Math

How to find x- value of vertex of quadratic formula and y vertex

x vertex = -b/2a and then plug back into quadratic equation given

Casey is selling 1 orange for x dollars. How many oranges can you buy for 80 cents?

1 orange = x dollars so 1 orange/x dollars. 1 orange/x dollars *.8 collars = 0.8 oranges/x aka 4/5x

For b2-4ac to determine if roots are equal what does it mean if b2-4ac =0

1 real solution

1 m = _____ yd

1.1

Bottle can be filled completely by faucet in 10 sec. The bottle also has hole that can drain full soda bottle to empty in 30 seconds. If bottle initially empty when faucet turned on, how long will it take for bottle to be fully filled?

1/A time + 1/B time = 1/A + B working together 1/10 sec - 1/30 sec = 1/x 3/30 - 1/30 = 1/x x = 15 seconds

1 cm = ___ milimeters

10

Bob can chop down 10 trees in 30 min while Frank can chop down 15 trees in 30 min. Working together, how long will it take them to chop down 100 trees?

10 trees/30 min + 15 trees/30 minutes = 25 trees/30 min (create rate) then 100 trees*(30 min/25 trees) = 120 min

1 kg = ______ g

1000

1 m = _____ cm

1000

Corey pays $100 for order of 1000 stickers. if 20% of stickers get lost in shipping, how much should Corey charge per sticker to make 100% profit?

1000 stickers *.8 = 800 stickers --> 100% profit is $100 +$100 = $200 so 800x=200

1 foot = ___ inches

12

Average distance between each data point and mean (average)

Standard deviation

1 SD from data set = ___ % 2 SD from data set = ___ % 3 SD from data set = ____%

68, 95,99.7

3 Liters of 50% salt pollution mixed with 7 liters of 13% salt solution, what is % of salt in mixture?

(3)(.5) + (7)(0.14) = (10)(x) 10 is final volume (7+3)

Weighted average for grad point equation

(Letter grade * grade point) + ..../# of credits taking

(x^b)(y^b)

(XY)^b

John can mow a lawn in 2 hour sand Matt can mow same lawn in 3 hours. How long will it take for John and Matt to mow lawn working together, but John only works for 24 minutes?

-Johns rate of mowing lawns: (1 lawn/2 hours) = 0.5 lawns/hr.... in 24 minutes (first convert to hours so 24 min *1/60 = 0.4 hours) so 0.5 lawns/hr (.4 hours) = 0.2 lawns in 24 min hes working. -Matt needs to finish 1-0.2 = 0.8 -Matt's rate of mowing lawns: (1 lawn/3 hours) so 0.8 lawns * 1 lawn/3 hours = 2.4 hours *60 min/hr = 144 minutes

How to calculate sum of roots of quadratic functions?

-b/a

Bag of chips normally cost $2. If there is sale to get 12 bags of chips for price of 6 bags. Price per bag of chips taking into account 10% sales tax?

12 bags for $12. 12 x 1.1 (10% sales tax) = 13.20. 13.20/12 = $1.10 per bag

Jacob finds value of his home is 150,000. Calculates value of his car, increased by half, is 30% value of his home. What is value of his car?

150,000(.3) = x +0.5x

Sum of of internal angles of shape

180(n-2)

In Gainesville, one in four people have blonde hair. What is the chance that if four people are chosen at random from Gainesville, at least one person has blonde hair?

1:4 Possibility of choosing at least one blonde= 1 - probability of choosing zero blonde people P (at least one blonde person chosen)= 1 - P(zero blonde people chosen) Probabilty of choosing zero blonde: 3/4 * 3/4* 3/4 *3/4 = 81/256 P(at least one blonde person chosen): 1- 81/256 = 175/256

For b2-4ac to determine if roots are equal what does it mean if b2-4ac > 0

2 real solutions

1 kg = _____ lbs

2.2

1 inch = ___ cm

2.54

Perimeter of circle aka circumference

2pi(r)

1 yard = ____ feet

3

For compound interest formula, if interest compounded quarterly (every 3 month) and length of investment is 1 year then what is n?

4 periods

Sphere volume formula

4/3(pi)r^3

John has enclosure with 400 feet perimeter fence. Width of rectangular enclosure is 120ft greater than length. Assuming each chicken takes up approximately 1 square feet, how many chickens can farmer Joe fit in enclosure?

