Integrated Math 1 / Unit 1: Quantities and Expressions
1.2 Dimensional Analysis and Rates
Notes
Dimensional analysis means using units to guide a conversion. Write each conversion factor as a fraction so that unwanted units cancel.
- A rate compares two quantities with different units, such as $\displaystyle \frac{60 \text{ miles}}{1 \text{ hour}}$.
- Example: convert 72 km/h to m/s: $\displaystyle \frac{72 \text{ km}}{1 \text{ h}} \cdot \frac{1000 \text{ m}}{1 \text{ km}} \cdot \frac{1 \text{ h}}{3600 \text{ s}} = 20 \text{ m/s}$.
- If the units do not cancel to what you want, flip a conversion factor.
- Check that your final answer has the right unit and a reasonable size.
Practice
A car travels 30 miles on 1 gallon of gas. Gas costs $4 per gallon. What is the gas cost for a 210-mile trip?
$\displaystyle \frac{210}{30} = 7$ gallons, and $7 \cdot 4 = 28$ dollars.
Which conversion factor correctly changes feet to inches?
Feet should cancel, so feet go in the denominator and inches in the numerator.
Convert 72 km/h to meters per second.
$72 \cdot 1000 = 72{,}000$ m per hour, and dividing by 3600 s gives 20 m/s.
Ready for more? Take the Unit 1 test.