Math Tips for the APES Exam

1. 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

2. 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.

3. 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?

4. 5 goats X 3 sheep X 2 pigs X 27 canaries X $3.25 = $877.50 1 goat 1 sheep 3 pigs 1 canary

5. 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).

6. Be able to convert within the metric system

7. 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%

8. 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

9. 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

10. 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

11. Calculate Percent Change • 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%

12. 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 tocalculate 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

13. 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

14. 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

15. Percentages • 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%

16. 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