GRE Math Tips

using a ratio to determine an actual number (ex. ratio of boys to girls is 3:4. if there are 135 boys, how many girls are there?)

3/4 = 135/x 3x = 135 x 4 3x = 540 x = 180

absolute value |x| = 3 Ex. |x-3| < 2, what is the range of values for x?

distance from zero x = -3 or 3 1. |x-3| < 2 2. (x-3) < 2 and (x-3) > -2 3. [x-3 < 2] and [x-3 > -2] 4. solve for x x < 5 and x > 1 5. 1 < x < 5

even/even =

even, odd, or a non-integer

solving sequence problems ex. what is the positive difference between the 5th and 4th terms in the sequence 0, 4, 18,... whose nth term is n^2(n-1)?

ex. a4 = fourth term in the sequence a5 or 5th term = 5^2(5-1) = 25 x 4 = 100 a4 = 4^2(4-1) = 16 x 3 = 48 difference between the 4th and 5th terms: 100 - 48 = 52

compound interest problems Ex. if $10k is invested at 8% annual interest, compounded semi-annually, what is the balance after 1 year?

final balance = principal x (1 + interest rate/C)^[time x C] A = P(1 + r/n) ^ nt C = # of times interest is compounded annually 1. A = 10k (1 + 0.08/2)^ 1x2 2. A = 10k x (1.04)^2 3. final balance = $10,816

How to solve a function problem ex. what is the minimum value of f(x) in the function f(x) = x^2 - 1?

function notation: f(x) since x^2 cannot be negative, the lowest number you can input for X is 0 f(0) = 0^2 - 1 = 0 -1 = -1 minimum value of function is -1 > (0, -1)

how to recognize multiples of 5

last digit is 5 or 0

how to recognize multiples of 4

last two digits are a multiple of 4

adding and subtracting decimals

line up the decimal points beforehand

Line m on a coordinate plane = -2x+3y = 6 A. slope of line parallel to line m B. slope of line perpendicular to line m which is greater, equal, or can't determine?

must be put in slope intercept form = -2x+3y = 6 3y = 2x +6 y = 2/3x + 2 A. slope of line parallel to m is the same slope = 2/3 B. perpendicular slope would be -3/2 since B is a negative number, A is greater

12^34 is divisible by all of the following except... a. 8 b. 28 c. 64 d. 81 e. 144

must factor tree out 12 and factor tree out each answer 12 = 2 x 2 x 3 a. 8 = 2 x 2 x 2 = 2 is contained within 12 = divisible b. 28 = 7 x 2 x 2 > contains 7 which 12 is not divisible by = NO c. 64 = 2^6 = yes d. 81 = 3^4 = yes because 3 is within the prime factors of 12 e. 144 = 3 x 3 x 2 x 2 x2 x2 = all contained not divisible is 28! B

how to factor polynomials (a^2 + 2ab + b^2)

(a^2 + 2ab + b^2) = (a+b)^2

A: 0.0125% of 1000 B: 1.25 Which is greater, equal, or can't determine?

0.0125% = divide by 100 = 0.000125 of 1000 equals multiply by 1000 or move right 3 spaces = 0.125 0.125 vs 1.25 B is greater

A. angles 1 + 3 B. Angles 2+5 which is larger, equal, or can't determine?

1+3 are complementary and equal 180 2 = 6 = 7 = 3 2 is the same as 3 so 2+5 is also 180 they are equal!

a negative number raised to an odd integer =

negative

even + odd =

odd

even - odd =

odd

odd - even =

odd

odd x odd =

odd

odd^3 =

odd

odd/odd =

odd or a non-integer (fraction)

percent problems setup

part = percent x whole

a negative number raised to an even integer =

positive

a pigeon is in a experiment with a square and round button. If it pecks the square button, there is a 50% chance that it'll get a treat. If it pecks the round button, there is a 75% chance of a treat. The pigeon is just as likely to peck the round button as the square. A. the probability that the pigeon will get a treat after it pecks one of the buttons B. 5/8 which is greater, are they equal, or can't determine?

probability of pecking square = 1/2 prob. of pecking round = 1/2 prob. of pecking Square and getting a treat: 1/2 x 1/2 = 1/4 prob. of pecking round and getting treat = 1/2 x 3/4 = 3/8 probability of pecking square OR round and getting a treat: 1/4 + 3/8 = 5/8 A = B

determining a ratio

ratio = of/to 20 oranges to 12 apples = 20/12 or 5/3 or 5:3

multiplying decimals

1. Multiple numbers as if whole numbers. 2. Then count total number of decimal places in the factors and add them (Ex: 3.77 and 2.8 has three total decimal places) 3. Add the total number of Decimal places to product, counting right to left (Ex: 377*28 = 10556 --> 10.556 final answer)

solving problems where a graph is shown and a picture is drawn on it ex. parabola given and you have to choose the equation that is responsible for the parabola

1. choose points you see on the graph (x,y) 2. plug them into the answer choices to see which ones don't work 3. choose the one that does work

how to find X "in terms of" y

1. isolate y on one side 2. move everything on the side with X

Predicting whether a sum/difference/product will be EVEN or ODD (Ex. if "M" is even and "N" is odd, is the product MN even or odd?)

1. pick numbers for each

2x^2 + 4x - 30 = 0 A: product of the roots of the equation B: sum of the roots of the equation which is greater, equal, or can't determine?

1. reduce to get x^2! divide by 2 > x^2 + 2x - 15 = 0 2. what multiplies to -15 (opp signs) and adds to +2 (3 and 5) (x-3)(x+5) 3. roots are x = 3, -5 4. product = -15 sum = -2 5. B is greater

What is the probability of rolling a number greater than 2 twice in a row on a fair six-sided die, with each of the numbers 1- 6 on each side?

rolling number greater than 2 = 4/6 4/6 x 4/6 = 16/36 = 4/9

[r√s] x [t√u] =

rt√su

solving a digits problem ex. if A, B, C, D represent distinct digits in the addition problem below, what is the value of D? AB + BA = CDC

1. two 2-digit numbers will add up to at most something in the 100's so C = 1 2. A+B in the ones column has to equal to 1 > carry the one so B+A = 11 3. which means D = 2 47+74 = 121 OR 83+38 = 121

area of a square with vertices labeled on a graph

1. use distance formula to figure out length of each side 2. square = all 4 sides are equal in length multiply one side by another to get area

A line whose slope is -1/4 passes through the points (4,3) and (x,1). What is the value of x?

1. use slope formula y2-y1/x2-x1 (4,3) and (x,1) x1 y1 x2 y2 1 - 3/ x-4 = -1/4 -2/x-4 = -1/4 > 4 is + cross multiply -8 = -1(x-4) -8 = -x + 4 -12 = -x x = 12

[16√90] / [8√10] =

16/8 = 2 90/10 =9 2√9 = 2x3 = 6

adding a positive number and a negative number together

subtract the smaller number from the bigger number the answer will have the same sign as the bigger number Ex. 3 + (-9) = 9-3 = 6 > -6 because 9 is bigger

average

sum of all terms/ number of terms 3 + 4 + 5/3 = 4

how many ways can 5 books be arranged on a shelf?

5! = 5 x 4 x 3 x 2 x 1 = 120 ways

(3x)^(2/5)

5√3x^2

[2x√6y] x [3y√x] =

6xy√6xy

if x = (3^8)(81^5) and y = (9^7)(27^12), then xy = ?

81 = 9^2 9 = 3^2 81 = 3^(2x2) 81^5 = 3^(2x2x5) = 3^20 x = 3^20 x 3^8 = 3^28 9^7 = 3^14 27^12 = 3^36 y = 3^50 xy = 3^28 x 3^50 xy = 3^78

[8x^2 √25y] / [2x√5y] =

8x^2/2x = 4x 25y/5y = 5 4x√5

A: [2√3]/6 B: 1/√3 which is greater, equal, or can't be determined?

