Section 4 Understanding Growth and Decay
Part 1 Working with Percentages (REVIEW)
Percentages are probably the most widely used tool for determining the level of growth or
decay of quantities over time. So, we begin today’s session by reviewing percentage
calculations.
First, let’s take a moment to recall what a percentage actually means. Complete the
following table:
Percentage
Meaning
Decimal Representation
95%
95 out of every 100
23.75%
23.75 out of every 100
or 0.2375
0.015%
0.015 out of every 100
or 0.00015
45.66%
45.66 out of every 100
0.4566
225%
225 out of every 100
2.25
or 0.95
As you can see from your above work, the decimal representation of a percentage is
found by dividing the percentage by 100 or, equivalently, moving the decimal point two
places to the left.
What is 5% of 600? Perhaps you remember HOW to perform this calculation, but do you
remember WHY? The next example illustrates the reasoning behind this percentage
calculation:
Example 1: Suppose you have a population of 600 individuals and 5% of the individuals
have an infectious disease. How many individuals actually have the disease?
The number of diseased individuals equals 5% of 600. Let’s use unit
analysis to estimate this number:
The relevant numerical information are:
Notice that, if the above quantities are multiplied, the resulting units are
diseased individuals, which is what we’re trying to estimate. So, we calculate
the estimate as follows:
Consequently,
.
In general,
.
Example 2: Determine the following percentages.
(a) 46% of 400
(b) 0.6% of 30
Interpreting percentages larger than 100% requires some extra thought. Let’s consider
150% of 60. What does this mean?
On the one hand,
On the other hand,
.
So, “150% of 60 = 90” means
• 90 is 1.5 times 60.
• 60 plus an additional 50% more equals 90. (i.e. 90 is 50% larger than 60.)
Example 3: Increase or decrease the number 350 by the given percentages.
(a) Increase by 20%
To increase 350 by 20%, we want 100% of 350 plus an additional 20%. In other
words, we need to calculate 120% of 350:
(b) Increase by 230%
To increase 350 by 230%, we want 100% of 350 plus an additional 230%. In
other words, we need to calculate 230% of 350:
(c) Decrease by 19%
To decrease 350 by 19%, we want 100% of 350 minus 19%. This will leave 81%
remaining. So, we need to calculate 81% of 350:
Note: In the last example, the numbers 1.20 and 3.30 are referred to as growth factors,
while 0.81 is called a decay factor. They are the decimal representations of 120%, 330%
and 81%. The word “factor” implies multiplication, and that’s exactly what is done with
these numbers:
•
•
•
To increase by 20%, multiply by 1.20.
To increase by 230%, multiply by 3.30.
To decrease by 19% (so that 81% remains), multiply by 0.81.
Example 4: Use growth/decay factors to increase or decrease the number 60 by the given
percentages.
(a) Increase by 13.5%
Increasing by 13.5% is the same as multiplying by 1 + 0.135 = 1.135.
The result is (60)(1.135) = 68.1.
(b) Decrease by 34.8%
Decreasing by 34.8% is the same as multiplying by 1 – 0.348 = 0.652.
The result is (60)(0.652) = 39.12.
Example 5: Determine the growth or decay factor in each of the following. Then
determine the corresponding percentage increase/decrease.
(a) Increase from 45 to 70.
Let c represent the growth factor that we’re looking for. We know that
. So,
. This corresponds to an increase of 55.6%.
(b) Decrease from 7000 to 5800.
Now, let c represent the decay factor that we’re looking for. We know that
. So,
. This tells us that 82.9% of 7000 equals
5800. This corresponds to a decrease of 100% - 82.9% = 17.1%.
Part 2 Measuring Change
Our world is a very dynamic place. Change is something that we are growing accustomed
to seeing and reading about everyday. For example,
•
In the last 150 years, the concentration of methane in the atmosphere has
increased 148%. (Source: IPCC (2007) Climate Change 2007: The Physical
Science Basis)
•
From 1991 to 2004, the number of internet servers in the U.S. has increased at an
average annual rate of 21.46 million servers per year. (Source: New Atlas of
Planet Management)
•
From 1990 to 2005, U.G. greenhouse gas emissions increased by 16% at an
average annual rate of 1%. (Source: U.S. Environmental Protection Agency
(2007) Inventory of U.S. Greenhouse Gas Emissions and Sinks: 1990-2005.)
