US College SAT-Test Math

8/9/2019 US College SAT-Test Math

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US Sat-Test Math

SAT Math Practice Questions (2015)

Multiple Choice

Information:

1.)A special lottery is to be held to select the student who will live in the only deluxe room in a

dormitory. There are seniors, juniors, and sophomores who applied. Each

senior's name is placed in the lottery times; each junior's name, times; and each sophomore's

name, time. hat is the probability that a senior's name will be chosen!

• "A)

• "#)

• "$)

• "%)

• "E)



To determine the probability that a senior's name will be chosen, you must determine the totalnumber o& seniors' names that are in the lottery and divide this number by the total number o&

names in the lottery. ince each senior's name is placed in the lottery times, there are

seniors' names. (iewise, there are juniors' names and

sophomores' names in the lottery. The probability that a senior's name will be

chosen is .

*.)

The table above shows the temperatures, in de+rees ahrenheit, in a city in -awaii over a onewee

period. /& represents the median temperature, represents the temperature that occurs most

o&ten, and represents the avera+e "arithmetic mean) o& the seven temperatures, which o& the

&ollowin+ is the correct order o& , , and !

• "A)

• "#)

• "$)

• "%)

• "E)

To determine the correct order o& , , and , it is help&ul to &irst place the seven temperatures inascendin+ order as shown below.

The median temperature is the middle temperature in the ordered list, which is , so .

The temperature that occurs most o&ten, or the mode, is , so . To determine the avera+e,

you can add the seven numbers to+ether and divide by . -owever, you can determine the

relationship between the avera+e and the median by inspection. The three numbers +reater than

are closer to than are the three numbers smaller than . There&ore, the avera+e o& the seven

numbers will be less than . The correct order o& , , and is .



0.)

The projected sales volume o& a video +ame cartrid+e is +iven by the &unction

where is the number o& cartrid+es sold, in thousands; is the price per cartrid+e, in dollars; and

is a constant. /& accordin+ to the projections, cartrid+es are sold at per

cartrid+e, how many cartrid+es will be sold at per cartrid+e!

• "A)

• "#)

• "$)

• "%)

• "E)

or cartrid+es sold at per cartrid+e, "since is the number o&

cartrid+es sold, in thousands) and . ubstitutin+ into the euation yields

.

olvin+ this euation &or yields

.

ince is a constant, the &unction can be written as . To determine how many

cartrid+es will be sold at per cartrid+e, you need to evaluate .

ince is +iven in thousands, there will be cartrid+es sold at per cartrid+e.

2.)



/n the plane above, line contains the points and . /& line

"not shown) contains the point and is perpendicular to , what is an euation o& !

•"A)

• "#)

• "$)

• "%)

• "E)

ince the coordinates o& two points on line are +iven, the slope o& is . (ine ,

which is perpendicular to will have a slope o& since slopes o& perpendicular lines are

ne+ative reciprocals o& each other. The euation o& can be written as . ince line

also contains point , it &ollows that .

There&ore, an euation o& line is .

3.)

/& two sides o& the trian+le above have len+ths and , the perimeter o& the trian+le could bewhich o& the &ollowin+!

.

.

.



• "A) only

• "#) only

• "$) only

• "%) and only

• "E) , , and

/n uestions o& this type, statements , , and should each be considered independently o& theothers. 4ou must determine which o& those statements could be true.

• tatement cannot be true. The perimeter o& the trian+le cannot be , since the sum o& the

two +iven sides is without even considerin+ the third side o& the trian+le.

• $ontinuin+ to wor the problem, you see that in , i& the perimeter were , then the third

side o& the trian+le would be , or . A trian+le can have side len+ths o& ,

, and . o the perimeter o& the trian+le could be .

• inally, consider whether it is possible &or the trian+le to have a perimeter o& . /n this

case, the third side o& the trian+le would be . The third side o& this

trian+le cannot be , since the sum o& the other two sides is not +reater than . #y theTrian+le /neuality, the sum o& the len+ths o& any two sides o& a trian+le must be +reater than

the len+th o& the third side. o the correct answer is only.

5.)

/& and , what is the value o& !

• "A)

• "#)

• "$)

• "%)

• "E)

ince can be written as and can be written as , the le&t side o& the euation is

. ince , the value o& is .



6.)

/& is divisible by , , and , which o& the &ollowin+ is also divisible by these numbers!

• "A)

• "#)

• "$)• "%)

• "E)

ince is divisible by , , and , must be a multiple o& , as is the least common

multiple o& , , and . ome multiples o& are , , , , and .

• /& you add two multiples o& , the sum will also be a multiple o& . or example,

and are multiples o& and their sum, , is also a multiple o& .

