Quick study academic SAT tips math 600dpi

English language skills • Critical Reading Understanding and interpreting written materials • Writing Ability to use basic standard writing skills Mathematical Basic number skills ~~~ S

WORLD'S #1 ACADEMIC OUTLINE

SAT TIPS:

This guide reflects the changes in the March 2005 SAT

~~

-­

guidance and examples to familiarize you with the test,

and offer helpful hints to achieve the best results

J:~~ ~; ~~~~~~ S

English language skills

• Critical Reading (Understanding and interpreting

written materials)

• Writing (Ability to use basic standard writing skills)

Mathematical (Basic number skills)

~~~ S :f:}~~~

See QuickStudy«' guide "SAT Tips: Verbal" for more detailed information

• Critical Reading

• A section of short reading passages followed by comprehension questions

• A long (400-900 words) reading passage testing the same

• Sentence completions: Incomplete sentences with multiple-choice answers

• Writing

• Multiple-choice questions based on a short written passage that test knowledge of basics (grammar, spelling, diction, agreements, etc.)

• A brief persuasive essay to test writing skills

• Regular math questions, covering basic arithmetic, geometry, and algebra through H.S Algebra II

• Grid-ins: Ask you to solve a problem and enter the answer itself in the grid

~

,

m

Remember this is only another test

Desig ed to see what you know

./ Do not spend too

much time on any

time, you should do pretty well

stumped, move on

and time allowing Where the SAT diHers from other tests come back to that

is in the time it takes, how the test is question later set up, and how you answer the

questions ./ Do not be "thrown"

if one section is

• The test is 3 h urs 45 minutes lo g; with

the exception of the writing questions ./ Do not try to "make

(35 minutes) no section lasts more than up time" in one

time in the one

before it

cannot skip ahead to another section

ou cannot return to a section once the time for that section is over

• The answer grid sheet contains Should you guess1 colunms of numbers corresponding to - I point is given for every correct answer

while 1/4 of a point is deducted for a

the test questions next to colunms

wrong answer containing open ovals with the letters

- No points are lost for an answer left blank,

A - E beneath them

• Blacken in the appropriate oval on the

A sealed test booklet, Inclucling the line corresponding to

ue and space for writIn It is

theeaay

45 with the answer to question 44

- At best, you may get one unfairly

A pencil

marked wrong answer

- At worst, this may lead to all

An aMWW .rid t

subsequent answers being wrong!

MIIt8rI.I you will b given

- If you have no idea of the correct answer,

and all of the choices seem just as valid,

don't guess

- You may discover you do not know the right answer, but you can eliminate all the

wronganswers

lI1lJC()c1llc:iIl~

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The Scholastic Aptitude Test (SAT): One of the

measures used by colleges and universities to

determine who gets in These pages will give you

•two %$

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•••••••••••••••••••••••

- -

{ -4 -2 O 2 4 6 ) I

{ -3 -1 1.3.5.7 ) I

I

I • Ifone choice can be made in m ways and another can be made in n ways

I then there are mn ways to make both

I • This principle extends to situations where several sequential choices are to

I be made;

I - Ex: A diner offers a lunch special in which customers can choose a soup

I a sandwich and a beverage; if they offer 2 kinds of soup 5 sandwiches