•
Since 1950, the average size of a new U.S. single-family house has grown by
148%. At the same time, the average number of occupants in a household has
decreased by 22% (Sources: National Association of Home Builders (2007)
Housing Facts, Figures and Trends, U.S. Census Bureau and Wilson, A. and J.
Boehland (2005) “Small is Beautiful, U.S. House Size, Resource Use, and the
Environment.” Journal of Industrial Ecology. Vol. 9, No. 1-2, 277-287.)
•
From 1992 to 2005, installations of photo-voltaic systems (i.e. solar panels) have
grown by 20% in the U.S. (Source: International Energy Agency, PV Power
Systems Programme (2005) “Cumulative installed PV power.”)
•
Currently, the world’s population is about 6.8 billion and is increasing by
approximately 225,000 people each day. (Source: The New Atlas of Planet
Management)
In order to describe and evaluate the changes we see occurring, we commonly try to
quantify the changes. There are two ways in which change is commonly quantified:
(1) total change—the actual amount by which a quantity grows or decreases
(2) percentage change—the percentage by which a quantity grows or decreases
Example 6: The table below gives the initial and later amounts of various quantities.
Complete the table. Be sure to provide units where ever appropriate!
Initial
Amount
Later
Amount
Total Change
(positive for increases
and
negative for decreases)
Growth/Decay Percentage
Factor
Change
$34,000
$42,000
$8000
1.235
600 bison
375 bison
-225 bison
0.625
6.9 meters
15.2 meters
8.3 meters
2.203
56,000 acres
45,000 acres
-11,000 acres
0.804
23.5%
increase
37.5%
decrease
120.3%
increase
19.6%
decrease
Example 7: The table below shows per capita energy consumption for several geographic
regions in the years 1990 and 2005 measured in kilograms of oil equivalent (kgoe) per
person. Use the table to answer the following questions. (Source: International Energy
Agency (IEA) Statistics Division. 2007. Energy Balances of OECD Countries (2008
edition) and Energy Balances of Non-OECD Countries (2007 edition). Paris: IEA.)
Asia
Europe
Middle East & North
Africa
North America
South America
1990
775.8
4080.4
2005
1051.5
3773.4
1184.6
1765.5
7686.3
970.1
7942.9
1151.2
(a) Which regions have the largest and smallest per capita energy consumption?
In both 1990 and 2005, Asia has the lowest per capita energy consumption
and North America has the highest.
(b) Which of the regions experienced the largest total increase in per capita
energy consumption? What was the total increase in consumption for this
region?
The Middle East and North Africa experienced the largest increase (an
increase of 580.9 kgoe).
(c) Which of the regions experienced the largest percentage increase in per capita
energy consumption? What was the percentage growth in this region?
The Middle East and North Africa experienced the largest percentage increase
(an increase of 49%).
(d) Consumption in Europe declined during this time period. By what percentage?
The decay factor for Europe is 0.925. Therefore, energy consumption declined
by about 7.5%.
(e) Using complete sentences, summarize what you’ve learned about energy
consumption trends in these regions.
The Middle East and North Africa as a whole experienced the largest increase
in energy consumption between 1990 and 2005. Per capita consumption in
Europe declined. Overall, North America still uses far more energy than the
other geographic regions.
Example 8: Suppose a population decreases by 20% every 4 years. The current
population size is 500.
(a) What is the decay factor?
To determine the amount remaining after 4 years, we take 80% of 500, or
rather (0.80)(500). Thus, the population decays by a factor of 0.8 every 4
years.
(b) Use the decay factor to complete the following table:
Time
Population
Current
500
4 years
400
8 years
320
12 years 20 years
256
164
For every 4-year period, we’ll need to multiply by 0.80. So, the values in
the table are determined as follows:
Population after 4 years =
Population after 8 years =
Population after 12 years =
Population after 20 years =
Example 9: For each of the factors given below, determine whether the corresponding
quantities are growing or decaying and give the percentage change.
