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) $0.0023 \mathrm{~m} \times 315 \mathrm{~m}$ (b) $84.25 \mathrm{~kg}-0.01075 \mathrm{~kg}$ (c) $(184.45 \mathrm{~g}-94.45 \mathrm{~g}) /(31.4 \mathrm{~mL}-9.9 \mathrm{~mL})$ (d) $(23.4 \mathrm{~g}+102.4 \mathrm{~g}+0.003 \mathrm{~g}) /(6.478 \mathrm{~mL})$ (e) $(313.44 \mathrm{~cm}-209.1 \mathrm{~cm}) \times 8.2234 \mathrm{~cm}$

Perform the following arithmetic and round off the answers to the correct number of significant figures. Include the correct units with the answers.
(a) $0.0023 \mathrm{~m} \times 315 \mathrm{~m}$
(b) $84.25 \mathrm{~kg}-0.01075 \mathrm{~kg}$
(c) $(184.45 \mathrm{~g}-94.45 \mathrm{~g}) /(31.4 \mathrm{~mL}-9.9 \mathrm{~mL})$
(d) $(23.4 \mathrm{~g}+102.4 \mathrm{~g}+0.003 \mathrm{~g}) /(6.478 \mathrm{~mL})$
(e) $(313.44 \mathrm{~cm}-209.1 \mathrm{~cm}) \times 8.2234 \mathrm{~cm}$

Key Concepts

Dimensional Analysis

Dimensional analysis is the process of keeping track of units during calculations to ensure that the resulting units are consistent with the physical quantity being measured. This concept is essential for verifying that the final answers are expressed in the correct units, which is as important as obtaining the correct numerical result.

Addition and Subtraction (Sig Figs)

For addition and subtraction, the result should be rounded off to the least number of decimal places in any of the measurements. This rule differs from the one used in multiplication and division and ensures that the precision of the final result does not exceed that of the least precise measurement in terms of decimal place accuracy.

Multiplication and Division (Sig Figs)

When multiplying or dividing, the result should be rounded off to the same number of significant figures as the measurement with the fewest significant figures. This rule helps to maintain the precision of the result based on the least precise measurement in the calculation.

Rounding Off

Rounding off involves reducing the number of digits in a number while keeping its value close to the original. In the context of significant figures, the result from an arithmetic operation is rounded to reflect the limited precision of the input quantities. Proper rounding is crucial in ensuring that the uncertainty in the measurements is not overstated in the final answer.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. This concept is important in scientific measurements and calculations because it reflects the uncertainty in the measured quantities. Only the reliable digits are used in calculations, and the final results must be rounded to the correct number of significant figures to accurately represent the precision of the original measurements.

Transcript

00:01 In this problem of scientific measurement we have to perform the following arithmetic and round of the answers of the correct number to significant figure and also we have to include the correct units with the answer.

00:16 So we have given first this is 0 .0023 meter is multiplied with 300 and 15 meter and then we have given 18 .25 kilograms minus this is kilograms minus 0 .01 075 kilograms and then we have given 184 .45 so this is here 184 .45 grams minus 984 .45 grams minus 94 .45 so this is here 94 .45 grams and this is divided with this is divided with 13 this is 31 .4 this is milliliter minus 99 point this is 9 .9 milliliter and then we have given 23 .4 grams plus so this is 102 .4 grams plus 0 .003.

01:40 This is 0 .003 grams.

01:45 And this is divided with 6 .478 milliliters.

01:51 So this is 6 .478 milliliter.

01:56 And then we have given this is 313 .44 centimeters.

02:07 Minus 209 .1 centimeter and this is multiplied with 8 .2234 centimeter and now we have to evaluate it.

02:23 And now we have to do the multiplication so 0 .0023 is multiplied with 315 is equals to 0 .725.

02:36 So this would be here 0 .725 meter square.

02:43 And now we have to see the significant figure.

02:47 So here we have only two significant figure and here we have three significant figure.

02:52 So in multiplication we have to see the minimum significant figure.

02:56 So as you can see here, this is only two significant figure.

03:00 So we have to write only two significant figure.

03:03 So this would be 0 .7 to 45.

03:06 Now this is 4 so this can be here eliminated so this would be only 0 .72 meter square so we can say the right answer is 0 .72 meter square.

03:18 Similarly here this is 84 .25 minus 0 .01075 kgs so this is equals to here this is equals to so, 84 point, this is 84 point, 239, this is 23925 and this would be kg.

03:48 And now we have to see the significant figure.

03:50 So here we have 1, 2, 3, 4 significant figure and here we have also 4 significant figure.

03:56 So we have to keep only 4 significant figure in the answer.

04:00 So this would be 1, 2, 3 and 4 and now we have to do the rounding of.

04:04 So this is 3 and after 3, this is 9.

04:06 So this has to be increased by 1.

04:09 So this can be here increased by 1.

04:12 So this would be equal to 84.

04:15 This is 24 kilograms.

04:18 So this is 84 .24 kilograms.

04:22 And now here we have given 184 .45 minus 94 .45.

04:32 And this is in gram.