The Unit Factor and Dimensional Analysis
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1 WORKSHEET 31 The Unit Factor and Dimensional Analysis The measurements you take in science class, whether for time, mass, weight, or distance, are more than just numbers they are also units. To make comparisons between measurements, it is convenient to have your measurements in the same units. A mathematical tool called a unit factor is used to convert back and forth between different kinds of units. A unit factor is a ratio that is equal to 1. Because it is equal to 1, multiplying a measurement by a unit factor changes the measurement s units but does not change its value. The skill of converting with a unit factor is known as dimensional analysis. Read on to see how it works. Part 1: Converting with a Unit Factor PROCEDURE: To convert units with a unit factor, determine the conversion factor between the units you have and the units you want to convert to. Then create the unit factor by making a ratio, in the form of a fraction, between the units you want to convert to in the numerator and the units you already have in the denominator. Finally, multiply your measurement by this unit factor to convert to the new units. SAMPLE PROBLEM A: Convert 3.5 km to millimeters. Step 1: Determine the conversion factor between kilometers and millimeters. 1 km Step 2: Create the unit factor. Put the units you want to convert to in the numerator and the units you already have in the denominator. 1 Step 3: Multiply the unit factor by the measurement. Notice that the original unit of the measurement cancels out with the unit in the denominator of the unit factor, leaving the units you are converting to. 3.5 km 3,500,000 mm On Your Own! 1. Convert the following measurements using a unit factor: a cm? mm b ml? L c. 12 kg? g 44 HOLT SCIENCE AND TECHNOLOGY
2 Part 2: Working with Square Units Many times in your science class, you will work with units of two dimensions, such as square centimeters (cm 2 ) or square kilometers (km 2 ). Dimensional analysis is especially useful when working with these types of units because it can help you to avoid confusing the different dimensions of your units. Carefully follow the steps in Sample Problem B to see how it works. SAMPLE PROBLEM B: 1 km 2 is how many square meters? Step 1: Simplify the units you are converting. 1 km 2 1 km 1 km Step 2: Create the unit factor for converting meters to kilometers. As in Sample Problem A, put the units you are converting to in the numerator. 1 1 km Step 3: Multiply the measurement you are converting by the unit factor. Because 1 km 2 1 km 1 km, you will need to multiply the measurement you are converting from by two unit factors. Notice that the original unit of measurement cancels the units in the denominator. This leaves the units you are converting to. 1 km 2 1,000,000 m m 1 km 1 km 1 km 2 1,000,000 m 2 Practice Your Skills! 2. Convert the following measurements: a. 3 cm 2? m 2 b. 12,000 m 2? km 2 c. 980 cm 2? mm 2 3. An Olympic-sized soccer field has an area of km 2. How many square meters does a soccer field cover? FOR SCIENCE 45
3 Working with Cubic Dimensions Because volume can be measured by multiplying length times height times width, volume is expressed in units of three dimensions, or cubic units. Volume is often expressed in cubic millimeters (mm 3 ) or cubic centimeters (cm 3 ), but larger volumes may be expressed in cubic meters (m 3 ) or cubic kilometers (km 3 ). A cubic centimeter (cm 3 ) is equal to one milliliter (ml), and a cubic decimeter (dm 3 ) is equal to one liter (L). Doing dimensional analysis with cubic units is much like doing dimensional analysis with square units, except that with cubic units you will multiply the measurement you are converting by three unit factors instead of two. Follow the steps in Sample Problem C to see how it is done. SAMPLE PROBLEM C: A certain plant needs about 525 cm 3 of soil to grow properly. How many cubic meters of soil is this? Step 1: Simplify the units you are converting. cm 3 cm cm cm Step 2: Create the unit factor for converting centimeters to meters, putting the units you are converting to in the numerator. 100 cm Step 3: Multiply the measurement you are converting by the unit factors. Because cm 3 cm cm cm, you will need to multiply the measurement you are converting from by three unit factors. 525 cm m m m 100 cm 100 cm 100 cm 525 cm m 3 Try It Yourself! 4. Convert the following measurements: a. 30 m 3? cm 3 b mm 3? m 3 c. 4 km 3? m 3 Challenge Yourself! 5. The Mississippi River has an average water discharge of 17,000 m 3 per second. How many cubic kilometers of water does the river discharge in 1 hour? Show your work. 46 HOLT SCIENCE AND TECHNOLOGY