(a) c = 1.24
A factor of 1.24 corresponds to an increase of 24%. So, the corresponding
quantity has grown by 24%.
(b) c = 0.70
A factor of 0.70 corresponds to 70% remaining, or rather, a 30% decrease.
So, the corresponding quantity has decayed by 30%.
Part 3 Change versus Rate of Change
Now that we understand different ways that change can be measured, we are ready to
discuss the rate at which something changes. Intuitively, rates measure how fast or slow
something is changing. To measure a rate of change, you need to know two pieces of
information:
(1) the amount of change that occurred (total or percentage)
(2) the amount of time required for that amount of change to occur
For example, suppose a population grew by 20%. Is this population growing fast or slow?
Well, it depends on how long it took to grow by 20%. If it took 3 days, then you might
think its growing fairly rapidly. If it took 3 years, you would say that the population is
growing less rapidly.
We determine the average rate at which a quantity is changing by determining the
average amount of change over one time unit. For example, we might determine the
average change over 1 second, or 1 day, or 1week, etc. Consider the next example.
Example 10: Determine and interpret the average rate of change of each of the following
quantities:
(a) A population increases from 46,000 to 52,000 over a 5-year period.
The total change in the population over these 5 years is
people. So, the average change per year is
found by dividing by 5:
So, during this 5-year period, the population grew at an average rate of 1200
individuals each year.
(b) A radioactive chemical substance decayed from 90 grams down to 77 grams
during a period of 150 days.
The total change in the amount of substance during these 150 days is
grams. So, the average change per day is found by
dividing by 150 days:
So, during this 150-day period, the substance decayed at an average rate of
0.867 grams per day.
The last example illustrates the general approach for calculating (average) rates of
change:
If a quantity changes in size by an amount
over a time period of length
, then
(Average) Rate of change =
Two Notes:
(1) The symbol Δ used above is the Greek letter delta. Mathematicians and scientists use
this letter to represent the total change in a variable. For example, if the letter A
represents some varying quantity, then ΔA represents the total change in the quantity.
(2) Suppose a population grows by 30% over 5 years. To determine the average yearly
percentage rate, you might think that you take 30% and divide by 5 to get 6%. This turns
out to be incorrect. In truth, the annual percentage rate turns out to be approximately
5.4%! We’ll see why in a few weeks. So, for now, we will use only total change to
calculate average rates of change.
630.1
680.8
79
65.1
Carbonated
Beverages (Gallons)
Red Meat (lbs)
269.8
263.8
Added Fats & Oils
(lbs)
Vegetables (lbs)
33.1
32.3
Broilers (lbs)
(Chicken)
Fruit (lbs)
62.73 240.81 22.54
62.41 202.93 32.67
Eggs (lbs)
Sugar (lbs)
45.2
56.2
Cheese (lbs)
High Fructose Corn
Syrup (lbs)
1985
2007
Milk & cream (lbs)
Example 11: The data below illustrate trends in U.S. per capita food consumption.
(Source: USDA, Economic Research Service.)
50.5
84.6
67.31
86.71
41.2
48.8
Vegetables (lbs)
Fruit (lbs)
Eggs (lbs)
Cheese (lbs)
Milk & cream (lbs)
Rate
0.63
1.55
0.88
Of
lbs/yr
lbs/yr
lbs/yr
Change
decrease increase increase
Percentage 17.6%
67.5%
28.8%
Change
decrease increase increase
Carbonated
Beverages (Gallons)
Added Fats & Oils
(lbs)
Broilers (lbs)
(Chicken)
0.5
0.01
1.72
0.46
0.04
0.27
2.3
lbs/yr
lbs/yr
lbs/yr
lbs/yr
lbs/yr
lbs/yr
lbs/yr
increase decrease decrease increase decrease decrease increase
24%
0.5%
15.7%
45%
2.4%
2.2%
8%
increase decrease decrease increase decrease decrease increase
Red Meat (lbs)
Rate
of
Change
Percentage
Change
Sugar (lbs)
High Fructose Corn
Syrup (lbs)
For each food item, determine the average annual rate of change and the percentage
change. Summarize your observations.