A = √3/3 must match same denominator = multiply by √3/√3 = 3/3√3 reduce = 1/√3 equal to B

how to count consecutive numbers (ex. number of integers from A to B inclusive)

A to B inclusive = B - A + 1 integers from 73 through 419 inclusive 419 - 73 + 1 = 347 integers

A = greatest prime factor of 243 B. smallest prime factor of 243 which is bigger, equal, can't determine

A. 243 = find it's prime factorization = 81 x 3 > 3 is the greatest B. 273 smallest prime = 3 x 91 = 3 equal because the greatest for one is equal to the smallest for the other

Age: 14 15 16 17 18 Number of students: 4 8 7 1 2 what is the mean?

total number of students = 4+8+7+1+2 = 22 students 2. find sum of students 14(4) + 15(8) + 16(7) + 17 + 18(2) = 341 3. average = sum of students/number of students = 341/22 average = 15.5

Similar Triangles

triangles that have the same shape but not necessarily the same size 1. corresponding angles are equal 2. corresponding sides are proportional 3. asked to find the length of a side or measure of an angle

x > or = to 0 A: 5^5x B: 5^x+5 which is bigger, they're equal, or can't determine?

try plugging numbers tried 1 > B is greater tried 10 > A is greater = cannot determine

0 raised to the power of 0

undefined

how to find the volume of a cylinder ex. r = 6 and h = 3

volume = area of the base x height or volume = pi x r^2 x height volume = pi x 6^2 x 3 = 108 pi

how to find the volume of a rectangular solid

volume = length x width x height

which of the following ratios is greater than 1:2? a. 200:700 b. 110:200 c. 240:700 d. 200:440 e. 25:50

B and E a = ~1/3 is smaller than 1/2 b = > 1/2 c = ~1/3 smaller than 1/2 d = 1/2... < 1/2 e = 1/2

x = odd and positive y = even and positive A = (-1)^2xy B = 0.5 which is bigger, equal, can't determine?

x is odd and y is even odd x even = even 2(even) = even x even = even -1 raised to an even power = positive pick numbers to determine if it will be bigger than 0.5 (-1)^2(1)(2) -1^4 = 1 A is bigger

if x^a times x^b = 1 and x is not 1 or -1, what is a+b?

x^a times x^b = a+b (add exponents when mutliply and because same base) if power is equal to 1, that means that either the bases are 1 or -1 or the exponents are 0. since we know the base x is not 1 or -1, we know that the exponents are 0 meaning a+b = 0. 0

exponents with a base of zero

zero raised to any exponent is 0

converting fractions to decimals

Divide numerator by denominator

3-4-5 triangle

If a right triangle's leg-to-leg ratio is 3:4, or if the leg-to-hypotenuse ratio is 3:5 or 4:5, it's a 3-4-5 triangle and you don't need to use the Pythagorean theorem to find the third side

The value of a certain stock rose by 40 percent from March to April, and then decreased by 30 percent from April to May. The stock's value in May was what percent of its value in March?

Let stock = $100 rise 40% = $100 + 40 = $140 Then dec. by 30% 140 x 0.3 = 42 140-42 = 98 98/100 = 98%

dividing decimals

Move the decimal to the right on both numbers until the smaller number is a whole number and move that same amount of spaces on the bigger number answer should have the decimal placed where the bigger number originally had a decimal place

probability keywords

OR = add the probabilities AND = multiply them Ex. blue and then red marble = multiply square or round button = add

Ex. Rhonda is 5x older than Frank is right now. A. Rhonda's age in 13 years B. 3 times Frank's age in 13 years

R = r + 13 F = 3(13 x F) Relationship cannot be determined because you don't know what their ages are

John buys R pounds of cheese to feed N people at a party. If N+P people come to the party, how many more pounds of cheese must John buy in order to feed everyone at the original rate?

R/N pounds of cheese per person x P extra people = RP/N

Distance =

Rate x Time

How to find the surface area of a rectangular solid

SA = 2(length x width) + 2(width x height) + 2(length x height)

how to find the surface area of a cylinder ex. r= 3 and h = 4

SA = 2pi r^2 + 2 pi r height = 2 pi 3^2 + 2 pi 3 x 4 = 18 pi + 24pi 42 pi

surface area of a cube

SA = 6s^2 s = length of one side

a store sells a watch for a profit of 25% of the cost. What percent of the selling price is the profit?

Say that watch is $100 profit = 25% = $25 Selling price = cost to the store + profit $100 + $25 = $125 1. figure out what percent of the selling price is the profit $25/$125 = 1/5 of selling price = 20% 20%

Youth club has 12 children. They get split into 3 groups of 4 and one of the kids in the group is designated leader. When the children line up, they have to line up by group and the leader has to be at the front. How many different ways can the children line up?

Since the leader has to be in the front, only the arrangement of the three remaining kids matters. 3 kids arrangement = 3! = 3x2x1 = 6 ways do it for all three groups = 6x6x6 = 216 Also there are three groups so 3 different arrangements for the 3 groups = 3! = 6 6x216 = 1296 ways to arrange the 3 groups

finding the range

Subtract smallest from largest number

sum of consecutive numbers (ex. sum of the integers from 10 through 50 inclusive)

Sum = (Average) X (Number of Terms) (so must find average and the number of terms and add them together) 1. find average [ 10 + 50/2 = 30] 2. number of terms > 50 - 10 + 1 = 41 terms 3. sum = 30 x 41 = 1230

how to recognize multiples of 12

Sum of digits is a multiple of 3, and last two digits are a multiple of 4

how to recognize multiples of 6

Sum of the digits is a multiple of 3 and the last digit is even.

A cylinder has a surface area of 22pi. If the cylinder has a height of 10, what is the radius?

Surface area = 2pi(r)(h) + 2pi(r^2) 22pi = 2pi r (h + r) 22pi = 2pi r (10 + r) 22pi/2pi = 2pi r (10 + r)/2pi 11 = r(10 + r) 11 = 10r + r^2 r^2 + 10r -11 = 0 (r - 11)(r + 1) = 0 has to be positive so r = 1 radius = 1

hypotenuse

The side opposite the right angle in a right triangle. usually the longest side of the right triangle

how to recognize multiples of 3

The sum of the digits is a multiple of 3

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. gives you the range of numbers the missing side could be if given only two sides, add them = total is greater than X if you subtract the two lengths, that total is less than x now you have a range x could be between

Converting a decimal to a fraction

To convert 0.75 to a fraction 1. Write down the decimal "over" the number 1 0.75 / 1 2. Multiply top and bottom by 10 for every number after the decimal point (0.75 has two numbers after the decimal so multiply by 100) 0.75 × 100 / 1 × 100 = 75 / 100 3. Simplify the fraction to 3 / 4

supplementary angles

Two angles whose sum is 180 degrees

complementary angles

Two angles whose sum is 90 degrees

Raghu rolls two fair six-sided dice numbered 1 through 6. A. probability that he rolls at least one number lower than a 5 B. probability he does not roll the same number twice which is greater, equal, or can't determine?

Use probability rule of P(A) = 1 - P(not A) 1. find probability of rolling 5 or higher two times 5 or higher = 2/6 = 1/3 two times=two dice = 1/3 x 1/3 = 1/9 2. probability of rolling number lower than 5 two times 1 - 1/9 = 8/9 A = 8/9 3. probability of rolling the same number twice probability of rolling the same number again = 1/6 4. probability of not rolling the same number = 1 - 1/6 = 5/6 B = 5/6 A is greater (8/9)

[a√b] / [c√d] =

[a/c] x [√b/d]

equilateral triangle

a triangle with 3 congruent sides

isosceles triangle

a triangle with at least two congruent sides and two equal angles

finding angles formed by intersecting lines degrees of a, b, c, d

a=c b = d ones on a line = 180 a + b = 180 c + d = 180 a+b+c+d = 360

how to factor polynomials (ab + ac)

ab + ac = a(b+c)

x^0

always 1

odd/even =

always a fraction = non-integer

any number raised to the exponent of 0

any nonzero number raised to an exponent of 0 = 1

If a square of side "x" and a circle of radius "r" have equal areas, what is the ratio of x/r?

area of square = x^2 area of circle = pi r^2 x^2 = pi r^2 divide by r^2 x^2/r^2 = pi sq rt x^2/r^2 = sq rt pi x/r = sq rt pi sq rt pi

quadratic equation

ax² + bx + c = 0

multiplying and dividing positive and negative numbers

both numbers have the same sign = positive answer both numbers have different signs = negative answer

even/odd =

can be even or a non-integer (fraction)

[b/a^c] =

c√a^b from bottom > to base > to the top

percent decrease equation

% decrease = amount decrease/original whole x100%

percent increase equation

% increase = amount of increase/original whole x 100%

area of a hexagon

( 3S^2 √3 ) / 2 hexagon has 6 (30-60-90) triangles height = x√3

how to factor polynomials (a^2 - b^2)

(a^2 - b^2) = (a-b)(a+b)

2|-3y+4| = 38 |-5y| = 25 A. Y B. 5 which is greater, equal, can't be determined?

- Given the two equations for absolute value = both equations must be true - need to figure out what "y" is - start with the easier one 5y = 25 5y = -25 y= 5, -5 - then the harder one 2(-3y+4) = 38 y = -5 2(-3y+4) = -38 y = 7.7 < not one of the answers for y so y = -5 since it worked for both equations A = -5 B = 5 B is greater than A

how to find the max and min lengths for a side of a triangle ex. the length of one side of a triangle is 7. the length of another side is 3. what is the range of possible lengths for the 3rd side?

1. 3rd side is greater than the difference of the two other sides 7 - 3 = 4 meaning that the 3rd side is > 4 2. 3rd side is also less than the sum of the other two sides 7 + 3 = 10 meaning the 3rd side is < 10 range of possible lengths is between 4 and 10

|8x-7| > 3x + 8 which are true? a. 1 < x < 3 b. -1 < x < 0 c. 2 < x < 4 d. x = 3.5 e. x = -1

1. 8x-7 > 3x + 8 x > 3 2. 8x-7 < -3x -8 x < (-1/11) a. x>3 so NO b. x < 0 so YES c. x>3 so it's > 2 and can be <4 YES d. x>3 so YES e. x can equal -1 because it's less than -1/11 or -0.09 all true but A

percent problems = what is ___% of #?

