GMAT Exam Preparation: Comprehensive Guide with Solutions and Strategies, Exams of Mathematics

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GMAT Exam with complete solutions 2024

Sections k- kANSWER ✓✓ -1. kAnalytical kwriting k(analysis kof kan kargument)- k 30 kmin. kType kessay kon kkeyboard. kScored kon ka kscare kof k0-6.

2. kIntegrated kreasoning k(multi-source kreasoning, ktable kanalysis, kgraphics kinterpretation, ktwo-part kanalysis)- k 30 kmin. kGet kcomputer kcalculator. kMultiple kchoice. k 12 kquestions. kTake kbreak kafter.
3. kQuantitative k(problem ksolving kand kdata ksufficiency)- k 75 kmin. kNo kcalculator. kMultiple kchoice. kScored kon ka kscale kof k0-60. k 37 kquestions. kTake kbreak kafter. k
4. kVerbal k(reading kcomprehension, kcritical kreasoning, ksentence kcorrection) k- k 75 kmin. kMultiple kchoice. kScored kon ka kscale kof k0-60. k 41 kquestions. No knotes/scratch kpaper kallowed, kbut kthey kgive ka kdry kerase kboard kand kmarker. Total kGMAT kscore kis kbetween k 200 kand k800, kbased kon kverbal kand kquantitative ksections. kDetermined kby k# kquestions kanswered, k# kanswered kcorrectly/incorrectly, kand kthe klevel kof kdifficulty kand kother kcharacteristics kof kthe kquestions. kIf kyou kanswer keasy kquestions kcorrectly, kget ka kchance kto kanswer kharder kones, kmaking ka khigher kscore kpossible. kAWA kand kIntegrated kreasoning kscores kreported kseparately kfrom kthe krest kof kthe ktest kand khave kno keffect kon kverbal, kquantitative, kor ktotal kscore. General kTest-taking ktips k- kANSWER ✓✓ -1. kUse ktime kwisely- k 1 k3/4 kmin kper kverbal kQ, k 2 kmin kper kquantitative kQ. k 2 k1/2 kmin kper kIR kQ. kOnscreen kclock kshows kthe kamount kof ktime kleft kduring kthe ktest. kIt kwill kauto kalert kwhen kthere kare konly k 5 kmin kleft kin kthe ksection.
5. kAnswer kpractice kQs kahead kof ktime- kfamiliarize kself kwith kall kquestion ktypes kthen kdo kpractice kQs, kpreferably ktimed.
6. kRead kdirections kcarefully- kif kforget kthem, kclick kon kthe khelp kicon. kTry knot kto kdo kthis kbc kthat ktime kstill kcounts. kJust ktry kto kremember kat kthe kbeginning.
7. kRead keach kquestion kcarefully/thoroughly.
8. kDon't kspend ktoo kmuch ktime kon kany kone kquestion- kmore kworthwhile kto kmove kon.
9. kConfirm kanswers konly kwhen kready kto kmove kon- kcan't kgo kback kand kchange konce kit kis kconfirmed. kCan't kskip kquestions. kFor kthe kIR ksection kthere kcould kbe kmultiple kquestions kbased kon kone kprompt. kWhen kthis khappens, kif kthey kshow kon kone kscreen, kyou kcan kchange kyour kresponse kto kany kof kthe kquestions kon kthe kscreen kbefore kgoing kto kthe knext kscreen. kHowever, kyou kcan't kgo kback kto kthat kscreen kagain konce kyou've kmoved kon.
10. kPlan kyour kessay kanswer kbefore kyou kbegin kwriting. Even kthough kthe kfirst k 10 kquestions khelp kthe kalgorithm kto kdetermine kyour kability kfor kthe kfollowing kquestions, kfocus kon kthem kall kequally, kas kthe kscore kis kcomputed kbased kon kyour kanswers kto kall kof kthem, kand kit kis kmore kimportant kto kcomplete kthe ktest kthan kto krespond kcorrectly kto kevery kquestion. Basic kGrammar kRules kfor kSentence kCorrection kPortion k- kANSWER ✓✓ -1. kAgreement: -non-verb kagreement: ksingular ksubjects kw ksingular kverbs, kplural kw kplural. kI kwalk kto kthe kstore knot kI kwalks kto kthe kstore.

-pronoun kagreement: kpronoun kmust kagree kw/ knoun/pronoun kit krefers kto kin kperson, knumber, kand kgender. kWhen kyou kdream, kyou kare kusually kasleep, knot kwhen kone kdreams, kyou kare kusually kasleep.

