Math Tips for the APES Exam
Lisa Turner
Thomas Jefferson High School
Adapted from R. Green, Hermitage HS
from J. Gardner, Glen Allen High School
Show ALL of your work
 Show ALL of your units
 Be proficient at unit manipulation,
also called dimensional analysis, factor
label, or train tracks. This is one of the
most important math skills, because you
will have to fit the numbers with units
together through multiplication and
division to get desired results.

Dimensional Analysis

A farmer started with 5 goats. He traded all of
his goats for sheep at an exchange rate of 3
sheep for 1 goat. He then traded his sheep for
pigs at a rate of 1 sheep for 2 pigs. Next, he
traded his pigs for canaries. For every three pigs
he received 27 canaries. He then sold all the
canaries for a rate of $3.25 per canary. How
much money did the farmer make?

5 goats
X
3 sheep
1 goat
X
2 pigs
1 sheep
X
27 canaries
3 pigs
X $3.25
= $877.50
1 canary
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Add, subtract, multiply, and divide
comfortably without a calculator.
Remember to show the proper placement of
numbers
Develop good “math sense” or “math
literacy”. The answers should make sense. If
you calculate a cost of $50 billion per gallon
of water, does that seem right?
Know simple conversion factors such as
the number of days in a year (365), the
hours in a day (24), the US population (310
million), and the world population (7 billion).
Be able to convert within the
metric system
Understand common statistical terms.
The mean is the mathematical average.
The median is the 50th percentile, which
is the middle value in the distribution of
numbers when ranked in increasing order.
The mode is the number that occurs
most frequently in the distribution.
 Be comfortable working with
negative numbers. Going from -8 °C to
+2 °C is a 10° change.
 Be able to calculate percentages.
Example 80/200 = 40/100 = 0.4 = 40%

Put very large or very small
numbers into scientific notation

Often in environmental science we use very
large numbers (146,000,000,000 kilograms
of biomass = 1.46 X 1011) or very small
numbers (7 ppm of Mercury that has
contaminated an aquifer = 7 X 108). Being
able to convert number into scientific
notation and feeling comfortable
manipulating them will increase your success
on the exam.
◦ 310,000,000 = 3.1 X 108
◦ 0.000000000000097 = 9.7 X 10-14
Know how to work with scientific
notation
Multiplication: add exponents, multiply bases
(3 X 103)(4 X 105) = 12 X 108 or 1.2 X 109
 Division: subtract exponents, divide bases
(5.2 X 104) / (2.6 X 102) = 2 X 102
 Addition: convert both numbers to the same
exponent, then add bases; exponents stay the same
(3.0 X 106) + (1.4 X 105) = (3.0 X 106) + (.14 X
106) = 3.14 X 106
 Subtraction: convert both numbers to the same
exponent, then subtract bases; exponents stay the
same.
(2.0 X 103) – (1.0 X 102) = (2.0 X 103) – (0.1 X
103) = 1.9 X 103

Know growth rate calculations

Growth rate = [Crude Birth Rate +
immigration] – [Crude death rate +
emigration]
◦ CBR = Crude birth rate = # births per 1,000,
per year
◦ CDR = Crude death rate = # deaths per
1,000, per year
◦ (CBR-CDR)/10 = percent change
Calculate Percent Change

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The rate of change (percent change, growth rate)
from one period to another = [(V final - V initial /
V initial] *100 (where V=value)
Annual rate of change: take an answer from the
previous step and divide by the number of years
between past and present values.
Example: A particular city has a population of
800,000 in 1990 and a population of 1,500,000 in
2008. Find the growth rate of the population of
this city. Growth Rate = (1,500,000 –
800,000)/800,000 * 100 = 87.5% Annual Growth
Rate = 87.5%/18 years = 4.86%
Know the rule of 70 to predict
doubling time. Doubling time = 70/annual
growth rate (in %, not a decimal!) Example: If
a population is growing at a rate of 4%, the
population will double in 17.5 years. (70/4 =
17.5)
 Be able to calculate half-life. Amount
remaining = (Original amount)(0.5)x where x
= the number of half-lives. X = time/half-life
 Know that per capita means per person or
per unit of population
 Graphing tips: include a title and a key, set
consistent increments for axes, and label
axes

Metric Conversions

Use dimensional analysis in order to
convert the following problems from one
unit to another.
10 cm= 1 X 10 -7 Mm
 6 mm = 6.0 X 10-6 km
 1.5 km= 1.5 X 106 mm
 8 watts= 8.0 X 10-6 MW
 5.4 mm= 0.54 cm or 5.4 X 10-1

Scientific Notation
One billion= 1.0 X 109
 Twenty three thousand= 2.3 X 104
 .0000676= 6.76 X 10-5
 Five hundred billion times thirty five
thousand= (5.0 X 1011) (3.5 X 104) = 17.5
X 1015 or 1.75 X 1016
 300 billion divided by 6 thousand = (3.0 X
1011) / (6.0 X 103) = (3/6) X 10(11+(-3)) =
0.5 X 108 or 5 X 107

Percentages
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An area of forest is 5000 acres. 45% of the area
will be developed. How many acres will be
preserved as forest area? 2250 acres
A natural gas power plant is 60% efficient. If one
cubic meter of natural gas provides 1000 BTUs of
usable electricity, how many BTU’s of waste heat
were produced? 400 BTU’s
If the concentration of mercury in a water supply
changes from 70 ppm to 42 ppm in a ten-year
period, what is the % change? 40%
What is the % change if the concentration of
carbon dioxide increases from 14 ppm to 63
ppm? 350%
Other
How many seconds are in 3 years? 9.5 X
107 seconds / 3 years
 If oil use in the US is 22 barrels per capita,
how much oil is used in the United
States? 6,820,000,000 or 6.82 X 109
 How much oil would be used applying
that same figure to per capita global use?
1.54 X 1011