• /& you add a multiple o& to a number that is not a multiple o& , the sum will not be a

multiple o& . or example, is a multiple o& and is not. Their sum, , is

not a multiple o& .

• The uestion ass which answer choice is divisible by , , and ; that is, which answer

choice is a multiple o& . All the answer choices are in the &orm o& 7 plus a number.7

8nly choice "%), , has added to a multiple o& . The sum o& and is

also a multiple o& , so the correct answer is choice "%).

u!"er and #perations$

The %ollo&in' concepts are coered on the SAT$

• Arithmetic word problems "includin+ percent, ratio, and proportion)

• 9roperties o& inte+ers "even, odd, prime numbers, divisibility, etc.)

• :ational numbers

• ets "union, intersection, elements)

• $ountin+ techniues

• euences and series "includin+ exponential +rowth)

• Elementary number theory



/nte+ers, 8dd and Even umbers, 9rime umbers, %i+its

nte'ers< . . . , 2, 0, *, 1, =, 1, *, 0, 2, . . ." Note: zero is neither positive nor negative.)

Consecutie nte'ers< /nte+ers that &ollow in seuence; &or example, **, *0, *2, *3. $onsecutiveinte+ers can be more +enerally represented by n, n >1, n > *, n > 0, . . .

#dd nte'ers< . . . , 6, 3, 0, 1, 1, 0, 3, 6, . . . , * k > 1, . . . , where k is an inte+er

*en nte'ers< . . . , 5, 2, *, =, *, 2, 5, . . . , * k , . . . , where k is an inte+er " Note: zero is an even integer.)

Pri!e u!"ers< *, 0, 3, 6, 11, 10, 16, 1?, . . ."ote: 1 is not a prime and 2 is the only even prime.)

+i'its< =, 1, *, 0, 2, 3, 5, 6, @, ?" Note: the units digit and the ones digit refer to the same digit in a number. For example, in the

number 12, the is !alled the units digit or the ones digit .)

9ercent /ncrease and %ecrease

Percent

9ercent means hundredths, or number out o& 1==. or example, 2= percent means

.

Pro"le! 1< /& the sales tax on a 0=.== item is 1.@=, what is the sales tax rate!

Solution<

.

Percent ncrease,+ecrease

Pro"le! 2< /& the price o& a computer was decreased &rom 1,=== to 63=, by what percent was the price decreased!

Solution< The price decrease is *3=. The percent decrease is the value o& n in the euation

. The value o& n is *3, so the price was decreased by *3B.

ote<

Avera+e peed

Pro"le!< CosD traveled &or * hours at a rate o& 6= ilometers per hour and &or 3 hours at a rate o& 5=

ilometers per hour. hat was his avera+e speed &or the 6hour period!Solution< /n this situation, the avera+e speed is



.

The total distance was

.

The total time was 6 hours. Thus, the avera+e speed was

.

ote< /n this example, the avera+e speed over the 6hour period is not the avera+e o& the two +ivenspeeds, which would be 53 ilometers per hour.

euences

Two common types o& seuences that appear on the AT are arithmetic and +eometric seuences.

An arith!etic seuence is a seuence in which successive terms di&&er by the same constantamount.or example< 0, 3, 6, ?, . . . is an arithmetic seuence.

A 'eo!etric seuence is a seuence in which the ratio o& successive terms is a constant.or example< *, 2, @, 15, . . . is a +eometric seuence.

A seuence may also be de&ined usin+ previously de&ined terms. or example, the &irst term o& aseuence is *, and each successive term is 1 less than twice the precedin+ term. This seuencewould be *, 0, 3, ?, 16, . . .

8n the AT, explicit rules are +iven &or each seuence. or example, in the seuence above, you

would not be expected to now that the 5th term is 00 without bein+ +iven the &act that each term is1 less than twice the precedin+ term. or seuences on the AT, the &irst term is never re&erred to asthe eroth term.

1.)

hat is the result when is rounded to the nearest thousand and then expressed inscienti&ic notation!

• "A)

• "#)

• "$)

• "%)

•"E)



/n the number , the thousands di+it is and the hundreds di+it is . ince the

hundreds di+it o& is more than , roundin+ to the nearest thousand +ives .

hen is expressed in scienti&ic notation, the result is , which is choice"%).

*.)

The positive inte+er is not divisible by . The remainder when is divided by and the

remainder when is divided by are each eual to . hat is !

