Section 3 Working with Units
Part 1 Fractions (REVIEW)
Fractions can be added, subtracted, multiplied and divided. Today, however, we focus on
multiplication and division of fractions. These operations are performed as follows:
(a)
Multiplication of Fractions
(b)
Division of Fractions
Example 1: Perform the indicated operations without using a calculator.
(a)
(b)
Fractions can be simplified whenever the numerator and denominator share a common
factor. (Factors are quantities that are multiplied.) For example, the numerator and
denominator of the fraction
share a common factor of c. In this case, the common
factors c may be canceled because the fraction
equals 1. In other words,
.
Example 2: Simplify the following fractions by factoring the numbers in the numerator
and denominator and then canceling common factors.
(a)
(b)
2
Example 3: Perform the indicated operations and simplify without using a calculator.
(a)
First, we’ll convert the division into multiplication. Then, we’ll simply the result.
(b)
Rather than calculating
and
, it is simpler to list out all of the common
factors in the numerator and denominator and cancel them. Consequently,
Example 4: Perform the indicated operations and simplify by canceling common factors.
The letters x and u represent numerical values.
(a)
(b)
Part 2 Units
Numerical quantities arising in everyday life are typically accompanied by units. For
example,
• 25 miles per gallon
• 16 days
• 0.04 tons per acre
• 1.1 million people per 5 years
Notice the word “per” above. The word “per” means “for each”. So “25 miles per gallon”
literally means “25 miles for each gallon”.
Quantities that have “per” in their units can be represented as fractions:
25 miles per gallon
0.04 tons per acre
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Example 5: Write each of the following quantities as a fraction.
(a) 365 days per year
(b) 12 inches per foot
(c) 1.1 million people per 5 years
(d) 1 year per 365 days
Example 6: Each of the following letters represents a variable quantity. Give the units of
each variable and write each variable as a fraction.
(a) Let E represent the fuel efficiency of a car, given in miles per gallon.
E miles per gallon
(b) Let R represent the number of average number of riders in a car at any time.
R riders per car
(c) Let D represent the average number of miles that a car is driven each day.
D miles per car per day
Part 3 Arithmetic with Units
In part 1, you learned how to multiply and divide fractions, and how to cancel common
factors. In part 2, you learned how quantities with units containing “per” can be written
as fractions. In this section, you will combine these concepts in order to perform
calculations with quantities having units.
Motivating Example: Suppose a bicyclist drives for 6 hours each day at an average
speed of 12 miles per hour. Estimate the miles driven by the bicyclist each day.
Driving at an average speed of 12 miles per hour means that a person drives 12 miles, on
average, during each hour. If the person drives 6 hours each day, then he/she will drive
.
We can also write this using fractions as follows:
4
Observe how the units were manipulated in the above calculation. What happened to the
“hours” unit?
When multiplying quantities that involve units, the units are effectively treated as factors
and are therefore capable of being canceled. In the last example, the “hours” in the
numerator canceled the “hour” in the denominator.
Example 7: The expression below shows the units of four quantities multiplied together.
Determine the units of the resulting quantity.
meters miles seconds minutes miles
⋅
⋅
⋅
=
second meter minute
hour
hour
Example 8: Simplify the following expression. Be sure to determine the units of your
final answer.
€
Suppose a person’s car has an average fuel efficiency of M miles per gallon. What would
the above calculation tell this person?
If this person drove 30 miles a day, on average, and the price of gasoline was a constant
$1.98 per gallon, then this person spends approximately
on gasoline. So, for example, if the person’s fuel efficiency were 17 miles per gallon,
then he/she would spend
on gasoline.
Example 9: The equation below shows only the units of each quantity involved.
Determine the units of the missing quantity.
 pounds
acres
acres 
tons
days
=
⋅ 
⋅
⋅
person ⋅ year
ton 
pound
 person ⋅ day year
€
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Part 3 Making an Estimate: ‘Unit Analysis’ Approach
In example 8, you estimated the amount of money a person spends on gasoline each year.
In this section, you will practice making similar kinds of estimates, using the units of the
given quantities to help you. This approach is referred to as unit analysis.
Example 10: Suppose a person runs at a speed of S feet per second. There are 5280 feet
in 1 mile. Estimate the number of miles M that the person runs in 1 hour.
Step 1: What are the units of the quantity you want to estimate?