400 = 2x + 2(x+120) x= 40 so l is 40 and w is 160. SQUARE FEET indicates area so 40*160 = 6400 square feet

1 lb = ____ g = _____ oz

454, 16

How to simplify 48^5/16^6

48*48*48*48*48/16*16*16*16*16*16 is really 3*3*3*3*3/16 so get 3^5/24

Surface area of sphere

4pi(r)^2

Deck of cards (total # cards, suits, cards per suit)

52 total cards, 4 suits, 13 cards per suit

1 mile = ___ feet

5280

Surface area of cylinder

A=2πrh+2πr2 (2 comes from both ends of cylinder)

If given x^4 = 9 how do you solve?

Aka X= 9^1/4 and x = 3^2(1/4) so x = 3^(1/2) so x = sq root of 3

Regular hexagon has perimeter 12. what is area of hexagon? 2 ways to solve --> one with formula

Area of hexagon: 3sqrt3/2 * s^2 or hexagon divided into 6 equilateral triangles (30/60/90 triangeles) and find area of one and times by 6. Center angle is 360/6 = 60 so split is 30

Taxi driver drives halfway to airport, a distance of 30 miles going 50 mph. How fast must he go for second 30 miles of drive in order to meet average speed of 60 mph?

Average rate = total distance/totla time total time= total distance/average rate Average at 50 mph: 30 miles/50milesperhour = 0.6 hours= 36 minutes Average at 60 mph: 60miles/60milesperhour= 1 hour = 60 minutes Get 60-36 = 24 minutes and 24 min = 0.4 hours Average speed = 30/ 0.4 = 75 mph

Larry drive 20 miles at 40 miles per hour, then 60 miles at 80 miles per hour. What is his average speed for whole trip?

Average speed: total distance/total time Distance: rate*time Time = distance/rate First: 20/40 = 1/3 Second: 60/80 = 3/4 Average speed: 80/(2/3 +3/4) = 64 mph

Average speed of skier on slalom is 23 mph. Skier knows her average speed for first 1/3 of slalom was 17 mph and average speed for second 1/3 was 15 mph. What was skiers average speed for final 1/3 of journey?

Average: x= skiers speed over final 1/3 of journey so (17 + 15 + x)/3 = 23 and get x = 37

How to bind area of parallelogram

Base x height or side length 1 * side length 2 * sin (angle) which is angle in between

How to calculate product of roots of quadratic functions?

C/a

Jack drives scooter 10 miles at 20 miles per hour and other 18 miles at 15 miles per hour. If Jack drove at a constant speed over entirety of his trip, what speed would allow him to make trip in same amount of time?

Calculate total time josh spent driving and divide total distance by total time Average rate: total distance/total time Total time= total distance/average rate 10 miles/20milesperhour = 1/2 hours 18 miles/15milesperhour= 6/5 hours 1/2 + 6/5 = 17/10 hours Constant speed: 10 + 18/(17/10) = 16 mph

What happens if to standard deviation if add additional data points at mean?

Compresses data set around mean and decreases standard deviation

Equation for distance?

Distance = rate(time)

What to do if (4x^11/6)^2

Distribute square to both terms so get 16x^22/6 (multiply)

While riding his bike, Mike travels 10 miles at 30 mph, then an additional 20 miles at 10 mph. What is Mike's approximate average velocity for entire trip?

Do time = distance/speed individual so 10 miles/ 30mph = 1/3 hours and 20 miles/10mph= 2 hours so total time is 1/3 + 2 = 7/3 hours. Do average velocity so: 20 + 10/(7/3) = 13 mph.

If add any number of data points directly at the average, what happens to mean value?

Does not change

Dylan can paint wall in 4 hours, while Brett can paint same wall in 3 hours. If Dylan paints wall for 2 hours, how much time will it take Brett to finish painting wall?

Dylans rate of painting: 1 wall/4 hours and in 2 hours 1 wall/4 hrs * 2 hours = 0.5 walls -means that Brett has 0.5 walls to finish and his rate is 1 wall/3 hours. So to finish he does 0.5/(1/3) = 1.5 hours

If have x:z is 1:8 and y:z is 2:3 and want to relate x:y

Find common number for z so 3(x:z) = 3(1:8) --> 3: 24 8(y:z)= 8(2:3) --> 16: 24 so get 3:16:24 so 3:16

Jim has $4. With $4, Jim can buy exactly 16 apples. 4 apples cost $0.20 more than 5 oranges. Which quantity is bigger: # oranges Jim can buy or 25?

Find cost of apples first so do 4 =16x and get x =0.25. Find out how much 4 apples cost ---> 4 (0.25) = 1. So 5O (5 oranges) =0.80 and get O = $0.16. Then do 4 = 0.16x and get x =25 so both quantities equal.