• "A)

• "#)

• "$)

• "%)

• "E) /t cannot be determined &rom the in&ormation +iven.

ince the positive inte+er leaves nonero remainder when divided by , it can be written as

, where is a nonne+ative inte+er and is eual to one o& the values , , , ,

, or . ince leaves the same nonero remainder when divided by , it can be written as

, where is a nonne+ative inte+er and has the same value. /t is also true that

, which can be written as . ince

, it &ollows that and leave the same remainder when

divided by .

e now that is eual to one o& the values , , , , , or . e can see which o& these

values &or has the property that and leave the same remainder when divided by . /&

, then , which leaves remainder when divided by . Thus, is a possible value

&or . #ut we are not done yet; since one o& the answer choices is "E), /t cannot be determined&rom the in&ormation +iven, we must continue and chec , , , , and as possible values &or

.

/& , then , which leaves remainder when divided by . /& , then

, which leaves remainder when divided by . /& , then , which

leaves remainder when divided by . /& , then , which leaves remainder

when divided by . /& , then , which leaves remainder when divided

by .There&ore, the only possible value o& is , which is choice "A).



0.)

The set consists o& all multiples o& 5. hich o& the &ollowin+ sets are contained within !

. The set o& all multiples o& 0

. The set o& all multiples o& ?

. The set o& all multiples o& 1*

• "A) only

• "#) only

• "$) only

• "%) and only

• "E) and only

To solve this problem, consider the set in each o& , , and separately and determine whether

that set is contained within set .

or , consider whether the set o& all multiples o& is contained within set . This will be the case

i& every multiple o& is also a multiple o& . ince is a multiple o& but is not a multiple o&

, it &ollows that every multiple o& is not necessarily a multiple o& . Thus the set o& all

multiples o& is not contained within .

or , consider whether the set o& all multiples o& is contained within set . This will be the

case i& every multiple o& is also a multiple o& . ince is a multiple o& but is not a

multiple o& , it &ollows that every multiple o& is not necessarily a multiple o& . Thus the set o&

all multiples o& is not contained within .

or , consider whether the set o& all multiples o& is contained within set . This will be the

case i& every multiple o& is also a multiple o& . Every multiple o& can be written as

&or some inte+er , and a number is a multiple o& i& it can be written as &or some inte+er

. /& is any multiple o& , then ; since is an inte+er, it &ollows that

can be written as , where is an inte+er, and so is also a multiple o& . Thus the

set o& all multiples o& is contained within .

There&ore, o& the three sets +iven in , , and , only the set in is contained within . Thus

the correct answer is "$).



2.)

At the be+innin+ o& *==5, both Alan and %ave were taller than #oris, and #oris was taller than

$harles. %urin+ the year, Alan +rew inches, #oris and %ave each +rew inches, and $harles

+rew inches. 8& the &ollowin+, which could 8T have been true at the be+innin+ o& *==6!

• "A) Alan was shorter than #oris.

• "#) Alan was shorter than $harles.

• "$) #oris was shorter than %ave.

• "%) %ave was shorter than Alan.

• "E) %ave was shorter than $harles.

$onsider the choices in turn. At the be+innin+ o& *==5, Alan was taller than #oris; durin+ the year,

Alan +rew inches and #oris +rew inches. ince Alan +rew less than #oris, it is possible thatAlan was shorter than #oris at the be+innin+ o& *==6. o choice "A) is not the correct answer.

At the be+innin+ o& *==5, Alan was taller than #oris, who was taller than $harles; thus, Alan was

taller than $harles. %urin+ the year, Alan +rew inches and $harles +rew inches. ince Alan+rew less than $harles, it is possible that Alan was shorter than $harles at the be+innin+ o& *==6. ochoice "#) is not the correct answer.

At the be+innin+ o& *==5, %ave was taller than #oris; durin+ the year, #oris and %ave each +rewinches. Thus #oris was still shorter than %ave at the be+innin+ o& *==6. o choice "$) is not thecorrect answer.

At the be+innin+ o& *==5, Alan and %ave were each taller than #oris, but we cannot determinewhether Alan was shorter than %ave or %ave was shorter than Alan, nor what the di&&erence in their

hei+hts was. o even thou+h %ave +rew inches durin+ the year while Alan +rew only inches,it is possible that %ave was shorter than Alan at the be+innin+ o& *==6. o choice "%) is not thecorrect answer.

At the be+innin+ o& *==5, %ave was taller than #oris, who was taller than $harles; thus, %ave was

taller than $harles. %urin+ the year, %ave +rew inches and $harles +rew only inches, so %averemained taller than $harles. Thus it could not be true that %ave was shorter than $harles at the

be+innin+ o& *==6. There&ore, choice "E) is the correct answer.

3.)

/& is an inte+er and i& is a positive inte+er, which o& the &ollowin+ must also be a positiveinte+er!

• "A)

• "#)

• "$)

• "%)

• "E)



/& is an inte+er and i& is a positive inte+er, then is the sum o& two

inte+ers, and must there&ore be an inte+er. ince is a positive inte+er, it &ollows that .