1. Percent % = make into a fraction Ex. 12% = 12/100 2. Whole: multiply by the # given 12/100 x 25 = 300/100 3. reduce ex. 3 Looking for "part" part = percent x whole

a,b,c,d,e = positive integers (no fractions, decimals) (a+b)^2 = 25 c x d x e = 6 A. the range of abcde B. 4 which is larger, equal, or can't determine

1. a + b could be 4+1 or 3+2 2. cde could be 2x2x2 or 3x2x1 or 6x1x1 3. just can't know what the range would be it could be a lot of possibilities Can't determine!

f(x) = x^3 - 9x + 17 A: f(3) B: f(-3) which is greater, equal, can't determine

1. f(3) = 27 - 27 + 17 -27 + 17 = -10 27 - 10 = 17 2. f(-3) = -27 + 27 + 17 = 0 + 17 = 17 Remember PEMDAS equal!

(k/5) - (j/3) = 7/15 2k + 3j = -8 A = J B = K which is bigger, equal, or can't determine?

1. if there's fractions, multiply by a number to cancel out the fraction and "k" (need both to to be 6 to cancel - want to get either k or j to equal each other in both equations so they can be cancelled out) 30 x (k/5) - (j/3) = 7/15 > 6k - 10j = 14 3(2k + 3j = -8) 2. 6k - 10j = 14 - 6k + 9j = -24 subtract the two equations > -19j = 38 j = -2 3. plug j into an equation to solve for k 2k + 3(-2) = -8 k = -1 k is bigger than j so B is bigger

Jerome is buying a new computer. He's choosing among 3 CPUs, 4 RAM, and 3 hard drives. Jerome must have one and only one of each type. How many different configurations?

3 CPUs 4 RAM 3 HDs 3 x 4 x 3 = 36 combos

in 1830, population density of NY was 44 people per square mile area of NY has stayed constant since - what was approximately the population of NY back in 1830? Given: 1920 population = 10 mill 1920 density = 217.9 people per sq mile

44 people per sq mile : 217.9 approximately 50:250 1/5 of the population 1/5 of 10 million = 2million people

fractional powers Ex. 4^1/2

4^1/2 = sq. root 4 = 2

√7x^2 =

7x square root and power cancel out

area of a triangle

A=1/2 x base x height draw perpendicular line from the base and create a right angle base and height are perpendicular to each other

A store offers a variety of discounts that range from 10-25%, inclusive, If a book is discounted to a price of $2.40, what was its greatest possible price originally?

Biggest discount = biggest original price 25% discount = $2.40 is 75% of the original price (p) $2.40 = 0.75p p = $3.20

The coordinates of point P are (a,b). Point P', with coordinates (c,d), is the reflection point over the x-axis. A. b B. -d

Let's say Point P = (x,y) reflection over the x-axis would be (x,-y) if reflection over x-axis, point P' is (x, -y) so P' = (a, -b) so -d = b and b = -d the quantities are equal

handling negative powers ex. 5^-1

a number raised to the exponent -x is the reciprocal of that number raised to the exponent of x 5^-1 = 1/5^1 = 1/5 5^-3 = 1/5^3 = 1/5 x 5 x 5 = 1/125

selling price

cost + profit

even + even =

even

solving f(g(x)) or g(f(x))

given two equations one is f(x) and one is g(x) 1. start from right to left so if it's f(g(x)) > plug the value of x into g(x) first and solve then whatever that answer is, plug into f(x)

A. The measure of the greatest interior angle of a pentagon B. 105 degrees Which is greater, equal, or can't determine?

greatest interior angle of a polygon = (n-2)180 deg. pentagon = 5 sides so n =5 (5-2)180 = 540 degrees 540 vs 105 A is greater

multiplying fractions

multiply numerators then denominators remember to reduce when you can or reduce before multiplying

value with an exponent raised to an exponent

multiply the exponents

one raised to any power =

one

the number of collisions between birds and planes near the airport increased by approximately what percent between 1990 and 2010? 1990 = 12 collisions 2010 = 50 collisions

percent change = amount of change/original whole x 100% 50 - 12 = 38 percent change = 38/12 x 100 = 317% change

someone buys a stock for $600 and sells it for $400. What is the percent change?

percent change = amount of change/original whole x 100% 600-400 = 200 change 200/600 x 100 = 33%

a is a positive integer A: -3a+15 B: -3(a+5) which is bigger, they're equal, or can't determine?

plug numbers A is always bigger

unique prime factors

prime factors that show up - not how many there are 450 = 3 x 3 x 2 x 5 x 5 x 9 9 is not because it is not divisible by only 1 and itself unique factors = 5, 3, 2

how to calculate probability ex. possibility of rolling a 5 on a fair six-sided die?

probability = # of desired outcomes/# of total possible outcomes 1 outcome of getting a 5 / 6 possible numbers = 1/6

Probability Equation

probability = (Number of desired outcomes) / (Total number of outcomes)

pt < 0 tq < 0 qr > 0 A: r/p B: (q/t)^3 which is greater, are they equal, or cannot be determined

pt = one is a negative # tq = one is negative qr = either both are negative or both are positive plug in numbers if p is positive then t is negative and vice versa if r is positive then q is also positive; same for negative 1st round > A is bigger 2nd round B is bigger = cannot determine

how to find the area of a sector

sector = a section of the circle n = degrees of the interior angle that is given area of sector = n/360 x pi r^2

Slope of parallel line vs perpendicular line ex. slope of line M is 2/3

slope of line parallel to line M is 2/3 = same slope perpendicular = -3/2

The perimeter of a square is 48 inches. The length of its diagonal is:

square = all sides equal 48/4 = 12 inch per side 45-45-90 triangle x-x-x√2 if x is 12, the diagonal is 12√2

The ratio of bluegills to minnows to sunfish to bass in a pond is 5:9:4:1. These are the only kinds of fish in the pond and the total number of fish in the pond is between 100 and 200. A. smallest possible number of minnows in the pond B. the largest possible number of bluegills in the pond which is larger, equal, or can't be determined

total amount of fish must be a multiple of 5+9+4+1 = 19 1. smallest number of fish possible in the pond what is multiple of 19 that is greater than 100? 19 x 6 = 114 So for minnows since it's ratio is 9 9 x 6 = 54 minnows = A 2. largest number of fish in pond multiple of 19 less than 200 19 x 10 = 190 5 x 10 = 50 bluegills = B A is larger!

adding and subtracting fractions

find common denominator first

simple interest problem Ex. if $12k is invested at 6% simple annual interest, how much interest is earned after 9 months?

interest = principle x rt r = interest rate per payment period t = number of payment periods 1. interest = 12k x [0.06 x 9/12] = $540

dividing with exponents

subtract the exponents

if p##q = (p-5)(q+3) and 2##x = -30, what is the value of x?

(p,q) = (p-5)(p+3) (2,x) = -30 plug in 2 and x into the (p,q) equation and set it to -30 2 = p x = q (2-5)(x+3) = -30 -3(x+3) = -30 x+3 = 10 x = 7 x = 7

distance between 2 points on a coordinate plane if either x or y are the same Ex. what is the distance from (2,3) to (-7,3)?

.y coordinate is the same so just subtract the x's 2 - (-7) = 9 distance is always positive

if a fair six-sided die with faces numbered 1 through 6 is tossed 3 times, what is the probability of getting a 1 or a 2 on al three tosses?

1 or a 2 = 2 chances total numbers = 6 2/6 chances x 3 times 2/6 x 2/6 x 2/6 = 8/216 = 1/27

Full Force's annual dance recital had 205 people attend. Children tickets = $3.50. Adult tickets = $5. The total revenue from tickets was $828.50. how many adults attended?

1. $3.50c + $5a = $828.50 2. c + a = 205 people 3. solve for either c or a c = 205 - a 4. plug c into the top equation to solve for "a" $3.50(205 - a) + $5a = 828.50 5. solve for a 1.5a = $111 a = 74 adults attended

In company X, no employee is both a technician and an accountant. Also, in company X, 2/5 of the employees are technicians, and 5/16 of the remaining employees are accountants. What fraction of the total number of employees at company X are neither technicians nor accountants?

1. 2/5 are technicians meaning 3/5 of remaining employees are not 2. 5/16 of remaining employees are accountants = 3/5 remaining employees x 5/16 accountants = 15/80 accountants 3. total employees has to equal 1 4. there are people who are neither techs or accountants (W) 5. add it up 2/5 = 32/80 32/80 techs + 15/80 accountants + w neither = 1 w = 33/80

in how many different ways can the letters ABCDEF be arranged if A and B must be next to each other and EF must be next to each other?

