Here are some examples of rates:
[beautiful math coming... please be patient] $\displaystyle\frac{\$5}{\text{hr}}$, also commonly seen as [beautiful math coming... please be patient] $\,\$5/\text{hr}\,$, and read as “five dollars per hour”
[beautiful math coming... please be patient] $\displaystyle\frac{40\text{ miles}}{3\text{ days}}$, read as “forty miles per three days”
[beautiful math coming... please be patient] $\displaystyle\frac{10\text{ kg}}{{\text{ in}}^3}$, read as “ten kilograms per cubic inch”
A rate is a mathematical expression, and like all types of mathematical expressions,
rates have lots of different names. For example,
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$\displaystyle
\frac{\$5}{\text{hr}}
= \frac{\$10}{2\text{ hr}}
= \frac{\$2.50}{30\text{ min}}
= \frac{50\text{ cents}}{6\text{ min}}
$
Notice that each of these names has a unit of currency (a money unit) in the numerator,
and a unit of time in the denominator.
A rate can be renamed to a new desired name,
providing the type of units in the numerator and denominator (e.g., length, time, volume, mass/weight) remain the same.
For example, suppose a rate has a unit of length in the numerator,
and a unit of volume in the denominator.
Then, it can only be renamed to rates that have length in the numerator
and volume in the denominator.
Multiplying by
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$\,1\,$ to rename in an appropriate way
provides a beautiful way to unify the solution of all kinds of rate problems,
as shown in the next example.
[beautiful math coming... please be patient] $\displaystyle \frac{24\text{ in}}{60\text{ min}} = \frac{7\text{ in}}{x\text{ min}} $ | (set up a proportion) |
[beautiful math coming... please be patient] $\displaystyle 24x = 420 $ | (cross-multiply) |
[beautiful math coming... please be patient] $ \displaystyle x = \frac{420}{24} = 17.5 $ | (solve for $\,x\,$) |
An object travels $\,5\text{ ft}\,$ in $\,8\text{ min}\,$.
Feel free to use scrap paper and a calculator to compute your answers.
In this problem set, you will not type in your answers.
You will just compare your answer with the one given here.
Depending on how you do the unit conversion,
you may get a slightly different answer than
the answer reported here.
Do not despair! If your answer is close, then you're fine!
For this exercise, use only the conversion information given in the Unit Conversion Tables to compute your answers.
All answers are either exact, or rounded to six decimal places.
It is possible to get $\,0.000000\,$ as an answer.