Math Tips for the APES Exam

Lisa TurnerThomas Jefferson High SchoolAdapted from R. Green, Hermitage HS

from J. Gardner, Glen Allen High School

Show ALL of your workShow ALL of your unitsBe 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 AnalysisA 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 X 2 pigs X 27 canaries X $3.25 = $877.50

1 goat 1 sheep 3 pigs 1 canary

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 notationOften 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 notationMultiplication: 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 calculationsGrowth 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 ChangeThe 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 ConversionsUse 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 km1.5 km= 1.5 X 106 mm8 watts= 8.0 X 10-6 MW5.4 mm= 0.54 cm or 5.4 X 10-1

Scientific NotationOne 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

PercentagesAn 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%

OtherHow 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