1. AB C D EF 4 different things the position of one depends on the position of the one before 2. If AB can take any of the 4 positions, it can be arranged in 4 different ways so C can take up only 3 different positions after AB, D can take 2, and EF can take 1 = 4x3x2x1 = 24 ways 3. AB and EF can be written as BA and FE so that's 2 different ways for each 2x2 = 4 24 x 4 = 96 ways

A motorist travels 90 miles at a rate of 20 mph. If she returns at the same distance at a rate of 40mph, what is the avg speed for the full trip on mph?

1. Figure out amount of hours it takes for each trip 90 miles/20 miles per hour = 9/2 hours 90 miles/40mph = 9/4 hours 2. figure out total hours 9/2 hours + 9/4 hours = 18/4 + 9/4 = 27/4 hours 3. figure out avg speed for the total 180 miles 180 miles/ (27/4 hours) = 180 miles/(4/27) = 720/27 = 80/3 mph 80/3 mph

In a group of 60 workers, the average salary is $160/per day/per worker. If some of the workers earn $150 a day and the rest of them earn $200 a day, how many workers earn $150 per day?

1. Figure out sum of the money per day all 60 workers make 160 = x/60 x = $9600 (all the money total that the 60 workers make per day) 2. Create equation to figure out amount of people who make $150/day 150x + 200(60-x) = 9600 x = 48 48 people make $150/day

The average of 6 positive numbers is 5. If the average of the least and greatest of these numbers is 7, what is the average of the other 4 numbers?

1. Find the sum of all 6 numbers, find the sum of the least and greatest numbers, and then subtract one from the other to get the final average of the 4 numbers 2. sum of all six numbers 5 = x/6 x = 30 30 is the sum of all the 6 numbers 3. sum of the least and greatest numbers 7 = x/2 x = 14 14 is the sum of the least and greatest numbers in the set 4. Find overall average to find avg of remaining numbers (divide by 4 to find sum of the remaining 4 numbers) x = 30-14/4 x = 4 4 = avg of the 4 numbers

viada's board game collection consists of 3 bidding games, 5 worker placement games, and 7 area control games. There is no overlap among the categories. A. number of ways to select 3 board games such that no two of them belong to the same category B. number of ways to select 4 board games such that exactly 2 are bidding games

1. For A, for it to work that 2/3 games are not from the same category, each of the 3 games chosen is each from a different category multiple principle = 3 bidding games x 5 worker placement games x 7 area control = 105 ways 2. ways to select 4 games where 2 of them are bidding games non-bidding games = 7+5 = 12 figure out combo of getting 2 bidding games/12 total games 12C2 = 12!/2!(12-2)! = 12!/2(10!) = 12 x 11 x 10!/2 x 10! = 12x11/2 = 66 possibilities 3 bidding games to choose from x 66 possibilities = 198 ways A= 105 B = 198 B is greater

Given f(x) = -(x+4)^2 - 3, which of the following must be true? A. f(x) is min. when x=4 B. f(x) is max. when x=4 C. f(x) is min. when x= -4 D. f(x) is max. when x= -4 E. f(x) has no max

1. anything squared will be (+) or 0 but it will be (-) of that so the squared portion will be either 0 or (-) 2. the greatest that (x+4)^2 could be is 0. So that can only happen if x is -4 3. so f(x) is max when x=-4 D

A: the sum of the integers from 19 to 31 inclusive B: the sum of the integers from 22 to 32 inclusive which is bigger, they're equal, or can't determine?

1. calculate average of each range 19+31/2 = 25 22+32/2 = 27 2. find # of terms = range/1 + 1 (divide by one because its including all terms; if just even #'s, divide by 2) (31-19)/1 + 1 = 13 terms (32-22)/1 +1 = 11 terms 3. multiply average x terms A = 25 x 13 = 325 B = 27 x 11 = 297 A is bigger

how to solve probability problems where probabilities need to be multiplied - CONDITIONAL PROBABILITY ex. if 2 students are chosen at random to run an errand from a class of 5 girls and 5 boys, what is the probability that the chosen students are both girls?

1. conditional probability = the probability of one chance is affected by another probability 10 total probability of a girl = 5/10 or 1/2 4 girls and 5 boys left of 9 students left - probability of the second student being a girl is 4/9 probability that it's both girls = 1/2 x 4/9 = 4/18 = 2/9 - conditional because the probability of the 2nd being a girl was affected by another girl being chosen first

Two cars travel away from each other in opposite directions at 24mph and 40mph respectively. If the first car travels for 20 mins and the second car for 45 mins, how many miles apart will they be at the end of their trips?

1. convert hours to minutes 1st car = 24mi/hr = 24miles/60 mins 2nd car 40mph = 40miles/60 mins 2. figure out how many miles each car travels 1st car = 24mi/60mins x 20 mins = 8 miles 2nd car = 40mi/60mins x 45 mins = 30 miles 3. total miles they are apart from each other they started at same place and one traveled 8 miles and one traveled 30miles = 30+8 miles = 38 miles

Solving Multiple Equations ex. if 5x - 2y = -9 and 3y - 4x = 6, what is the value of x + y?

1. don't need to figure out what x or y is individually, just rearrange to combine 5x - 2y = -9 +[-4x + 3y = 6] = x + y = -3

What is the value of a when ab + ac = 21 if b+c = 7?

1. factor out a from the equation ab+ac = 21 a(b+c) = 21 2. replace b+c with 7 a(7) = 21 a = 3 a = 3

A certain mule travels at 2/3 the speed of a certain horse. If it takes the horse 6 hours to travel 20 miles, how many hours will the trip take the mule?

1. figure out the speed of the horse speed = distance/time = 20miles/6 hours = 10miles/3 hours 2. multiple it by 2/3 to get the speed of the mule 10/3 x 2/3 = 20miles /9 hours = It takes the mule 9 hours to travel 20 miles 9 hours

Janine wants to increase the percent profit on her cookie sales. Her percent profit when the cookies are priced at $5 a box is 20%, so she decides to increase the price of her cookies by $1 per box. Her new percent profit is p. A: p B: 25%

1. figure out what she's making per box $5 x 20% = $1 means cost for one box is $4 2. use percent change formula amount of change/original whole x 100 $6 - $4/$6 = $2/$6 = 33 1/2% A. 33 1/3% is greater than 25% A is greater

how many positive integers less than 60 are equal to the product of a positive multiple of 5 and an even number?

1. find all multiples of 5 that are less than 60 2. easier way is that all multiples of 5 are multiples of 10 so how many numbers that are multiples of 10 are between 1 and 59? 10, 20, 30, 40, 50 = 5 numbers 5 positive integers

Polygon P has "n" sides Polygon Q has 2 fewer sides than polygon P n is greater than or equal to 5 A: average measure of an interior angle in polygon Q B: [180n/n-2] - [700/n-2]

1. find the sum of the angles in Q sum of the interior angles of a polygon = (n-2) x 180 Q has n-2 sides [n-2-2] x 180 = Q 180(n-4) = Q 180n - 720 = Q 2. Average = sum / # of terms [180n/n-2]- [720/n-2] 3. it's almost identical to B but B is subtracting only 700 instead of 720 meaning the total will be greater B >A B

If 20 students in one class had an average grade of 94 percent and 18 students from another class had an average grade of 92 percent. What is the average grade for all 38 students across both classes?

1. find the total sum of the grades for each class 94 = x/20 x = 1880 92 = x/18 x = 1656 2. add them up to get total sum of scores for all 38 students 1880 + 1656 = 3536 3. divide total score by number of students to get average 3536/38 = 93.05% average score average = sum of scores/number of students

The average of John's test scores in his economics class is 80%. Only one test remains, and it is weighted at 25% of the final grade. If his final grade is based entirely on his test scores, what is the lowest percent John can receive on the final to earn an 85% in the class? A. 96% B. 97% C. 98% D. 99% E. 100%

1. if his last test is weighted 25%, the other tests he took are worth 75% of his grade. (80% x 0.75) + (X x 0.25) = 85% average 0.6 + 0.25x = 85 0.25x = 0.25 x = 1 x 100 x = 100% he needs 100% on his final to get an 85% average

isolating a variable in a linear equation [Ex. 28 = -3x - 5]

1. isolate the x [add 5 to the 28] 28 + 5 = -3x 2. divide both sides by -3 [33/-3 = -3x/-3] 3. solve for x [x = -11]

Remainders problem Ex. when N is divided by 7, the remainder is 5. What is the remainder when 2n is divided by 7?

1. just pick a number that fits the conditions 2. find a number that leaves a remainder of 5 when divided by 7 > take a multiple of 7 and add 5 to it. 3. example 7+5 = 12 if n =12, 2n = 24 divided by 7 = remainder of 3

simplifying radicals Ex. sq rt 48

1. look for factors under the radical that are perfect squares 2. simplify as much as possible - no more perfect squares left sq rt 48 = sq rt 16 x sq rt 3 = 4 sq rt 3

how to find the dimensions or area of an inscribed or circumscribed figure ex. if the area of a square is 36, what is the circumference of the circle?

