Math

When multiplying 2 or more fractions, you can cancel any numerator with any denominator. Always cancel before you multiply. To divide by a fraction, we multiply by its reciprocal We can separate a fraction into 2 fractions by addition or subtraction in the ___ (numerator / denominator) We (can / can not) ___ separate a fraction into 2 fractions by addition or subtraction in denominator. In addition and subtraction, denominator stays unchanged. Fraction times denominator equals numerator.

Numerator

Basic patterns for sequences

n^2 = sequence of perfect square 7n = all positive multiples of n n = all positive integers 3^n = all powers of 3 2n-1 = all positive odd integers

Fractional Exponents 2^(2/3)

1) A square root of a quantity equals that quantity to the ^ (1/2) 2) The power B ^ 1/n = the nth root of B 3) B^ (m/n) can be expressed as nth root to the m power

How to multiply big numbers? (25 x 8)

If you double one side, half the other. 25 x 8 = 50 x 4. = (5 x 4) and add a 0 = 200

Factorials

N! Factorials is the product of N and all integers less than N. N! is divisible by any number less than N and also any ! less than N N different items can be arranged in N! different ways

What to do when subtracting (mental math) and second number set has bigger second digit than first number (i.e., 65-47)?

Add until second digit ends with a 0, and whatever you added to second number, also add to first number. So, 47 + 3 = 50, then 65 + 3 = 68. 68 - 50 = 18

7x8 = 8x5 = 6x8=

56 40 48

Eliminating Fractions with Addition or Subtraction 7x/6 + 2/3 = 13/2. What is X?

7x/6 + 2/3 = 13/2. GCF = 6. Multiply each fraction by 6. 7x + 4 = 39. Solve for x = 5.

How to get fraction to decimal when you have multiple of 10 in denominator? (i.e., 1/20). Try one as fractions multiplied and one as 10 to the power of (^) Now try 1/40.

1/20 = 1/10 x 1/2. 1/2 = .5 x 10^-1 = .05 Another way is 1/20 = 5/100 (multiply each side by 5) = .05 Answer to 1/40 = .025

Laws of Exponents

7^2 * 7^3 = 7^(2+3) . Add exponents together. 7^3 / 7^2 = 7^1 . Subtract exponents. (7^2)^3 = Multiply exponents. Base must be the same to do this. No rule when base is different. Also, no rules for adding or subtracting base + exponents. A base to the negative power is the reciprocal of same base to positive power. so b^-3 = 1/ b^3. Also, (p/q) ^ -3 = (q/p) ^3. Remember to treat numbers separately from exponents. You CAN NOT distribute exponents over addition or subtraction. you CAN distribute exponents over multiplication or division. ( Legal: (ab)^3 = a^3 x b^3 ^3 = a^3 / b^3 Legal: (a/b) ILLEGAL: (a+b)^3 DOES NOT EQUAL a^3 + b^3.

Work Equation

A=RT (like art and A is the amount of work you're doing (think: any type of work is really an art) the combined work rates of people / machines working together is the sum of the individual work rates.

What is equal to 8^24?

Break it down to the root of what equals 8, so (2^3)^24 = 2^72. Then play around with that root. So 4 is (2^2)^(72/2) = 4^36, and so on.

Concentration

Concentration = Solvent / Solution (Solution = solvent + water) * 100% (always expressed as a percent) If two solutions are unknown, have to set up simultaneous equations. One equation about total amount of mass, second is about amount of solute.

Rate, time, distance

D=RT formula (Dirt) Distance = Rate x Time The units for Rate = units for distance and time. So if rate is MPH, rate = miles, time = hour. Interestingly, if you have m/s * s = miles (eliminate the seconds). And if you have ft/s x ft, you cross out both feet to get seconds). Unit conversions can be written as fractions equal to 1. You can multiply and divide by them.

Trick for dividing by 5 (475/5)

Double and divide by 10. So, 475 / 5 = 950 / 10 = 95

How to square numbers ending in 5? (25 squared)

Remove 5 (all answers will now end in 25). Take the digit before the 5 (n) and multiply it by (n+1). Add 25 to end of product. I.e., 25 squared. Remove 5, so 2. Multiply 2x3 = 6. Add 25 = 625.

