Question

Carry out the following operations and express the answers with the appropriate number of significant figures. (a) \(14.3505+2.65\) (b) \(952.7-140.7389\) (c) \(\left(3.29 \times 10^{4}\right)(0.2501)\) (d) \(0.0588 / 0.677\)

Short answer

(a) \(17.00\) (b) \(812.0\) (c) \(8.22 \times 10^{4}\) (d) \(0.0869\)

Step-by-step solution

(a) Addition

To perform the addition \(14.3505 + 2.65\), we need to align the numbers by the decimal point and add them. However, we must also maintain the least number of decimal places in the final answer due to significant figures rules. \(14.3505\) \(+\) \(2.6500\) \(=\) \(17.0005\) Since the lowest number of decimal places in the numbers is 2 (in 2.65), the final answer should have 2 decimal places. Therefore, the final answer is: \(17.00\)

(b) Subtraction

To perform the subtraction \(952.7 - 140.7389\), we need to align the numbers by the decimal point and subtract them. As with addition, we must also maintain the least number of decimal places in the final answer due to significant figures rules. \(952.7000\) \(-\) \(140.7389\) \ (=\) \(811.9611\) Since the lowest number of decimal places in the numbers is 1 (in 952.7), the final answer should have 1 decimal place. Therefore, the final answer is: \(812.0\)

(c) Multiplication

To perform the multiplication \(\left(3.29 \times 10^{4}\right)(0.2501)\), first multiply the numbers. \(3.29 \times 10^{4} \times 0.2501 = 82232.9\) When multiplying numbers, the final answer should have the same number of significant figures as the original number with the least significant figures. In this case, 3.29 has 3 significant figures, while 0.2501 has 4. So, the final answer should have 3 significant figures. Therefore, the final answer is: \(8.22 \times 10^{4}\)

(d) Division

To perform the division \(0.0588 / 0.677\), simply divide the numbers. \(0.0588 / 0.677 = 0.08686352\) When dividing numbers, the final answer should have the same number of significant figures as the original number with the least significant figures. In this case, both 0.0588 and 0.677 have 3 significant figures. Thus, the final answer should have 3 significant figures. Therefore, the final answer is: \(0.0869\)

Key concepts

Operations with Significant Figures

Working with significant figures is a fundamental aspect of scientific measurement and calculation. When performing operations with numbers, it's not only the value that matters but the precision specified by the number of significant figures. This precision represents how accurately a number has been measured.

For instance, when we add or subtract numbers, we must align them by their decimal points and then round off the result to the least number of decimal places provided in any number. During multiplication or division, the answer should have the same number of significant figures as the number with the fewest significant figures used in the calculation. This process ensures that the result is neither overestimated nor underestimated in precision.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a concise form. It is written as a product of a number between 1 and 10 and a power of 10. For example, the number 6500 could be written as \(6.5 \times 10^{3}\).

This notation is particularly useful when dealing with significant figures because it makes the identification of significant digits straightforward. Any zeros after the decimal point and before the first non-zero digit are not considered significant. For example, in \(0.00052\), only the 5 and 2 are significant.

Addition and Subtraction with Significant Figures

In addition and subtraction, the rule of thumb is to look at the decimal places rather than the total number of digits. The result should be rounded off to the least precise measurement used in the calculation. For example, when we added \(14.3505 + 2.65\), we rounded the sum, \(17.0005\), to two decimal places, giving \(17.00\) because \(2.65\) had the least number of decimal places, two.

This approach maintains the integrity of the precision throughout the computation. Not adhering to these rules may result in an inaccurate representation of the data's precision.

Multiplication and Division with Significant Figures

The significant figures concept applies differently in multiplication and division compared to addition and subtraction. Here, it's the count of significant digits that's important, not the position of the decimal point. The rule we follow is that the final result should have the same number of significant figures as the factor with the fewest significant figures.

For instance, in the multiplication problem \(3.29 \times 0.2501\), we calculated the answer and then rounded it to three significant figures, the same as in the least precise number \(3.29\), resulting in \(8.22 \times 10^{4}\). A similar process is used for division, emphasizing accuracy in significant digits throughout all operations.