Solving Significant Figures Problem What Is 12.0 Divided By 7.11

In the realm of chemistry, accuracy and precision are paramount. When performing calculations, it's not enough to simply arrive at a numerical answer; we must also express that answer with the appropriate number of significant figures. Significant figures are the digits in a number that carry meaning and contribute to its precision. They indicate the certainty with which a quantity is known. This comprehensive guide will delve into the rules for determining significant figures and how they apply to calculations, specifically focusing on the given problem: ${\frac{12.0}{7.11}=}$? We will explore the concept of significant figures, learn how to identify them, and apply the rules for calculations involving significant figures to arrive at the correct answer.

Significant figures are crucial in scientific calculations as they reflect the precision of measurements. They include all the digits that are known with certainty plus one estimated digit. The rules for identifying significant figures are as follows:

1. Non-zero digits are always significant. For example, in the number 123.45, all five digits are significant.
2. Zeros between non-zero digits are significant. For instance, the number 1002 has four significant figures.
3. Leading zeros are not significant. They serve only as placeholders. The number 0.0052 has only two significant figures (5 and 2).
4. Trailing zeros in a number containing a decimal point are significant. The number 12.230 has five significant figures.
5. Trailing zeros in a number not containing a decimal point may or may not be significant. This is where scientific notation can be helpful for clarity. For example, 1200 could have two, three, or four significant figures depending on the context. If written as 1.2 x 10^3, it has two significant figures; as 1.20 x 10^3, it has three; and as 1.200 x 10^3, it has four.

Understanding these rules is the first step in mastering calculations with significant figures. Ignoring these rules can lead to misrepresentation of the accuracy of your results, which is a significant concern in scientific disciplines.

When performing calculations, the number of significant figures in the result must reflect the precision of the measurements used. The rules vary slightly depending on the operation:

• For multiplication and division, the result should have the same number of significant figures as the measurement with the fewest significant figures. For example, if you divide 12.0 (3 significant figures) by 7.11 (3 significant figures), the answer should also have 3 significant figures.
• For addition and subtraction, the result should have the same number of decimal places as the measurement with the fewest decimal places. For instance, if you add 12.1 (one decimal place) and 3.12 (two decimal places), the result should be rounded to one decimal place.

These rules ensure that the calculated result does not imply a greater degree of precision than the original measurements allow. Always remember to apply these rules meticulously to maintain the integrity of your scientific work.

Now, let's apply these rules to the problem at hand: ${\frac{12.0}{7.11}}$. We need to divide 12.0 by 7.11 and express the answer with the correct number of significant figures. Following the rules, we first perform the division:

${\frac{12.0}{7.11} ≈ 1.687763713}$

Using a calculator, the result is approximately 1.687763713. However, we must consider significant figures. The number 12.0 has three significant figures, and 7.11 also has three significant figures. According to the rules for multiplication and division, our answer should have the same number of significant figures as the number with the fewest significant figures, which in this case is three.

To round 1.687763713 to three significant figures, we look at the fourth digit (7). Since it is 5 or greater, we round up the third digit (8). Therefore, the correct answer to three significant figures is 1.69.

It is important to note that among the given options, only one adheres to the rules of significant figures. Now, let's analyze the options provided to determine the correct answer.

We have calculated the result of ${\frac{12.0}{7.11}}$ to be approximately 1.69 when considering significant figures. Now, let’s evaluate the given options:

A. 1.688 B. 1.60 C. 1.7 D. 1.70

• Option A, 1.688, has four significant figures. This is incorrect because our calculation requires the answer to be rounded to three significant figures.
• Option B, 1.60, has three significant figures, but it is not the correct rounding of our calculated result. Although it adheres to the significant figure rule, the digits are incorrect.
• Option C, 1.7, has two significant figures. This is also incorrect because we need three significant figures.
• Option D, 1.69, has three significant figures and accurately reflects the result of our calculation rounded correctly. This is the most appropriate answer.

In summary, the solution to the problem ${\frac{12.0}{7.11}}$ expressed to the correct number of significant figures is 1.69. This comprehensive explanation highlighted the importance of understanding and applying the rules for significant figures in calculations. Mastery of these concepts is essential for accurate scientific work, ensuring that results reflect the true precision of measurements. Always remember to identify the number of significant figures in each measurement, apply the appropriate rules for calculations, and round your final answer accordingly. In this case, the correct answer was obtained by adhering to the rules of significant figures, reinforcing the necessity of this practice in chemistry and other scientific disciplines.

Therefore, the most appropriate answer, adhering to the rules of significant figures, is 1.69.