The inte+er will be positive i& . ince , it &ollows that , and so

. There&ore, i& is a positive inte+er, must also be a positive inte+er.

/& is a positive inte+er, then the expressions in the other choices must be inte+ers, but they neednot be positive inte+ers<

• or , i& , then , which is not positive.

• or , i& , then , which is not positive.

• or , i& , then , which isnot positive.

• or , i& , then ,which is not positive.

There&ore, the correct answer is "#).

Al'e"ra and .unctions

The %ollo&in' concepts are coered on the test$

• ubstitution and simpli&yin+ al+ebraic expressions

• 9roperties o& exponents

• Al+ebraic word problems

• olutions o& linear euations and ineualities

• ystems o& euations and ineualities

• Fuadratic euations

• :ational and radical euations

• Euations o& lines• Absolute value

• %irect and inverse variation

• $oncepts o& al+ebraic &unctions

• ewly de&ined symbols based on commonly used operations



.actorin'

4ou may need to apply these types o& &actorin+<

x* > *x G x "x > *)

x* H 1 G "x > 1) "x H 1)

x* > *x > 1 G "x > 1) "x > 1) G "x > 1)*

*x* > 3x H 0 G "*x H 1) "x > 0)

.unctions

A &unction is a relation in which each element o& the domain is paired with exa!tly one element o&the ran+e. 8n the AT, unless otherwise speci&ied, the domain o& any &unction I is assumed to bethe set o& all real numbers x &or which I"x) is a real number.

or example, i& I"x) G , the domain o& I is all real numbers +reater than or eual to H*. or

this &unction, 12 is paired with 2, since I"12) G G G 2.

ote$ the symbol represents the positive, or principal, suare root. or example, G 2, notJ2.

*/ponents

4ou should be &amiliar with the &ollowin+ rules &or exponents on the AT.

or all values o& a, b, x, y<

or all values o& a, b, x K =, y K =<

Also, . or example, .

ote< or any nonero number x, it is true that .

ariation

+irect ariation< The variable y is directly proportional to the variable x i& there exists a noneroconstant k such that y " kx.

nerse ariation< The variable y is inversely proportional to the variable x i& there exists a nonero

constant k such that or xy " k .



A"solute alue

The absolute value o& x is de&ined as the distance &rom x to ero on the number line. The absolutevalue o& x is written as L xL. or all real numbers x<

1.)

At a snac bar, a customer who orders a small soda +ets a cup containin+ ounces o& soda, where

is at least but no more than . hich o& the &ollowin+ describes all possible values o&!

• "A)

• "#)

• "$)

• "%)

• "E)

/& is at least but no more than , then describes all possible values o&

. These ineualities are euivalent to , or

. #y the de&inition o& absolute value, the latter ineualities are euivalent to

the ineuality . There&ore, , which is choice "E), describes all

possible values o& .



*.)

/& and are real numbers and the suare o& is eual to the suare root o& , which o& the&ollowin+ must be true!

.

.

.

• "A) only

• "#) and only

• "$) and only

• "%) and only

• "E) , , and

/t is +iven that the suare o& is eual to the suare root o& . /n symbols, this means ,

or . uarin+ both sides o& the latter euation +ives . There&ore, statement ,

, must be true.

ince has a suare root, must be a nonne+ative number. There&ore, statement , ,must be true.

tatement need not be true. or example, i& and , then it is true that the suare

o& is eual to the suare root o& , yet statement , , is nottrue.

Thus, only and must be true, which is choice "#).



0.)

or the &irst part o& his bie trip, %a+ rode down a hill at miles per hour &or hours. or the resto& his trip, %a+ rode up a hill at hal& that speed &or twice as lon+. hat was %a+'s avera+e speed, inmiles per hour, &or his entire trip!

• "A)

• "#)

• "$)

• "%)

• "E)

or the &irst part o& his trip, %a+ rode &or hours at miles per hour, so the distance he rode on

this part o& the trip was miles. or the rest o& his trip, he rode at hal& the speed o& the &irst part o&

the trip, and &or twice as lon+. There&ore, he rode at a speed o& miles per hour &or hours, and

thus he covered a distance o& miles in the rest o& his trip. %a+'s avera+e speed, inmiles per hour, &or the entire trip is eual to the total distance o& his trip divided by the total time o&

the trip. The total distance o& his trip was miles, and the total time o& his trip was

hours. There&ore, %a+'s avera+e speed &or the entire trip was miles perhour, which is choice "#).

2.)

A convenience store sells small bottles o& juice &or each and lar+e bottles o& juice &or each.

$iara bou+ht bottles o& juice at this store and paid . ome o& the bottles o& juice that she bou+ht were lar+e, and the rest were small. -ow many small bottles o& juice did $iara buy!

