Rate Problems

DEFINITION rate

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

Here are some examples of rates:

$\displaystyle\frac{\$5}{\text{hr}}$, also commonly seen as $\,\$5/\text{hr}\,$, and read as “five dollars per hour”

$\displaystyle\frac{40\text{ miles}}{3\text{ days}}$, read as “forty miles per three days”

$\displaystyle\frac{10\text{ kg}}{{\text{ in}}^3}$, 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,

$\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 $\,1\,$ 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.

EXAMPLE:

Question:
A snail crawls about

$\,2\,$ feet in one hour.
How many minutes will it take to crawl $\,7\,$ inches?

Solution #1(one-sentence solution):

Step 1: Identify the original rate. Answer: $\displaystyle\frac{2\text{ ft}}{1\text{ hr}}$
Step 2: Identify the name for the rate that you want.
Use $\,x\,$ for any number that is unknown. Answer: $\displaystyle\frac{7\text{ in}}{x\text{ min}}$
Step 3: Check that the original rate and its desired new name are “compatible”;
i.e., same types of units in both numerator and denominator.
Answer: Both rates have units of length upstairs (feet and inches),
and units of time downstairs (hours and minutes). They're compatible.
Step 4: Turn the original rate into the new name by multiplying by $\,1\,$ in appropriate ways.

Answer: $\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}} \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.

Notice the single complete mathematical sentence.
This approach is fast and efficient, once you get used to working with fractions within fractions.
There are lots more details on this process in the text.

Solution #2(two-sentence solution):

Step 1: Convert the given rate to the desired units.

Answer: $\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}} $

Step 2: Set up and solve a proportion.
Answer:

$\displaystyle \frac{24\text{ in}}{60\text{ min}} = \frac{7\text{ in}}{x\text{ min}} $	(set up a proportion)
$\displaystyle 24x = 420 $	(cross-multiply)
$ \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

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.