0.35
gal/yr
increase
18.4%
increase
The greatest increases during this 22-year period were seen in the following categories:
broilers, cheese, added fats & oils, high fructose corn syrup and carbonated beverages.
The largest decreases were in the categories of red meat and milk & cream. Modest
increases occurred in consumption of vegetables, while consumption of eggs and fruits
decreased by comparable percentages. Overall, the data show a general trend toward
greater consumption of high fat, high starch foods.
Section 4 Homework Assignment
1. Consider the following growth and decay factors: 1.33, 0.9, 0.59, 2.1, and 1.03.
(a) Which factors describe a growing quantity? What are the corresponding
percentage changes?
(b) Which factors describe a decaying (i.e. decreasing) quantity? What are the
corresponding percentage changes?
2. Suppose a population decreases from a size of 650 down to 480 over a 2-year
period. Answer each of the following questions. Don’t forget to include units!
(a) What is the total change in the population?
(b) By what percentage has the population decreased?
(c) What is the (average) rate of change of the population over these 2 years?
(d) By what factor did the population decay over these 2 years?
3. Suppose a population increases from a size of 4700 up to 6200 over a 6-week
period. Answer each of the following questions. Don’t forget to include units!
(a) What is the total change in the population?
(b) By what percentage has the population increased?
(c) What is the (average) rate of change of the population over these 6 weeks?
(d) By what factor did the population grow over these 6 weeks?
4. Suppose a population decreases from a size of 1.24 million down to 900,000 over
a 15-year period. Answer each of the following questions. Don’t forget to include
units!
(a) What is the total change in the population?
(b) By what percentage has the population decreased?
(c) What is the (average) rate of change of the population over these 15 years?
(d) By what factor did the population decay over these 15 years?
5. Suppose a bacteria population increases from a size of 3.4 million up to 4.1
million over a 4-month period. Answer each of the following questions. Don’t
forget to include units!
(a) What is the total change in the population?
(b) By what percentage has the population increased?
(c) What is the (average) rate of change of the population over these 4 months?
(d) By what factor did the population grow over these 4 months?
6. Given below is information about the growth or decline of different populations
over specified periods of time. When percentage information is given, determine
the corresponding growth/decay factors and use the factors to complete the tables.
When information about the total change of the population is given to you,
determine the average rate of change of the population and use it to help you
complete the table. Let’s assume that current population size in all cases is 300.
(a) 3.2% growth over every 5-year period
Time
Population
Current
300
5 years
10 years
20 years
(b) 22% growth over every 3-week period
Time
Current 3 weeks 6 weeks
Population
300
(c) 13% decay over every 2-month period
Time
Current 2 months
Population
300
(d) 14 individuals lost annually
Time
Current
Population
300
1 year
30 years
15 weeks
4 months
18 weeks
6 months
12 months
2 years
5 years
8 years
(e) 100% increase over every 4-year period
Time
Current
4 years
8 years
Population
300
16 years
20 years
(f) 35 additional individuals over every 5-day period
Time
Current
5 days
15 days
Population
300
16 days
19 days
7. For each of the scenarios, determine the growth or decay factor and the annual
percentage change.
(a) a population doubles each year
(b) a population triples each year
8. Suppose a population grows from a size of 3500 to 5600 over a period of 15
years.
(a) Show that the population grew by 60% during this 15-year period.
(b) What is the average rate of change of the population over these 15 years? Be
sure to give units!
(c) When the population grows from 3500 to 5600, this corresponds to a 60%
increase. If the population then drops from 5600 back down to 3500, this does
not correspond to a 60% decrease. Is this surprising to you? Determine the
true percentage decrease.
9.
“Both et al….examined the timing of peak caterpillar populations and arrival
dates of the flycatchers in nine populations in the Netherlands. They found that
due to warming, the peak availability of prey-caterpillar populations was
occurring earlier in the season, and that by the time the birds arrived, they often
did not have enough food for their nestlings. According to Both and colleagues,
this led to a 40% population decline of the flycatcher over the past 20 years.”
(Source: WRI 2006 Climate Report)
By what factor did the flycatcher population decrease over these 20 years?
10. Between 1990 and 2002, greenhouse gas emissions in South Korea grew by 97%
whereas emissions in the U.S. grew by 18%. (Navigating the Numbers, WRI.)
