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# 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.