RATE PROBLEMS

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.)

DEFINITION rate
A rate is a comparison of two quantities that are measured in different units.

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.

Multiplying by [beautiful math coming... please be patient] $\,1\,$ to rename appropriately provides a beautiful way
to solve all kinds of rate problems, as shown in the next example.

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!

EXAMPLE:
Question:
A snail crawls about [beautiful math coming... please be patient] $\,2\,$ feet in one hour.
How many minutes will it take to crawl $\,7\,$ inches?

Both Versions of the Solution Start Off the Same Way:

Solution #1(one-sentence solution):

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 $\,1\,$!

$$ \frac{2\text{ ft}}{1\text{ hr}} \ =\ \overbrace{\frac{2\cancel{\text{ ft}}}{1\bcancel{\text{ hr}}} \cdot \frac{12\text{ in}}{1\cancel{\text{ ft}}} \cdot \frac{1\bcancel{\text{ hr}}}{\vphantom{\bcancel{\text{ hr}}}60\text{ min}}}^{\text{get desired units}} \cdot \overbrace{\frac{\frac1{2\,\cdot\, 12}}{\frac1{2\,\cdot\, 12}}}^{\text{get rid of undesired #s}} \cdot \underbrace{\frac{7}{7}}_{\text{introduce desired #}} \ =\ \frac{7\text{ in}}{60\cdot\frac{1}{24}\cdot 7\text{ min}} \ =\ \frac{7\text{ in}}{17.5\text{ min}} $$ Here it is again, in the most compact form.
This is the form you'll use when you get really good at this method! $$ \frac{2\text{ ft}}{1\text{ hr}} = \frac{2\text{ ft}}{1\text{ hr}} \cdot \frac{12\text{ in}}{1\text{ ft}} \cdot \frac{1\text{ hr}}{60\text{ min}} \cdot \frac{\frac1{24}}{\frac1{24}} \cdot \frac{7}{7} = \frac{7\text{ in}}{17.5\text{ min}} $$

Thus, it takes 17.5 minutes for the snail to crawl 7 inches.
There are lots more details on this one-sentence solution in the text (starting on page 172).
Solution #2(two-sentence solution):

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.

First sentence:   Convert the given rate to the desired units.

Answer:   [beautiful math coming... please be patient] $\displaystyle \frac{2\text{ ft}}{1\text{ hr}} = \frac{2\text{ ft}}{1\text{ hr}} \cdot \frac{12\text{ in}}{1\text{ ft}} \cdot \frac{1\text{ hr}}{60\text{ min}} = \frac{24\text{ in}}{60\text{ min}} $

Second sentence:   Set up a proportion to solve for the unknown number.
Answer:
[beautiful math coming... please be patient] $\displaystyle \frac{24\text{ in}}{60\text{ min}} = \frac{7\text{ in}}{x\text{ min}} $
Now, solve for $\,x\,$.
Ignore all the units; just work with the numbers.
[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 $\,x\,$)

Thus, it takes 17.5 minutes for the snail to crawl 7 inches.
EXAMPLES:

An object travels $\,5\text{ ft}\,$ in $\,8\text{ min}\,$.

Question: How many $\,\text{ft}\,$ will it travel in $\,20\text{ min}\,$?
Answer: $12.5$
Question: How many $\,\text{min}\,$ will it take to travel $\,43\text{ cm}\,$?
Answer: $2.293333$
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Work Problems

 
 

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.

An object travels
in
.