• "A) Two

• "#) Three

• "$) our

• "%) ive

• "E) ix

(et be the number o& lar+e bottles o& juice $iara bou+ht, and let be the number o& small

bottles o& juice she bou+ht. ince $iara bou+ht a total o& bottles o& juice, the euation



is true.

The small bottles o& juice cost dollars each, and the lar+e bottles o& juice cost dollars each.

ince $iara bou+ht lar+e bottles o& juice and small bottles o& juice and paid a total o&

dollars, the euation is true.

ince , it is true that . ubstitutin+ this expression &or in

+ives . impli&yin+ +ives , or . Thus, $iara bou+ht sixsmall bottles o& juice, which is choice "E).

3.)

/& , which o& the &ollowin+ must be true!

• "A)• "#)

• "$)

• "%)

• "E) or

The ineualities are euivalent to the ineualities ,

or . %ividin+ each term o& the latter ineualities by the positive number does not

chan+e the direction o& the ineualities, so is euivalent to . /& is addedto each term o& , the euivalent ineualities result. There&ore, i&

, it must be true that , which is choice "$).



eo!etry and Measure!ent


• Area and perimeter o& a poly+on

• Area and circum&erence o& a circle

• Molume o& a box, cube, and cylinder

• 9ytha+orean Theorem and special properties o& isosceles, euilateral, and ri+ht trian+les

• 9roperties o& parallel and perpendicular lines

• $oordinate +eometry

• Neometric visualiation

• lope

• imilarity

• Trans&ormations

eo!etric .i'ures

i+ures that accompany problems are intended to provide in&ormation use&ul in solvin+ the problems. They are drawn as accurately as possible EO$E9T when it is stated in a particular problem that the &i+ure is not drawn to scale. /n +eneral, even when &i+ures are not drawn to scale,the relative positions o& points and an+les may be assumed to be in the order shown. Also, linese+ments that extend throu+h points and appear to lie on the same line may be assumed to be on thesame line. A point that appears to lie on a line or curve may be assumed to lie on the line or curve.

The text 7ote< i+ure not drawn to scale7 is included with the &i+ure when de+ree measures maynot be accurately shown and speci&ic len+ths may not be drawn proportionally. The &ollowin+examples illustrate what in&ormation can and cannot be assumed &rom &i+ures.

*/a!ple 1

ince and are line se+ments, an+les #$% and &$' are vertical an+les. There&ore, you canconclude that x G y. Even thou+h the &i+ure is drawn to scale, you should 8T mae any otherassumptions without additional in&ormation. or example, you should 8T assume that #$ G $& orthat the an+le at vertex ' is a ri+ht an+le even thou+h they mi+ht loo that way in the &i+ure.



*/a!ple 2$

A uestion may re&er to a trian+le such as #%$ above. Althou+h the note indicates that the &i+ure isnot drawn to scale, you may assume the &ollowin+ &rom the &i+ure<

#%& and &%$ are trian+les.

& is between # and $ .

#, &, and $ are points on a line. The len+th o& is less than the len+th o& .

The measure o& an+le #%& is less than the measure o& an+le #%$ .

4ou may not assume the &ollowin+ &rom the &i+ure<

The len+th o& is less than the len+th o& .

The measures o& an+les %#& and %&# are eual.

The measure o& an+le #%& is +reater than the measure o& an+le &%$ .

An+le #%$ is a ri+ht an+le.

Properties o% Parallel ines

1./& two parallel lines are cut by a third line, the alternate interior an+les are con+ruent. /n the &i+ureabove,

! G x and ( G d

*. /& two parallel lines are cut by a third line, the correspondin+ an+les are con+ruent. /n the &i+ure,

a G (, b G x, ! G y, and d G z

0. /& two parallel lines are cut by a third line, the sum o& the measures o& the interior an+les on thesame side o& the transversal is 1@= . /n the &i+ure,��

! > ( G 1@= and d > x G 1@=



An'le 3elationships

1. The sum o& the measures o& the interior an+les o& a trian+le is 1@=P. /n the &i+ure above,

x G 6= because 5= > 3= > x G 1@=

*. hen two lines intersect, vertical an+les are con+ruent. /n the &i+ure,

y G 3=

0. A strai+ht an+le measures 1@=P. /n the &i+ure,

z G 10= because z > 3= G 1@=

2. The sum o& the measures o& the interior an+les o& a poly+on can be &ound by drawin+ alldia+onals o& the poly+on &rom one vertex and multiplyin+ the number o& trian+les &ormed by 1@=P.

ince this poly+on is divided into 0 trian+les, the sum o& the measures o&

its an+les is 0 x 1@=P, or 32=P.