1. need to find the diameter or radius to get the circumference - circle's diameter = the square's diagonal 2. square is two 45-45-90 triangles meaning that the diagonal is 6 sq rt 2 because the area of the square is 36 diameter = 6 sq rt 2 3. circumference = diameter x pi = 6 pi sq rt 2

how to find the least common multiple of 2 numbers ( Ex. LCM of 28 and 42?)

1. prime factorization for each number 28 = 7 x 2 x 2 42 = 7 x 3 x 2 2. multiply each prime number by the max number of times it appears 28: 2 is used twice and 7 is used once 42: 7 is once, 3 is once, 2 is once 2 = 2x 3= 1x 7 = 1x LCM = 2 x 2 x 3 x 7 = 84

Solving quadratic equations ex. x^2 + 6 = 5x ex. x^2 = 9

1. set equation equal to zero x^2 - 5x + 6 = 0 2. reverse foil to find x what when multiplied together equals +6 and when added together equals -5 - numbers must have same sign for the 6 to be positive (x - 3)(x - 2) = 0 3. solve for x x = 3 or 2 ex. x^2 = 9 x = -3 or 3

x^2 + 5x + 6 = 0 A: the square of the sum of the roots of the equation B: 25 which is greater, equal, or can't determine?

1. solve for x what multiplies to be +6 (have same sign) and adds up to +5 (3 + 2) (x+3)(x+2) x = -2, -3 A: the roots are -3 and -2 add them up to get sum = -5 square it (-5)^2 = 25 A = B

4x+3y=12 > LINE A. value of x-intercept of the line B. the y-intercept of the line

1. solve x intercept by setting y=0 and solve for X 0 = -4/3x - 4 -4 = -4/3x -4(-3/4) = x 12/4 = 3 x = 3 B. y-intercept = 4 y = -4/3x + 4 A =3 B = 4 B is greater

rob received scores of 96, 89, and 85 on 3 quizzes. if the exam is weighted twice as heavily as each of the three quizzes, what is the lowest score Rob can get on the exam to have a final average of at least 90%? a. 88 b. 89 c. 90 d. 91 e. 92

1. total scores are 3 quizzes and an exam that is weighed twice as heavy as quizzes quiz 1 = 1/5 grades = 20% of grade quiz 2 = 1/5 = 20% quiz 3 = 1/5 = 20% exam = 2/5 = 40% of grade 2. multiply grades so far by 0.2 to get his average so far in the class 96 x 0.2 = 19.2 89 x 0.2 = 17.8 85 x 0.2 = 17 = 54% so far 3. he wants 90% so look at all options and plug in to see what number times 40% added to 54% will give at least 90% c = 90 90 x 0.4 = 36 36 + 54 = 90 He needs at least a 90 to have a final average of 90% in the class

Solving an inequality (Ex. rewrite 7-3x > 2in its simplest form)

1. treat like normal equation 2. add, subtract, multiply, and divide the same on both sides 3. reverse inequality sign if you multiply or divide by a negative number 7-3x > 2 -7 -3x > -5 divide by -3 and reverse inequality sign x < 5/3

49, 50, 61, x average = 52 what is X?

1. use average formula average = sum of terms/number of terms 52 = 49 + 50 + 61 + x/4 2. multiply by 4 on both sides 208 = 160 + x 3. subtract 160 48 = x

how to find the diagonal of a rectangular solid ex. what is the length of AG?

1. use pythagorean theorem twice unless you spot special triangles 2. find the diagonal length for the top and bottom (which are the same) ABC is a 3:4:5 triangle so AC = 5 3. ACG is a 45-45-90 triangle so AG = 5 sq rt 2 45-45-90 is x-x-x sq rt 2 so since x is 5, x sq rt 2 is 5 sq rt 2

The figure shown represents a square garden. If each side is increased in length by 20 percent, by what percent is the area of the garden increased ?

1. want to find change in area between the first size garden and the increased in size garden 2. say the length now is 10. if it increases by 20% > 10 x 0.2 = 2; 10 + 2 = 12 new size 3. compare area area before = 10 x 10 = 100 new area = 12 x 12 = 144 4. look at change in area 144 - 100/1 = 44 > 44% increase

3x^(2/3) + 2(3√x) - 8 = 0 A: x B: 64/25 which is greater, equal, or can't determine?

1. x^2/3 = (3√x)^2 2. the second part also has a 3√x so let 3√x = m 3. new equation = 3m^2 + 2m - 8 = 0 4. want to make two equations so what adds up to 2? 3m^2 + 6m - 4m - 8 = 0 5. now can take out a common thing = 3m for first part and 4 for the second part 3m(m+2) - 4(m+2) = 0 6. separate the numbers outside of parentheses (3m-4)(m+2) = 0 7. solve for m m = 4/3 or -2 8. figure out 3√x 3√x = 4/3 or -2 x = (4/3)^3 or (-2)^3 x = 64/27 or -8 B = 64/25 which is larger than 64/27 and -8 B is greater!

Machine A can produce 1/6 of a ton of paintbrushes in one hour. Machine B can produce 1/14 of a ton of paintbrushes in one hour. Working together at their individual rates, how long would it take the two machines to produce 2 tons of paintbrushes?

1/6 + 1/14 = 1/t 10/42 = 1/t 42/10 = time it took to make 1 ton Since we are looking for how long it takes them both > 42/10 x 2 tons = 84/10 hours 84/10 = 8.4 = more than 8 hours 8.4 or 8 hours and 24 mins!

A truck driver needs to complete her route with an avg of 56 mph or more to receive a bonus. She drives the first 108 miles at 54 mph and the next 168 miles at 56mph. At least how fast does she need to drive for the last 228 miles to earn the bonus?

108mi @ 54mph = 2 hrs 168 mi @ 56mph = 3 hrs total miles of the trip = 108 + 168 + 228 = 504 miles total 1. use speed formula to figure out total time since we know Rate and Distance avg speed = total distance/total time 56mph = 504 miles/(x) x = 9 hrs total for the trip from above, she's already driven 5 hrs > 4 hrs left for trip 2. use distance equation to figure out Rate for the remaining distance needed distance = rate x time 228miles = x (4) x = 57mph needed to earn the bonus

a vault holds only 8 oz gold coins and 5oz silver coins. If there are 130oz of gold and silver total, what is the greatest amount of gold that can be in the vault in ounces?

130 oz = total weight silver coins are 5oz each 1. how many 8oz gold coins can fit into 130? 8x16 = 128 but that only leaves 2oz for silver and we need at least 5oz 2. 8x15 = 120oz which leaves 10oz for silver = 2 silver coins 3. greatest amount of gold is 120 oz of gold 120

In a bag, there are 6 red marbles, 7 blue marbles, and 3 black marbles. if two marbles are drawn without replacing the first, what is the probability of drawing a blue and then a red marble?

16 marbles total blue = 7/16 red since not replacing = 6/15 multiply 7/16 x 6/15 = 42/240 = 7/40

In a set of numbers, the mean is 12 and 17 is 1.25 standard deviations from the mean. What is the standard deviation?

17-12 =5 1.25 x standard deviation = 5 5/1.25 = 4 4 is the standard deviation

sum of all angles of a triangle

180 degrees

When Percy plays darts, the probability of getting a bullseye increases by 40% with each subsequent throw. The likelihood of getting a bullseye on his first throw is 1 in 5. What is the probability that he throws 3 bullseyes in a row?

1st throw: 1 in 5 = 0.2 = 20% 2nd throw = add on 40% of 20% 0.2 x 1.4 = 0.28 = 28% 3rd throw = 0.28 x 1.4 = 0.392 = 39.2% add it all together 0.2 x 0.28 x 0.392 = 0.022 = 2.2% probability

a jar contains only pink, blue, and white gumballs. Of these, 15 are pink. The number of blue gumballs is 3 more than twice the number of white gumballs. The probability of pulling a pink or white gumball out of the jar is 2/3. How many blue gumballs are in the jar?

2/3 = 15 + w b = 3 + 2w total = 15+w+b 1. 2/3 = (15+w)/(15+w+b) 2. plug 3+2w as b into 2/3 = 15 + w/ 15 +w+3+2w 2/3 = 15+w/18 +3w 2/3 = 15+w 3. Cross multiply 2(18+3w) = 3(15+w) w= 3 4. figure out blue by plugging in b = 3+ 2(3) b = 9 9 blue marbles in the jar

On Sundays, the local farmer's market sells all produce without the usual 8% sales tax and an additional 20% off discount. On Saturdays, the brussel sprouts were $4.50/lb, the tomatoes were $3/lb, and the cucumbers were $2.50/lb. what is the percent decrease in price between Saturday and Sunday if you by 3 lbs brussel sprouts, 5 lbs tomatoes, and 2 lbs cucumbers?