Arithmetic Sequence (Need more practice)

The nth term of any arithmetic sequence is a>n = a>1 + d(n-1)

Absolute value. How to solve for variable within absolute value sign? |x-s| = 4

Write 2 equations. |x-s| = 4 |x-s| = - 4

properties of integers

make sure problem states integer, or even odd or prime, all integers 0 is even promise are positives remainders are only used with positive numbers 2 is the only prime number and only even prime number. if questions mentions multiples, divisors, etc. and number is over 100, best to do prime factorization. 0^2 = 0

Inclusive counting

regular subtraction excludes the lower endpoint. If you want to include that endpoint (# of pages in book), you must subtract and add 1

Fractions: When adding two fractions, what must the denominator be? When multiplying or dividing fractions, what must the denominator be?

when adding or subtracting, denominator must first be made the same, then the addition or subtraction occurs. Denominators can be multiplied (don't have to be the same), however, in division, flip the order of the second number.

fraction to decimal: 1/10, 1/100, 1/1000

.1. .01. .001

Circular cylinder Surface area of cylinder

Area of circle x height so v=pie r squared x height Surface area - think of areas of both circles 2(pie r square) + 2(pie r height)

Average Velocity

Velocity is a rate. So use formula D=RT. Velocity = D / T. You can't simply take average. Figure out DTR for Leg 1, then DTR for leg 2, and then add both legs of the trip for each D = D1 + D2

Percent: a) What is 1% in decimal form? b) What is .01% in decimal form? c) What is the rule of thumb to converting a number to a percent? d) what percent of 150 is 12.9 translates to: e) 30% of 350

a) .01 b) .0001 c) multiply # by 100 d) 12.9/ 150 e) 350 x 0.3

Inequalities subtraction and addition Multiplication or Division When all numbers in an inequality ratio situation (1/x > 2/3), we can simply cross-multiply.

do to both sides, does not change the sign Multiplication or division by a positive number, you can do to both sides and keep as is. However, by a NEGATIVE number, reverses the sign of the inequality.

Weighted averages so group 1 = 70% earn 40K, group 2: 20% earn 80k and group 3: 10%

mutiply each part by its proportion. = then .7(40) + .2(80) + .1(120) = 56K

Exponents: 1) 1 ^ n = 2) 0 ^ n = 3) A negative to any even power is ___, while a negative to an odd power is ___ 4) an equation with an expression to the even power equal to a negative number ___(does / does not) have a solution while an equation with an expression to the odd power equal to a negative number ___(does / does not) have a solution. Growth or decline? 5) Positive # >1 = 6) Pos # < 1 (i.e., 1/2) = 7) Neg # > -1 = (-1/2) =

1) 1 2) 0 3) positive, negative 4) does not, does 5) exponential growth 6) gets smaller 7) Exponential growth 8) gets smaller

When subtracting two negatives, what can you do? (-22-61)

Factor out the negative and rewrite as # + #. Put - sign in front of sum. So, -22 - 61 = - (22+61) = -83

Multiple Travels (or Trip)

Each traveler or leg requires its own D=RT equation. Use techniques of substitution or elimination to solve unknown variables. (If you can get D = for one equation, sub that for D of the other equation so that you are eliminating D). If you are having trouble with the algebra, sub 0 for one of the rates. That will mean that leg 0 they traveled 0 distance. so then leg 2 will equal total trip time. This will help eliminate some answer choices.

Distance between two points

For slanted distances, draw or imagine the slope triangles and use the Pythagorean theorem. (Remember pathogen triplets.) For a circle in the x-y axis, each slanted radius is a slope triangle with an equal hypotenuse. The equation for a circle where R is at (0,0) is X squared + Y squared = Radius squared

When should you not pick answer choice D in Quant Comparison?

If there is no variable (x) but all numbers, never pick D. Numbers will always give you 'enough' information.

