Solving Math Problems With Significant Figures A Step-by-Step Guide

In the realm of mathematics and scientific calculations, precision is paramount. When solving complex problems, it's not just about arriving at a numerical answer; it's about expressing that answer with the appropriate level of certainty. This is where the concept of significant figures comes into play. Significant figures are the digits in a number that carry meaning contributing to its precision. They dictate the accuracy of the result we present, ensuring that we don't overstate or understate the reliability of our calculations. In this comprehensive guide, we will delve into the intricacies of solving the mathematical expression $\frac{105}{22} \times(1.251-0.620)$, paying close attention to the rules of significant figures every step of the way. By mastering these principles, you'll be equipped to tackle a wide range of mathematical and scientific problems with confidence and accuracy. This article will not only provide a step-by-step solution to the given problem but also offer a thorough explanation of the underlying concepts. We will break down the calculation into manageable parts, highlighting the importance of significant figures in each operation. From subtraction to division and multiplication, we'll explore how the rules of significant figures apply and how to ensure your final answer reflects the true precision of your calculations. Whether you're a student learning the ropes or a professional seeking a refresher, this guide will serve as an invaluable resource. So, let's embark on this journey of mathematical precision and unravel the intricacies of significant figures together.

When addressing the mathematical expression $\frac{105}{22} \times(1.251-0.620)$, the first step involves tackling the subtraction within the parentheses: 1.251 - 0.620. This seemingly simple operation is crucial, as it sets the stage for the rest of the calculation and directly influences the accuracy of our final answer. When dealing with subtraction and addition, the guiding principle for significant figures is to consider the numbers' decimal places. The result should be rounded to the same number of decimal places as the number with the fewest decimal places involved in the operation. In our case, we have 1.251, which has three decimal places, and 0.620, which also has three decimal places. Performing the subtraction, we get:

1.  251
-
2.  620
------
3.  631

The result, 0.631, has three decimal places, which aligns with the rule we've established. Since both numbers in the subtraction had three decimal places, our result maintains that same level of precision. It's important to note that the number of significant figures in each number plays a critical role here. The number 1.251 has four significant figures, and 0.620 has three significant figures. However, in addition and subtraction, we prioritize the decimal places over the total number of significant figures. This is because addition and subtraction are concerned with the absolute uncertainty of the measurements, rather than the relative uncertainty. For example, if we were measuring lengths with a ruler that had millimeter markings, our uncertainty would be in the thousandths place. Therefore, when adding or subtracting these measurements, we need to ensure our result is also accurate to the thousandths place. Now, with the result of the subtraction (0.631) in hand, we can move on to the next step of our calculation, keeping in mind that this value, with its three decimal places and three significant figures, will influence the final answer. This meticulous attention to detail is what allows us to maintain mathematical integrity and express our results with the appropriate level of certainty. By understanding and applying the rules of significant figures in each step, we ensure that our final answer accurately reflects the precision of the original measurements and calculations.

Having successfully completed the subtraction, the next step in solving the expression $\frac{105}{22} \times(1.251-0.620)$ is to tackle the division: 105 / 22. This operation introduces a new facet to our consideration of significant figures, as the rules for division (and multiplication) differ from those for addition and subtraction. In division and multiplication, the guiding principle is to focus on the total number of significant figures in each number involved. The result should be rounded to the same number of significant figures as the number with the fewest significant figures. Let's break down our division: 105 has three significant figures, and 22 has two significant figures. Therefore, according to the rule, our result should be rounded to two significant figures. Performing the division, we get:

105 / 22 ≈ 4.772727...

This result has a string of digits after the decimal point, but we need to round it to two significant figures. The first two significant figures are 4 and 7. To determine whether we need to round up or down, we look at the next digit, which is 7. Since 7 is greater than or equal to 5, we round up the 7 to an 8. Thus, the result of the division, rounded to two significant figures, is 4.8. It is crucial to understand why we prioritize the total number of significant figures in division and multiplication. These operations deal with relative uncertainties. For example, if we are calculating the area of a rectangle by multiplying its length and width, the relative uncertainty in the area will depend on the relative uncertainties in the length and width. The number with the fewest significant figures has the largest relative uncertainty, and therefore, it limits the precision of the result. By rounding our division result to two significant figures, we acknowledge this limitation and avoid overstating the accuracy of our calculations. Now, with the result of the division (4.8) and the result of the subtraction (0.631) in hand, we are well-prepared to proceed to the final step: multiplication. We will carry forward the understanding of significant figures gained in this division step, ensuring that our final answer accurately reflects the precision of all the operations we have performed.

