Quantitive Reasoning

when 3x^3 -7x+7 is divided by x+2, the remainder is what?

-3

solve for X 8^x = 16^(x-1)

2^3x= 2^(4x-4) 3X=4X-4 X= 4

simplify: 5sin²+5cos²

5(sin²+cos²) *sin²+cos²=1* plug this back in 5x1= 5

how many ways can you arrange ASCENT?

6!= 720

if 19/(5x+17) = 19/31, what is x?

cross multiply x= 2.8

what is arcsin(1/2) equal to? -90<y<90

where is y= +1/2 on the unit circle in the 1st or 4th quadrant? at 30º

whats the value of X+Y if: 2x-9y=11 and x+12y= -8

x+y=1 2x-9y=11 x+12y=-8 _____________ add 3x+3y=3

What is sin(0)? what is cos(0)?

0 1

pyramid volume

1/3Bh

a machine produces toy cars in repeating cycle of Blue, red, green, yellow, and black. If a string of 6 consecutively produced cars is selected at random, what is the probability of two of the cars being red?

1/5

parallelgram area what is a rhombus

A= BxH rhombus= 4 equal sides

Circle Area and Circumference

Area= πr² Circumference= 2πr

company Z spends 1/4 of its revenue on marketing and 1/7 of the remainder on maintenance. What fraction of the original revenue did company Z have after marketing and maintenance expenditures? A. 5/14 B. 1/2 C. 17/28 D. 9/14 E. 41/56

D 3/4 x 1/7= 3/28 spent on maintenance 3/28 + 1/4= 10/28 spent 1-10/28= 18/28= 9/14

multiple vs factor

multiple: ex 15 is a multiple of 5 factor: ex 5 is a factor of 15

if a>1, which of the following is equal to 2a+6/a²+2a-3? A. a B. a+3 C 2/a-1 D. 2a/a-3 E. a-6

C. Note: when picking numbers, check ALL answers plug in 2 to a and solve for answers; could be A or C so then pick an additional number and solve

p, q, and r are positive integers. q is a factor of r and r is a multiple of p. which of the following must be an integer? A. p+q/r B. p/q C. r(p+q)/pq D. r+p/q

C. set r=6; q=3; p=2

3D rectangular solid volume cube solid volume cylinder volume

rectangle: L•w•h cube: e³ cylinder: πr²h

What's X if: a= (b+x)/(c+x)

ac+ax= b+x x-ax= ac-b x(1-a)=ac-b x=ac-b/1-a

a restaurant is hanging 7 tiles in a row. how many arrangements if there are 3 white and 4 blue tiles?

*total!/repeats!* 7!/3!4!= 35

name 6 pythagorean triples

3, 4, 5 5, 12, 13 6, 8, 10 8, 15, 17 9, 12, 15 9, 40, 41

hexagon area

3s²(SQRT3)/2 s= side length

6x²-x-2/(4x²-1)

3x-2/2x-1 (3x-2)(2x+1)/(2x-1)(2x+1)

how many radians in 270º? how many degrees in 5π/6 radians?

3π/2 150

lateral area of a cone volume of a cone

lateral area= area from vertex to base, excluding the base *LA: 1/2cl* where c= circumference and l= slant height (note: circumference of a circle is 2πr) *volume= 1/3πr²h* (note: 1/3 of the volume of a cylinder)

trapezoid area

1/2h(b1+b2)

solve: tan(sin^-1(7/SQRT4))

DRAW OUT; find the triangle for sin7/sqrt4 tanx= 7/5

what are the 3 squared trig functions?

Sin²+Cos²=1 tan²+1=sec² cot²+1=csc²

how many ways can you arrange REASSESS?

total letters!/number of repeats! 8!/2!4!= 840

tan²(x)= 6[tan(x)]-9 what is the measure of x? A. 30º B. 45º C. 75º D. 90º E. 105º

C. 75º factor tan²x-6tanx+9 (tanx-3)(tanx-3) tan(x)=3 tan= sin/cos; note: since the value is larger than one, the sine (y) value should be larger than cos (x) look at unit circle and approximate

if theres 3 boys and 3 girls, how many ways can they be lined up alternating genders?

bgbgbg OR gbgbgb 3•3•2•2•1•1 + 3•3•2•2•1•1 36+36= 72 Note: Or= ADDITION

combo vs permutation

combo= order DOESNT matter *nCk= n!/k!(n-k)!* where n>k permutation= order matters *nPk= n!/(n-k)!*

how many integers in |4x+3|<8

four -8<4x+3<8 intigers in the range are: -2, -1, 0 and 1

From a group of 5 men and 7 women, in how many different ways can a team of 2 men and 3 women be formed?

order doesn't matter so use combo: *nCk= n!/k!(n-k)!* 5C2=5!/2!(5-2)! = 35 7C3= 10 10x35= 350 total ways

Law of Sines

sinA/a=sinB/b=sinC/c where uppercase= sides and lowercase= angles

volume of a sphere Surface area of a sphere

volume= 4/3πr³ SA= 4πr²

if 3^(x²) x 9^x x 3= 27, what is x?

x= .73 3^x² x 3^2x x 3^1 = 3^3 x²+2x+1= 3 (x+1)²=3

in the list: 3, 4, 5, 5, 5, 5, 7, 11, 21 what is the fraction that is less than the mode?