Finding the percent increase or decrease. 1) The price of an item increase from $60 to $102. What was the percentage increase? 2) Price decreased from $250 to $200. What's the percent decrease? 3) Price of item increase from $200 to $800, what is the percent increase? 4) At beginning of year, purchased at 30% increase, then 40% discount from new price. The price paid was what percent below original price? 5) Price of stock increase 20% in Jan., decrease 50% in Feb., and increase 40% in March = what's total percent of change for 3 months?

New / Old = 102/60 = 1.7. Then you have to think what percentage increase does 1.7 equal? It's 70% 2) 200/250 = .8, .8 means item decrease by 20% 3) 800/200 = 4. 4-1 = 3, so 300% 4) X * 1.3 * .60 = .78. .78 is a 22% decrease. 5) 1.2 x .5 x 1.4 = .84, which is a 16% decrease. Note: if a number doubles - used to be 150, now 300 so 300/150 = 2. Then it's a 100% increase.

Lines and angles

Opposite angles always have same measurement. A transversial intersects two parallel lines. Remember how the opposite angles equal one another on both lines. A bisector line means angles on either side are equal to one another. Cuts it exactly down the middle. Supplementary - together they make 180 degrees Vortex is the dot connecting two sides of the triangle. Median - goes from vertex to midpoint of triangle.

Prime factors of a square

The exponents of the prime factors of a square must all be even. Thus, if we see an unknown number in its prime factorization form and the exponents are even, we know the number must be a perfect square. So K= 2^2, 3^4, 4^2, since all the exponents are even, must be a perfect square. When we try to figure out what K is, we know K must be a factor times by itself (since exponents are even), so we can take the factorization and divide by 2, so K = 2^2, 3^2 4^4, so 2 x 3 x 4^2 Since the powers of a prime factor of a square are all even, when you add 1 to every exponent, then all the numbers become odd. Thus, a perfect square always has an odd number of factors. Perfect square are ONLY integers with an odd number of factors

Absolute value

When multiplying something by a negative number in between absolute value signs, always multiply by the positive value of that number, once you've completed all operations inside the brackets

Adding or Subtracting Odds and Evens:

When we add or subtract "likes" (2 odds or 2 evens), we get an even number. When we add or subtract "unlikes" (even plus odd), we get an odd.

Growth or decay

follow calculations step by step, one [change interval] at a time. Bacteria multiplying every 4 hours by 2.5, at 9 am there are 24 B, how much by 1 pm? 9 am = 24 B, 1 pm, 24 / 2.5 = 60

When you see x^2 - y^2, you should know:

(x+y)(x-y)

In equalities, when you multiply or divide by a negative number, you must:

Reverse the sign of the inequality.

Roots

Anything under square root sign has a positive output. If we take square root ourselves (so # started as n^2), we must consider both positive and negative roots. If b > 1 then the square root of b is smaller than B. If 0 < B < 1, then square root of B is BIGGER than B. Cannot take square root of a negative. Square root of -3 = no solution. Any root of 1 equals 1, any root of 0 = 0 Odd roots (cubed 3), can take negative numbers. Even roots (2,4,6) cannot take a negative number. w/ # > 1, the higher the order of the root, the smaller the number. w/ 0 < B < 1, the higher the order of the root, the closer the result is to 1. SO inverse - the higher the root number, the bigger the root. Roots distribute over multiplication and division. Roots do not distribute over addition and subtraction.

How to find the greatest common factor

1) Apply prime factorization 2) Write down factors and exponents for each number, 3) For each factor they have in common, write down highest number of exponents they have in common 4) Multiply So, 360 is 2^3, 3^2, 5. 800 is 2^5, 5^2 Now, what do they have in common? 2^3, 5= 40

Memorize prime numbers under 50. GRE loves to test that 2 is the only even prime # and 1 is NOT a prime number. To figure out if a prime number is divisible by another number (bigger than 10), do the prime factorization of divisible number, see if the factorization primes align to larger # primes.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, 37, 41, 43, 47, 53, 59 1 is NOT a prime #. 2 is the only even prime #