With the subtraction and division steps completed, we arrive at the final operation in the expression $\frac{105}{22} \times(1.251-0.620)$: multiplication. We now need to multiply the result of the division (4.8) by the result of the subtraction (0.631). As we established in the previous step, multiplication follows the same significant figures rules as division. This means that the final answer should have the same number of significant figures as the number with the fewest significant figures in the multiplication. In this case, we are multiplying 4.8 (which has two significant figures) by 0.631 (which has three significant figures). Therefore, our final answer should be rounded to two significant figures. Let's perform the multiplication:

4.  8 * 0.631 ≈ 3.0288

This gives us the result 3.0288. However, we need to round this to two significant figures. The first two significant figures are 3 and 0. To determine whether to round up or down, we look at the next digit, which is 2. Since 2 is less than 5, we round down, keeping the 0 as it is. Therefore, the final answer, rounded to two significant figures, is 3.0. The process of rounding to significant figures is not merely an exercise in mathematical precision; it's a crucial step in communicating the certainty of our results. By rounding to two significant figures, we are acknowledging that our final answer has a level of precision that is consistent with the least precise value used in the calculation. This prevents us from making claims about the accuracy of our result that are not supported by the data. In scientific and engineering contexts, this is especially important, as decisions are often made based on the results of calculations. Misrepresenting the precision of a result could lead to flawed decisions and potentially serious consequences. In summary, by meticulously applying the rules of significant figures throughout the calculation, we have arrived at the final answer, 3.0. This answer not only solves the mathematical problem but also accurately reflects the precision of the operations involved. This careful attention to detail is the hallmark of a skilled mathematician and scientist.

In conclusion, the journey of solving the mathematical expression $\frac{105}{22} \times(1.251-0.620)$ has highlighted the critical importance of significant figures in mathematical and scientific calculations. We've meticulously navigated through each step, from subtraction to division and multiplication, emphasizing the rules that govern how we maintain and express precision in our results. The final answer, 3.0, is not just a numerical solution; it's a testament to the careful application of these principles. Significant figures are not arbitrary digits; they are the digits in a number that carry meaning and contribute to its precision. They tell us the degree of certainty we have in our measurements and calculations. By adhering to the rules of significant figures, we ensure that our results accurately reflect this level of certainty, preventing us from overstating or understating the reliability of our findings. Throughout this guide, we've seen how the rules for significant figures differ depending on the operation. In addition and subtraction, we focus on the number of decimal places, while in multiplication and division, we focus on the total number of significant figures. These distinctions are crucial for maintaining accuracy and avoiding errors. The application of significant figures extends far beyond academic exercises. In real-world scenarios, particularly in science, engineering, and finance, the accuracy of calculations can have significant implications. Whether it's calculating the dosage of a medication, designing a bridge, or managing financial assets, precise calculations are essential for making informed decisions. Understanding and applying the principles of significant figures is a fundamental skill for anyone working with numbers. It's a skill that promotes clarity, accuracy, and integrity in calculations, ensuring that our results are not only correct but also meaningful. As we conclude this detailed guide, remember that the mastery of significant figures is an ongoing process. It requires practice, attention to detail, and a deep understanding of the underlying principles. By embracing these principles, you'll be well-equipped to tackle a wide range of mathematical and scientific challenges with confidence and precision. The next time you encounter a calculation, remember the lessons learned here, and strive to express your answer with the correct number of significant figures. Your accuracy and clarity will be greatly enhanced, and your work will stand as a testament to your commitment to precision.

• Significant Figures
• Mathematical Calculations
• Precision
• Subtraction
• Division
• Multiplication
• Rounding
• Accuracy
• Problem Solving
• Mathematics
• Scientific Calculations