I and 7 beverages how many different specials can be composed? Since

•

•

**

• Set: A collection of elements; relations between sets include:

•

•

:

Union: (U): All elements that belong to either set or both

• Prime number: Divisible by only 1 and itself:

• Composite number: An integer that is not

prime; every integer can be factored into a

product of primes in only one way

• Digits: The numerals O 1.2.3.4.5.6 7 8 and 9

• Arithmetic sequence: A constant number is

added to produce the next term; the

following sequence adds 4 between

consecutive terms: 3 7 11 15 19.23

• Geometric sequence: A constant number is

multiplied to produce the next term; in the

sequence 2.4.8 16 24.48 the next term

is obtained by multiplying by 2

• Rational numbers: Can be expressed as a

ratio of integers m In; in decimal form such a

number either terminates or repeats; Ex:

3/8 = 0.375 2/3= 0.66666 • 5!g= 0.55555 • 1 h = 12

• Irrational numbers: Cannot be expressed

as a ratio of integers; in decimal form

irrational numbers neither terminate nor

reJ?eat; Ex: j3 = 1.7320508

./2 =1.414213 1t =3.14159

• Percent refers to hundredths: 35% = 0.35 =

35/100 = 7/20; Ex: What is 40% of 80?

80x 0.40 =32

• Percent increase or decrease: = increase or

decrease 1 original quantity; Ex: An item

usually priced at $400 is given a sale price

of $340; what is the percent decrease?

Decrease = $400 - $340 = $6 0 percent

decrease = $60 1 $400 = 0.15 = 15%

• Average speed: Total distance/total time;

do not simply average the speeds! Ex:

Jennifer travels for 4 hours at 50 mph then

for 3 hours at 60 mph; what is her average

speed? Total distance = 4 x 5 0 + 3 x 60 = 200

+ 180 = 380 miles total tim e = 4 + 3 = 7

hours average speed = 380 miles I 7 hours =

54 2 hmph

~ there are 2 ways to choose a soup 5 ways to choose a sandwich and 7 ways

I to choose a beverage, there are a total of2 x 5 x 7 = 70 combinations

._ -­

The number of ways in which x objects can be

• This is the product of all the integers between x and I: xl =x(x-l)(x-2)

- Ex:

possible for the order in which they finish? Since there are 6 players who

could come in first, then 5 who could come in second, then 4 who could come

in third, and so on, the total number ofpossible rankings is 6! = 6 x5 x4 x3

x2x1=720

• If only x objects out of a larger group of size n are being arrang number of arrangements is nll(n -x)

- Ex: If, out often runners in a race, awards are to be given for 1st• 2nd and 3

place, how many ways can they be aSsigned? The r e ar e 10 run ne rs who could win 1 st place, and then 9 who could win 2"d, and the n 8 who co uld win

3"", so the total number ofarrangements is 10 x 9 x 8 =1O! I (10 - 3)f =720

• When counting the number of choices ofx things out of a group of n regard to order the number of choices is n!/(n - x)!xl since the arrangements of each group chosen are counted as on

- Ex: Three people must be chosen from among the five in the accountin department to attend a meeting; how many choices are possible? Th e re a re Sf! (5 - 3)f 3! = Sf I 2f 3! =5 x 4 x 3 x 2 x 1 12 x 1 x 3 x 2 x J = 10 ways to c1wo se

the three

- Or, call the five employees A B C D and E

• There are five choices for the first then four for the second then three for the third employee to attend the meeting

• Bu this takes into account the order which doesn't matter here: there are six sequences for every possible group

• That is, ABC, ACB BAe BCA, CAB, and CBA all consist of the same

employees

• So the 5 x 4 x 3 that we would obtain should be divided by the 3 x 2 x 1 ways to arrange the members yielding 60/6 = 10 groups

2