Combinations

When you have to draw a small group from a large group and the order doesn't matter (i.e., you have 6 clothing options and you get to pick 3). you put number of total options in numerator and divide by number they are asking you to draw: so numerator (6x5x4) / denominator (3x2x1) = 20

Absolute value. How to solve for variable within absolute value sign when there is in inequality sign? |x-s| < 4

Write two equations but flip the sign for the negative equation: |x-s| < 4 |x-s| > -4

Similar shapes

Have same angles in corresponding triangle As long as they share 2 angles, they are similar. The ratio of either side of one triangles equal the ratio of the corresponding side in the other triangle. Scale factor - factor by which all lengths in smaller figure are multiplied to equal lengths in larger figure. If all lengths are multiplied by scale factor, the area is multiplied by scale factor ^ 2. Any right isosceles triangle has a hypotenuse that's square root of 2 x one of the legs. We can refer to these right isosceles triangles as 45-45-90 degree triangles, 1 or 1, square root of 2 triangle Make an equilateral triangle a right triangle for calculations! But if you think about it, it's not 45-45-90, because the equilateral used to have all 60 degree angles, so now one angle remains 60, and the other was chopped in half so it's 30. So isosceles triangle is also called the 1, square root of 3, 2 triangle, or a 30, 60, 90 triangle

How to square numbers ending in 0? (20 squared)

Remove 0 and square remaining digits, then add 00 to answer. 20 squared = 2 squared = 4, add 2 00s = 400

Lowest common multiple. Note, if A is a factor of R, then the LCM of A & R must be R. For example, the LCM of 8 and 24 is 24.

1) Find prime factorization and the Greatest Common Factor 2) Write each number in the form GCF * ? 3) the LCF is the product of these 3 factors. So 24 = 2^3 * 3 and 32 =2^5 GCF = 8 Use GCF as one of those numbers and multiply by another to get numbers. 24 = 8 *3 32 = 8 *4 LCM = 8 * 4 * 3 OR LCM = P * Q (both numbers given) / GCF

Trapezoids

Trapezoids has exactly one pair of parallel sides. Parallel sides are called bases and non parallel sides are called legs. The two angles on a leg are supplementary (together equal 180 degrees) To find base of trapezoid: (base 1 + base 2) / 2

Probability

0 < P < 1 Total # of successes / Total # of Outcomes Probability of something NOT happening is 1 - it happening P(A) OR means add, AND means multiply The AND and OR roles when things are mutually exclusive (one doesn't impact the other) is straight forward. 'With replacement'. And = P(A) x P(B) while OR = P(A) + P(B). Examples include cards, coins, dice. However, when things are not mutually exclusive (instead, dependent on another), formulas get more tricky. 'Without replacement' OR Rule = P(A) + P(B) - (P(A) x P(B)) AND Rule: P(A) x (P(B) / given A) AT Least Rule: (what's the probability of getting at least one head)? "At least 1", meaning 1 or more. So take out the probability of it being 0. Probability(that it's not A) = 1 - (Probability that it's A)

If problem calls for % increase, what must you always do? An item originally cost $800.The price increased by 20%. What is the new price? If problem calls for % decrease, what must you do

1 (representing the whole) + % increase as multiplier. (1 + .n) 1+.2 = 1.2, so $800 x 1.2 = $960 1 (representing the whole) - multiplier of % (decimal form)

Compound interest. 1) How would you set up this problem: you are investing $4000 at a 6% interest for 8 years 2) You invest $1000 in account that yields 5% annually, compounded quarterly. How much do you have after 6 years?

1) 4000 (1.06)^8 2) 5 % / 4 = 1.25 %, then make into decimal = .0125, then add it to 1 = 1.0125. Take number of years and multiply by periods (6x4=24) equation is $1000 (1.0125)^24

1) As we move decimal to the right, # is (what) by 10? 2) As we move decimal to the left, # is (what) by 10? Decimals to exponents. Any # to the right of the decimal translates to 10 to the power of (-1) 3) So do the following: .1, .01, .001., .0001 4) Adding or subtracting decimals:

1) Divided 2) Multiplied 3) .1 = 10 to the power of -1 .01 = 10 to the power of -2 .001 = 10 to the power of -3 .0001 = 10 to the power of -4 4) just line up the decimal and do the math!