M miles per hour
or
Step 2: What quantities do you need to know in order to obtain this estimate?
• S feet per second
or
• 5280 feet per mile
or
• 60 seconds per minute
or
• 60 minutes per hour
or
Step 3: Combine the quantities in step 2, using multiplication, so that you obtain the
quantity you are trying to estimate (given in step 1). Then, simplify.
So, suppose a person runs at an average speed of 3 feet per second. How many miles
will the person run over an hour?
In this case, S = 3. So, M = 0.6818 (3) = 2 miles per hour. So, the person will run
2 miles over an hour.
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Example 11: In 2007, there were 1,129,000 acres of potatoes planted in the U.S. which
yielded approximately 22,457,800 tons of potatoes. (Source: USDA
www.nass.usda.gov ) Per capita consumption of potatoes in the U.S. is
approximately 55 kilograms. (Note: 1 kilogram is approximately 2.2 pounds
and 1 ton equals 2000 pounds.) Use unit analysis to estimate how many
acres of potato crop land are needed to feed the current U.S. population
(approximately 301 million people) for a year.
Step 1: Determine the units of the quantity being estimated.
We want to estimate the number of acres required to feed the entire U.S.
population. So, the units are acres per U.S. population or
.
Step 2: List all of the needed information.
•
•
•
•
•
Step 3: Combine the known information using multiplication to obtain the units of
the estimated quantity from step 1. Simplify.
So, the U.S. currently requires about 915,480 acres of potato crop land to support its
population. Since 1,129,000 acres were planted in 2007, domestic production is
sufficient to meet the current needs of the U.S. population.
Step 4: Go back to step 2 and restate the assumptions that underlie the estimate.
The ratio
represents the average potato yield in the U.S. in the year
2007. Since the yield varies each year, this would impact the estimated amount of
land. The estimate also assumes a per capita consumption of 55 kg per person.
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Example 12: In 2005, the average person living in the U.S. generated and disposed of 4.5
pounds of solid waste per day. (Source: U.S. Environmental Protection
Agency (2006) Municipal Solid Waste Generation, Recycling and Disposal
in the United States: Facts and Figures 2005) Use unit analysis to estimate
the total tons of solid waste G disposed of by one person during a period of
T years. (Note: Recall that there are 2000 pounds in 1 ton.)
Step 1: G represents the total amount of waste a person generates, which can be
written as
.
Step 2: The information needed to obtain this estimate is:
•
T years or
•
•
•
.
Step 3: Combine the information from step 2 using multiplication to obtain the
estimated quantity.
Simplify:
or
Step 4: Restate underlying assumptions.
This equation assumes that a person disposes of 4.5 pounds of solid waste per day
on average. If this amount should change, then our equation would change.
Now, use your equation to determine how many tons of solid waste the average
person generates and disposes of over the course of a lifetime. (Life expectancy in the
U.S. is currently about 78 years.
We want to know the value of G when T=78. Replacing T with 78 yields
G = 64.0575. In other words, the average person living in the U.S. generates and
disposes of 64 tons of solid waste over their lifetime.
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Now, use your equation to determine how long it will take for a person to dispose of 1
ton of solid waste.
Now, we want to know the value of T when G = 1. Replacing G with 1 yields the
equation
. Consequently,
. In other words, a
person disposes of 1 ton of solid waste in approximately 1.2 years (about 14.5
months).
Summary Using unit analysis to obtain an estimate:
(Step 1) Determine the units of the quantity being estimated.
(Step 2) Determine all potentially relevant numerical information and write it in
fraction form (with units). Sometimes the information will be given to you,
sometimes you’ll have to do some research to obtain values, and sometimes
you may need to make an educated guess about some of the values.
(Step 3) Combine the known information using multiplication to obtain the units of
the estimated quantity from step 1. Simplify.
(Step 4) Reconsider the assumptions made in step 2.
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Section 3 Homework Assignment
1. Perform all indicated operations and simplify by factoring the numerators and
denominators and canceling common factors. No calculators here!
(a)
(i)
(b)
(j)
(c)
(k)
(d)
(l)
(e)
(m)
(f)
(n)
(g)
(o)
(h)
2. (a) Show why
.
(b) Using the same reasoning as in part (a), find equivalent ways of writing
.
