Some Math to Get Us Started

• We will be doing some math problems this semester and I would like to review a basic practice.
• The Factor-Label method.
  • I have always called this dimensional analysis.
  • It generally occurs when you have a quantity in one unit, and you want it in another unit.
  • This will solve a majority of math problems in this class.
• A simple problem
  • What is $\frac{245}{33}\times \frac{11}{35}$?
  • We can factor each
  • What is $\frac{7^2\times 5}{11 \times 3 }\times \frac{11}{7 \times 5}$?
  • We then cancel the common factors in the numerator and denominator
  • What is $\frac{7\cancel{^2}\times \cancel{5}}{\cancel{11} \times 3 }\times \frac{11}{\cancel{7} \times \cancel{5}}$?
  • What is $\frac{7}{3 }\times \frac{1}{1} $
  • Or $\frac{7}{3}$
• In this method, we do the same thing, but with units.
• A simple example.
  • I have a measurement in inches, and I would like to know how many feet that is.
  • I have: 475 inches.
  • I want: feet.
  • I need the conversion factor, 12 inches = 1 ft
  • So I set up the following:
    • $ 475 \text{in} \times \frac{1 \text{ft}}{12 \text{in}} $
    • $ 475 \cancel{\text{in}} \times \frac{1 \text{ft}}{12 \cancel{\text{in}}} = \frac{475}{1}\text{ft} $
  • Notice, we "cancel" the units.
• This method:
  • Makes it so we don't have to memorize many formulas.
  • It helps us discover what information is needed to solve a problem
  • It allows us to check our work so we don't do this.
  • It allows us to understand others work.
• A more complex example
  • I want to drive to my mothers house in Ohio, what will that cost me (in gas)?
    • I have: ?
    • I want: $
    • The cost of gas is $3.99 per gallon.
      • What does the per mean?
      • $\frac{\$3.99}{1 \text{gallon}}$
      • Or $\$3.99 = 1 \text{gallon}$.
    • So can I find something related to my trip involving gallons?
    • My truck gets 23.5 miles per gallon
      • $\frac{23.5\text{miles}}{1 gallon}$
    • If I start to multiply these I get
    • $\frac{\$3.99}{1 \text{gallon}} \times \frac{23.5\text{miles}}{1 text{gallon}}$
    • $= \frac{\$3.99 \times 23.5\text{dollar-miles}} {1 \text{gallon} \times 1 \text{gallon}} $
    • $ = \frac {3.99 \times 23.4 \text{dollar-mile}} {1 \text{gallon}^2}$
    • There seems to be something wrong here
      • What is a dollar-mile
      • Why do I have square gallons?
    • If we flipped the second fraction, could things work out better?
    • $\frac{\$3.99}{1 \text{gallon}} \times \frac{1 \text{gallon}}{23.5\text{miles}}$
    • $ = \frac{\$3.99}{1 \cancel{\text{gallon}}} \times \frac{1 \cancel{\text{gallon}}}{23.5\text{miles}}$
    • $ = \frac{\$3.99}{23.5\text{miles}}$
  • But can I do this?
    • $\frac{\$3.99}{1 \text{gallon}}$
    • Or $\$3.99 = 1 \text{gallon}$.
    • So divide both sides by $\$3.99$
    • $ 1 = \frac{1 \text{gallon}}{\$3.99}$
  • Now I have a value of dollars/mile.
    • Is it rational? A good thing?
    • Is this helpful?
    • What do I need to finish this problem?
    • I want to make the miles go away.
    • Do I have another piece of information?
      • It is about 300 miles round trip to visit my mom.
      • So we have $\frac{1 \text{trip}} {300 \text{miles}}$
      • Or $\frac{300 \text{miles}}{1 \text{trip}}$
      • Which do I want and why?
  • So the final solution is:
    • $\frac{\$3.99}{1 \text{gallon}} \times \frac{1 \text{gallon}}{23.5\text{miles}} \times \frac{300 \text{miles}}{1 \text{trip}}$
    • $\frac{\$3.99}{1 \cancel{\text{gallon}}} \times \frac{1 \cancel{\text{gallon}}}{23.5 \cancel{\text{miles}}} \times \frac{300 \cancel{\text{miles}}}{1 \text{trip}} $
    • $= \frac{3.99 \times 300 }{23.5} \frac{\$}{\text{trip}}$
    • $=\$50.94$
• In general:
  • You need the units you want in the end.
  • Either work backwards, or forwards if you know your starting units.
  • Pay attention to the current units to derive the new units.
  • In the end, you want the equivalent of this equation:
  • $\frac{\$3.99}{1 \cancel{\text{gallon}}} \times \frac{1 \cancel{\text{gallon}}}{23.5 \cancel{\text{miles}}} \times \frac{300 \cancel{\text{miles}}}{1 \text{trip}} = \frac{\$3.99 \times 300 }{23.5 \text{trip}} =\$50.94$
• Notice you can
  • Check this for accuracy
  • Update it if the conversion factors change
  • Understand it if you didn't do it.
  • This is a way to document your calculations.