Quadratic Equations What is 5x^2 - 10 x - 27 = 13

1) Get everything to one side, set equal to O (5x^2 - 10 x - 40 = 0) 2) Factor out any GCF x^2 - 2 x - 8 = 0) 3) Factor (x-4) (x +2) 4) Create two linear equations and solve. X = 4 or X = -2

Divisibility Rules: 1) 4's: 2) 3's 3) 9's 4) 6's

1) If last two digits of number are divisible by 4 2) If sum of all digits is divisible by 3 3) If sum of all digits is divisible by 9 4) to be divisible by 6, must be divisible by both 2 and 3

Strategy for units digit questions

1) Only units matter, so in 28 * 31 (you multiply just 8 and 1 = 8. Your answer will have units digit of 8. 2) Look for repeating pattern. So if question is 87^123, what is the units digit? Only focus on multiples of 7 to find pattern. 1) 7^1 = 7 2) Carry over 7 x 7 = 49 3) Carry over 9. 9 x 7 = 63 4) Carry over 3. 3x7 = 21. 1) Carry over 1. 1x7 = 7 2) Carry over 7....etc. So we have a pattern of 4, meaning every number divisible by 4 will end in 1. So now if we want to figure out 87^123. 120 is a multiple of 4 so that ends in 1. 121 would then end in 7 122 would end in 9 123 would end in 3.

How to count factors of large numbers.

1) Perform prime factorization and write out exponents of each of the primes, for those with no exponent, consider that an exponent of 1 2) Add 1 to every exponent 3) Then multiply exponents together 4) The product of that, is how many factors there are, including 1 and itself. to find the odd factors, we would repeat this procedure but ignore the factors of 2. We have no direct way to calculate the even number of factors. We have to calculate total, calculate, odd, and then subtract to find # of equal.

Exponents: Positive or negative? 1) Negative raise to even power is always _ 2) Negative raised to odd power is always _ 3) without parenthesis, -3 (squared) = 4) while with parenthesis, (-3) squared = 5) Anything squared to 0 that is non-zero = 6) 0 squared to 0 is 7) Anything squared to the negative power

1) Positive 2) Negative 3) Exponent is applied before the negative sign, so -9 4) -3 x -3 = 9 5) 1 6) undefined 7) Is 1 / number squared

Simplifying Expression. Try this problem: (2xy)(3xz)(4yz) = Remember, you can combine like terms. Like terms have same variable and exponents but not necessarily coefficients. You can add X

1) We can add or subtract like terms (so different powers of y won't work) 2) With multiplication, you can switch the order (so 3ab^2 is same as 3ba^2) and groupings (AB)(CD) = A (BCD) 3) When addition sign appears before parenthesis, we can remove the parenthesis 4) When a subtraction sign appears in front of the parenthesis, reverse all signs within parenthesis 5) Distributive Law - multiplication distributes over addition or subtraction but does not distribute over multiplication (so 3 (a +b) = 3A + 3B but 3(ab) is 3ab. 6) If we multiply a number times a monomial (with a variable, constant multiplies the coefficient 7* (5x^2) = 35x^2 7) X * X = X^2 and X^2 * X = X^3 -- when we multiply two powers, we add them Solution to problem: 24x^2y^2z^2

Remainders

1) When we divide an integer by a divisor, the number is an integer & one of the numbers factors 2) the remainder must be less than divisor 3) Dividend / divisor = quotient + (remainder / divisor) 4) rebuilding the dividend, D = S x Q + R (If S is the divisor)

Consecutive integers:

1) a set of n consecutive integers will always contain one integer divisible by n. 2) if n is odd, the sum of a set of consecutive integers will always be divisible by n 3) In a set of 3 consecutive integers, you could have 2 evens (or odds) and 1 odd (or even). But in a set of 4 you'll have 2 odds and 2 evens.

comparing fraction sizes: 1) if denominator gets bigger and numerator stays same, fraction becomes ____ 3/5 --> 3/7 2) if numerator gets bigger and denominator stays same, fraction becomes ____ 2/8 --> 3/8 3) If num gets bigger and denominator becomes smaller, fraction becomes 5/8 --> 6/7 4) What if we add the same # to numerator and denominator?

