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Factor-Label Technique (aka Dimensional Analysis)

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

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Factor-Label Technique (aka Dimensional Analysis)

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  1. 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

  2. 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

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

  4. 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

  5. Conversion Factors….Part 3 • 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

  6. Conversion Factors….Part 4More 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

  7. Conversion Factors….Part 5Still 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

  8. Conversion Factors….Part 5Continued…………… • The right side of our equation becomes one because 4 quarts divided by 4 quarts is 1 • 1 gallon = 14 quarts • Rearranging this becomes 1 = 1 gallon 4 quarts

  9. Conversion Factors….Part 5Continued…………… • 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?

  10. 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

  11. Putting It TogetherHere’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

  12. 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

  13. The Answer…………… • 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-

  14. 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

  15. 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

  16. 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”

  17. 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

  18. 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

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