3 lbs Brussel sprouts = $4.50 x 3 = $13.50 2 lbs cucumbers = $2.50 x 2 = $5 5 lbs tomatoes = $3 x 5 = $15 total = $33.50 1. Saturday Sales tax = 8% = multiply by 1.08 $33.50 x 1.08 = $36.18 2. Sunday 20% discount = multiply by 80% or 0.8 $33.50 x 0.8 = $26.80 3. plug into change in percent equation percent change = amount of change/original whole x 100% change = $36.18 - $26.80 = $9.38 = $9.38/$33.50 x 100 = 28% difference

1/(8^3) =

3√8

In a certain school, the ratio of boys to girls is 5 to 13. If there are 72 more girls than boys, how many boys are there?

5:13 = boys:girls set as 5x : 13x 5x + 72 = 13x x = 9 5x = boys 5 x 9 = 45 boys total

a bag contains only green and blue marbles, and the ratio of green to blue marbles in a bag is 6:7. if there are 18 green marbles, how many total marbles are there?

6 green : 7 blue 18 green: x blue 18x7 = 6x 126 = 6x x = 21 Total marbles = 18 + 21 = 39 total marbles

sum of the interior angles in a polygon w/ "n" sides

= (n-2) x 180

cube roots ex. write cube root of -5 x -5 x -5

= cube root -125 = -5

In triangle ABC, A measured seven degrees less than half the measure of angle B. If angle C measures 49 degrees, what is the measure of angle A?

A = 0.5B - 7 C = 49 degrees triangle = 180 degrees B + (0.5B - 7) + 49 = 180 1.5B + 42 = 180 B = 92 degrees A = 0.5(92) -7 A = 39 degrees

vertical angles

A pair of opposite congruent angles formed by intersecting lines

both x and y are integers. which of the following must be the square of an integer? A. (x+y)(x-y)+ 8xy + 17y^2 B. 9x^4 - 12x^2y^2 + 4y^4 C. x^9 + 2x^3y^3 + y^9

A. (x+y)(x-y) = x^2 - y^2 so plug that into the equation x^2 - y^2 + 8xy + 17y^2 > reverse foil > (x+ 4y)(x+4y) = (x + 4y)^2 since x is an integer and y is an integer, 4y is an integer = everything integer so it is a square of an integer B. 9x^4 - 12x^2y^2 + 4y^4 > reverse foil = (3x^2 - 2y^2)^2 also a square of an integer C. x^6 + 2x^3y^3 + y^6 > reverse foil = (x^3 + y^3)^2 also a square of an integer ALL THREE

Which ones are positive and which ones are negative? x and y are greater than 0 and x > y A. (x-y)^2(y-x)^2 B. (x-y)^3(y-x)^3 C. x^2 - y^3 D. 2xy-x^2 E. -1(x-y)^2 (y-x) F. -5(x-y)(y-x)^2

A. no matter - or +, squared its positive B. since X is bigger than y, the result of x-y is positive. Cubed = positive. y-x is negative and cubed is negative so neg x pos = Negative C. positive - positive = could be positive or negative D. positive - positive = neg or pos E. -1(pos)(neg) = -1 x neg = positive F. -5(pos)(pos) = negative

area of a circle

A=πr² ex. r = 5 5x5 = 25 A = 25pi

Of the five people in a room, 3 wear glasses and 3 have a beard. Which of the following events cannot have a probability greater than 50%? A. one person selected at random wears glasses and has a beard B. one person selected at random neither has glasses nor beard C. one person selected at random wears glasses but does not have a beard D. 2 people selected at random have at least one beard and one pair of glasses between them (so each has either one or one has both and other has neither) E. 3 people selected at random that each have either glasses, a beard, or both

AKA which ones equal to less than 50% 5 people in room Scenarios: 1. 1 glasses, 1 beard, 2 both, 1 neither 2. 2 glasses, 2 beard, 1 both 3. 3 glasses and beards, 2 neither A. for scenario 1, it's a 2/5 chance (40%); Scenario 2 (1/5; 20%), Scenario 3 (3/5 is more than 50%) = NO B. Scenario 1 (1/5; 20%), S2 (0), S3 (2/5; 40%) = YES C. S1 (1/5; 20%); S2 (2/5; 40%); S3 (0) = YES D. S1 (4/5) = NO E. S1 (4/5) = NO B and C are all under 50% probability

Ann and Claire run laps around a 1/4 mile track. They run in the same direction, start at the same time and place and run for 36 mins. Ann runs at a constant speed of 8mph and Claire runs at 6mph. A: The number of time Ann passes Claire B: 4 which is greater, equal, can't determine

Ann ran 2 more miles per hour faster than Claire (8-6mph) 1/4mile track = 4 laps per mile Ann runs 2mph faster x 4 laps = 8 laps per hour faster 36mins ran/60 mins in an hr = 0.6 hrs so in tht 36 mins or 0.6 hours she ran > 0.6 x 8 = 4.8 laps more than Claire She ran 4 laps but not a full 5 laps more so 4 more laps than Claire A and B are equal!

the numbers in a data set have a mean of 0. A. number of data elements below the mean B.. number of data elements above the mean which is larger, equal, or can't be determined

Cannot be determined because you don't know how many numbers are in the data set so you can't know how many numbers are above or below the mean.

how to solve a combined word problem ex. if it takes Joe 4 hrs to paint a room and Pete twice as long to paint the same room, how long would it take the two of them working together to paint the same room?

Equation: 1/r + 1/s = 1/t r = 1st person/variable s = 2nd person/variable SOLVE FOR "T" 1/4 hrs + 1/8 hrs = 3/8 3/8 = 1/t 1. figure out "t" divide (3/8) by 1 = reciprocal t = 8/3 = 2 hrs and 40 mins 2 and 2/3 > 2/3 (divide 60 by 3 to get 20 mins and then 2/3s = 40 mins)

how to find an original whole before a percent increase/decrease? ex. after decreasing by 5%, the population is 57,000. what was the original population?

Ex. a 15% increase over x is equal to 1.15x 100% - 5% = 95% 0.95 x original population = 57k divide by 0.95 on both sides original population = 57000/ 0.95 = 60,000

how to find an average rate Ex. if the first 500 pages have an avg of 150 words per page, and the remaining 100 pages have an avg of 450 words per page, what is the avg. number of words per page for the entire 600 pages?

Ex. if the first 500 pages have an avg of 150 words per page, and the remaining 100 pages have an avg of 450 words per page, what is the avg. number of words per page for the entire 600 pages? Average A per B = Total A/Total B 1. find total words (500x150) + (100x450) = 120,000 words 2. find avg. words per page 120,000/600 pages = 200 words per page

The sum of the interior angles of a regular polygon is less than 540 degrees. Which could be the polygon? A. triangle B. quadrilateral C. pentagon D. hexagon

Formula for sum of the interior angles of a polygon = (n-2) x 180 n = number of sides set 180(n-2) < 540 to figure out number of sides 180n - 360 < 540 n < 5 the answer is a polygon with less than 5 sides > A,B - triangle and quadrilateral

PEMDAS

Parentheses, Exponents, Multiplication, Division, Addition, Subtraction Please Excuse My Dear Aunt Sally

The perimeter of a rectangle is 24 and its width is greater than or equal to 2. Which of the following could be the area of the rectangle? A. 12 B. 24 C. 30 D. 36 E. 40 F. 48

Perimeter = 2L + 2W = 24 2(L+w) = 24 L + W = 12 Length has to be longer than width so the max it could be is if width and length are both 6 W2, L10 W3, L9 W4, L8 W5, L7 W6, L6 Minimum = Width 2 = 10L so area is 20 min is 20 max is 6x6 = 36 pick areas between 20 and 36 Areas possible: B, C, D [24, 30, 36]

If the range of {9, -1, x, 3, 14, 5} is 17, which of the following could be the value of x? A. -17 B. -14 C. -3 D. 8 E. 16 F. 17

Range = highest number - lowest number highest = 14; lowest = -1 x must be between -1 and 14 1. X on the lower end: highest number = 14 14 - x = 17 14 - (-3) = 17 x can be -3 2. X on the higher end: lowest number = -1 x - (-1) = 17 x = 16 x = -3, 16

An aquarium keeps 72 fish in 4 different tanks. The second tank has 4 more fish than the first, the third has 4 more fish than the second, and the fourth has 4 more fish than the third. What is the probability that a given fish will be in the third tank?

