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  • 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% decrea