Factor-Label Technique

(aka Dimensional Analysis) This technique involves the use of conversion factors and writing all measurements with both numerical values and the unit of measurement A conversion factor is where you have the same amount (entity) represented by two different units of measurement with their corresponding numerical values

Conversion Factors

       Here are some examples 1 foot = __ inches 1 meter = ____ millimeters 1 inch = 2.54 centimeters 1 gallon = __ quarts 1 acre = 4840 square yards 1 day = ___ hours

Missing answers to previous slide   1 foot = 12 inches 1 meter = 1000 millimeters  1 gallon = 4 quarts  1 day = 24 hours

Conversion Factors….Part 2

 One member of a dinner party orders a 16 ounce steak and another orders a one pound steak- Compare the two steaks  They are the same since 16 oz dry wt. = 1 pound

Conversion Factors….Part 3

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In grade school we learned that 1 gallon contained 4 quarts or stating that relationship as an equality: 1 gallon = 4 quarts Since 1 gallon and 4 quarts represent the same amount, we have a

Conversion Factor

    Conversion Factors….Part 4 More About Gallons and Quarts Start with 1 gallon = 4 quarts Now divide each side by 1 gallon we get this equation

1 gallon = 4 quarts 1 gallon 1 gallon

Since 1 gallon divided by 1 gallon equals 1

 Our equality becomes:

1 = 4 quarts 1 gallon

Conversion Factors….Part 5

Still more about Gallons & Qts.

  Again start with 1 gallon = 4 quarts But this time we’ll divide each side of the equality by 4quarts  The resulting equation is 

1 gallon = 4 quarts 4 quarts 4quarts

Which becomes

Conversion Factors….Part 5

Continued……………

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

1 gallon = 1 4 quarts

 Rearranging this becomes 1 = 1 gallon 4 quarts

Conversion Factors….Part 5

Continued……………

    A mid-presentation summary We know that 1 gallon = 4 quarts Using a little mathematical magic

1 gallon = 1 and 4 quarts = 1 4 quarts 1 gallon

 Why is this an important concept?

Conversion Factors….Part 6

How Do They Work

      Now a little math review…………….

What is 5 x 1?

What is 5 x 2 ?

2 Both expressions give you the same answer why?

Because 2/2 equals 1 and therefore the second equation is just like the first and We did not change the initial value of 5

Putting It Together Here’s An Example

   How many quarts are in 15 gallons ?

Remember we do NOT want to change the amount represented by 15 gallons, only the units in quarts So we’ll use the conversion factor between gallons and quarts; that is 1 gallon = 4 quarts

Our Example

continued…….

  We set it up like this: 15 gallons x 4 quarts 1 gallon This works because we have already shown that 4 quarts divided by 1 gallon is like multiplying 15 gallons by 1

The Answer……………

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Solving this equation we get 60 quarts The 60 comes from 15 x 4 in our equation we get quarts because the gallons cancelled each other out So, 15 gallons and 60 quarts represent the same quantity-

Confused over the units ?

Or what happened to the gallons    We had: 15 gallons x 4 quarts 1 gallon Perhaps it will be more visible, err clear, if we change 15 gallons to a fraction by placing it over 1. Like below: 15 gallons x 4 quarts = 1 1 gallon

Continuing our Discussion

    The gallons cancel one another out just like the “a” in this algebraic equation 5a x 1 = 5 a There is one “a” in the numerator and an “a” in the denominator So the gallons cancelled leaving quarts

The Steps

    Preliminary chores: From the problem determine the following: What the Known quantity is (number and units) which is called the Given Identify what the Desired units are What do we know about the relationship between the two units of measurement “the conversion factor”

     Step 1: Write the initial (Given) quantity, both the number and its units Step 2: Write the times sign “x” after the Given Step 3: Draw the fraction line after the times sign Step 4: Write the unit of the Given under the fraction line forming the denominator of the conversion factor Step 5: Write the unit of the Desired above the fraction line creating the numerator of the conversion factor

    Step 6: Write in the appropriate numerical values thereby making a correct conversion factor Step 7: Cancel units (not the number in front of the units) and perform the necessary mathematical operations Step 8: If the resulting unit is not the one you need for the final answer, repeat steps 2 through 7 until you’re there Now it’s your turn solving the problems in the Factor-Label Problem Set