The
Unit Conversion Tables should be helpful for this section.
(The link will open in a new window, so you can keep this information close by.)
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,
[beautiful math coming... please be patient]
$\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.
Which name should you use? It depends!
If you're earning \$5 per hour, and want to know how much you'll earn in $\,30\,$ minutes, then $\displaystyle\,\frac{\$2.50}{30\text{ min}}\,$ is a good name.
You can get whatever name you need just by multiplying by one, as you'll see below!
Try not to be intimidated by the ‘fractions within fractions’.
If you give this method
a chance, you'll be solving an incredible variety of problems quickly and efficiently.
Two versions of the solution are given: a one-sentence
solution and a two-sentence solution.
Both solutions have the same beginning steps—to make sure you know
what you're starting with, and what you want to end up with!
This one-sentence solution is ultra-fast and efficient, once you get used to working with fractions within fractions.
First, you multiply by $\,1\,$ in appropriate forms to get the units you want.
Then, you get rid of the numbers you don't want.
Then, you introduce the number you do want!
All by just multiplying by $\,1\,$!
If you're not seeing what's going on here, there are lots more details in the text (starting on page 172).
$$ \frac{2\text{ ft}}{1\text{ hr}} \ =\ \overbrace{\frac{2\cancel{\text{ ft}}}{1\bcancel{\text{ hr}}} \cdot \frac{12\text{ in}}{1\cancel{\text{ ft}}} \cdot \frac{1\bcancel{\text{ hr}}}{\vphantom{\bcancel{\text{ hr}}}60\text{ min}}}^{\text{get desired units}} \cdot \overbrace{\frac{\frac1{2\,\cdot\, 12}}{\frac1{2\,\cdot\, 12}}}^{\text{get rid of undesired #s}} \cdot \underbrace{\frac{7}{7}}_{\text{introduce desired #}} \ =\ \frac{7\text{ in}}{60\cdot\frac{1}{24}\cdot 7\text{ min}} \ =\ \frac{7\text{ in}}{17.5\text{ min}} $$ Here it is again, in the most compact form.
Here's the two-sentence solution.
The first sentence renames the rate with the desired units.
The second sentence sets up a proportion, which is then solved for the unknown number.
[beautiful math coming... please be patient] $\displaystyle \frac{24\text{ in}}{60\text{ min}} = \frac{7\text{ in}}{x\text{ min}} $ |
[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.