- 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 $\$3.99 = 1 \text{gallon}$.

- What does the
- 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}} $
- $= \frac{3.99 \times 300 }{23.5} \frac{\$}{\text{trip}}$
- $=\$50.94$

- I want to drive to my mothers house in Ohio, what will that cost me (in gas)?
- 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.