3. Simplify each of the unit calculations below.
(a)
(b)
(c)
(d)
(e)
and
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(f)
4. Determine the missing units in each of the equations.
(a)
(b)
(c)
(d)
(e)
5. Write each of the following using fractions.
(a) 14 liters per day
(b) y dollars per foot
(c) L milligrams per minute
(d) Five tons of corn is produced on 1 acre of farmland.
(e) The water level is dropping approximately H feet every 6 hours.
(f) 20,000 head of cattle grazed on 200,000 acres of pasture.
6. Estimate each of the quantities below using unit analysis.
(a) Estimate the tons of solid waste disposed of each year by a manufacturing plant.
Suppose that the plant disposes of 350 pounds of solid waste each day (on
average). Note that there are 2000 pounds in a ton and 365 days in a year.
(b) Estimate the annual revenue of a fast food restaurant. Suppose that the average
revenue per person is $4.50 and that the restaurant serves an average of 3500
people each week. Note that there are approximately 52 weeks in a year.
7. (Food Footprint—U.S. Banana Consumption) In 2004, 4.96 million tons of
bananas were imported into the U.S. The worldwide average yield for bananas is
14,000 pounds per acre.
(a) Use unit analysis to estimate the amount of foreign land required to grow bananas
sold to the U.S over 1 year.
(b) Use your estimate from part (a) to estimate the amount of foreign land required to
grow bananas for one person living in the U.S for 1 year. Recall that there are
approximately 301, 000,000 people living in the U.S.
8. (Water Consumption) Use unit analysis to estimate the number of bath tubs of water
an average person living in the U.S. drinks in their lifetime, given the following
information:
• Average life expectancy in the U.S. is approximately 78 years.
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• A standard bath tub has a volume of approximately 19 cubic feet.
• One cubic foot is equivalent to approximately 7.5 gallons.
• There are 16 cups in a gallon.
Note: There are some additional quantities that you will need to know in order to
make this estimate. Can you determine the units of the missing quantities? (Think
back to your work in exercise 4!) You will probably need to make an educated guess
for the value of one of these quantities.
9. (Cardboard Consumption) The average person in the U.S. eats about 46 slices of
pizza each year. (Unverified source: packagedfacts.com). If you took all of the
cardboard (take-out and frozen) pizza boxes used in the U.S. over one year and
stacked them, one on top of the next, about how many miles would this stack reach?
Use unit analysis to answer this question. Round your answer to the nearest mile.
Note: There are 5280 feet in one mile. There are a few key pieces of relevant
information that are not stated for which you will need to make educated guesses.
Mount Everest stands about 29,000 feet or about 5.5 miles high. You will find that the
stack of pizza boxes is much higher! How many times the height of Mount Everest is
the stack? The average center-to-center distance between the Earth and the moon is
238,857 miles. How many years will it take for the stack to reach the moon?
10. (a) Assume that the volume of required landfill space grows at an average rate of
0.005 cubic yards per pound of solid waste collected. Use unit analysis to estimate
the volume of landfill space V (in cubic yards) required for one person’s waste for
a year, if the person disposes of P pounds of solid waste each week. (When
finished, you will have a function where V represents the output and P represents
the input.)
(b) Use your function from part (a) to determine the annual required landfill space
needed to dispose of one person’s solid waste, if that person created about 5
pounds of solid waste each week, on average.
11. (Fossil Fuel Consumption in Agriculture) Most farming operations today rely on
fossil energy (primarily from petroleum and natural gas) for operating farm
machinery, providing electricity, and creating chemical fertilizers. In this exercise,
you will use unit analysis to estimate the equivalent amount of oil required to produce
one bushel of a crop commodity like corn or wheat. You will then use this estimate to
estimate the amount of oil required to produce corn-feed beef cattle.
(a) Suppose a certain commodity crop yields B bushels per acre. Also, suppose that E
kilocalories of fossil energy are expended per hectare of cropland. Estimate the
equivalent gallons of oil G required to produce one bushel of this commodity
crop. (You will be creating a function for which G is the output.) Note: One
hectare of land is equivalent to approximately 2.47 acres. One barrel of oil
(equivalent to 42 gallons) will produce approximately 5,800,00 Btu (British
thermal units) of energy. (Source: Unit Conversions, Emissions Factors, and
12
Other Reference Data, Environmental Protection Agency (2004).) Also, 1
kilocalorie is equivalent to approximately 3.9657 Btu.