Qnless otherwise noted in the AT, the term 7poly+on7 will be used to mean a convex poly+on, thatis, a poly+on in which each interior an+le has a measure o& less than 1@=P.

A poly+on is 7re+ular7 i& all its sides are con+ruent and all its an+les are con+ruent.

Side 3elationships

1. 9ytha+orean theorem< /n any ri+ht trian+le, a* > b* G !*, where ! is the len+th o& the lon+est sideand a and b are the len+ths o& the two shorter sides.

To &ind the value o& x, use the 9ytha+orean Theorem.

x* G 0* > 2*

x* G ? > 15



*. /n any euilateral trian+le, all sides are con+ruent and all an+les are con+ruent.

#ecause the measure o& the unmared an+le is 5=P, the measures o& all an+les o& the trian+le areeual; and, there&ore, the len+ths o& all sides o& the trian+le are eual< x G y G 1=.

0. /n an isosceles trian+le, the an+les opposite con+ruent sides are con+ruent. Also, the sidesopposite con+ruent an+les are con+ruent. /n the &i+ures below, a G b and x G y.

2. /n any trian+le, the lon+est side is opposite the lar+est an+le, and the shortest side is opposite thesmallest an+le. /n the &i+ure below, a R b R !.

3. Two poly+ons are similar i& and only i& the len+ths o& their correspondin+ sides are in the sameratio and the measures o& their correspondin+ an+les are eual.

/& poly+ons #%$&'F and )*I+- are similar, then and are correspondin+ sides, so that



. There&ore, x G ? G *I .

ote$ means the line se+ment with endpoints # and F , and #F means the len+th o& .

Area and Peri!eter

3ectan'les

Area o& a rectan+le G len+th S width G S (

9erimeter o& a rectan+le G *" > () G * > *(

Circles

Area o& a circle G r * "where r is the radius)

$ircum&erence o& a circle G * r G d "where d is the diameter)

Trian'les

Area o& a trian+le G "base S altitude)

9erimeter o& a trian+le G the sum o& the len+ths o& the three sides

Trian+le ineuality< The sum o& the len+ths o& any two sides o& a trian+le must be +reater than thelen+th o& the third side.

olu!e

Molume o& a rectan+ular solid "or cube) G S ( S h

"l is the len+th, ( is the width, and h is the hei+ht)

Molume o& a ri+ht circular cylinder G r 2h "r is the radius o& the base, and h is the hei+ht)

Be familiar with the formulas that are provided in the Reference Information included with the

test directions.

Coordinate eo!etry

1. /n uestions that involve the x and yaxes, xvalues to the ri+ht o& the yaxis are positive and x



values to the le&t o& the yaxis are ne+ative. imilarly, yvalues above the xaxis are positive and y

values below the xaxis are ne+ative. /n an ordered pair " x, y), the xcoordinate is written &irst. 9oint / in the &i+ure above appears to lie at the intersection o& +ridlines. rom the &i+ure, you canconclude that the xcoordinate o& / is H* and the ycoordinate o& / is 0. There&ore, the coordinateso& point / are "H*, 0). imilarly, you can conclude that the line shown in the &i+ure passes throu+hthe point with coordinates "H*, H1) and the point "*, *).

*. lope o& a line G

A line that slopes upward as you +o &rom le&t to ri+ht has a positive slope. A line that slopesdownward as you +o &rom le&t to ri+ht has a negative slope. A horiontal line has a slope o& ero.The slope o& a vertical line is unde&ined.

9arallel lines have the same slope. The product o& the slopes o& two perpendicular lines is H1,

provided the slope o& each o& the lines is de&ined. or example, any line perpendicular to line

above has a slope o& .

The euation o& a line can be expressed as y G mx > b, where m is the slope and b is the yintercept.

ince the slope o& line is , the euation o& line can be expressed as , since

the point "H*, 1) is on the line, x G H* and y G 1 must satis&y the euation. -ence, so

and the euation o& line is .

0. A uadratic &unction can be expressed as y G a " x H h)* > k , where the vertex o& the parabola is atthe point "h, k ) and a =. /& a K =, the parabola opens upward; and i& a R =, the parabola opensdownward.



The parabola above has its vertex at "H*, 2). There&ore, h G H* and k G 2. The euation can be

represented by y G a " x > *)* > 2. ince the parabola opens downward, we now that a R =. To &indthe value o& a, we also need to now another point on the parabola. ince we now the parabola

passes throu+h the point "1, 1), x G 1 and y G 1 must satis&y the euation.

-ence, 1 G a"1 > *)* > 2,

so .

There&ore, an euation &or the parabola is .

1.)

/n the &i+ure above, side o& lies on line . hat is the value o& !

