Skills Beta

Math you need: the one-page sheet

Scientific Notation and Metric Prefixes

Scientific notation writes very large and very small numbers as a number from 1 to 10 times a power of ten, such as 2.5×10⁶ = 2.5 million = 2,500,000. To multiply, multiply the fronts and add the exponents; to divide, divide and subtract. Metric prefixes (kilo, mega, giga, milli, micro) are powers of ten too, so change them before you compare or combine numbers.

Scientific notation writes a number as (a number from 1 to 10) × a power of ten: $2.5 \times 10^{6}$ = 2.5 million = 2,500,000.
Multiply: multiply the fronts, add the exponents. Divide: divide the fronts, subtract the exponents. Then fix the front number if it is not between 1 and 10.
A metric prefix is a power of ten: kilo ( $10^{3}$ ), mega ( $10^{6}$ ), giga ( $10^{9}$ ), centi ( $10^{-2}$ ), milli ( $10^{-3}$ ), micro ( $10^{-6}$ ).
Scale words: thousand = $10^{3}$ , million = $10^{6}$ , billion = $10^{9}$ , trillion = $10^{12}$ .

the front number is between 1 and 10 that count is the exponent: left gives a positive power, right a negative one multiply the front numbers and add the exponents

scientific notation: A way of writing a number as a number from 1 to 10 times a power of ten, such as 2.5 × 10⁶ for 2,500,000.
metric prefix: A word part in front of a metric unit that multiplies it by a power of ten: kilo (10³), mega (10⁶), giga (10⁹), centi (10⁻²), milli (10⁻³), micro (10⁻⁶).

Rates and Per-Capita Values

A rate divides one quantity by another: liters per day, people per km², kilograms per person. A per-capita value divides a total by the number of people, so places of different sizes can be compared fairly. Write the setup with units: the word after "per" goes on the bottom, and the units of the answer tell you what you found.

A rate compares two quantities with different units: liters per day, people per km², kg per person.
A unit rate is a rate for one unit of the second quantity: 150 L per 1 person.
Per capita means per person: the total ÷ the population.
The word after "per" goes on the bottom: liters per km = L ÷ km.

you get a per-capita value you get a density or a rate per unit of time the answer’s unit tells you what you found: L per person, people per km²

unit rate: A rate for one unit of the second quantity, such as 150 liters per 1 person or 3 cm per 1 week.
rate: A comparison of two quantities with different units, found by dividing, such as liters per day or people per square kilometer.
per capita: Per person: a total divided by the number of people, used to compare places of different sizes.

Unit Conversions and Dimensional Analysis

Dimensional analysis converts units by multiplying by conversion factors, fractions equal to 1, arranged so unwanted units cancel. Write every factor with units in the setup, then the answer with its unit. Power (watts, kW) is a rate; energy (kWh, J, BTU, kcal) is an amount, and energy = power × time. Cost and emissions follow the same pattern: kWh × price per kWh, kWh × kg CO2 per kWh.

A conversion factor is a fraction equal to 1, such as 1 kWh / 3,400 BTU or 3.79 L / 1 gal.
Dimensional analysis: line up factors so each unwanted unit appears once on top and once on the bottom.
A watt is power (energy per second); a kilowatt-hour is energy. Energy = power × time.
Energy units: joule, kWh, BTU, kilocalorie. 1 kWh = 3.6 million J, about 3,412 BTU.

you know what has to cancel the old unit cancels and the new unit is left the number is right because each factor equals 1

dimensional analysis: A method of converting units by multiplying by conversion factors arranged so that unwanted units cancel, leaving only the units of the answer. Also called the factor-label method.
conversion factor: A fraction whose top and bottom are equal amounts in different units, such as 1 kWh / 3,412 BTU or 3.79 L / 1 gal, so multiplying by it does not change the amount.
kilowatt-hour: Units of energy: the joule (J), the kilowatt-hour (kWh, 3.6 million J), the British thermal unit (BTU, about 1,055 J) and the kilocalorie (kcal, about 4,184 J).
watt: The watt (W) is a unit of power: one joule of energy per second. A kilowatt (kW) is 1,000 W and a megawatt (MW) 1,000,000 W. Energy = power × time.

Percents and Percent Change

A percent is a part out of 100: part ÷ whole × 100. Percent of a quantity is (percent ÷ 100) × quantity. Percent change is (new - old) / old × 100: subtract inside brackets, divide by the starting value, multiply by 100, and keep the sign. Show this setup exactly; readers give a point for it.

A percent is a number out of 100: part ÷ whole × 100.
Percent of a quantity: (percent ÷ 100) × quantity. 18% of 4.0 million = 0.18 × 4,000,000 = 720,000.
Percent change = (new - old) / old × 100. Brackets first, divide by the old value, then × 100.
Percent changes from different starting values do not add: +25% then -10% is +12.5%, not +15%.

you get the change, with its sign you get the change as a fraction of where you started you get the percent change: positive for a rise, negative for a fall

percent: A number out of 100. A share as a percent is the part divided by the whole, times 100; a percent of a quantity is (percent ÷ 100) times the quantity.
percent change: How much a value changed compared with its starting value: (new - old) / old × 100. Positive for an increase, negative for a decrease.

Growth Rates, Doubling Time and the Rule of 70

A population growing by a fixed percent each year doubles in a fixed time. By the rule of 70, doubling time ≈ 70 ÷ growth rate in percent. Add the doubling time to the start year for "in what year"; multiply by 2 for each doubling to project the size. Doubling time depends only on the rate: twice the rate, half the time.

A growth rate is the percent change per year: 2% a year means 2 more for every 100 each year.
Rule of 70: doubling time ≈ 70 ÷ growth rate (in %, not as a decimal).
Doubling time depends on the rate, not on the starting size.
Each doubling multiplies by 2: after n doublings, multiply by 2 n times (2, 4, 8, 16, ...).

each year’s increase is larger than the last you get the doubling time in years you get the year the population doubles

growth rate: How fast a quantity grows, as a percent of its size per unit of time: a growth rate of 2% a year adds 2 for every 100 each year.
rule of 70: A shortcut for doubling time: divide 70 by the percent growth rate. At 2% a year, 70 ÷ 2 = 35 years.
doubling time: The time a quantity growing at a steady percent rate takes to double, about 70 divided by the percent growth rate.