(x.y) a =xa.ya Xa'Xb=X(a+b)

(xa) b X a • b X a/b= b /Xa

EX: S4/J = Jj84 = eM =24 = 16

Systems of Eq,uations

It is useful 10 bear in mind that an equivalent equation - that is, one with the same solution set - can be obtained by addinq, subtracting, multiplying, or dividing the entirety of both sides by the same expression

Unless noted otherwise, ["denotes the positive (principal)

• Ex: If5x + 3y = 11 and 3x- 6y = 30, solve for x and y

root That is, /9 = 3, not ± 3 - By substitution: Taking the second equation, we can add 6y to both

sides to produce 3x = 6y + 30

a

• By subsituting 2y + 10 for x in the first equation, we obtain 5(2y + 10)

• Distributing the 5 yields lOy + 50 + 3y = II; combining like terms gives 13y + 50 = 11

• Subtract 50 from both sides: 13y = -39; then y = -3

• Now we can substitute -3 for y in either equation to find x = 4, so the solution is y = -3, x = 4

.I Remove a common factor: 3x2+ 12x = 3x(x + 4)

Factoring

• By combination: Multiply both sides of the first equation by 2 to produce

.I Perfect square: 4x2+ 12x + 9 = (2x + 3) (2x +3) = (2x + 3)2 10x+ 6y= 22

.I Difference of squares: x 2 - 25 = (x + 5) (x - 5) • Then add the two equations together, combining all like terms: (lOx + 6}1

.I Quadratic: 3x2+ 13x -IO = (3x- 2)(x + 5) + (3x- 6}1 = 22 +30, so 13x = 52, or x = 4; then substitution yields y = -3

SolVing Eq,uations

• The solution set of an equation consists of all values of the variable

that make the equation true

• Equations can be solved by performing operations to transform into

equivalent equations the whole expressions on both sides of the

equation; preserves the solution set

• Work toward isolating the variable

• When multiplying or dividing, be careful to perform the

operation on every term of each side

• Always check your solution(s) against the original equation

• Linear equations have degree (the highest exponent of a variable) I;

they can be solved by simple arithmatic operations

• Ex: Solve for z: 3z+ 5 = 12; subtracting 5 from both sides produces 3z= 7

- Dividing both sides by 3 produces the solution z = 7/3

- Check:3(7/3)+5=7+5=12

• Ex: Solve for y: 4

Subtracting 4 from both sides, we have -Sly

-Multiplying by y, -5 = lOy; then y = _Sly) = - 1/

- Check: 4 -51 (-liz)

• Quadratic equations: Have a squared term; that is, they can be

expressed Ax2 + Bx+ C= 0, where A, B, and Care constant coefficients

• They can be solved by checking that they are set equal to zero, and

then either factoring the quadratic expression or applying the

quadratic formula: given a quadratic equation set equal to zero,

x= -B±jB2-4AC

2A

• Ex: Solve for x: 3x2 -7x=-2

-First, set the equation equal to zero: 3x2 -7x + 2 = 0

-Then appl the uadratic formula, with A = 3, B = -7, and C =2,

x =(7 ± 49 - 4x3x2) 16 = (7 ± j2s) 16 =(7 ± 5) 16 = 12 /6 or 216

SO the solution is x = 2 or x = 1/3

• When solving inequalities, you can perform the arithmetic

operations as with equations, but remember to change the direction

o/the inequalitywhen mUltiplying or dividing by a negative number

• Always check your answer against the original inequality with some

relevant x values

• Ex: Solve for x: 9 - x 5 7; -x5 -2, so x ~

-Check: x = 0 fails, x = 2 works, x = 5 wor

• The equation can also be solved by factoring: if 3x 2 - 7x + 2 =

0, then (3x-I)(x- 2) = 0

- Since the product will be zero only if one of the factors is equal to zero, 3 x - 1 = 0 or x-2 = 0

- Solving the first produces x = 113, the second, x = 2; the solution is x = 2 or x =

-Check x = 2: 3(2)2 - 7(2) = 3(4) -14 = 12 -14

- Checkx= 1/3: 3(1/3)2-7(1/ 3) = 1/3 -713 =-%

