Factor-Label TechniqueFactor-Label Technique(aka Dimensional Analysis)(aka Dimensional Analysis)

This technique involves the use of This technique involves the use of conversion factors and writing all conversion factors and writing all measurements with both numerical values measurements with both numerical values and the unit of measurementand the unit of measurement

A conversion factor is where you have the A conversion factor is where you have the same amount (entity) represented by two same amount (entity) represented by two different units of measurement with their different units of measurement with their corresponding numerical valuescorresponding numerical values

Conversion FactorsConversion Factors

Here are some examplesHere are some examples 1 foot = __ inches1 foot = __ inches 1 meter = ____ millimeters1 meter = ____ millimeters 1 inch = 2.54 centimeters1 inch = 2.54 centimeters 1 gallon = __ quarts1 gallon = __ quarts 1 acre = 4840 square yards1 acre = 4840 square yards 1 day = ___ hours1 day = ___ hours

Missing answers to previous Missing answers to previous slideslide

1 foot = 12 inches1 foot = 12 inches 1 meter = 1000 millimeters1 meter = 1000 millimeters

1 gallon = 4 quarts1 gallon = 4 quarts

1 day = 24 hours1 day = 24 hours

Conversion Factors….Part 2Conversion Factors….Part 2

One member of a dinner party orders One member of a dinner party orders a 16 ounce steak and another orders a 16 ounce steak and another orders a one pound steak- Compare the two a one pound steak- Compare the two steakssteaks

They are the same since 16 oz dry They are the same since 16 oz dry wt. = 1 poundwt. = 1 pound

Conversion Factors….Part 3Conversion Factors….Part 3

In grade school we learned that In grade school we learned that 1 gallon contained 4 quarts or 1 gallon contained 4 quarts or stating that relationship as an stating that relationship as an equality:equality:

1 gallon = 4 quarts1 gallon = 4 quarts Since 1 gallon and 4 quarts Since 1 gallon and 4 quarts

represent the same amount, we represent the same amount, we have a have a Conversion FactorConversion Factor

Conversion Factors….Part 4Conversion Factors….Part 4 More About Gallons and QuartsMore About Gallons and Quarts

Start with 1 gallon = 4 quartsStart with 1 gallon = 4 quarts Now divide each side by 1 gallon Now divide each side by 1 gallon we get this equationwe get this equation 1 gallon 1 gallon = = 4 quarts 4 quarts 1 gallon 1 gallon1 gallon 1 gallon Since 1 gallon divided by 1 gallon equals Since 1 gallon divided by 1 gallon equals

11

Our equality becomes:Our equality becomes: 1 = 1 = 4 quarts 4 quarts 1 gallon1 gallon

Conversion Factors….Part 5Conversion Factors….Part 5Still more about Gallons & Qts.Still more about Gallons & Qts.

Again start with 1 gallon = 4 quartsAgain start with 1 gallon = 4 quarts But this time we’ll divide each side of But this time we’ll divide each side of

the equality by 4quartsthe equality by 4quarts The resulting equation isThe resulting equation is 1 gallon 1 gallon = = 4 quarts 4 quarts

4 quarts 4quarts4 quarts 4quarts

Which becomesWhich becomes

Conversion Factors….Part 5Conversion Factors….Part 5Continued……………Continued……………

The right side of our equation becomes The right side of our equation becomes one because 4 quarts divided by 4 quarts one because 4 quarts divided by 4 quarts is 1is 1

1 gallon 1 gallon = 1= 14 quarts4 quarts

Rearranging this becomes 1 = Rearranging this becomes 1 = 1 gallon 1 gallon

4 quarts4 quarts

Conversion Factors….Part 5Conversion Factors….Part 5Continued……………Continued……………

A mid-presentation summaryA mid-presentation summary We know that 1 gallon = 4 quartsWe know that 1 gallon = 4 quarts Using a little mathematical magicUsing a little mathematical magic 1 gallon 1 gallon = 1 and = 1 and 4 quarts4 quarts = 1 = 1

4 quarts 1 gallon4 quarts 1 gallon

Why is this an important concept?Why is this an important concept?