(b) In the conventional production of commodity corn, approximately 405,000
kilocalories of energy are expended from gasoline use, 1,003,000 kilocalories are
expended from diesel use and 2,448,000 calories are expended in the production
of ammonium nitrate (fertilizer) per hectare of harvested land. (Source: Pimental,
D. and M. Pimentel. Food, Energy, and Society. (2008) CRC Press.) Recent corn
yields are around 160 bushels per acre. Use your function from part (a) to
estimate the equivalent gallons of oil required per bushel of harvested corn.
(c) In the conventional production of commodity wheat, approximately 352,000
kilocalories of energy are expended from gasoline use, 565,000 kilocalories are
expended from diesel use and 1,272,000 calories are expended in the production
of ammonium nitrate (fertilizer) per hectare of harvested land. (Source: Pimental,
D. and M. Pimentel. Food, Energy, and Society. (2008) CRC Press.) Recent
wheat yields are around 40 bushels per acre. Use your function from part (a) to
estimate the equivalent gallons of oil required per bushel of harvested wheat.
(d) Use unit analysis to estimate the equivalent gallons of oil required for the
production of one corn-fed beef cow. You will need to use your answer from part
(b). Note: One bushel of harvested corn weighs approximately 56 pounds. The
feed-to-meat conversion rate for cattle is about 7 pounds of grain (primarily corn)
to one pound of meat. One beef cow yields about 550 pounds of meat.
12. (Annual Fuel Costs) In this exercise, you will compare annual fuel costs for a
gas/electric hybrid sedan, a regular gas-powered sedan, and a large SUV. You will
use unit analysis to create three functions, one function for each type of vehicle. By
comparing the outputs of these functions, you will be able to determine the amount of
money you would save in choosing one vehicle type over another.
Let D represent the number of miles driven each day (for all vehicle types), and let C
represent the annual cost for fuel.
(a) Let’s assume that a typical large SUV has a fuel efficiency of 10 miles per gallon.
In addition, let’s assume that this SUV can use the cheaper E85 fuel (i.e. 85%
ethanol) which sells for $3.09 per gallon. Use unit analysis to estimate the annual
cost C in this scenario.
(b) Now, let’s assume that a typical 4-passenger sedan has a fuel efficiency of 26
miles per gallon. Currently, regular fuel sells for about $3.89 per gallon in the
Twin Cities. Use unit analysis to estimate the annual cost C in this scenario.
Note: Your work in part (a) will help you here.
(c) Finally, let’s assume that a gas/electric hybrid sedan has a fuel efficiency of 50
miles per gallon. Let’s assume, as in part (b), that this vehicle uses regular fuel
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sold at $3.89 per gallon. Use unit analysis to estimate the annual cost C in this
scenario.
(d) Now, let’s compare annual fuel costs for these three vehicle types. Use your
functions in parts (a-c) to complete the following table.
Miles Driven
Per Day
20
40
80
Hybrid Sedan
Annual Fuel Cost
Gas-powered Sedan
Large SUV
(e) Now, let’s explore cost savings. Use your work in part (d) to complete the
following table.
Miles Driven
Per Day
Hybrid Sedan
Versus
Gas-Powered Sedan
Annual Cost Savings
Hybrid Sedan
Gas-Powered Sedan
Versus
Versus
Large SUV
Large SUV
20
40
80
Section 3 Answers to Selected Homework Exercises
14
1. (a)
(c)
(e)
(g)
(i)
(k)
(m)
(o)
2.
(a)
3. (a)
(c)
(e)
4. (a)
(c)
(e)
5. Write each of the following using fractions.
(a)
(c)
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(d)
6. (a) Step 3:
7. (Ecological Footprint—U.S. Banana Consumption)
(a) Step 3:
So, 708,571 acres of foreign land are required each year to grow bananas for the
U.S.
(b) Dividing the number of acres from part (a) by the population of the U.S. yields the
per capita land area of 0.0023 acres of banana cropland.
8. (Water Consumption)
Step 1:
Step 2: We assume that the average person drinks approximately 8 cups of water a
day. With this assumption, the list of relevant information is as follows:
10. (a) Step 1: The units of V are
.
Step 2: The units of the relevant information are as follows:
Step 3:
(b) Determine the value of V when P = 5 pounds per week:
So, if a person disposes of about 5 pounds of solid waste to the landfill each week,
then that person would have needed about 1.3 cubic yards of landfill space.