=-• Rational equations: Contain variables in the denominators of rational expressions; solve them by mUltiplying to eliminate the denominators

• Ex: Solve for x: 3/x- 1/2= 7/4

- First, add liz to each side to produce 3/x= % Multiplying by x

produces 3 = (9/4)x Then dividing by % yields x = 3(4/9) =4/3

- Check: 3/( 4/3 ) - _ = 3(31 4 ) -_ = % -24 = 7/4

• Ex: Solve for w: 3/(w + 2) -lIw= 1I( 4w ); multiply each term

by the appropriate expression to obtain the common

denominator of 4w(w + 2)

-That is, [3/(w + 2)][4w14w) - [lIw)[4(w + 2) I 4 (w + 2)) = [11 (4w))[(w + 2) I (w + 2») so 12w I 4w( w + 2) - 4(w + 2) I 4w(w+ 2) =(w+ 2) 14w(w+ 2)

- Now that all terms have a common denominator, that

denominator doesn't matter, except that it must be nonzero

- That is, 4w(w+ 2) 0, so 4w 0 and (w+ 2) 0, so W" 0 and

-Or, we can mUltiply the entire equation by 4w(w + 2) which must be nonzero to give: 12w- 4(w + 2) =w + 2, so

12w- 4w-8 = w+ 2, 8w-8 = w+ 2, 7w = 10 so W= 1%

- Check: 3/(lOh+ 2) -lI(lOh) = 1/( 4 x IOh)

31(2 4h) -7/10= 7/40

7/8 - 7/10 = 7/40 35/40 - 28 / 40 = 7/40

• The solution to a system of inequalities is the intersection of

the solutions of the inequalities

• Ex: Solve for x: 2x + 8 5 12 and 7 - 3x 5 13

- x + 2, 4 5 6 and -3 x 5 6

- X 5 2 and x ~ -2; so the solution is -2 5 x 5 2

- Check: x =-3 fails the second inequality, x = -2 satisfies

both, x = 2 satisfies both, x = 4 fails the first

3

III

3/(x + 3), the

since

is undefined;

4, the domain

0; -x , if x< 0

An is a mefI8lIte used to describe data, usualIyMferrJDg to the .,Ic'ywedc

The mean ofa set of II numbers is defined 81 the sum oftbe numbers divided by II

• Ex: The mean of 550, 820, and 830 is: SSO + 'f +§30 _ lye» - 600

On the SAT, the word "average" refers to the artthmetic mean, except for probIBms

illuoluing average speed

The median ofa group of numbers is the middle value when the numbers are ordered

• The median aU, 4, 6, 11, 15, 19, aDd 23 is 11; ifthem are an even number ofvalues, 1M medianisthemeanofthetwomkldlenumbers:3,4,S, 11,15, 19 23, and30is: 15119 -17 The mode ofa data set is the most frequent value

• For instance, the numbers 88, 74, 82 88, and 94 have a mode of 88; the numbers 88

74.82.88, 94 and 74 have two modes: 88 and 74; the numbers 88 74 82 and 94 have

no mode, as none appear more than another

-_.,J

Perimeter and Area

Trianfles

b :

Algebra continued:

• Vertic:a1 IlJIIles are congruent 12; what is the length of the hypotenuse? GO =do bo =co eo =laO.I" =,

52 + 1~- 25 + 144 = 169-=CZ.soc= 1i69-13 • Corresponding anpea are

Variation

- Special RIght 'JrIangI.es: A 45· - 45° - goo congruent: flO =eo IJo = 1" c· =

• A function - denoted I(x) - is a

relation in which each element

of the domain is matched with

only one element of the range

• The domain consists of all

numbers x for which fix) is a

real number

• Ex: For I(x) =

domain is x 3,

division by 0

for fix) = j x ­

is x ~ 4, since the square root

of a negative is imaginary

•

Absolute Value

• The absolute value of x, denoted

lxi, is its distance from zero

• That is,lxl = x, if x~

• Equations involving absolute value

can be solved by taking the positive

and the negative of the expression

inside the absolute value

• Ex: Solve for x: 13x- 41 = 17

- (3x-4) =17 or-(3x-4) =-3x+

4=,17

- 3x= 21 or-3x= 13

- X= 7 or X= -13/3

• Inequalities involving absolute

value can be similarly solved, but

the direction of the inequality

must be switched when taking the

negative

A ">" inequality results in two

disjoint solutions, and a "<"

inequality results in one solution;

always check a range 01 x values

