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 or 0.95

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

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 Factor

Percentage Change

$34,000 $42,000 $8000 1.235 23.5% increase

600 bison 375 bison -225 bison 0.625 37.5% decrease

6.9 meters 15.2 meters 8.3 meters 2.203 120.3% increase

56,000 acres 45,000 acres -11,000 acres 0.804 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.)

1990 2005 Asia 775.8 1051.5

Europe 4080.4 3773.4 Middle East & North

Africa 1184.6 1765.5

North America 7686.3 7942.9 South America 970.1 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 Current 4 years 8 years 12 years 20 years Population 500 400 320 256 164

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. Example 11: The data below illustrate trends in U.S. per capita food consumption. (Source: USDA, Economic Research Service.)

Hig

h Fr

ucto

se C

orn

Syru

p (lb

s)

Suga

r (lb

s)

Milk

& c

ream

(lbs

)

Che

ese

(lbs)

Eggs

(lbs

)

Frui

t (lb

s)

Veg

etab

les (

lbs)

Red

Mea

t (lb

s)

Bro

ilers

(lbs

) (C

hick

en)

Add

ed F

ats &

Oils

(lb

s)

Car

bona

ted

Bev

erag

es (G

allo

ns)

1985 45.2 62.73 240.81 22.54 33.1 269.8 630.1 79 50.5 67.31 41.2 2007 56.2 62.41 202.93 32.67 32.3 263.8 680.8 65.1 84.6 86.71 48.8

For each food item, determine the average annual rate of change and the percentage change. Summarize your observations.

Hig

h Fr

ucto

se C

orn

Syru

p (lb

s)

Suga

r (lb

s)

Milk

& c

ream

(lbs

)

Che

ese

(lbs)

Eggs

(lbs

)

Frui

t (lb

s)

Veg

etab

les (

lbs)

Rate of

Change

0.5 lbs/yr

increase

0.01 lbs/yr

decrease

1.72 lbs/yr

decrease

0.46 lbs/yr

increase

0.04 lbs/yr

decrease

0.27 lbs/yr

decrease

2.3 lbs/yr

increase Percentage

Change 24%

increase 0.5%

decrease 15.7%

decrease 45%

increase 2.4%

decrease 2.2%

decrease 8%

increase

Red

Mea

t (lb

s)

Bro

ilers

(lbs

) (C

hick

en)

Add

ed F

ats &

Oils

(lb

s)

Car

bona

ted

Bev

erag

es (G

allo

ns)

Rate Of

Change

0.63 lbs/yr

decrease

1.55 lbs/yr

increase

0.88 lbs/yr

increase

0.35 gal/yr

increase Percentage

Change 17.6%

decrease 67.5%

increase 28.8%

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 Current 5 years 10 years 20 years 30 years

Population 300 (b) 22% growth over every 3-week period

Time Current 3 weeks 6 weeks 15 weeks 18 weeks Population 300

(c) 13% decay over every 2-month period

Time Current 2 months 4 months 6 months 12 months Population 300

(d) 14 individuals lost annually

Time Current 1 year 2 years 5 years 8 years Population 300

(e) 100% increase over every 4-year period

Time Current 4 years 8 years 16 years 20 years Population 300

(f) 35 additional individuals over every 5-day period

Time Current 5 days 15 days 16 days 19 days Population 300

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

1990 2003 Coal&Coal Products 458,304 531,169 Oil&Petroleum 770,250 921,413 Natural Gas 439,352 519,978 Hydroelectric 23,491 23,960 Solar,Wind&Wave 321 2,398 Nuclear 159,384 205,310 Geothermal 14,101 8,545 Solid Biomass 43,566 47,341

(a) Use the data above to complete the following table. Answers have been provided for the first row.

Average rate of

change in consumption over

this period

Factor by which consumption

changed over this period

Percentage change over this period

Coal & Coal Products

5,605 ttoe per year 1.16 (over 13 years) 16% (over 13 years)

Oil & Petroleum Natural Gas

Hydroelectric Solar, Wind &

Wave

Nuclear Geothermal

Solid Biomass (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.)

Use the data above to complete the following table. Be sure to include units!

Time Period Average rate of change

Factor by which population grew over this period

Percentage growth over this period

1963—1974 1974—1981 1981—1989 1989—1995 1995—2000 2000—2006

Year Population (in number of breeding pairs) 1963 487 1974 791 1981 1188 1984 1757 1986 1875 1987 2238 1988 2475 1989 2680 1990 3035 1991 3399 1992 3749 1993 4015 1994 4449 1995 4712 1996 5094 1997 5295 1998 5748 1999 6404 2000 6471 2005 7066 2006 9789

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)

(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.)

Females Males U.S. Population Black White Black White 1980 13,975,836 96,686,389 12,519,189 91,685,333 2006 20,419,202 122,313,220 18,639,632 120,326,022

2002 1961 2002 1961 Region/Classification Country

Asia (excluding Middle East) 27.8 4.8 Cambodia 13.9 4.9 Central America & Caribbean 46.9 22.8 China 52.4 3.8

Europe 74.3 .. Congo, Dem Rep 4.8 11.2 Middle East & North Africa 25.7 13.7 Ethiopia 7.9 19.8

North America 123.2 88.5 India 5.2 3.7 South America 69.7 39.3 Japan 43.9 7.6

Sub-Saharan Africa 13 14.2 Mexico 58.6 25.4 High Income Countries 93.5 55.9 Saudi Arabia 44.6 9.3 Low Income Countries 8.8 6.6 United States 124.8 89.2

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.

Females Males U.S. Population w/ Diabetes Black White Black White

1980 684,816 2,513,844 500,768 2,292,133 2006 1,837,729 5,993,348 1,565,729 6,497,605

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 20 years 30 years

Population 300 309.6 319.5 340.3 362.4

(c) Factor = 0.87 over every 2-month period. Time Current 2 months 4 months 6 months 12 months

Population 300 261 227.1 197.6 130.1 (d) Average rate of change is -14 individuals per year.

Time Current 1 year 2 years 5 years 8 years Population 300 286 272 230 188

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)

Average rate of change in

consumption over this period

Factor by which consumption

changed over this period

Percentage change over this period

Coal & Coal Products

5605 ttoe per year 1.16 (over 13 years)

16% (over 13 years)

Oil & Petroleum 11,628 ttoe per year 1.2 20% Natural Gas 6202 ttoe per year 1.18 18%

Hydroelectric 36 ttoe per year 1.02 2% Solar, Wind &

Wave 160 ttoe per year 7.47 647%

Nuclear 3533 ttoe per year 1.29 29% Geothermal -427 ttoe per year 0.61 -39%

Solid Biomass 290 ttoe per year 1.087 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 Average rate of change in per capita meat consumption

during 1961-2002

Percentage change in per capita meat consumption during 1961-2002

China 1.18 kg per person per year 1280% increase Ethiopia -0.29 kg per person per year 60% decrease

U.S. 0.87 kg per person per year 40% increase