Calculating 6 Times The Ceiling Of (4 - 9.4) A Step-by-Step Solution

This article delves into the step-by-step process of evaluating the mathematical expression $6\lceil 4-9.4\rceil$. We will break down each component of the expression, starting with the subtraction within the ceiling function, then applying the ceiling function itself, and finally, performing the multiplication. This exploration aims to provide a clear understanding of how to solve such expressions, emphasizing the significance of the ceiling function in mathematics. Whether you're a student tackling math problems or someone looking to refresh your knowledge, this guide will offer a comprehensive breakdown of the solution.

The problem at hand is to determine the value of the expression $6\lceil 4-9.4\rceil$. This expression involves several mathematical operations, including subtraction and the application of the ceiling function, before finally multiplying the result by 6. To accurately solve this, it's crucial to follow the correct order of operations and understand the function of the ceiling function. The ceiling function, denoted by $\lceil x \rceil$, returns the smallest integer that is greater than or equal to $x$. This means that for any real number, the ceiling function will round it up to the nearest integer. Before we dive into the step-by-step solution, let's recap the order of operations, often remembered by the acronym PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). By understanding these fundamental concepts and the order of operations, we can approach this problem methodically and arrive at the correct solution. This article will not only provide the solution but also ensure a clear understanding of the process involved, making it easier to tackle similar mathematical problems in the future.

1. Performing the Subtraction Inside the Ceiling Function

The first step in evaluating the expression $6\lceil 4-9.4\rceil$ is to perform the subtraction inside the ceiling function. We need to calculate $4 - 9.4$. This is a straightforward subtraction of two real numbers. When subtracting a larger number from a smaller number, the result will be negative. In this case, we are subtracting 9.4 from 4, which will give us a negative result. To find the result, we subtract the smaller number from the larger number and then apply the negative sign. So, we calculate $9.4 - 4$, which equals 5.4. Since we are subtracting 9.4 from 4, the result is $-5.4$. Therefore, the expression inside the ceiling function simplifies to $-5.4$. This step is crucial because the ceiling function will then be applied to this result. Understanding how to perform subtraction with real numbers, especially when dealing with negative results, is essential in mathematics. This initial step sets the stage for the next operation, which involves understanding and applying the ceiling function to the value we have just calculated. By correctly performing this subtraction, we ensure that the subsequent steps are based on an accurate foundation, leading us to the correct final answer.

2. Applying the Ceiling Function

After performing the subtraction inside the ceiling function, we have the expression $6\lceil -5.4\rceil$. The next crucial step is to apply the ceiling function to the value $-5.4$. The ceiling function, denoted by $\lceil x \rceil$, returns the smallest integer that is greater than or equal to $x$. In simpler terms, it rounds the number up to the nearest integer. When dealing with positive numbers, the concept is fairly straightforward. For example, the ceiling of 3.2 is 4 because 4 is the smallest integer greater than or equal to 3.2. However, when dealing with negative numbers, it's important to remember that “rounding up” means moving towards zero. For instance, consider the number $-2.7$. The integers around $-2.7$ are $-3$ and $-2$. The smallest integer that is greater than or equal to $-2.7$ is $-2$, not $-3$. Therefore, $\lceil -2.7 \rceil = -2$. Applying this concept to our problem, we have $-5.4$. The integers around $-5.4$ are $-6$ and $-5$. The smallest integer that is greater than or equal to $-5.4$ is $-5$. Hence, $\lceil -5.4 \rceil = -5$. This step is pivotal because it transforms the decimal value into an integer, which we can then use for the final multiplication step. A clear understanding of the ceiling function, especially its behavior with negative numbers, is vital for accurately solving this type of mathematical problem.

3. Performing the Multiplication

Having evaluated the ceiling function, we now have the expression $6 \times -5$. The final step in solving the problem is to perform this multiplication. This is a straightforward multiplication of two integers, one positive and one negative. When multiplying a positive number by a negative number, the result is always negative. In this case, we are multiplying 6 by -5. To find the result, we multiply the absolute values of the numbers and then apply the negative sign. The absolute value of 6 is 6, and the absolute value of -5 is 5. Multiplying 6 by 5 gives us 30. Since we are multiplying a positive number by a negative number, the result is negative. Therefore, $6 \times -5 = -30$. This final calculation gives us the solution to the original expression. It demonstrates the importance of following the order of operations and understanding the rules of multiplying integers with different signs. By accurately performing this multiplication, we complete the process of evaluating the expression $6\lceil 4-9.4\rceil$ and arrive at the final answer.

After meticulously working through each step, we have arrived at the final answer. First, we subtracted 9.4 from 4, which gave us -5.4. Then, we applied the ceiling function to -5.4, which resulted in -5. Finally, we multiplied 6 by -5, which yielded -30. Therefore, the value of the expression $6\lceil 4-9.4\rceil$ is -30. This solution highlights the importance of understanding the order of operations and the function of the ceiling function in mathematical expressions. Each step, from subtraction to the application of the ceiling function and finally multiplication, plays a crucial role in arriving at the correct answer. By breaking down the problem into manageable steps, we can ensure accuracy and clarity in our calculations. This comprehensive approach not only provides the solution but also reinforces the fundamental concepts of mathematics involved in the process.

In conclusion, the value of the mathematical expression $6\lceil 4-9.4\rceil$ is -30. This determination was reached through a step-by-step process that involved performing the subtraction within the ceiling function, applying the ceiling function itself, and then completing the multiplication. Each step is critical and requires a solid understanding of mathematical principles, including the order of operations and the behavior of the ceiling function. The ceiling function, in particular, plays a significant role by rounding a number up to the nearest integer, which is especially important when dealing with negative numbers. By following this structured approach, we can accurately solve complex mathematical expressions. This exercise not only provides a solution to a specific problem but also enhances our understanding of mathematical operations and functions, which is invaluable for tackling more complex challenges in the future. The ability to break down a problem into smaller, manageable steps is a key skill in mathematics, and this example serves as a practical demonstration of that skill. The final answer of -30 underscores the importance of precision and attention to detail in mathematical calculations.