against the original i n equality!

• Ex: Solve for x : 12x- 51> 9

- (2x-5) > 90r (2x-5) <-9

- 2x>4or2x<-4

- x> 2orx<-2

- Check: x = 5 works, x = 1 fails,

x=-3 works

• Ex: Solve for x: 13x + 41 :s 13

- (3x+ 4):s 13 and (3x+ 4) ~ -13

- 3x:s 9 and 3x ~ -9

-x:s3andx~-3

- So, -3 :s x:s 3

- Check: x = -4 fails, x = -1

works, x = 3 works, x = 4 fails

• Variables x and y vary directly (or

are "directly proportional") if there

is a constant a such that y = ax

• X and y vary inversely (or are

"inversely proportional") if there

is a constant b such that y = b/x, or

xy=b

• Its value is always between 0 and 1; if the probability of an event is O the event is impossible; ifprobability is 1 the event is certain

• If all outcomes are equally likely the probability of an event is the ratio:

Total numbu of possible outColfUS

• Ex: There are 14 girls and 11 boys in a kindergarten class; one is chosen at random;

what is the probability of choosing a girl? 14 t.: 11 - ~~ = 0.62

9" ses

• The interior angles of any triangle add up to 18C)o

• Congruent angles have the

• The triangle inequality: the sum ofany two sides of

same measure

a triangle must be greater than the third side

• That is, ifa, b, and care the sides of a triangle and

cis the longest side II +

• Right contains a right angle (goo) cro d by a

- Pythagoreanlbeorem: liZ + bZ = cZ where cis thirclline E\P

the hypotenuse, the side opposite the right G\H

angle and II and bthe other two sides

• Ex: A right triangle has sides oflength 5 and

triangle has sides of length %: %: /2x , do= laO

• A SOO - 600 - goo triangle has sides of length • Interior ugles are

%:/3% :2% supplementary: co + eo = 180'"

• Equilateral: 3 sides of the same length 3 angles flo +I"= laoo

of the same measure (8C)o) • Angles comprising a straight

• Isosceles: '!\vo sides with the same length line are also supplementary: opposite two angles of the same measure + IJo =lao-••• + g =18C)o etc

4

- Similar polygons have corresponding angles of the same

so b =3 + 213 =11/3, and the equation of th,e-+-+_

measure and have the same ratio for every pair of perpendicular line is y= 2/3x+ 11/3

corresponding sides

Sample Problems

Multiple Choice

The nth term of a sequence is

defined as 3n + 7 How much

greater is the 40th term than the

32nd?

A 8

B 9

C 24

D.31

E 40

A

B

E

Be sur

to ~

If y- 2 z = 16, how is Z

expressed in terms of y?

A Z= 16y-2

B Z=4y-l

C Z= 4y

C Z= 16y2

D Z= 16y

For the function f(x) = 3x + 5 which of the following is

equivalent to f(a + b)?

3a+ b+ 5 3(a+ b+ 5)

C 3a+ 3b+ 10

D 3a+ 3b+ 5 a+ b+ 5

o

get a good night's sleep and eat a healthy meal prior to '

taking the test

continued:

• It can be expressed in slope-intercept form as

Y = mx + b; the slope m represents the

change in y per unit change in x, or the

vertical "rise" over the horizontal "run"

• The y-intercept b represents the value ofy

when x = O or the point on the y-axis

crossed by the graph

• Parallel lines never meet in the coordinate

plane and have the same slope

• Ex: Find the equation parallel to y =7/4x+ 3

passing through the point (8 5)

- The parallel line must also have slope 7/4 ;

finding its complete equation requires

solving for the intercept b

- Since (8 5) is a point on the line x = 8 and

y = 5 must satisfy its equation

- That is y =7/4 X + b means that 5 =(1/ 4 )(8)

+ b so 5 = 14 + b and b = 5 - 14 = -9;

the equation then is y= 7/4x-9

• Perpendicu1ar lines meet at a right angle; iftwo

lines with slopes mi and "Iz are perpendicular

then ~=-1I"Iz and "Iz =-11~

• Ex: Find the equation of the line

