Worksheet 6 - Percentage Change
I. The Formula:
( New  Old )
 % (where the % is written as a decimal)
Old
II. Visually:
On day 1 a class has 12 happy students:
☺☺☺ ☺☺☺ ☺☺☺ ☺☺☺
Soon the news spreads that this is a good class. So, on the second day of class there
are now 15 happy students:
☺☺☺ ☺☺☺ ☺☺☺ ☺☺☺ ☺☺☺
By what percentage did the class increase from day 1 to day 2?
Answer: We can see visually what is going on. On the second day three more
students have been added.
☺☺☺ ☺☺☺ ☺☺☺ ☺☺☺ ☻☻☻
What percentage of the original class added? Note: This is just the same
part/whole question you have already been practicing:
(☻☻☻) / (☺☺☺ ☺☺☺ ☺☺☺ ☺☺☺) = 3 / 12 = .25
So, the class increased by 25%.
We just found the difference between the new and the old class (15-12 = 3) and then
compared this to the original whole (12). In general, you can think about this as:
( New  Old )
%
Old
What would the problem look like if the original class size of 12 dropped down to only 9
happy students? Give an illustration similar to the one above.
III. There are a few ways to write this relationship:
( New  Old )
%
Old
or
New
1  %
Old
Old  % * Old  New or
Old*(1+%)=New
IV. Deriving and Visualizing the Formulas
a. Figure out algebraically how to get the second two versions from the first.
b. Then make practical sense of these two versions using the example above.
c. Finally, let’s say you were given only the new value and a %. How could you
figure out the Old. In other words, find one more version of this relationship by
solving one of the above forms for Old.
V. Number Drills
In each problem state which is the new quantity (always the most recent date) and
which is the old quantity (always the most distant date). Then try to solve the problem
using any version of the relationship above.
1. A town has 999 residents in the year 2000. In 2005, the size of the town is
1023. By what percentage did the population of the town increase?
2. A town has 999 residents in the year 2000. In 2005, the size of the town is
768. By what percentage did the population of the town decrease?
3. A town has 999 residents in the year 2000 and increases by 13% in five
years. What is the population of the town in 2005?
4. A town has a population of 1050 in the year 2005. This represents a 26%
increase since 1980. What was the population in 1980?
VI. Applications
1. Chlorofluorocarbons (CFC’s) are chemicals used in air conditioners and
refrigerators. They are known to damage the earth’s protective ozone layer. In
1971, the US produced 341,600 metric tons of CFC’s. By 1992, production had
decreased to 216,200 metric tons. By what percent did CFC production
decrease?
2. Consumer debt has soared 39% in the last five years and is now $1.3 trillion.
What was consumer debt five years ago?
3. In 1994, Zaire’s per capita GNP was $125 (70% lower than it was in 1958). What
was Zaire’s per capita GNP in 1958?
4. During a 25% off sale, you buy an item for $22.50. What was the original price of
the item?
5. Cell Sites are the basic geographic unit of a wireless PCS or cellular system. A
city or county is divided into smaller "cell" each of which is equipped with a lowpowered radio transmitter/receiver. The cells can vary in size depending upon
terrain, capacity demands, etc. By controlling the transmission power, the radio
frequencies assigned to one cell can be limited to the boundaries of that cell.
When a wireless PCS or cellular phone moves from one cell toward another, a
computer at the switching office monitors the revenue generated by subscribers'
calls outside of their system. In 1995, there were 22,663 cell sites and that
number grew to 162,986 cell sites in 2003.
In 2003 the average cell phone bill was $49.91 and the average call length was
2.87 minutes. The number of subscribers increased from 33,786 thousand in
1995 to 158,722 thousand in 2003. The cellular telephone industry employed
21,382 people in 1990. That number rose to 184,449 people in 2000 and to
205,629 people in 2003.
a. If the average monthly cell phone bill was 38% lower in 2003 than it was in 1990,
what was the average monthly cell phone bill in 1990?
b. The number of cell sites tripled from 1995 to 1998. How many cell sites were there
in 1998?
c. How many cell sites were there per 10,000 subscribers in 1995? in 2003? Has the
number of cell sites per subscriber increased or decreased from 1995 to 2003? What
does the number of cell sites per 10,000 subscribers infer (if anything)? Show your
work.
d. The number of cellular telecommunications employees increased 10.4% from 2000
to 2001 and decreased 5.5% from 2001 to 2002. What was the overall percent change
in the number of employees from 2000 to 2002? How many cellular
telecommunications employees were there in 2002?