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# mathfacts (Math facts ** )

### Math Facts

• If x and y vary DIRECTLY one with another, then their relationship is defined by x / y = constant.  If x and y vary INVERSELY one with another, then their relationship is defined by x • y = constant.
• An ‘arithmetic sequence’ is a list of numbers in which there is a common difference between the consecutive numbers in the sequence.  For example, 3, 6, 9, 12, 15, 18 … is an arithmetic sequence in which the common difference is 3.  The nth term of an arithmetic sequence is an, which is equal to a1 + d(n – 1), where a1 is the first term and d is the difference between each of the terms in the sequence.  Thus,  an  = a1 + d(n – 1) .
• The sum of n terms of an arithmetic sequence is equal to half n times the sum of the first and last terms of the sequence.  Thus  Sn = n/2 (a1 + an) .
• The order of operations in algebraic equations is: 1) powers  2) roots  3) multiplication  4) division  5) addition  6) subtraction.
• Types of numbers:
• Natural (counting):  the numbers starting with 1 and going up by ones to infinity (1, 2, 3, 4, 5, ….).
• Whole:  the numbers starting with 0 and going up by ones to infinity (0, 1, 2, 3, 4, …).  The whole numbers differ from the natural numbers just by the number 0.
• Integer:  the positive and negative whole numbers and 0 (… -3, -2, -1, 0, 1, 2, 3, …).
• Rational: numbers that can be written as p/q, where both p and q are integers (3/4, 19/8, -5/21, etc.).  However, q cannot be equal to zero.
• Even: numbers evenly divisible by 2 (… -4, -2, 0, 2, 4, …).
• Odd: numbers not evenly divisible by 2 (… -3, -1, 1, 3, 5, …).
• Prime: numbers divisible only by 1 and themselves (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …).
• Composite: numbers that are not prime; that is, numbers that are evenly divisible by some number other than just 1 and themselves (4, 6, 8, 9, 10, 12, 14, 15, …).
• Plane geometry equations:
• The distance between two points (x1, y1) and (x2, y2) equation is:  d = √ (x2 – x1)2 + (y2 – y1) 2
• The midpoint of a line joining (x1, y1) and (x2, y2) is:  midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
• The slope of a line passing through points (x1, y1) and (x2, y2) is:  m = (y2 – y1) / (x2 – x1)
• The slope- y intercept equation is:  y = mx + b
• The point-slope equation is: ( y – y1) = m ( x – x1)
• Conversions:
• 1 meter = 39.37 inches
• 1 kilometer = 0.621 miles
• 1 liter = 1.057 quarts
• 1 gallon = 4 quarts
• 1 kilogram = 2.205 pounds
• 1 yard = 0.9144 meter
• 1 pound = 0.454 kilogram
• 1 cup = 0.2365 liter
• 1 rod = 16.5 feet
• 1 fathom = 6 feet
•  1 furlong = 220 yards
• 1 hand = 4 inches
• 1 league = 3 miles
• 1 pica = 12 points
• 1 mile per hour = 88 feet per minute
• 1 foot = 12 inches
• 1 yard = 3 feet
• 1 mile = 5,280 feet
• 1 square foot = 144 square inches
• 1 square yard = 9 square feet
• 1 square mile = 640 acres
• 1 cubic foot = 1,728 cubic inches
• 1 cubic yard = 27 cubic feet
• 1 quart = 2 pints
• 1 gallon = 4 quarts
• 1 gallon = 32 gills
• 1 bushel = 4 pecks
• 1 pound = 16 ounces
• 1 ton = 2,000 pounds
• 1 ton = 20 hundred-weights
• C° = 5/9 (F° – 32) or F° = (9/5 C°) + 32
• The standard formula for the volume of a sphere is V = (4/3) • π • r3 or  V = 4.19 • r3
• Interest formulas are of two types: simple interest and compound interest.  The simple interest formula is I = Prt . The I indicates how much interest your money has earned or how much interest you owe.  The P is the principal, how much money you invested or are borrowing.  The r represents the interest rate, or the percentage that gets changed to a decimal.  The t stands for time, which is usually a number of years.  The compound interest formula is  A = P ( 1 + r/n)m.  The A equals the total amount of money, or the principal plus the interest.  The r and t are the same as in simple interest.  The n represents the number of times each year that the interest is compounded.  This is usually 4.  Compounding interest means that you split up the rate of interest into a designated number of sub-intervals (every 3 months, twice a year, daily, etc.), figure the interest earned during that sub-interval, add the interest to the principal, and then figure the next interval’s interest on on the sum of the original principal plus the interest you have added.  You will earn more money with a compound interest account than with a flat interest rate account.
• The definition of probability is the number of desired outcomes divided by the total number of possible outcomes.  For example, if a bag contains 3 red marbles, 2 blue marbles, and 1 black marble, what is the probability that a marble drawn out of the bag will be red?  The number of desired outcomes is 3, and the total number of possible outcomes is 6.  Hence, the probability is 3/6, or 1/2, or 0.5 (50%).  The odds of an event is the number of favorable outcomes divided by the number of unfavorable outcomes.  Using the marble example, there are three favorable outcomes and three unfavorable outcomes.  Thus, the odds are 3:3, or 1:1, or 1 to 1.
• The probability of an event is a percentage between 0% and 100%, which is calculated by dividing the number of ways an event can happen by the total number of ways that all the events can happen.  For example, what is the probability that there is at least one daughter in a family of two children?  The possibilities are BB, BG, GB, and GG.  Three of the four possibilities contain at least one girl, so the probability is 3 ÷ 4 = 0.75 = 75%.  The odds of an event is the ratio of the number of ways that an event can happen to the number of ways that it cannot happen.  Thus the odds of having at least one daughter in a family of two children is 3:1.
• The easiest way to calculate the odds of an event when you are given the probability of the same event is to let the probability be represented by a p percent and write p to (100-p) and reduce the two numbers as if they were fractions.  For example, if the probability is 75 percent, write 75 to (100-75) which becomes 75 ÷ 25, which reduces to 3 to 1.  The easiest way to calculate the probability of an event when given the odds of the event is to write the fraction a ÷ (a + b), with a to b being the odds.  For example, if the odds are 4 to 1, then you would write 4 ÷ (4 + 1) which equals 4÷5 = 80%.