2I CAN solve mathematical conversions using Dimensional Analysis.
3Dimensional Analysis is an orderly and systematic approach used to make MATHEMATICAL CONVERSIONS from one unit to another easy to accomplish.Since measurements require both a NUMBER and a UNIT, this method is often referred to as the FACTOR LABEL method because UNITS MUST BE INCLUDED with each measurement.
4Using the Factor Label Method Let’s use a simple example to learn this method:If someone were to tell you that they had 275 eggs and asked you how many dozen eggs that represented, how would you find that answer?Before solving that problem, you would have to know that 12 eggs make up 1 DOZEN.Once you knew that, solving the problem would be easy.
5You would simple divide 275 by 12 to determine how many dozen eggs you have. Written out mathematically, your calculation would look something like this:275 eggs dozen = dozen12 eggsFrom this set up, do you see any UNITS that could be canceled DIAGONALLY?
6This is called the FACTOR LABEL METHOD: 275 eggs dozen = dozen12 eggsNotice the EGGS canceled diagonally, leaving the final unit of DOZENS as the only unit for the answer.This is called the FACTOR LABEL METHOD:Conversions involve NUMBERS with UNITS.What was the relationship between 12 eggs and 1 dozen?
7Since 12 eggs make 1 dozen, this relationship is called a CONVERSION FACTOR. A conversions factor represents a number of smaller units that make 1 of a larger unit, as in this case 12 eggs (smaller units) = 1 dozen (larger unit).Dimensional Analysis is written using a CROSS-BRACKET like this:
8This is usually what’s GIVEN IN THE PROBLEM. A dimensional analysis always begins with WHAT YOU ALREADY KNOW and is written in the upper corner of the frame:=XThis is usually what’s GIVEN IN THE PROBLEM.In our previous example, we were given 275 eggs.
9Generally NOTHING (put an X in it) is written in the bottom of the first frame: 275 eggs The SECOND FRAME contains the CONVERSION FACTOR:12 EGGS = 1 DOZEN
10How do you know which value GOES ON TOP and which value GOES ON BOTTOM? That’s easy….when everything is added to the frame, ALL UNITS EXCEPT THE FINAL ONE MUST CANCEL DIAGONALLY!If they do not, something is in the wrong place!
11EXAMPLE Now let’s set up our egg problem using DIMENSIONAL ANALYSIS: 275 EGGS1 DOZEN275 Dozen12=X12 EGGSOnce all the numbers are in place and UNITS ARE CANCELED…everything ACROSS the TOP LINE is MULTIPLIED, then everything ACROSS the BOTTOM LINE is MULTIPLIED, then the TOP is DIVIDED by the BOTTOM for the FINAL ANSWER.
12PRACTICE PROBLEMThere are inches in 1 meter. Set up a DIMENSIONAL ANALYSIS problem to convert 8.75 meters to inches.
138.75 meters39.37 inchesinches1=X1 meterOnce all the numbers are in place and UNITS ARE CANCELED…everything ACROSS the TOP LINE is MULTIPLIED, then everything ACROSS the BOTTOM LINE is MULTIPLIED, then the TOP is DIVIDED by the BOTTOM for the FINAL ANSWER.= INCHES
14YOUR TURN SHOW YOUR WORK as DIMENSIONAL ANALYSIS Problem #1 How many inches are there in 45 meters if each meter is inches?How many ounces are contained in 37.5 pounds assuming 1 pound is 16 ounces?
15Practice with CONVERSION FACTORS Below are some common CONVERSION FACTORS. If you do NOT know these, use Google or another search engine to find them.1. 1 foot = _____ inches meter = ______ centimeter3. 1 pound = ______ ounces pound = ______ grams5. 1 mile = ______ feet mile = _______ kilometers7. 1 liter = ______ ounces gallon = _______ ounces9. 1 ton = _______ pounds meter = ______ millimetersgallon = ______ milliliters inch = _____ centimeters