Can you determine which country is the larger emitter in 2002, or is there
additional information that you would need to know to answer this question?
11. The following energy consumption data are taken from the International Energy
Agency (IEA) Statistics Division (2006). All amounts are in thousands of tons of
oil equivalent (ttoe).
U.S. Energy Consumption
Coal&Coal Products
Oil&Petroleum
Natural Gas
Hydroelectric
Solar,Wind&Wave
Nuclear
Geothermal
Solid Biomass
1990
458,304
770,250
439,352
23,491
321
159,384
14,101
43,566
2003
531,169
921,413
519,978
23,960
2,398
205,310
8,545
47,341
(a) Use the data above to complete the following table. Answers have been
provided for the first row.
Coal & Coal
Products
Oil & Petroleum
Natural Gas
Hydroelectric
Solar, Wind &
Wave
Nuclear
Geothermal
Solid Biomass
Average rate of
change in
consumption over
this period
5,605 ttoe per year
Factor by which
consumption
changed over this
period
1.16 (over 13 years)
Percentage change
over this period
16% (over 13 years)
(b) Summarize your results from part (a). Which areas of energy consumption
have grown the most? Which areas have shown little growth and which areas
have decreased?
12. Until recent years, the bald eagle was listed as ‘endangered’ under the
Endangered Species Act in 43 of the lower 48 states and listed as ‘threatened’ in
Michigan, Minnesota, Oregon, Washington and Wisconsin. ‘Endangered’ means a
species is considered in danger of extinction throughout all or a significant portion
of its range. ‘Threatened’ is a less dire category, meaning a species is considered
likely to become endangered but not in danger of extinction. In 1995, 5 decades
after government passed the Bald Eagle Protection Act, the population of these
birds of prey had grown to the point that the U.S. Fish and Wildlife Service
reclassified it as ‘threatened’.
Shown in the table below are the recorded numbers of breeding pairs of eagles in
the lower 48 states for most years between 1963 and 2006 (the recovery period).
(Source: U.S. Fish and Wildlife Service, Bald Eagle Population Size: Chart and
Table of Bald Eagle Breeding Pairs in Lower 48 States.)
Year Population (in number of breeding pairs)
1963
1974
1981
1984
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2005
2006
487
791
1188
1757
1875
2238
2475
2680
3035
3399
3749
4015
4449
4712
5094
5295
5748
6404
6471
7066
9789
Use the data above to complete the following table. Be sure to include units!
Time Period
1963—1974
1974—1981
1981—1989
1989—1995
1995—2000
2000—2006
Average rate of
change
Factor by which
population grew
over this period
Percentage growth
over this period
13. Data in the table below show trends in per capita meat consumption (in kilograms
per person) by geographic region, income level and development status. Per
capita values for a sample of countries follow. (Source: Food and Agriculture
Organization of the United Nations (FAO), FAOSTAT on-line statistical service
(FAO, Rome, 2004), http://apps.fao.org)
2002
Region/Classification
Asia (excluding Middle East)
Central America & Caribbean
Europe
Middle East & North Africa
North America
South America
Sub-Saharan Africa
High Income Countries
Low Income Countries
27.8
46.9
74.3
25.7
123.2
69.7
13
93.5
8.8
1961
Country
Cambodia
China
Congo, Dem Rep
Ethiopia
India
Japan
Mexico
Saudi Arabia
United States
4.8
22.8
..
13.7
88.5
39.3
14.2
55.9
6.6
2002
1961
13.9
52.4
4.8
7.9
5.2
43.9
58.6
44.6
124.8
4.9
3.8
11.2
19.8
3.7
7.6
25.4
9.3
89.2
(a) What initial observations do you have?
(b) Complete the following table. Be sure to include units!
Country
Average rate of change in per
capita meat consumption during
1961-2002
Percentage change in per capita meat
consumption during 1961-2002
China
Ethiopia
U.S.
(c) Let Y represent annual per capita meat consumption within a particular country or
group of countries (measured in kilograms per person per year). Use unit analysis to
determine the equivalent daily per capita consumption D (measured in pounds per
person per day). (Recall that there are approximately 2.2 pounds in 1 kilogram.)