Tank 1 = x Tank 2 = x+4 Tank 3 = x+8 Tank 4 = x + 12 1.set all tanks to equal 72 x + (x+4) + (x+8) + (x+12) = 72 x = 12 2. plug x into the third tank to see how many fish are in it x+8 12+8 = 20 fish 3. proportion 20 fish/72 total fish = 5/18 chance the fish is in the third tank

There are 30 teachers in the mathematics and science departments at a certain school. Of these teachers, 20 are certified math teachers and 14 are certified science teachers. Not all teachers in both departments are necessarily certified in either subject matter. Which of the following could be the number of teachers certified in both math and science? A. 0 B. 5 C. 7 D. 10 E. 14 F. 16

Total = Group 1 + Group 2 + Neither - Both 30 = 20 + 14 + N - B 1. Lower limit - lowest number that are both certified in math and science (14 + 20)-30 = 4 2. Upper limit - highest number that are both certified in math and science 14 > if all science teachers were also certified in math 3. teachers are between 4 and 14 B, C, D, E (5,7,10,14)

prime numbers

a number which can only be divided by itself and 1 2,3,5,7,11,13,17,19, etc.

how to find the length of an arc in a circle

arc = fraction of a circle's circumference (part of the outline of a circle) 1. use one measure of an interior angle in the circle that is given ("n" degrees) 2. circle is 360 degrees 3. r = radius, from center to the edge of circle length of arc = n/260 x 2pi r

Comparing Quantity A to Quantity B Ex. comparing 2 fractions to an unknown fraction a/b = c/d = e/f

cannot determine if A is bigger or B is bigger > unknown because we can't know what a or b is individually

multiplying square roots ex. [2 sq rt 3] x [7 sq rt 5]

deal with the numbers outside the sq rt and numbers under the sq rt separately = 14 sq rt 15

dividing square roots ex. [10 sq rt 21] / [5 sq rt 3]

deal with the numbers outside the sq rt and numbers under the sq rt separately = 2 sq rt 7

How to solve an overlapping sets problem involving either/or categories Ex. at a certain professional conference with 130 attendees, 94 of the attendees are doctors, and the rest are dentists. if 48 of the attendees are women and 1/4 of the dentists are women, how many of the attendees are male doctors?

doctors: 94 male doctors: 55 female doctors: 39 dentists: 36 male dentists: 27 female dentists: 9 total men: 82 total women: 48 fill out a table figure out total men and women

even - even =

even

even x even =

even

odd + odd =

even

odd - odd =

even

odd x even =

even

How to find the NEW AVERAGE when a number is added or deleted ex. michael's avg score after 4 tests was 80. if he scores 100 on the 5th test, what is his new avg?

ex. michael's avg score after 4 tests was 80. if he scores 100 on the 5th test, what is his new avg? 1. find original sum of the first 4 tests 80 x 4 = 320 2. add the 5th score to the original sum to make the new sum 320 + 100 = 420 3. find the avg using the new sum 420/5 = 84

f(x) = (x-1)/4 if (w+1)/f(w) = -2 what is the value of w?

f(w) = (w-1)/4 plug into the original equation (w+1)/(w-1)/4 = 4/(w-1) x (w+1) = 4w+4/(w-1) = -2 multiply by w-1 on both sides 4w + 4 = -2w + 2 w= -1/3

How to find the sum of all the ANGLES of a POLYGON and one angle measure of a REGULAR POLYGON ex. what is the measure of one angle of a regular pentagon?

finding a degree measure of one angle for any regular polygon: [(n-2) x 180]/n n = number of sides regular = all angles in the polygon are equal ex. pentagon has 5 sides so n= 5 (5-2)180/5 = 108 degrees

the height of an equilateral triangle is √3/2 x (S) where S is the length of a side. The length of each side is 4√3. A. perimeter of the triangle B. the area of the triangle which is greater, equal, or can't determine?

height = √3/2 x 4√3 = 4√9/2 = 4x3/2 = 6 perimeter = 4√3 x 3 = 12√3 area = 1/2bh = 0.5 x 4√3 x 6 = 3 x 4√3 = 12√3 equal!

factorials Ex. 8!/6! x 2!

indicated by "!" n = integer greater than 1 n! = all integers from 1 to n ex. 0! = 1 4! = 4 x 3 x 2 x 1 = 24 also 6! can equal 6 x 5! Example: 8 x 7 x 6! / 6! x 2 x 1 cancel out the 6! 56/2 = 28

ab does not equal 0 A: (a^-2b^-2)^-1 B: (a^-1b^-1)^-2 which is bigger, they're equal, or can't determine?

multiply the exponents A = a^2 b^2 B = a^2b^2 equal

[√7x+3y] / √7x - √3y =

multiply top and bottom by √7x + √3y notice the sign difference! add not subtract Foil and reduce bottom is a difference of perfect squares = 7x - 3y top is foiled = 7x + √21xy + 3y√7x +3y√3y / 7x -3y

Adding roots ex. sq root 2 + 3 sq rt 2

only add if the numbers under the root are the same; if not, they can't be combined = 4 sq root 2

subtracting roots ex. [sq rt 2] - [3 sq rt 2]

only subtract if the numbers under the root are the same; if not, they can't be combined = [-2 sq rt 2]

♥x♥y♥ = (2x - y)^2 and ☻a☻b☻ = (a+2b)^2. If ☻5☻(n-1)☻ = 5n^2 + 4n and n<6, then ♥2n♥2♥ =

replace symbols with letters! h = hearts s = smileys h(x,y) = (2x - y)^2 s (a,b) = (a+ 2b)^2 trying to solve hearts and given info for smileys 1. plug in 5 for a and (n-1) for b into the smileys equation a = 5 b = n-1 s(a,b) = (5+ 2(n-1))^2 = (5 + 2n - 2)^2 = (2n +3)^2 = 4n^2 + 6n + 9 2. set it equal to 5n^2 + 4n and solve for n 4n^2 + 6n + 9 = 5n^2 + 4n n^2 - 8n - 9 = 0 n = (n-9)(n+1) n = 9, -1 3. given that n<6, the only one that works is n = -1 4. move to hearts! given that h(2n, 2), plug -1 into h(x,y) = (2x - y)^2 h(x,y) = (2(2n) - 2)^2 = (2 (2 x-1) - 2)^2 = (2 x -2 - 2)^2 = (-4 -2)^2 = 36 answer is 36

slope of a line Ex. slope if it contains points (1,2) and (4, -5)?

rise/run or y2 - y1/ x2 - x1 (1,2) and (4, -5) > y2 = -5 y1 = 2 x2 = 4 x1 = 1 -5 - 2/4-1 -7/3

how to use the average to find the sum (Ex. 17.5 is the average of 24 numbers. What is the sum?)

sum = average x (number of terms) sum = 17.5 x 24 sum = 420

corresponding angles

the angles that occupy the same relative position at each intersection where a straight line crosses two others. If the two lines are parallel, the corresponding angles are equal.

how to recognize multiples of 2

the last digit is even

finding the mode

the number that appears most often in the list

a quiz has 10 true/false questions. if each question is answered with "true" or "false" and none of them are left bank, in how many ways can the quiz be answered?

two choices = 2 10 questions 2^10 = 1024 possible ways to answer

how to find a weighted average Ex. the girls average score is 30. the boys average score is 24. if there are twice as many boys as girls, what is the overall average?

weighted avg = [(1 x 30) + (2 x 24)]/3 > 78/3 = 26 3 is used because 1+2

If a dealer had sold a stereo for $600, he would have made a 20% profit. Instead, he sold it for a 40% loss. At what price was the stereo sold for?

x = cost to the dealer selling price = $600 1. find cost to dealer use cost price = (100/profit % + 100) x selling price x = (100/20% + 100) x 600 x = 100/120 x 600 x = $500 2. Figure out price of stereo sold at 40% loss $500 x 0.4 = $200 $500 - $200 = Sold at $300

x is even and is between or equal to -3 and 3 y is odd and between or equal to -1 and 1 A = y^x B = y^(x/y) which is bigger, equal, can't determine?

x is even so it can be -2, 0, 2 y is odd so it can be only -1 and 1 A = -1 or 1 raised to an even power = +1 B = plug in numbers and try combos to see what the answers would be. 1 raised to any power is 1 so B is always 1 A = B

finding x and y intercepts of a line

x-intercept = where the line crosses the x-axis/horizontal line and the value for y = 0 y-intercept = where line crosses y-axis/vertical line and the value for x = 0

linear equations ex. given graph and asked to find the equation of the graph shown

y = mx + b m = slope [rise/run] b = y-intercept ( point where line crosses the y-axis) picture: point crosses y-axis at (0, 1) b = 1 going from the y-intercept to the second point = up 2 going right to reach the 2nd point = 1 slope = 2/1 = 2x 1. plug in obvious points on graph and plug into given equations to pick out the right one

number of possibilities problems Ex. how many 3-digit numbers can be formed with the numbers 1, 3, and 5, each only used once?

1. 100 30 5 1st digit, 1, is in the hundreths place and can be used 3 times 2nd digit is in the tenths place, can be used 2 times 3rd digit can be used once 3 possibilities x 2 x 1 = 6 possibilites

how to solve probability problems where probabilities need to be multiplied ex. if a coin is tossed 4 times, what is the probability that at least 3 of the 4 tosses will be heads?