2/9

A game is played by rolling a pari of 6 sided dice and moving forward the number of spaces indicated by the sum of the dice. A player is frozen if her opponent's piece lands on the same space as her piece. If player A is about to roll and is currently 6 spaces behind player B, what is the probability that player B will be frozen after player A rolls?

1, 5; 5, 1 2, 4; 4, 2 3, 3 5/36

a watch has 60 second marks and an single hand. If the hand is randomly set and allowed to count exactly 10 seconds, what is the probability that the hand will stop less than 10 marks on either side of the 53 second mark?

19/60 less than 10 so take 9 seconds of each side+ the 1 53 mark which equals 19

what are the sides for 45º, 45º, 90? 30º, 60º, 90º?

1:1:SQRT2 (isosceles) 1:SQRT3:2

What are the side lengths of a 30-60-90 triangle? 45-45-90?

1:sqrt3:2 1:1:sqrt2

A school has 50 students assigned to 5 distinct classes so the number of students in the class are consecutive and each student is assigned to only 1 class. What is the probability that a student is selected at random is one of the two largest classes?

46% *consecutive= KEY WORD* it means that each number in the series is separated from the other by a FIXED amount we know each class is separated by ONE Classes: x, x+1, x+2, x+3, x+4 --> 5x+10=50 --> x= 8 classes: 8, 9, 10, 11, 12 11+12/59= 46%

Line: PQRS PS= 2x+12; PQ= 6X-10; If R is the midpoint of QS, what is the length of segment PR?

4x+1

a code is made by arranging 7 letters. 3 of the letters used will be the letter A, 2 of the letters will be B, and 1 of the letters will be C, and one will be D. If there is only 1 way to present each letter, how many different codes are possible?

7!/3!x2!= 420

solve for x: 2^x + 2^x + 2^x + 2^x A. 2^(x+2) B. 2^(x+4) C. 2^3x D. 2^4x E. 8^x

A plug in 2 for x: 2²=4 --> 4x4= 16

a car company charges for milage as follows: X dollars per mile for the first n miles and X+1 dollars per mile over n miles. how much will the first mileage charge be, in dollars, for a journey of d miles, where d>n? A. d(x+1)-n B. xn+d C. xn+d(x+1) D. x(n+d)-d E. xd+n

A plug in x=4, n=3, and d=5 or any other simple set of integers

if sinx=1/2cosx and 0< x< π/2, the value of 1/2 sinx is: A. .22 B. .25 C. .45 D. .5 E. .75

A square: sin²= (1/4)cos² factor in cos²= 1-sin² and solve for X sin²x=1/5 sinx=.2 and you're looking for 1/2 sin(x) so multiply .2 by 1/2= .22

which of the following equals cos⁴-sin⁴/cos²-sin²? A. 1 B. 2 C. cos+sin D. cos-sin E. cos²+sin²

A. 1 factor the top out

Area of sector vs length of sector

Area= (n/360)πr² Length= (n/360)2πr n= arc's central angle

if a biclyst increases speed by 30% and then increases again by 10%, what is the percent of increase from original? A. 40 B. 43 C. 64 D. 140 E. 180

B

if 2 is the remainder when m is divided by 5, what is the remainder when 3m is divided by 5? A. 0 B. 1 C. 2 D. 3 E. 4

B remainder= how many numbers are left after division m/5=2 --> set m=7 3(7)/5= 4 1/5 (remainder of 1)

the positive difference between sam and lucy's ages is a and the sum of their ages is z. if lucy is older than sam, which of the following represents lucy's age? A. z-a/2 B. z+a/2 C. a- (z/2) D. a-z/2 E. (a+2z)/2

B set Lucy=4 and sam=3 L-s= 1= a S+L= 7= z

what is the y-coordinate of the point at which the graph of y= 2sinx- cos2x intersects the y-axis? A. -2 B. -1 C. 0 D. 1 E. 2

B. -1 when does anything intersect the y axis? when x is set to ZERO 2sin(0)-cos2(0) *sin0= 0 but cos0=1*

2 triangles have exactly the same angles; the triangles correspond in a 2:1 ratio however. How many times larger is the area of the bigger triangle?