What to do if given equation for parabola (quadratic) and line and want to know which one is greater.

Find out if line intersects parabola --> if does not intersect: parabola always greater than line but if does intersect at 1 or 2 points: relationship cannot be deteremined --> set equal to each other

How to set up equations if says Grace is twice as old as her sister. And grace was three times as old as her sister 2 years ago.

G= 2s G -2= 3(S-2) take into account 2 years ago for her sister as well

If 8≤ l 4x - 12 l ≤ 48 how do u find solution for x?

Have to write out two inequalities for both possibilities so 8≤4x-12≤48 and -8≤4x -12≤-48 -flip inequality signs if multiply or divide by negative number

Bird launches into flight from point on ground. Bird can fly 8 feet per second horizontally and 6 feet per second vertically. After 10 seconds, how far has bird flow relative to initial spot on ground.

Horizontally: 8ft/sec * 10 sec = 80 feet Vertically: 6 ft/sec * 10 sec = 60 feet Use Pythagorean theorem to find c^2

Interest formula

I = Prt P: principal (initial) amount r: interest rate t: time the money is invested for

Manager notices Julio can change tire in 5 mintues while Jaden takes 10 minutes. How long would it take both brothers working simultaneously to change one tire if Julio only works 2 minutes?

Julio: 1 tire/5 minutes =0.2 tires/min 1tire/5min *2 minutes = 0.4 tires which leaves 0.6 left for Jaden 0.6/(1/10) = 6 minutes

Find solution set for following inequality x² + 3x > 0

Keep on left side so x(x+3) > 0

Given: lxl > 1 , y= -x, z = -y Which one bigger? Quantity A: xyz^2 Quantity B: x

Know z= -y and y= -x so z= x and plug in

How to tell what following data sets has largest standard deviation? A. {1,3,5,7,9} B. {1,2,3,4,5} C. {1,3,5,7,10} D. {2,3,5,7,9} E. {1,3,6,7,9}

Larger standard deviation =larger spread of values and larger range so C. but if other answer {1,5,5,5,10} compare if numbers are closer (these numbers closer so smaller deviation)

Longer line that goes through ellipse?

Major axis

Shorter line that goes through ellipse?

Minor axis

Formula for compound interest

N = # times compounded each time period T= number time periods

If start with even negative number and end with even number what type of number? (i.e -28*-27....*-4)

NEGATIVE

How many ways can letter APPALOOSA be arranged?

No repeating letters than 9! but if repeating letters must divide by factorials of number of repeats: -repeats: (a=3, p=2, O=2) (9!/3!x2!x2!)

Ellipse defined by x²/9 + y²/16 = 1. In diagram point a is intersection of minor axis and the ellipse and point b is at intersection of the major axis and the ellipse. What is the length of line segment AB?

Numbers in denominator give you numbers of major and minor axis. So square root of 9 is 3 (minor) and sq root of 16 is 4 (major). makes 3-4-5 triangle so AB is 5

What does intersection (upside down "u" mean)?

Only data that exists in both

Calculating percent decrease?

P = 1 - M/a ( aka slope of line from greater value over smaller)

Four dice are rolled. Probability of getting exactly three 4s and 2 in any order.

P(4442) = 1/6*1/6*1/6*1/6 = 1/1296 but could also have P(4244) or P(2444) or P(4424) so add 1/1296(4) = 4/1296

If start with odd negative number and end even what type of number? (i.e -43...-42....-12)

POSITIVE

What does it mean when ax² + bx + c > 0

Parabola is concept up and doesn't cross x axis so A must be positive

#'s that can't be divided by anything but themselves and 1

Prime factors

Farmer redistributes cows among three pastures. First has 10 cows less than half the total, second has 8 cows less than one third the total, and the third has 6 less than one fourth the total. How many cows does he have?

Set up equation (1/2x - 10) + (1/3x-8) + (x/4-6) = x

The sum of two numbers is 168. If the larger number is divided by the smaller number, the quotient is 6 with a remainder of 7. what is the smaller number?

Smaller number: x Larger number: 6x +7 so get x + (6x +7) =168 and x = 23

Given x > 3 and quantity a is y =x^2 + 3x + 2 and quantity b is y= 4x + 2 which quantity bigger?

Subtract equation in quantity b from quantity a to determine if difference is: + = quantity A greater 0= Quantities equal -= Quantity B greater get y = x^2 -x and factor to get x =0 and x=1. Parabola dips below x axis between these points so for x> 1 and x < 0 difference is positive so quantity a larger

If comparing value of quantity a: 3x/8y and quantity b: 15x/39y what do you do?