4 WORKSHEET 31 The Unit Factor and Dimensional Analysis The measurements you take in science class, whether for time, mass, weight, or distance, are more than just numbers they are also units. To make comparisons between measurements, it is convenient to have your measurements in the same units. A mathematical tool called a unit factor is used to convert back and forth between different kinds of units. A unit factor is a ratio that is equal to 1. Because it is equal to 1, multiplying a measurement by a unit factor changes the measurement s units but does not change its value. The skill of converting with a unit factor is known as dimensional analysis. Read on to see how it works. Part 1: Converting with a Unit Factor PROCEDURE: To convert units with a unit factor, determine the conversion factor between the units you have and the units you want to convert to. Then create the unit factor by making a ratio, in the form of a fraction, between the units you want to convert to in the numerator and the units you already have in the denominator. Finally, multiply your measurement by this unit factor to convert to the new units. SAMPLE PROBLEM A: Convert 3.5 km to millimeters. Step 1: Determine the conversion factor between kilometers and millimeters. 1 km Step 2: Create the unit factor. Put the units you want to convert to in the numerator and the units you already have in the denominator. 1 Step 3: Multiply the unit factor by the measurement. Notice that the original unit of the measurement cancels out with the unit in the denominator of the unit factor, leaving the units you are converting to. 3.5 km 3,500,000 mm On Your Own! 1. Convert the following measurements using a unit factor: a cm? mm b ml? L c. 12 kg? g 10 mm 1 cm 1 L L 1000 g 1 kg 23.4 mm L 12,000 g 44 HOLT SCIENCE AND TECHNOLOGY
5 Part 2: Working with Square Units Many times in your science class, you will work with units of two dimensions, such as square centimeters (cm 2 ) or square kilometers (km 2 ). Dimensional analysis is especially useful when working with these types of units because it can help you to avoid confusing the different dimensions of your units. Carefully follow the steps in Sample Problem B to see how it works. SAMPLE PROBLEM B: 1 km 2 is how many square meters? Step 1: Simplify the units you are converting. 1 km 2 1 km 1 km Step 2: Create the unit factor for converting meters to kilometers. As in Sample Problem A, put the units you are converting to in the numerator. 1 1 km Step 3: Multiply the measurement you are converting by the unit factor. Because 1 km 2 1 km 1 km, you will need to multiply the measurement you are converting from by two unit factors. Notice that the original unit of measurement cancels the units in the denominator. This leaves the units you are converting to. 1 km 2 1,000,000 m m 1 km 1 km 1 km 2 1,000,000 m 2 Practice Your Skills! 2. Convert the following measurements: a. 3 cm 2? m 2 b. 12,000 m 2? km cm 1 km c. 980 cm 2? mm 2 10 mm 1 cm m km 2 98,000 mm 2 3. An Olympic-sized soccer field has an area of km 2. How many square meters does a soccer field cover? Unit factor: ; km 2 1 km 1 km 1 km 7776 m 2 FOR SCIENCE 45
6 Working with Cubic Dimensions Because volume can be measured by multiplying length times height times width, volume is expressed in units of three dimensions, or cubic units. Volume is often expressed in cubic millimeters (mm 3 ) or cubic centimeters (cm 3 ), but larger volumes may be expressed in cubic meters (m 3 ) or cubic kilometers (km 3 ). A cubic centimeter (cm 3 ) is equal to one milliliter (ml), and a cubic decimeter (dm 3 ) is equal to one liter (L). Doing dimensional analysis with cubic units is much like doing dimensional analysis with square units, except that with cubic units you will multiply the measurement you are converting by three unit factors instead of two. Follow the steps in Sample Problem C to see how it is done. SAMPLE PROBLEM C: A certain plant needs about 525 cm 3 of soil to grow properly. How many cubic meters of soil is this? Step 1: Simplify the units you are converting. cm 3 cm cm cm Step 2: Create the unit factor for converting centimeters to meters, putting the units you are converting to in the numerator. 100 cm Step 3: Multiply the measurement you are converting by the unit factors. Because cm 3 cm cm cm, you will need to multiply the measurement you are converting from by three unit factors. 525 cm m m m 100 cm 100 cm 100 cm 525 cm m 3 Try It Yourself! 4. Convert the following measurements: a. 30 m 3? cm 3 b mm 3? m 3 c. 4 km 3? m 3 1 km 100 cm 30,000,000 cm 3 m m 3 4,000,000,000 m 3 Challenge Yourself! 5. The Mississippi River has an average water discharge of 17,000 m 3 per second. How many cubic kilometers of water does the river discharge in 1 hour? Show your work. 1 km 1 km 1 km 1 km Unit factor: ; 17,000 m 3 /s km 3 /s; km 3 /s 60 s/min km 3 /min; km 3 /min 60 min/hr km 3 /hr 46 HOLT SCIENCE AND TECHNOLOGY
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