Perform the following arithmetic and round off the answers...

Perform the following arithmetic and round off the answers to the correct number of significant figures. Include the correct units with the answers. (a) $3.58 \mathrm{~g} / 1.739 \mathrm{~mL}$ (b) $4.02 \mathrm{~mL}+0.001 \mathrm{~mL}$ (c) $(22.4 \mathrm{~g}-8.3 \mathrm{~g}) /(1.142 \mathrm{~mL}-0.002 \mathrm{~mL})$ (d) $(1.345 \mathrm{~g}+0.022 \mathrm{~g}) /(13.36 \mathrm{~mL}-8.4115 \mathrm{~mL})$ (e) $(74.335 \mathrm{~m}-74.332 \mathrm{~m}) /(4.75 \mathrm{~s} \times 1.114 \mathrm{~s})$

Perform the following arithmetic and round off the answers to the correct number of significant figures. Include the correct units with the answers.
(a) $3.58 \mathrm{~g} / 1.739 \mathrm{~mL}$
(b) $4.02 \mathrm{~mL}+0.001 \mathrm{~mL}$
(c) $(22.4 \mathrm{~g}-8.3 \mathrm{~g}) /(1.142 \mathrm{~mL}-0.002 \mathrm{~mL})$
(d) $(1.345 \mathrm{~g}+0.022 \mathrm{~g}) /(13.36 \mathrm{~mL}-8.4115 \mathrm{~mL})$
(e) $(74.335 \mathrm{~m}-74.332 \mathrm{~m}) /(4.75 \mathrm{~s} \times 1.114 \mathrm{~s})$

Key Concepts

Significant Figures

Significant figures represent the digits in a number that carry meaning contributing to its precision. They include all the non-zero digits, any zeros between them, and, where applicable, trailing zeros after the decimal point. In measurements and calculations, significant figures indicate the degree of confidence in the reported value and determine how results should be rounded.

Rounding

Rounding involves reducing the number of significant digits in a number while keeping its approximate value. This process is essential to ensure that the final result does not imply greater precision than the measurements allow. Rounding should be done according to standard rules, such as rounding up if the next digit is 5 or greater.

Multiplication and Division (Significant Figures Rules)

When performing multiplication or division, the rule for significant figures states that the result should have as many significant figures as the measurement with the fewest significant figures used in the calculation. This rule ensures that the precision of the final result is consistent with the least precise measurement.

Addition and Subtraction (Decimal Place Rules)

For addition and subtraction, the final answer should be rounded off to the same number of decimal places as the term with the smallest number of decimal places. This rule is applied instead of counting significant figures and helps maintain the consistency of the measurement's precision in the summation or difference.

Units and Unit Analysis

Including units in calculations is crucial for defining the physical quantity represented by numerical values. Unit analysis, or dimensional analysis, involves checking that the units correctly cancel or combine in a calculation to produce a result in the appropriate unit. This practice ensures the physical validity and consistency of the computed answer.

Transcript

00:02 For these problems, we need to perform the calculation, round to the correct number of significant digits, and include the proper units.

00:10 For problem a, we're dividing.

00:13 When we divide for significant digits, we're going to count the number of significant digits.

00:18 For this first one, we have three significant digits and the denominator has four significant digits.

00:29 So in our answer, we're going to round to the fewest, which is three significant digits.

00:34 So if we put it on the calculator, you get 2 .05865...

00:41 It continues on.

00:43 We want to round to the correct number of significant digits, which is only three.

00:47 So we're going to round 2 .06.

00:50 So the eight after the five rounds it up to six.

00:54 Our units, we're just going to follow the units that were in the problem.

00:58 So grams per milliliter.

01:00 For the second one, when we add, we look at decimal places.

01:05 So 4 .02 has two numbers after the decimal place.

01:09 0 .001 has three numbers after the decimal place.

01:14 We're going to take the shortest.

01:15 So when you put it on the calculator, you would get 4 .021, but we're going to turn that into having only two places after the decimal point, which is 4 .02.

01:32 Our unit milliliters plus milliliters is just milliliters.

01:36 It's like combining like terms in math.

01:39 All right.

01:40 For c, we have a numerator, which we are going to subtract.

01:45 And both of these have only one decimal place.

01:48 So we can round our answer to only one decimal place, which is 14 .1 grams on the top for the numerator.

01:56 For the denominator, both of them have three decimal places.

02:01 So we are going to keep three decimal places in our answer there.

02:06 So 1 .140 milliliters.

02:09 Include that zero to show that it is significant.

02:12 Now, when we're dividing, we count all our significant digits.

02:16 So this has three significant digits.