Dimensional Analysis (or factor label method) is a way of mathematically approaching many types of mathematical problems in science.  Common applications include unit conversions, problems using ratios, and solving stoichiometry problems in chemistry.

DEFINE - Factor label method involves the multiplication of a given value by a series of fractions arranged to yield the desired value to answer a problem.  The fractions are arranged in such a way so that units cancel leaving the correct unit in the final answer.  This process is also called dimensional analysis.  

Factor label method is most often encountered by science students as a method of unit conversion (usually for conversions other than metric unit prefixes).  Dimensional analysis is also used as a way to check manipulated equations to see if they are correct before doing the math.  If you are the type of student who is often not sure whether to multiply or divide, then these techniques will work for you.

Consider the formula for density and common units for each of the quantities.

Since the units for mass are grams, and for volume the units are milliliters, the formula tells us that the unit for the calculated density must be g/mL.  If you see this then you are well on your way to understanding dimensional analysis.

USE - Let's say you wanted to calculate the mass of 720 mL of vegetable oil and you knew the density of the oil was 0.85 g/mL.  To do this calculation you need to rearrange the formula to solve for mass.  There are three possibilities:

Only one of the equations can be correct.  If you are good at algebra you probably already know that the correct equation is the third choice.  LOOK at the units though.  The third choice is also the only one where the units on the right side of the equal sign cancel to give grams, the same as on the left side.  This can be very useful to check your work or as a strategy for when you get stuck on more complex problems.

Now we can move on to factor label method.

To convert between units a conversion factor is required.  Here are some examples:

1 minute = 60 seconds

1 hour = 60 minutes

1 inch = 2.54 cm

1 imperial gallon = 4546 mL

Each of these conversion factors is an equality.  One minute of time is the same amount of time as 60 s for example.  This means if a conversion factor is arranged as a fraction (either way) it has a value of 1.  This means you can multiply a number by any conversion factor fraction, and you will not change the overall value ONLY THE UNITS.

EXAMPLE - Let's say we want to convert 4.5 inches to centimeters.  Start by writing the given information (4.5 inches) and then determine the desired result (centimeters).  Arrange the conversion factor fraction so that the units cancel to give the desired result.  Then do the math: 

EXAMPLE - Convert 1.25 days to seconds.  In this example, multiple conversion factors are arranged as fractions to get the desired unit (seconds) in the final answer.  See if you can follow how the units cancel (dimensional analysis) to get the correct answer.

Applying factor label method - a very basic introduction :  

APPLY - Practice questions 

Applying factor label method - same idea, more complex problem as it deals with derived units - this is one of the best teaching videos I have ever seen :  

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