perpendicular to 3x + 2y = 12 that passes

through the point (-1.3)

- First put the given equation in slope­

intercept form: 2y= -3x+ 12 so y= -3/2x+

12 so the slope of the line is -%

- The perpendicular line then must have

slope 2/3 , so its equation is y = 2/3 X + b; to

solve for b use the coordinates of the

o point (-1 3) which must be in the

solution of the equation: 3 = (2/3 )(-1) + b

• Points can be plotted and equations can be graphed on the coordinate plane

• Points are represented as ordered pairs (x, y) where the x value represents the position on the horizontal x-axis and the y value represents the vertical position on the y-axis

• The x-axis and the y-axis meet at the origin represented (0.0)

• Some special formulas for pairs of points (~, YI) and (x 2• Y2):

[Note: it does not matterfor these formulas which point is considered to be (xl' YI) and which is (x 2• Y2)]

• The slope of the line segment connecting two points: m =(YI - Y2) I(Xl - Xz)

• The distance between two points: d = ./(XI - X2) 2+(YI - Y2) 2

• The midpoint between two pointsis the coordinate pair «~+x2)/2) «Yl +Y2)/2)

• Ex: Find the slope distance and midpoint for the points (1 -3) and (-2 1)

- Slope: m = (-3 -1)/(1- (-2» =-4/-3 = 4/3

- Distance: d=j(l-(-2»2+(-3-1)2=

- Midpoint: «1 - 2)/2) «-3 + 1)/2) = (-112,-1)

• A polygon is a closed, plane geometric figure

• Aregu1ar polygon has all sides ofequal length and all angles ofequal measure

• The total sum of interior angles of a polygon can be found by drawing diagonals from one vertex to the other vertices dividing the polygon into triangles, and multiplying the number of triangles by lSOO

• Ex:

sum of interior angles must be lSOO x 4 =

regular polygon any exterior angle is equal to 36()0

- Ex:

360° I 5 = 72°, and interior angles of lBOO - 72° = 1080

Sample problems continued: n

Student Produced Response

1

z

2 If x is an integer, such that 5x + 9 < 55 1 (e) Since the nth term of a sequence is defined as 3n + 7, the 40th term is 3(40) + 7 = and 27 - 3x < 2, what is the value of x? 127, and the 32nd term is 3(32) + 7 = 103; the difference is 127 -103 = 24

3 If 28% of 300 is 24% of x, then what is 2 (B) "Squaring the product ofx and 4" means (4X)2, and "squaring the sum of x and 9"

the value ofx?

4 IfYand x vary inversely, and y = 7 when

used; note also that (x+ 9)2 = (x+ 9)(xv+ 9) = X2 + 18x+ 81, not X2 + 92= X2 + 81

x = 12, what is the value ofywhen x = 8?

5 In the figure below, the circle inscribed

X2 + 3x-108 = 0, so X= (-3 ±v'(3L 4(1)(-108))/2(1) = (-3 ±v'441)/2 = -12 or 9

or x = 9; since -12 is not a possible value for width, x = 9

five fewer than Carmen If all of them 6 (D) Substituting a + b for x in the function producesf(a + answer back into the

many were sold by the one who sold

seconds to make sure you

A

C

2.9; the inequality 5x + 9 < 55 means 5x < 46, so x < 9 1/5; similarly, 27 - 3x < 2, so -3x < -25, and, switching the direction of the

• For x = 8, Y= 84/8 = 2112 or 10.5

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NOTE TO STUDENT

~~ Commonly Misspelled and Confused Words U se this QUICICSTUDY ~ gu ide as a resou rce t o help you Improv e you r t t

score s, but not as a substitut e f o dHigent st udy, home wo rk and cl ass attendance

A l l rlghts re&erved N o part o this publication may be reproduced o r t ransmitled In any form o r 911~ lll,lli ~ ~ II ~~ll1Jl1l1 1 11 1il1 11r Il i l ll

: SAT Verbal by an y means , electronic or mechanical, i~tuding phOtocop y r ecording or any informati(l n

o s tora g e and retrleval system, w ithout wrtnen permiSSion from the pu bl isher

6