheses or a horizontal line as a fraction bar to clearly indicate the order of operations. For instance, the correct interpretation could be:
(3 ÷ 4) × (3 4) (3/4) × 7 2.25 × 7 15.75
or
3 ÷ (4 × (3 4)) 3 ÷ (4 × 7) 3 ÷ 28 0.1071 (repeating)
By ensuring the correct order of operations and using proper notation, we can avoid misinterpretations and arrive at the accurate solution.
Conclusion
In conclusion, understanding the order of operations, specifically adhering to PEMDAS, is crucial for evaluating mathematical expressions accurately. The expression 3 ÷ 4 × 3 4 evaluates to 6.25 when properly solved according to the rules of PEMDAS. By following these guidelines, you can simplify complex expressions and arrive at the correct solutions.