1) smaller 2) bigger 3) bigger 4) If fraction is < 1, then the addition brings it closer to 1, so fraction gets bigger. If the fraction is >1, then the addition brings it close to 1, which makes fraction smaller.

Memory: What's 1/6 and 5/6 in decimal form? 1/7 1/8 3/8 1/9 and then 7/9

1/6= .16666 repeating 5/6 = .8333 repeating 1/7 = .143 1/8 = .125 3/8 = .375 1/9 is a bunch of 1's repeating .1111 so therefore 7/9, is a bunch of .77777 repeating

Circles

A =(pie)R^2 C = 2(pie)R diameter is longest possible chord in a circle (chord connecting 2 places on circumference of circle) Any inscribed angle that intercepts a diameter (one leg is the diameter), has to be a right angle. A tangent line touching the circumference, is perpendicular to radius (90 degree angle). If 2 sides are a radii, the triangle is isosceles. A central angle has same measure as arc it intercepts. The measure of an inscribed angle is half of the measure of the arc that it intercepts equal length cords equal equal arches. We find arc length and area of sector by setting up part to whole ratios. Angle / 360 = arc length over 2pieR When you have a triangle inscribed in a circle, angle that touches circumference is always 90 degrees.

Unit conversion

Always use the conversion given. Put unit you're trying to convert x (desired unit / unit it's currently in). If you're doing an area problem, that's squared, so you have to multiply by desired unit twice. If you're doing a volume problem, that's cubed, so you have to multiply by desired unit 3 times. Example: Pool that's 5*8*3 meters has to be filled with sand, how much sand? (1 meter = 1,000 mm) So 80 m cubed (1,000 mm / 1 m) * (1,000 mm / 1 m) * (1,000 mm / 1 m) = 8 Billion mm cubed.

write in equation form 50 more than B 50 less than B twice A is 100 less than 3 times B A is 50 percent of B A is 50 percent greater than B

B + 50 B - 50 2A = 3B - 100 A = .5B OR 2A = B A = 1.5B

Counting

And means Multiply and OR means Add. Suppose we have 6 books to place on a shelf, in what variety of orders could we place them? Spot 1 (6 options) x Spot 2 (5 options) x Spot 3 (4 options), and so on... so 6x5x4x3x2x1 = 720. OR 4 apps, 3 entrees, 2 desserts, how many total combos of dinner? 4 x 3 x 2 = 24 As general rule, if we have to arrange N different items in orders, we multiply N times every number smaller than it. When a counting problem has restrictions, always do the most restrictive stages first. Example: 4 classes in summer school: but science must always follow science lab. Science can not be fourth period because lab can't follow, so science can be in 3/4 slots = 3, SL can only be in 1 slot (following S) = 1, then that leaves 2 classes uninhibited so 2, 1 = 3x1x2x1 = 6 Counting with identical items. When there are N items and P are identical, and you have to arrange them in combinations, N = N! / P! So 4 different books and 3 identical dictionaries, how many orders? 7! Items / 3! Identical = 840 If there are several groups within a group of identical items: N! / A! x B! x C! (Mississippi rule) When we count, we must eliminate repetitions. We do so by dividing by the number of each arrangement is repeated. Working with identical items:

Multiplying Odds and Evens: NO RULES FOR DIVISION because often we don't get whole number!!! Always use 1 and 2 when testing even and odd numbers.

Even x Even = Even Odd x Odd = Odd Odd x Even = Even As long as there is even one even factor in multiplication, the product will be even. Only if all factors are odd, will the product be odd.