• "A)

• "#)

• "$)

• "%)• "E) /t cannot be determined &rom the in&ormation +iven.



ince the an+les o& are o& measures , , and , and the sum o& the measures o&

the an+les o& a trian+le is , it must be true that . There&ore, ,

or .

ince side o& lies on line , the an+le measures and must add up to

. That is, . ince , it &ollows that . There&ore, ,which is choice "A).

*.)

/n the &i+ure above, is a radius o& the circle, is tan+ent to the circle at point , and the

area o& is . hat is the area o& the circle!

• "A)

• "#)

• "$)

• "%)

• "E)

To &ind the area o& the circle, we need to &ind its radius. is a radius o& the circle. ince is

tan+ent to the circle at point , it &ollows that is a ri+ht an+le and, there&ore, is a

ri+ht trian+le. The area o& is , so area o& . ince

, it &ollows that . There&ore, the circle has radius , and the area o& the

circle is thus which is choice "$).



0.)

The rectan+ular solid above has sur&ace area suare &eet. hat is the volume, in cubic &eet, o&the solid!

• "A)

• "#)

• "$)

• "%)

• "E)

The sur&ace area o& the rectan+ular solid is the sum o& the areas o& its six &aces. The rectan+ularsolid has two &aces with area suare &eet each, two &aces with area suare &eet each, and

two &aces with area suare &eet each. There&ore, its sur&ace area is suare

&eet. ince the sur&ace area is +iven to be suare &eet, it &ollows that

, and . Thus the width o& the rectan+ular solid is &eet.

The volume o& the solid is +iven by the product o& its len+th, width, and hei+ht. There&ore, the

volume is " &eet)" &eet)" &eet), or cubic &eet. Thus the correct answer is "$).

2.)

/n the &i+ure above, the circle has center and the &our shaded semicircles each have area .hat is the circum&erence o& the circle!

• "A)

• "#)

• "$)

• "%)

• "E)



The circum&erence o& the circle is eual to , where is the diameter o& the circle. Thediameter o& the circle is the sum o& the diameters o& the &our semicircles. The &our semicircles have

eual area, so they must also have eual radii. (et be the radius o& one o& the semicircles. Each o&

the &our semicircles has area , so . olvin+ &or +ives . There&ore, the

sum o& the diameters o& the &our semicircles is . Thus is also thediameter o& the circle, and the circum&erence o& the circle is , which is choice "%).

3.)

/n the &i+ure above, is similar to . The ratio o& to is to . hat

is the ratio o& the area o& to the area o& !

• "A) to

• "#) to

• "$) to

• "%) to

• "E) to

To compare the areas o& and , it is help&ul to draw an altitude o& &rom

point and an altitude o& &rom point , as shown in the &i+ure above. Then area o&

and area o& . /t is +iven that . o to

&ind , you need to &ind the value o& . 4ou can use the &act thatcorrespondin+ altitudes o& similar trian+les are in the same ratio as correspondin+ sides to conclude



that . Alternatively, you could notice that and each have a ri+ht an+le

and share a common an+le at to conclude that they are similar. ince and are

correspondin+ sides o& and , respectively, it &ollows that . ince

and are also correspondin+ sides o& and , it &ollows that the similarity ratio o&

these two trian+les is also and thus .

There&ore, . Thus the ratio

o& the area o& to the area o& is to , which is choice "$).

+ata Analysis4 Statistics4 and Pro"a"ility


• %ata interpretation "tables and +raphs)

• %escriptive statistics "mean, median, and mode)

• 9robability

Measures o% Center

An aera'e is a statistic that is used to summarie data. The most common type o& avera+e is thearith!etic !ean. The avera+e "arithmetic mean) o& a list o& n numbers is eual to the sum o& thenumbers divided by n.

or example, the mean o& *, 0, 3, 6, and 10 is eual to

.

hen the avera+e o& a list o& n numbers is +iven, the sum o& the numbers can be &ound. orexample, i& the avera+e o& six numbers is 1*, the sum o& these six numbers is 1* x 5, or 6*.

The !edian o& a list o& numbers is the number in the middle when the numbers are ordered &rom+reatest to least or &rom least to +reatest. or example, the median o& 0, @, *, 5, and ? is 5 becausewhen the numbers are ordered, *, 0, 5, @, ?, the number in the middle is 5. hen there is an evennumber o& values, the median is the same as the mean o& the two middle numbers. or example, themedian o& 5, @, ?, 10, 12, and 15 is the mean o& ? and 10, which is 11.

