Question

Round off the following quantities to the indicated number of significant figures. (a) \(132.505 \mathrm{~g}\) (four significant figures) (b) \(298.693 \mathrm{~cm}\) (five significant figures) (c) \(13.452 \mathrm{lb}\) (two significant figures) (d) 345 oz (two significant figures)

Short answer

Question: Round the following quantities to the specified number of significant figures: (a) \(132.505 \mathrm{~g}\) to four significant figures, (b) \(298.693 \mathrm{~cm}\) to five significant figures, (c) \(13.452 \mathrm{lb}\) to two significant figures, and (d) 345 oz to two significant figures. Answer: (a) \(132.5 \mathrm{~g}\), (b) \(298.69 \mathrm{~cm}\), (c) \(13 \mathrm{~lb}\), (d) 350 oz.

Step-by-step solution

(a) Round \(132.505 \mathrm{~g}\) to four significant figures.

When rounding off to four significant figures, we need to look at the fifth digit. In this case, the fifth digit is \(5\). Since it is equal to or greater than \(5\), we will round the fourth digit up by one. Our final result will be \(132.5 \mathrm{~g}\) with four significant figures.

(b) Round \(298.693 \mathrm{~cm}\) to five significant figures.

We need to look at the sixth digit to round off to five significant figures. In this case, the sixth digit is \(3\). Since it is less than \(5\), we will keep the fifth digit as it is. So, the final result remains \(298.69 \mathrm{~cm}\) with five significant figures.

(c) Round \(13.452 \mathrm{lb}\) to two significant figures.

The third digit is what we need to look at when rounding to two significant figures. In this case, the third digit is \(4\). Since it is less than \(5\), we will keep the second digit as it is. We are left with \(13 \mathrm{~lb}\) as the final result, with two significant figures.

(d) Round 345 oz to two significant figures.

To round to two significant figures, we need to look at the third digit. In this case, it is \(5\). Since it's equal to or greater than \(5\), we round the second digit up by one. The final result is 350 oz, with two significant figures.

Key concepts

Rounding Off Numbers

Understanding how to round off numbers is crucial in various scientific fields, including chemistry, where precision can significantly impact the results. To round a number, you first need to determine how many significant figures are required for your calculation or measurement. Significant figures are the digits that carry meaning contributing to its precision.

Rounding involves looking at the digit immediately to the right of the last significant figure you want to keep. If this digit is five or greater, you increase the last significant figure by one. If it is less than five, you leave the last significant figure as it is.

For example, properly rounding the number 132.505 to four significant figures would result in 132.5, since the digit after the fourth significant figure is 5. Similarly, rounding 345 ounces to two significant figures would lead to 350 ounces, as explained in the step by step solution. Applying these rules ensures that the numerical value you are working with is as close as possible to the original number, while still conforming to the needed degree of precision.

Significant Figures in Measurements

In scientific measurements, significant figures are vital because they indicate the precision of the measurement being taken. They help communicate the certainty, or the lack thereof, in the reported numbers. The more significant figures a number has, the more precise the measurement. However, it's important not to overstate the precision; it should reflect the actual capability of the measuring instrument used.

For instance, when measuring length with a ruler that has millimeter marks, the measurement can be precise up to a millimeter. If you measure something as 298.693 centimeters with such a ruler, you may need to consider significant figures based on the context of your experiment. As detailed in the steps, rounding this to five significant figures retains that level of precision, resulting in 298.69 centimeters.

It's essential to understand that the number of significant figures in a given measurement also dictates the significant figures in the result of a calculation. This ensures that the result of a calculation is not artificially more precise than the measurements that were used to obtain it.

Accuracy in Chemistry Calculations

In chemistry, accuracy refers to how close a measured or calculated value is to the true value. When performing chemistry calculations, accuracy is partly achieved by properly managing significant figures throughout the process. This begins with measuring substances correctly, continues with using the correct number of significant figures in calculations, and ends with rounding the final result appropriately.

For example, if we mix 13.452 pounds of a substance with another measured quantity, and we want to keep only two significant figures for a rough calculation, we would use 13 pounds, as described in the solution steps. When dealing with chemical reactions where the ratios of reactants matter greatly, knowing how to maintain the accuracy of your calculations through significant figures is paramount.

An accurate result, properly rounded to reflect its significant figures, ensures the reliability of the data and fosters confidence in the conclusions drawn from experiments. Therefore, mastering the use of significant figures is not just about following mathematical rules; it's also about ensuring the integrity and reproducibility of scientific work.