What to do when subtracting small positive from big positive? (or small positive + big negative)

Factor out the negative sign to reverse the order of subtraction. (you get to flip the order of the numbers but not the sign)

Quadrilaterals

For every quad, sum of all 4 internal angles equals 360 degrees. Can all be divided into two triangles. Parallelograms: opposite sides are parallel, opposite sides are equal, opposite angles are equal, diagonals bisect each other. If any of these facts are true, the rest are true (comes as a package) Rhombuses have all the same properties as parallelograms but also have 2 unique additional properties: all 4 sides are equal and diagonals are perpendicular (90 degree angles on inside) These 2 unique additional properties do not always come together. Rectangles: have 4 90 degree angles. Also parallelograms but have the following 2 attributes: each angle is 90 degrees, diagonals are congruent. Both of these attributes don't have to be true, just 1 to make it a rectangle A square has all the rectangle, rhombus, and parallelogram properties

Triangles

If polygon has n sides, it can be divided into n-2 triangles. Sum of measure of polygon with n sidesThe largest angle is always opposite of largest side (visa versa) The sum of any two sides must be bigger than the 3rd side. Any side of a triangle must be greater than the difference of the two other sides and less than the sum of the two other sides. The sum of the total of 3 angles must be 180 degrees. Isosceles - 2 (or more) sides are equal, thus 2 angles opposite those lines are equal (2 way theorem). Isosceles triangles are also equilateral triangles. In isosceles triangle, line that bisects vertex also marks midpoint of base and forms right angle. So it plays many important roles. 1,1, square root of 2 Equilateral - 3 equal sides with 3 equal angles. 1,square root of 3, 2 Area = 1/2bh Right triangle - 2 sides that make up the 90 degrees are called legs and the opposite long side is called the hypotenuse. If you are using the Pythagorean theorem - a^2 + b^2 = c^2, and the numbers are too big. find GCF, divide both sides by it. Then do Pythagorean with smaller numbers. Then take final answer and multiply by GCF to scale back up. Common Pythagorean triplets: 3,4,5; 5,12,13; 8,15,17; 7,24,25 Sum of measure of internal angles of polygon is n sides minus 2 (ie number of triangles within polygon) x 180 degrees (ie degrees of a triangle)

Two Travelers

If they are moving in opposite direction, we ADD speeds. If travels are moving in same direction, we SUBTRACT speeds. If gap is obviously shrinking or expanding, set up D=RT for gap itself! If we figure out the speed which gap is shrinking or expanding, we can relate two individual speeds by subtraction or addition.

XY Coordinates

If you are given two sets of x and y (1,-3), (2, 4), you can always plot the line. All you need are two sets of point. Any horizontal line has a firm y-axis (y-axis never changes but x-axis does) so y = k always. Any vertical line has a firm x-axis which doesn't change, so X=K Any two points that share same y-coordinate must be a horizontal line. Any two points that share same x-coordinate, must be vertical line. Slope of 2 (m = 2), could mean (-2, -1) because if you divide -2/-1 = 2. Or it could mean (2,1) If given two sets of points and you want to find slope, subtract both Ys as your 'rise' and subtract both x's as your 'run' for rise / run. If you have a slope of 2, it means that everytime you add x (or -x) to the x-coordinate, you're adding 2 to the y-coordinate. So if you have the points (-3,-1), then (-2, 1) must also be a coordinate. Slope of -2/3 could also mean left 1 unit and up 2/3 of a unit (-1 + 2/3) Lines with slope of 1 or -1 makes 45 degree angles. So angle they would form is 45, 45, 90. Parallel lines have equal slope. Perpendicular lines have opposite signed slopes (-1 and 1 and you flip the numbers) so slope of -(1/2) becomes slope of positive 2 for the reciprocal. Doesn't matter what b equals, just the slope. Only lines that intercept at the origin have an x & y axis of 0. x-intercept is where line intersects x axis, y - intercept when y intercepts y axis. When given an equation, to find x we plug y=0, when we want to find the y, we plug x=0. Intercepts can be classified as points: (s, 0) = x-intercept, (0,S) can be classified as y-intercept;. The x-coordinate of any y-intercept is always equal to 0. To find y-value always replace x with 0! If we are given the point (0, 2m) and asked to populate y=mx+b. We know b is 2m since 2m is y-intercept when x = 0. Anytime a y passes through the origin, the y-axis will be 0. To find slope of a line with coordinates (7, -4) that passes through origin. Consider (0, 0) as second set of points and do rise (0 - (-4) - run ( 0 - 7) = 4/-7