The !ode o& a list o& numbers is the number that occurs most o&ten in the list. or example, 6 is themode o& *, 6, 3, @, 6, and 1*. The list *, 2, *, @, *, 2, 6, 2, ?, and 11 has two modes, * and 2.

ote< 8n the AT, the use o& the word average re&ers to the arithmetic mean and is indicated by



7avera+e "arithmetic mean).7 The exception is when a uestion involves avera+e speed. Fuestionsinvolvin+ median and mode will have those terms stated as part o& the uestion's text.

Pro"a"ility

9robability re&ers to the chance that a speci&ic outcome can occur. hen outcomes are eually

liely, probability can be &ound by usin+ the &ollowin+ de&inition<

or example, i& a jar contains 10 red marbles and 6 +reen marbles, the probability that a marbleselected &rom the jar at random will be +reen is

/& a particular outcome can never occur, its probability is =. /& an outcome is certain to occur, its

probability is 1. /n +eneral, i& p is the probability that a speci&ic outcome will occur, values o& p &allin the ran+e = U p U 1. 9robability may be expressed as either a decimal, a &raction, or a ratio.

1.)

The bar +raph above shows the number o& employees at $ompany &or each o& the years &rom1??5 throu+h *===. 8ver which o& the &ollowin+ periods was the avera+e rate o& increase in the

number o& employees at $ompany +reatest!

• "A) rom 1??5 throu+h 1??@

• "#) rom 1??5 throu+h 1???

• "$) rom 1??6 throu+h 1???

• "%) rom 1??@ throu+h 1???

• "E) rom 1??@ throu+h *===

4ou can estimate the rate o& increase &or each o& the &ive choices to &ind which is the +reatest.



rom 1??5 throu+h 1??@, $ompany +ained rou+hly employees over years, &or an

avera+e rate o& increase o& about employees per year.

rom 1??5 throu+h 1???, $ompany +ained rou+hly employees over years, &or an


rom 1??6 throu+h 1???, $ompany +ained rou+hly employees over years, &or an


rom 1??@ throu+h 1???, $ompany +ained rou+hly employees over year, &or a

rate o& increase o& about employees per year.

rom 1??@ throu+h *===, $ompany +ained rou+hly employees over years, &or an


8& the choices, the avera+e rate o& increase in the number o& employees at $ompany was

+reatest &or the period &rom 1??@ throu+h 1???.There is also a uicer way to &i+ure out the answer. $hoice "%), &rom 1??@ throu+h 1???, is thesin+le lar+est yearly increase in the number o& employees. There&ore, the avera+e rate o& increase

&or any other time period must be less than that &or this year period.

*.)

The amount that a plumber char+es &or a service call that is hours lon+ is shown in the +raphabove. The char+es &or a service call consist o& an initial amount plus a char+e &or each hour o&wor. Accordin+ to the +raph, what is the hourly char+e!

• "A)

• "#)

• "$)

• "%)

• "E)



The plumberVs char+e &or hour is the sum o& the initial amount and the hourly char+e, and the

plumber's char+e &or hours is the sum o& the initial amount and twice the hourly char+e. -ence

the hourly char+e is eual to the char+e &or hours, or , minus the char+e &or hour, ,

which is eual to .

0.)

The scatterplot above shows the area, in suare miles, versus the population, in thousands, &or six

states. -ow many o& these states have a population under and an area over

suare miles!

• "A) 8ne• "#) Two

• "$) Three

• "%) our

• "E) ive

This is a scatterplot in which each point represents a state. The is the area in

suare miles, and the is the population in thousands. or example, the le&tmostand lowest point has approximate coordinates " ). :emember that the population in

the plot is in thousands and that is eual to thousand. (oo &or points

that have an +reater than and a less than

. There are three points with an +reater than ; two o& these

three points have a less than .



2.)

The scatterplot above shows the enrollment in 1?@= a+ainst the enrollment in *=== &or twentycolle+es. hich o& the labeled points represents the colle+e that had the smallest percent increase inenrollment &rom 1?@= to *===!

• "A)

• "#)

• "$)

• "%)

• "E)

This is a scatterplot in which each point represents a colle+e. The is the

enrollment in 1?@=, and the is the enrollment in *===. 9oint has approximate

coordinates , so the colle+e represented by this point had an enrollment that

increased only between 1?@= and *===. All the otherlabeled points represent colle+es with +reater percent increases in enrollment.



3.)

At the $ross treet $onvenience tore between 11 a.m. and 1 p.m. on riday, a total o& itemswere purchased. The +raph above shows the distribution o& the number o& items purchased in each

o& &ive cate+ories. /n terms o& , how many more sandwiches than pastries were purchased!

• "A)

• "#)

• "$)

• "%)

• "E)

8& the items purchased, were sandwiches and were pastries. o& is eual

to , and o& is eual to . There&ore, , or moresandwiches than pastries were purchased.

Documents

US College SAT-Test Math