BxHx1/2 assign a number- Base and heigh of 2 for the small triangle and 4 for the large triangle 2x2x1/2= 2 4x2x1/8= 8 FOUR times larger

carol spent 1/4 of her savings on a stereo, and then spends 1/3 less than she spend on the stereo or a TV. what is the fraction of savings she spent on both stereo and TV? A. 1/4 B. 2/7 C. 5/12 D. 7/12 E. 3/4

C 100x1/4= 25 dollars spent on stereo 25x1/3=--> 25-8= 17 dollars spent on TV 25+17= 42 42/100= percent of savings spent

a crate of apples has one bruised for every 30 apples in the rate. 3 out of 4 bruised apples are considered not fit to sell. If there are 12 apples not fit to sell, how many total plus are there in the crate? A. 270 B. 360 C. 480 D. 600 E. 720

C 3/4 --> 12/16 --> 16x30= 480

each writer for a local newspaper is paid as follows: a dollars for each n stores each month and a+b dollars for each story thereafter, where a>b. How many more dollars will writer who submits n+a stories in a month earn than a writer who submits n+b stories? A. (a-b)(a+b+n) B. a-b C. a²-b² D. n(a-b) E. n(a+b)

C set a= 3; b= 2; n= 4 the first person earns 5 more dollars than the second

which slope has no point of intersection with line y=4x+5? A. y= 1/4x-5 B. y= -1/4x-5 C. y= 4x+1/5 D. y= -4x+1/5 E. y=-4x-1/5

C. no intersection= lines are PARALLEL

Two trains are approaching each other from opposite directions with the speeds 10 miles per hour and 20 miles per hour, and heading for a collision. If they are a quarter mile apart, how long will it take for them to collide? A. 45 seconds B. 20 seconds C. 30 seconds D. 40 seconds E. 1 minute

C. 30 seconds .25 miles x 1hr/30mph x 60min/1hr x 60sec/1min

Amanda goes to the store to buy 1 ball and 3 board games. If there's 3 types of balls and 6 types of board games, how many different selections of the 4 items can she make?

COMBO 3C1: 3 6C3: 20 20x3= 60

Country X holds an annual math competition and sends a team of 4 students. If school A has 6 boys and 7 girls who's grades qualify them and the competition rules stat that the team must consist of 2 girls and 2 boys, how many different teams could school A send to the competition?

COMBO: nCk= n!/k!(n-k)! 7C2= 21 6C2= 15 15x21= 315

a Company is selecting 4 members of its board of directors to sit on a committee. If the board has 9 members, how many different selections could they make?

COMBO: nCk= n!/k!(n-k)! 9!/4!(9-4)!= 126

ron begins reading a book @ 4:30 and reads 30 pages per hour. Michelle reads the same good at 6:00 and reads 50 pages an hour. If mitchell started 5 pages behind ron, what time will they be on the same page? A. 7:00 B. 7:30 C. 8 D. 8:30 E. 9

D pick B or D first since they are in the middle and then determine if you need to go up or down

A father's age is twice the sum of the ages of his four sons. In 30 years, the total of the sons' ages will be twice the age of their father. How old is the father now?

F= 2x 2(F+30)= x + 4(30) 4x + 60= x + 120 3x= 60 x=20 --> 2x is the fathers age now therefore, the father is 40 years old

solve for X: (3X² -11X+6)/(9-X²)

factor X= (2-3x)/(x+3)

a machine is made of components A and B. Each component either works or fails and is independent of the other component. The machine works if at least one of the components works. If the probability that each component works is 2/3, what is the probability that the machine works?

probability that one part doesn't work: 1/3 so take that prob and square it: 1/9 probability that the whole machine doesn't work, therefore 8/9 probability that it works

what is the range of the set? {2, 4, 26, 10, 20}

rage= high-low 26-2= 24

What is Arsine?

sin^-1 use this to determine ANGLE

what are the values of x and y? 4x+ky=15 x-ky=-25

x=-2; y= 3

what is the value of x 5/(x+3)=1/x + 1/2X

x=9/7

what is the value of y? x=y+a²/(y-4a)

y= a²+4ax/(x-1)

what does cos, tan, and sin represent in terms of x and y?

y= sin x= cos y/x= tan= sin/cos

what's the solution set of |2x-3|<7

{-2<x<5} -7<2x-3<7

how to convert degrees to radians vs radians to degrees

º x π/180= radians radians x 180/π= º

how many radians in 180º? 360º?

π 2π