Calculating 0.05 + 1.5 × 5 ÷ 16 × 0.5 A Step-by-Step Guide
Understanding the Order of Operations
To accurately calculate the value of the expression 0.05 + 1.5 × 5 ÷ 16 × 0.5, it's crucial to understand and apply the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This set of rules dictates the sequence in which mathematical operations should be performed to arrive at the correct answer. Ignoring the order of operations can lead to drastically different and incorrect results.
In the given expression, we have addition, multiplication, and division. According to PEMDAS, multiplication and division should be performed before addition. Moreover, when multiplication and division appear in the same expression, they are carried out from left to right. This is a critical point to remember, as altering this sequence will change the outcome. Similarly, addition and subtraction are performed from left to right.
Therefore, our first step involves tackling the multiplication and division operations in their order of appearance. We begin by multiplying 1.5 by 5, followed by dividing the result by 16, and finally multiplying by 0.5. Only after completing these operations can we proceed to the addition part of the expression. Let's break down each step to ensure clarity and accuracy.
Step-by-Step Calculation
First, we perform the multiplication: 1. 5 × 5. This equals 7.5. This is a straightforward multiplication, but it's essential to double-check to avoid any minor errors that can cascade through the rest of the calculation. Next, we take this result and perform the division: 7. 5 ÷ 16. This yields 0.46875. Division can sometimes be trickier, especially with decimals, so it’s a good idea to either use a calculator or perform long division carefully. Now, we multiply this result by 0.5: 0. 46875 × 0.5. This gives us 0.234375. This multiplication finalizes the multiplication and division sequence in the expression.
With the multiplication and division operations completed, we are left with a single addition operation. We now add 0.05 to our previously calculated result: 0. 234375 + 0.05. This final addition gives us 0.284375. This is the value of the entire expression when adhering to the order of operations. The process highlights how each step builds upon the previous one, emphasizing the importance of accuracy in each operation.
Importance of Precision
In mathematical calculations, especially those involving decimals, precision is paramount. Even small rounding errors at intermediate steps can accumulate and lead to a significant difference in the final result. For example, if we had rounded 0.46875 to 0.47 prematurely, our final answer would be slightly off. Therefore, it's best practice to maintain as many decimal places as possible throughout the calculation and only round the final answer if necessary. This ensures the highest level of accuracy and reliability in the result.
In practical applications, such as engineering or finance, even minor discrepancies can have significant consequences. Therefore, a meticulous approach to calculations, with a focus on precision and adherence to the order of operations, is essential. This not only yields the correct answer but also instills a discipline that is crucial in many professional fields.
Detailed Breakdown of the Calculation
Let's delve deeper into each step of the calculation to ensure a comprehensive understanding. This detailed walkthrough will further clarify the application of the order of operations and highlight potential pitfalls to avoid. By breaking down each operation, we can better appreciate the mechanics of the calculation and minimize the chances of errors.
Step 1: Multiplication of 1.5 and 5
The first operation we encounter is the multiplication of 1.5 and 5. This is a relatively straightforward calculation, but it's crucial to perform it accurately as it sets the stage for the subsequent steps. When we multiply 1.5 by 5, we get 7.5. This can be verified through manual multiplication or using a calculator. The key here is to ensure that the decimal point is correctly placed in the result. An incorrect placement of the decimal point would lead to a wrong answer, affecting the rest of the calculation.
Step 2: Division of 7.5 by 16
Next, we divide 7.5 by 16. This operation yields a decimal result, which requires careful calculation. Performing this division gives us 0.46875. This step often benefits from using a calculator, especially if you are aiming for high precision. If performing long division manually, each step needs to be meticulously executed to avoid mistakes. The result, 0.46875, is a crucial intermediate value that will be used in the next operation.
Step 3: Multiplication by 0.5
We then multiply 0.46875 by 0.5. This is equivalent to finding half of 0.46875. The multiplication results in 0.234375. Again, attention to detail is essential here. A slight error in this multiplication can throw off the final result. The use of a calculator can help ensure accuracy, but understanding the underlying arithmetic principles is equally important.
Step 4: Final Addition
Finally, we add 0.05 to 0.234375. This is the last operation in the expression, and it combines the initial value with the result of the multiplication and division sequence. Adding these two numbers gives us 0.284375. This is the final value of the expression, and it represents the solution to our original problem.
Common Mistakes to Avoid
One common mistake in such calculations is neglecting the order of operations. For instance, someone might be tempted to add 0.05 and 1.5 first, which would lead to an entirely different and incorrect result. Another mistake is rounding off intermediate results too early. This can introduce errors that accumulate and affect the final answer. Therefore, it's best to maintain as much precision as possible throughout the calculation and only round the final answer if necessary.
Alternative Approaches and Verification
While following the order of operations is the standard approach, there are alternative ways to verify the result. One method is to use a calculator with built-in order of operations functionality. Inputting the expression directly into such a calculator should yield the same result, providing a check on the manual calculation.
Another approach is to break the problem into smaller parts and solve each part separately before combining the results. This can make the calculation more manageable and reduce the likelihood of errors. For instance, one could calculate 1.5 × 5 first, then divide the result by 16, multiply by 0.5, and finally add 0.05. This segmented approach can help in identifying and correcting mistakes more easily.
Furthermore, estimation can be a useful tool for verifying the reasonableness of the answer. For example, one could approximate 1.5 × 5 as 7.5, and 7.5 ÷ 16 as roughly 0.5. Then, 0.5 × 0.5 is 0.25, and adding 0.05 gives approximately 0.3. Comparing this estimate to the calculated result of 0.284375 gives confidence that the answer is in the right ballpark.
The Final Answer
Therefore, after carefully following the order of operations and performing each calculation with precision, we arrive at the final answer: 0.284375.
In conclusion, calculating the value of the expression 0.05 + 1.5 × 5 ÷ 16 × 0.5 requires a thorough understanding of the order of operations (PEMDAS) and meticulous attention to detail. Each step, from multiplication and division to addition, must be performed in the correct sequence to ensure an accurate result. The final answer, 0.284375, is a testament to the importance of precision and the application of mathematical principles. By breaking down the calculation into manageable steps and verifying the result through alternative approaches, we can confidently arrive at the correct solution. Remember, mathematical accuracy is not just about getting the right answer; it's about fostering a systematic and disciplined approach to problem-solving, a skill that is invaluable in many areas of life.