Take out x/y because same so just compare 3/8 to 15/39 and try to get same denominator by timing 3/8 * 5/5 and will see quantity b bigger

Samantha's pie stand has 17 average visitors per day for past 6 days. How many people must visit her stand on the seventh day to maintain her quota of 20 visitors per day?

Total # visitors she needs per day to maintain quota: 20 visitors/day * 7 days = 140 visitors Visitors so far: 17 visitors/day * 6 days = 102 visitors so do 140-102 and need 38 remaining visitors

How many ways can 4 identical footballs and 3 identical basketballs be ordered on a shelf?

Total possible # of order if football and bacsketballs different: 7! = 5,040 orders so 7!/(3!4!)= 35

Train A leaves city traveling 100 mph. Train B leaves from same city in same direction of train A, traveling at 150 mph. If Train A leaves 3 hours before Train B, how long will it take for Train B to catch up to train A?

Train A: 100h + 300 Train B: 150h and do 100h + 300 = 150h and get h= 6 which is hours it take to catch up

Volume of a cylinder

V=πr²h

Price of wheat increases 10% in one week, decreases 30% next week, the increases 20% following week. Net change in price of wheat?

Week 1: prince increase 10% = new price 110% initial wheat (1.1 * initial price) Week 2: price decrease 30% = new price 70% of week 1 (.7 * week1 price) Week 3: price increase 20%= new price is 120% week 2 (1.2 * week 2) DO (1.1 * 0.7 * 1.2)x = 0.924x which Is 92.4% of what it was originally so do 100 - 92.4% = 7.6%

Standard deviation equatino

X with thing over top is average and N is total numbers

Two ways to solve quadratic formula and horizontal line to determine which is bigger?

b2-4ac or finding y vertex

log(x^b)

blogx

What happens when you take sq root of number

can always get + or -

If given sqroot of (8a^7b^2/2ab^4) what can you do first

can divide terms so get sqrt of 4a^6/b^2 and then solve

If have logxY^10=20 what is value of logxY^2?

can do logxY^2(^5) and get 5logxY^2 = 20 so answer is 4

Distance formula between two points

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Whats another way to write x^(2/3)

denominator becomes taking triple root of x2

Total time equation

distance/speed

When comparing quadratic formula equation and horizontal line equation how do you decide which one is bigger?

find y vertex if concave up parabola (coefficient A is (+)) and if greater than the horizontal line then it is bigger

What does "u" union mean>

includes all data but excludes duplicate values

log(xy)=

logx+logy

log(x/y)

logx-logy

Given: f(x) = 5x^2 + 14x -19 Quantity A: values of x such that 2f(x) = 0 Quantity B: values of x such that -3f(x) = 0

multiplied by constants so can do (2f(x)= 0)/2 and (3f(x)=0)/3 and get the two quantities are equal

Permutation equation (order DOES matter)

n!/(n-r)! i.e how many distinct ways can 2 people sit in 5 chairs: 5!/(5-2)!

Combination formula (order DOES not matter)

n!//(n-k)!(k!) n= number of objects from which you can choose k= # of objects to be chosen

For b2-4ac to determine if roots are equal what does it mean if b2-4ac <0

no real solution

Slopes for perpendicular line

slopes are opposite reciprocals

How to determine is sqroot 8x or sqroot x + 2sqroot x is bigger

sqroot x + 2 sqroot x is same as 3 sqroot x which can be written as sqroot 9 *sqroot x which can overall be written as sqroot 9x so this is bigger

Average velocity equation

total distance/total time traveled

Given: m > 3 Which quantity bigger? Quantity A: m Quantity: B: m+5/2

treat equation algebraically and simplify them A: 2m B: 2(m+5/2) A: 2m B: m + 5 Subtract m from both sides A: m B: 5 relationship cannot be determined

For standard deviation curve if give mean of 4 what are numbers to left or right

usually subtract and add one standard deviation

Commuter drives 30 miles each way between home and work every day. When traveling home from work, traffic causes his commute to take triple the amount it takes compared to trip driving to work. If traveling to and from work takes total of 2 hours. What is commuters average speed traveling to work?

x= amount of time it takes for commuter to travel to work x + 3x = 2 so x= 0.5 hours to travel to work Average speed = total distance traveled/ total time =30 miles/0.5 hours = 60 mph

If points on perimeter of circle, what equation what must they satisfy?

x² + y² = r² i.e if x² + y² = 4 then radius is 2

2 ways to write line equation

y=mx+b or y2-y1=m(x2-x1)