How can we tell if two quantities are directly proportional? QUESTION: Given: 1 inch is exactly equivalent to 2.54 cm. A measurement of 7.00 cm is equivalent to how many inches? A. 2.76 in., B. 3 in., C. 17.8 in., D. 20 in. In general, measurements expressed in different units are directly proportional quantities. How can we tell if two quantities are directly proportional? When one quantity is zero, the other is also zero When one quantity changes, the other changes by the same factor (example: if one quantity is doubled, the other quantity is also doubled) Their ratio is a constant. (L in in. / L in cm) = constant Given that one inch is exactly equivalent to 2.54 centimeters, a measurement of 7.00 cm is equivalent to how many inches? PAUSE CLICK In general, measurements expressed in different units are directly proportional quantities. A great deal of chemical calculations, not just unit conversions, involve directly proportional quantities. It is, therefore, important you are able to recognize this type of relationship. So what does it mean when we say two quantities are directly proportional? CLICK First of all, when one quantity is zero, then the other must also be zero. You can see than if the length in inches is zero, then the length in centimeters is also zero. Furthermore... when one quantity changes, the other quantity also changes by the same factor. For example, we know that if the length in inches doubles, the length in inches also doubles. Mathematically, we can say that the ratio of the two quantities is a constant. CLICK Continued on next slide

L1 in. 1 in. L2 cm L2 cm = 2.54 cm L2 cm In other words,... If L1 is the length in inches CLICK and L2 is the length in centimeters Then the ratio of L1 to L2 is equal to... The ratio of 1 inch to 2.54 cm L1 inch is to L2 centimeter HIGHLIGHT ratio on the left as 1 inch is to 2.54 centimeters. HIGHLIGHT ratio on the right If we multiply both sides by L2 centimeters, We can cancel L2 centimeters on the left hand side. CLICK CLICK What does this mean?... If we know the length in centimeters... HIGHLIGHT L2 Then, all we have to do is multiply it by the ratio of the known equivalents... HIGHLIGHT 1 in. / 2.54 cm in order to calculate the length in inches. CONTINUED ON NEXT SLIDE

General strategy for unit conversion: L1 in. 1 in. L2 cm L2 cm = 2.54 cm L2 cm L2 cm 2.54 cm L1 in. L1 in. = 1 in. L1 in. Similarly,... If we know the length in inches, HIGHLIGHT L1 in. in second equation we just multiply it by the ratio 2.54 centimeters over 1 inch...... HIGHLIGHT 2.54 cm / 1 in. in order to calculate the length in centimeters. CLICK In other words, the general strategy for converting units is to take the known amount and multiply it by a ratio of equivalent amounts. We call this ratio of equivalent amounts a conversion factor. Note that we set up our conversion factor so that the the original unit cancels out when we do the calculation. If we want to convert centimeters to inches, we want centimeters to cancel out CLICK CLICK If we want to convert inches to centimeters, we want inches to cancel out CLICK CLICK CLICK CONTINUED ON NEXT SLIDE General strategy for unit conversion: (known amount) x Conversion factor Conversion factor = ratio of equivalent amounts

QUESTION: Given: 1 inch is exactly equivalent to 2.54 cm. A measurement of 7.00 cm is equivalent to how many inches? A. 2.76 in., B. 3 in., C. 17.8 in., D. 20 in. L1 in. 1 in. L2 cm 7.00 cm L2 cm = 2.54 cm L2 cm Let’s go back to the question... We are given the length in centimeters, HIGHLIGHT 7.00 cm So, we multiply the length in centimeters CLICK by a conversion factor that allows us to cancel the centimeter unit. CLICK Punching these numbers into a calculator gives us an answer of 2.7559055 inches. We round the answer to three significant digits. HIGHLIGHT 2.76 in. The correct answer is A. Let’s review why the answer should only have three significant digits? There are 3 significant figures in 7.00 centimeters. Since 1 inch and 2.54 are, by definition, exact. HIGHLIGHT “exactly” in the question they do not have any uncertainty. Therefore, the least precise term in this calculation is 7.00, which has 3 significant figures. In general, when doing unit conversions, the numbers in conversion factors are exact. So the number of significant figures in the answer should be the same as in the original measurement. PAUSE 2 seconds END RECORDING = 2.7559055 in.  2.76 in.

Video ID: 1A-5-1 © 2008, Project VALUE (Video Assessment Library for Undergraduate Education), Department of Physical Sciences Nicholls State University Author: Glenn V. Lo Narrator: Funded by Louisiana Board of Regents Contract No. LA-DL-SELECT-13-07/08