2 possible outcomes for each toss - heads or tails total possible outcomes = 2 x 2 x 2 x 2 = 16 1. list all outcomes that heads happens 3 times: HHHT HTHH HHTH THHH HHHH = 5/16

standard deviation ex. for the 5-day period, which city had the greater standard deviation in high temps? City A: 54, 61, 70, 49, 56 City B: 62, 56, 60, 67, 65

= how spread out the set of numbers is from the mean greater spread = greater standard deviation A has the greater spread in temperature because it goes as low as 49 to as high as 70

How to use the ORIGINAL AVERAGE and NEW AVERAGE to figure out WHAT WAS ADDED OR DELETED Ex. the average of five numbers was 2. after one number is deleted, the new average is -3. What number was deleted?

Ex. the average of five numbers was 2. after one number is deleted, the new average is -3. What number was deleted? Number added = new sum - original sum Number deleted = original sum - new sum 1. find original sum from the original average. 5 x 2 = 10 2. find new sum [4 because one number was deleted] 4 x -3 = -12 3. find number deleted 10 - [-12] = 22

How to solve an overlapping sets problem involving both/neither Ex. of the 120 students, 65 are studying french, 51 are studying spanish, and 53 are studying neither. how many are studying both french and spanish?

Group 1 + Group 2 + Neither - Both = Total 65 + 51 + 53 - both = 120 169 - both = 120 both = 49 students taking both french and spanish

how to recognize multiples of 9

The sum of the digits is a multiple of 9.

How to solve a mixture problem: ex. if 5 lbs of raisins that cost $1/lb are mixed with 2lbs of almonds that cost $2.40/lb , what is the cost per pound of the resulting mix?

ex. if 5 lbs of raisins that cost $1/lb are mixed with 2lbs of almonds that cost $2.40/lb , what is the cost per pound of the resulting mix? raisins > $5 almonds > $4.80 [$2.40 x 2 lbs almonds] = $9.80 for the whole mix total lbs = 5 + 7 = 7 lbs total cost per pound = $9.80/7 lbs = $1.40 per lb

dividing fractions

keep first fraction the same but flip the second multiply the first fraction by the reciprocal

how to recognize multiples of 10

last digit is 0

how to solve a dilution problem Ex. how many liters of a solution that is 10% alcohol by volume must be added to 2 liters of solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume?

need to balance the weakest and strongest solutions (% difference between the weakest and the desired solution) x (amount of weaker solution) = (% difference between the stronger solution and the desired solution) x (amount of stronger solution) n = amount of weaker solution in liters n( 15% desired - 10% weakest) = (50% stronger - 15% desired) x 2 liters stronger solution 5n = 70 n = 14 liters of the 10% solution is needed and added to the original stronger solution to create a 15% alcohol by volume solution

special right triangles

ratios of the side lengths for triangles seen on test > 3:4:5 5:12:13 1.in a 30/60/90 triangle, side lengths are multiples of 1, sq. rt 3, and 2 2. in 45/45/90 triangle, side lengths are multiples of 1, 1, and sq. rt 2

how to determine a combined ratio ex. the ratio of a to b is 7:3. the ratio of b:c is 2:5. what is the ratio of a to c?

since both have b, multiply each set of ratios by the other b's value 1. b is 3 in the first set > 3x2 = 6 a is 7 > 7x2 = 14 a:b = 14:6 2. b is 2 in 2nd set > 2x3 = 6 c is 5 > 5x3 = 15 b:c = 6:15 since both b's now match; a:b:c = 14:6:15 a:c = 14:15

percent increase example The price goes up from $80 to $100. What is the percent increase?

% increase = amount of increase/original whole x 100% 1. % inc. = 20/80 x 100% 2. 0.25 x 100% 3. = 25%

How to find a COMMON FACTOR of two numbers (Ex. what factors >1 do 135 and 225 have in common?)

1. Break both numbers down to their prime factors to see which they have in common. 2. multiply the shared prime factors to find all common factors. 1. 135 = 3 x 3 x 3 x 5 2. 225 = 3 x 3 x 5 x 5 3. share 3 x 3 x 5 in common 3. common factors = all possible products of 3 x 3 x 5 common factors = 3, 5, 15, 45, 9

percent problems = ___ is ___% of what number? ex. 15 is (3/5)% of what number?

1. Find the WHOLE (part = percent x whole) 2. ex. 15 is (3/5)% of what number? 3. 15 = [(3/5)x(1/100)] x Whole 4. 15 = (3/500) x whole 5. Multiply by reciprocal > whole = 15x(500/3) 6. reduce > 7500/3 = 2500

Finding the Median

1. First arrange the values from least to greatest 1. If the number of data values is odd, the median is the number in the exact middle of the list. 2. If the number of data values is even, the median is the mean/average of the two middle numbers.

percent problems = # is what percent of #? ex. 45 is what percent of 9?

1. find the percent part = percent x whole 2. 45 is what percent of 9? 3. 45 = percent/100 x 9 4. 45x100 = percent x 9 5. 4500/9 = % 6. % = 500

how to find the missing angle of a triangle

1. given 2 angles of the triangle 2. sum of all angles = 180 deg. 3. add the ones known and subtract from 180 to find X

How to solve a PERMUTATION problem ex. 5 runners rub in a race. the runners who come in 1st, 2nd, and 3rd will win gold, silver, and bronze respectively. how many possible outcomes for gold, silver, and bronze medal winners are there?

1. how many different ways to arrange formula: nPk = n!/(n-k)! n = number in the large group k= number you're arranging any of the 5 can come in first leaving 4 who can come in second and 3 who could come in third n = 5 total runners k = 3 possible results 5P3 = 5!/(5-3)! = 5!/2! 5 x 4 x 3 x 2!/2! = 60 possible outcomes

distance between 2 points on a coordinate plane if x and y are different Ex. distance from (2,3) to (-1, -1)?

1. if x's and y's are both different, plot the points and make a right triangle 2. use pythagorean theorem (a^2 + b^2 = c^2) or recognize the special triangles (3,4,5 or 5,12,13) if it's a 3, 4, 5 > can figure out the missing side

how to determine combined percent increase/decrease when no original value is specified ex. a price rises by 10% one year and 20% the next. What's the combined percent increase?

1. just say original price is $100 since it isn't given 2. 100 + (10% of 100) = $110 3. 110 + (20% of 100) = $132 4. from 100 > 132 = 32% increase!

simplifying binomials Ex. (3x + 5)(x-1)

1. use the foil method (3x + 5)(x-1) 3x^2 - 3x + 5x - 5 3x^2 + 2x - 5

how to factor polynomials (a^2 - 2ab + b^2)

(a^2 - 2ab + b^2) = (a-b)^2

finding angles formed by a transversal across parallel lines

1. all acute angles are equal 2. all obtuse angles are equal 3. any acute + obtuse angle = 180 degrees e = g = p = r f = h = q = s e + q = 180 g + s = 180

How to solve a COMBINATION problem ex. how many different ways are there to choose 3 delegates from 8 possible candidates

1. arranging a smaller group within a larger group formula > nCk = n!/k!(n-k)! n = number in larger group k = # you're choosing nCk = 8!/3!(8-3)! = 8!/3! x 5! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1/ 3 x 2 x 1 x 5 x 4 x 3 x 2 x 1 = 8 x 7/1 = 56 different combos

finding the average of consecutive numbers Ex. Find average of all integers from 13 to 77

1. consecutive = numbers one right after the other 2. average of all integers from A > B = the average of A and B Ex. 13 and 77 13 + 77/2 = 90/2 = 45

using actual numbers to determine a rate (ex. Anders typed 9450 words in 3.5 hours. what was his rate in words per minute?)

1. convert hrs to mins > 3.5 hrs = 210 mins 2. set up ratio words/minute 9450 words /210 mins = 45 WPM

Area of a trapezoid

A=1/2 x height x (b1+b2) or A = avg of parallel sides x height trapezoid: has only 2 parallel lines 1. break into a triangle and a rectangle - figure out area of rectangle - figure out area of triangle - add together 2. add together the parallel sides and divide by 2 - then multiply by the height

Area of a parallelogram

A=base x height 1. draw line to make a right triangle > line is the height 2. base is the whole length shown, not just one side of the line

multiplying with exponents ex. (x^a)( x^b)

add the exponents = x^a+b

adding two negative numbers

add the numbers like normal the answer will always be negative

how to find an average speed & equation ex. rosa drove one way at an average speed of 40 mph and returned the same 120-mile way at an avg. speed of 60 mph. What was her avg. speed for the entire 240-mile trip?

average speed = total distance/total time total distance = 240 miles find time it takes each way 1. 120miles/40 miles per hour = 3 hrs 2. 120 miles/60mph = 2 hrs 3 total time = 5 hrs avg. speed = 240 miles/5 hrs = 48mph was her avg speed

Circumference of a circle

circumference = 2πr or pi x diameter r = radius r x 2 = diameter ex. r = 3 2 x pi x 3 = 6 pi