Conversion Factors….Part 6 Conversion Factors….Part 6 How Do They WorkHow Do They Work

Now a little math review…………….Now a little math review……………. What is 5 x 1?What is 5 x 1? What is 5 x What is 5 x 2 2 ? ? 22 Both expressions give you the same Both expressions give you the same

answer- why?answer- why? Because 2/2 equals 1 and therefore the Because 2/2 equals 1 and therefore the

second equation is just like the first andsecond equation is just like the first and We did not change the initial value of 5We did not change the initial value of 5

Putting It TogetherPutting It TogetherHere’s An ExampleHere’s An Example

How many quarts are in 15 gallons ?How many quarts are in 15 gallons ? Remember we do Remember we do NOTNOT want to want to

change the amount represented by change the amount represented by 15 gallons, only the units in quarts15 gallons, only the units in quarts

So we’ll use the conversion factor So we’ll use the conversion factor between gallons and quarts; that isbetween gallons and quarts; that is

1 gallon = 4 quarts1 gallon = 4 quarts

Our Example Our Example continued…….continued…….

We set it up like this:We set it up like this:

15 gallons x 15 gallons x 4 quarts 4 quarts

1 gallon1 gallon This works because we have already This works because we have already

shown that 4 quarts divided by 1 shown that 4 quarts divided by 1 gallon is like multiplying 15 gallons gallon is like multiplying 15 gallons by 1by 1

The Answer……………The Answer……………

Solving this equation we get 60 Solving this equation we get 60 quarts quarts

The 60 comes from 15 x 4 in our The 60 comes from 15 x 4 in our equation equation

we get quarts because the gallons we get quarts because the gallons cancelled each other outcancelled each other out

So, 15 gallons and 60 quarts So, 15 gallons and 60 quarts represent the same quantity-represent the same quantity-

Confused over the units ?Confused over the units ?Or what happened to the gallonsOr what happened to the gallons

We had: 15 gallons x We had: 15 gallons x 4 quarts 4 quarts 1 gallon1 gallon Perhaps it will be more visible, err Perhaps it will be more visible, err

clear, if we change 15 gallons to a clear, if we change 15 gallons to a fraction by placing it over 1. Like fraction by placing it over 1. Like below:below:

15 gallons 15 gallons x x 4 quarts 4 quarts = = 1 1 gallon1 1 gallon

Continuing our DiscussionContinuing our Discussion The gallons cancel one another out The gallons cancel one another out

just like the “a” in this algebraic just like the “a” in this algebraic equationequation

5a x 5a x 1 1 = 5 = 5

aa There is one “a” in the numerator and There is one “a” in the numerator and

an “a” in the denominatoran “a” in the denominator So the gallons cancelled leaving So the gallons cancelled leaving

quartsquarts

The StepsThe Steps

Preliminary chores: From the problem Preliminary chores: From the problem determine the following:determine the following:

What the Known quantity is (number What the Known quantity is (number and units) which is called the Givenand units) which is called the Given

Identify what the Desired units areIdentify what the Desired units are What do we know about the What do we know about the

relationship between the two units of relationship between the two units of measurement- “the conversion factor”measurement- “the conversion factor”

Step 1: Write the initial (Given) quantity, Step 1: Write the initial (Given) quantity, both the number and its unitsboth the number and its units

Step 2: Write the times sign “x” after the Step 2: Write the times sign “x” after the GivenGiven

Step 3: Draw the fraction line after the Step 3: Draw the fraction line after the times signtimes sign

Step 4: Write the unit of the Given under Step 4: Write the unit of the Given under the fraction line forming the denominator the fraction line forming the denominator of the conversion factorof the conversion factor

Step 5: Write the unit of the Desired above Step 5: Write the unit of the Desired above the fraction line creating the numerator of the fraction line creating the numerator of the conversion factorthe conversion factor

Step 6: Write in the appropriate numerical Step 6: Write in the appropriate numerical values thereby making a correct values thereby making a correct conversion factorconversion factor

Step 7: Cancel units (not the number in Step 7: Cancel units (not the number in front of the units) and perform the front of the units) and perform the necessary mathematical operationsnecessary mathematical operations

Step 8: If the resulting unit is not the one Step 8: If the resulting unit is not the one you need for the final answer, repeat steps you need for the final answer, repeat steps 2 through 7 until you’re there2 through 7 until you’re there

Now it’s your turn solving the problems in Now it’s your turn solving the problems in the Factor-Label Problem Setthe Factor-Label Problem Set