DIMENSIONAL ANALYSIS

Dimensional Analysis is a problem solving technique that is indispensible to the sciences. It relies on the relationship between two quantities. For example, a $5km$ road race is also a certain distance in miles. In order to figure out how many miles are in $5km,$ we need to know the relationship between kilometers & miles. These relationships are referred to as conversion factors. A conversion factor is a fraction that expresses the relationship between two numbers and can be used to convert from one quantity to another.

Example: $"1km=0\mathrm{.}621$ miles $"$ can be written as $\frac{1(km).}{0\mathrm{.}621mi}$ or $\frac{0\mathrm{.}621mi}{1(km)}$

PROCEDURE FOR WORKING PROBLEMS BY DIMENSIONAL ANALYSIS $($UNIT CANCELLATION$)$:

Step $1.$ Write the provided value or measurement on the left.

Step $2.$ Put the target unit on the right.

Step $3.$ Choose a conversion factor or set of conversion factors and arrange them in such a way that the given unit is algebraically canceled and converted to the target unit. This requires brainstorming and may involve looking up a particular conversion factor or making your own from relationships given in the problem.

Ex: $($algebra$)250a**\frac{1b}{100a}=25b,($units$)250cm**\frac{1(m)}{100(cm)}=25m$

Step $4.$ Multiply numerators. Divide by denominators to get an answer.

Step $5.$ Round off to the correct number of significant digits in your problem.

given unit $x\frac{\text{t}\text{a}\text{r}\text{g}\text{e}\text{t}\text{u}\text{n}\text{i}\text{t}}{\text{g}\text{i}\text{v}\text{e}\text{n}\text{u}\text{n}\text{i}\text{t}}=$ target unit

Conversion Problem Examples

a$)$ How many miles are in $5km?$

b$)$ How many kilometers are in $500$ miles?

a$)5\mathrm{.}0km\frac{0\mathrm{.}621\text{m}\text{i}\text{l}\text{e}\text{s}}{1(km)}=3\mathrm{.}1$miles

b$)500mi\frac{1(km)}{0\mathrm{.}621mi}=800km$

c$)$ How many fluid ounces are in a gallon of milk?

c$)1\mathrm{.}0$galx$\frac{3\mathrm{.}7854(L)}{1\text{g}\text{a}\text{l}}\frac{1000(mL)}{1(L)}\frac{1\text{f}\text{l}\text{o}\text{z}}{29\mathrm{.}574(mL)}=127\mathrm{.}99757$floz$=130$floz

Useful Conversion Factors

$\backslash$table$[[$Length$,1in=2\mathrm{.}54cm,3\mathrm{.}2808ft=1m,3ft=1yd,5280\mathrm{.}0ft=1mi$