(d) Use the function that you created in (c) to determine and compare daily per capita
meat consumption in the U.S., China and Ethiopia in the year 2002.
14. The tables below display population demographics for black and white males and
females, and the number of individuals in each of these demographic groups that have
diabetes. (Source: U.S. Census, and the Division of Diabetes Translation, National Center
for Chronic Disease Prevention and Health Promotion, Centers for Disease Control and
Prevention.)
U.S. Population
1980
2006
Females
Black
White
13,975,836
96,686,389
20,419,202
122,313,220
Males
Black
12,519,189
18,639,632
White
91,685,333
120,326,022
U.S. Population
w/ Diabetes
1980
2006
Females
Black
684,816
1,837,729
Males
White
2,513,844
5,993,348
Black
500,768
1,565,729
White
2,292,133
6,497,605
Use either average rates of change or percentage changes to analyze trends in the
prevalence of diabetes over time within each of the four demographic groups. Write a
paragraph which incorporates your quantitative results and summarizes your findings.
Section 4 Answers to Selected Homework Exercises
1. (a) Factors describing growing quantities are 1.33 (33% growth), 2.1 (110%
growth), and 1.03 (3% growth).
(b) Factors describing decaying quantities are 0.9 (10% decrease) and 0.59 (41%
decrease).
2.
(a) Total change = -170 individuals which represents a decrease of 170
individuals during these 2 years.
(b) Percentage change = -0.26 which corresponds to a 26% decrease over these 2
years.
(c) Average rate of change = -85 individuals per year which represents a decrease
of 85 individuals each year, on average.
(d) Decay factor = 0.74 over this 2-year period.
5.
(a) Total change = 700,000 individuals which represents an increase of 700,000
individuals during these 4 months.
(b) Percentage change = 0.206 which corresponds to a 20.6% increase over these
4 months.
(c) Average rate of change = 175,000 individuals per month which represents an
increase of 175,000 individuals each month, on average.
(d) Growth factor = 1.206 over this 4-month period.
6.
(a) Factor = 1.032 over every 5-year period.
Time
Current
5 years 10 years
Population
300
309.6
319.5
20 years
340.3
(c) Factor = 0.87 over every 2-month period.
Time
Current 2 months
4 months
Population
300
261
227.1
30 years
362.4
6 months
197.6
(d) Average rate of change is -14 individuals per year.
Time
Current
1 year
2 years
5 years
Population
300
286
272
230
12 months
130.1
8 years
188
7.
(a) Growth factor = 2; annual percentage growth = 100%
(b) Growth factor = 3; annual percentage growth = 200%
9.
The decay factor was 0.6 during this time period.
10. You can not determine which country is the larger emitter from the information
provided.
17
11.
(a)
Coal & Coal
Products
Oil & Petroleum
Natural Gas
Hydroelectric
Solar, Wind &
Wave
Nuclear
Geothermal
Solid Biomass
Average rate of
change in
consumption over
this period
5605 ttoe per year
11,628 ttoe per year
6202 ttoe per year
36 ttoe per year
160 ttoe per year
Factor by which
consumption
changed over this
period
1.16 (over 13
years)
1.2
1.18
1.02
7.47
3533 ttoe per year
-427 ttoe per year
290 ttoe per year
1.29
0.61
1.087
Percentage change
over this period
16% (over 13 years)
20%
18%
2%
647%
29%
-39%
8.7%
(b) Energy use in all categories, except geothermal, is growing. Solar, wind and
wave generated energy has seen the greatest percentage growth, but this is
deceiving given that the actual amounts of solar, wind and wave energy used
is significantly less than all other categories. The rates of change of oil and
petroleum, coal, and natural gas consumption are significantly larger than any
of the other categories. These are nonrenewable sources of energy, which is
perhaps the most noteworthy feature displayed in the table.
13. (b)
Country
China
Ethiopia
U.S.
Average rate of change in per
capita meat consumption
during 1961-2002
1.18 kg per person per year
-0.29 kg per person per year
0.87 kg per person per year
Percentage change in per capita meat
consumption during 1961-2002
1280% increase
60% decrease
40% increase