Mean (Average)

Sum of all terms of N / N = Mean, which means that Mean * N = Sum of terms. Which is also important to note. Median - middle number in ordered list. If 2 middle numbers, take average. Mode: number occurring most frequently. Can have no mode. In evenly spaced list (all multiples of a # or consecutive integers, mean and median are the same. OR when list is entirely symmetrical. In any arithmetic sequence, the mean = median.

two equations: 2 variables When to use substitution and when to use elimination? If the test asks you to solve for an expression, NOT the individual variables, there will always be an elegant shortcut, you do not need to use either substitution or elimination. If Xs or Ys are eliminated when you go to solve, infinite number of solutions! If you are lead to a never true equation (15 = 8), then there is no solution. (this would be the case of parallel lines) Always try to isolate or eliminate a variable. If you have 3 variables and 3 expressions: Note: you can only solve if you are given 3 equations! 1) Pick 2 of the 3 equations and using either substitution or elimination, eliminate 1 variable. (elimination is easier) 2) pick another pair, eliminate same variable, so now you are down to 2 variable, 2 equation case. 3) Now use elimination or substitution for 2 variables and then plug those 2 numbers into 1 of the 3 original equations to find last variable.

Use substitution if coefficient for either x or y is 1. Otherwise very cumbersome - many fractions! Use elimination if you can make coefficient in both equations equal in number but one positive, one negative.

Combining inequalities

We can combine inequalities in the same direction (a < b & b < c = a < c) We can ADD inequalities in the same direction (a < b + c <d = a + c < b + d) We can subtract inequalities of opposite directions. (a < b - c > d = (a-c < b=d) - keep same inequality as first (or top) number No rule for division or multiplication of inequalities any positive > any negative Adding a positive makes it bigger, subtracting makes it smaller You can cross multiply when you are sure variable is positive (so if you have X^2, you know variable is positive)

Age questions (Right now, Steve's age is twice of Tom's age). In 8 years, twice Tom's age will be will be 5 more than 3 times Steve's age)

When ages are specified at different times, use addition or subtraction to represent ages at other times (but do it to both sides) 2S=T 2 (2S + 8) = 3 (S+8) +5

Operations with Roots

You cannot distribute root for addition or subtraction (Square root of 2-3 does not equal square root of 2 minus square root of 3). you can distribute roots over multiplication or division. When multiplying, treat whole numbers and things under root sign separately. When we raise a radical to an expression, we distribute the exponent to each factor. To undo a radical equation, we have to square both sides. We have to isolate the radical before squaring (move other number to other side of equation) Much check answers back within the original equation.

a) How to multiply with decimals: (0.03) ^ 3 b) How to divide with decimals? .56 / .0007 c) remember that 39.85 x .1 is the same as dividing by 10, so answer is 3.985 When we multiply an number by the positive power of 10, we move decimal to the right.

a) 1) 0.03 x 0.03 x 0.03 2) count how many #s to the right of the decimal = 6. 3) Remove all decimals, so 3 x 3 x 3 = 27. Then place 6 0's to the right of 27 = .000027 b) denominator must be whole number so move decimals to the right in both numerator and denominator until bottom number is whole. Divide. 5600 / 7 = 800

slope

as you move to the right, if line goes up, then it's a positive slope. If it moves down, it's a negative slope. Slope = rise / run (change in y / change in X) Change in y means difference between the 2 y-coordinate values of the two points on the line. To determine the slope of a line, put into formula y=mx+b where the slope is m.

One divided by any fraction is equals , so 1/ (7/3) = You can not divide by a variable or a variable expression because:

the reciprocal of that fraction; 3/7 if the variable is 0, you can't have 0 in denominator., unless the variable is squared or cubed (because you know that regardless number will be positive)