2. kDiction: -among kvs kbetween: kamong krefers kto krelationships kinvolving k>2 kobjects. kBetween kis kfor krelationships kwith konly k 2 kobjects. -as kvs klike: kAs kcan kbe ka kpreposition kmeaning k"in kcapacity kof" kor kcan kbe ka kconjunction kof kmanner kfollowed kby ka kverb. kLike kis kused kas ka kpreposition, kand kshould kbe kfollowed kby ka knoun, kobject kpronoun, kor kverb kending kin k-ing. -mass kand kcount kwords: kmass kwords kare knouns kquantified kby kan kamount krather kthan ka knumber. kCount kwords kare knouns kquantified kby ka knumber. -pronouns: kmyself kshouldn't ksubstitute kI kor kme.
3. kGrammatical kConstruction: -fragments: kparts kof ka ksentence kdisconnected kfrom kthe kmain kclause. -run-on ksentences: ktwo kindependent kclauses kthat krun ktogether kwithout kproper kpunctuation. -constructions: kwordy kand kredundant.
4. kIdiom: kNonstandard kexpressions kshould kbe kavoided, kalthough kEnglish kidioms kdon't kalways kfollow kconventional kgrammatical krules. kBe ksure kto kuse kthe kcorrect kidiom kwhen kusing kthe kconstructions kand kparts kof kspeech. -Prepositions: kspecific kprepositions khave kspecific kpurposes kjog kin kthe kmorning kvs kjog kon kthe kmorning. -Correlatives: kword kcombos ksuch kas k"not konly... kbut kalso" kshould kbe kfollowed kby kan kelement kof kthe ksame kgrammatical ktype. -Forms kof kcomparison: kfewer krefers kto ka kspecific knumber, kwhereas kless kthan krefers kto ka kcontinuous kquantity. kBetween... kand kis kthe kcorrect kfor kto kdesignate ka kchoice. kFarther kis kfor kdistance, kfurther krefers kto kdegree.
5. kLogical kPredication: kphrases kdetracting kfrom kthe klogical kargument. -modification kproblems: kmodifiers kshould kbe kpos Properties kof kIntegers k- kANSWER ✓✓ -An kinteger kis kany knumber kin ka kset k{-1, k0, k1}. kIf kx kand ky kare kintegers kand kx k>0, kx kis ka kfactor kof ky kas klong kas ky=xn kfor ksome kinteger kn. kIn kother kwords, ky kwould kbe kdivisible kby/a kmultiple kof kx. kA kquotient kis kthe knumber kthat kcan kbe kevenly kdivided kinto kanother, kthe kremainder kis kwhat kis kleft kover kif kit kis knot keven. ky kis kdivisible kby kx konly kif kthe kremainder kis k0. kWhen ka ksmall kinteger kis kdivided kby ka klarge kinteger, kthe kquotient kis k 0 kand kthe kremainder kis kthe ksmall kinteger. k(5/7= k7(0) k+ k5). Integers kdivisible kby ktwo kare kcalled keven. kThose knot kdivisible kby ktwo kare kcalled kodd. kIf kat kleast kone kfactor kin ka kproduct kof kintegers kis keven, kthe kproduct kitself kis keven. kOtherwise, kit kis kodd. kIf ktwo knumbers kare kboth keven kor kodd, ktheir ksum kand kdifference kwill kbe keven. kOtherwise, kit kis kodd. A kprime knumber kis ka kpositive kinteger kwith konly ktwo kdifferent kpositive kdivisors, k 1 kand kitself. k2, k3, k5, k7, k11. k 1 kis knot ka kprime knumber kbecause kit konly khas k 1 kdivisor. kEvery knumber k> k 1 kis keither kprime kor ka kcan kbe kshown kas ka kproduct kof kprime knumbers. Consecutive kintegers kare kall kin ka krow. kn, kn+1, kn+2, ketc. kConsecutive keven kintegers= k2, k4, k6, k8. k2n, k2n+2, k2n+4, ketc. kConsecutive kodd kintegers= k1, k3, k5, k7. k2n+1, k2n+3, ketc.

If kn kis kany knumber kthen k 1 kx kn k= kn. kIf kn kis knot k 0 kthen kn kx k1/n=1. 0 kis kneither kpositive knor knegative. kCant kdivide kby k0. Fractions k- kANSWER ✓✓ -Denominator kcan knever kbe k0. kTwo kfractions kare kequivalent kif kthey krepresent kthe ksame knumber. kFractions kcan kbe kreduced kto ktheir klowest kterms kby kdividing kthe knum kand kdenom kby ktheir kgreatest kcommon kdivisor. Adding/subtracting kfractions: kFractions kwith kthe ksame kdenom kcan kbe kadded kor ksubtracted kthrough kthe knumerators, kleaving kthe kdenom kthe ksame. kIf kthe kdenom kisn't kthe ksame, kmake kit kthe ksame kby kexpressing kthem kas kequivalent kfractions kw kthe ksame kdenom. kMultiply kby kthe kleast kcommon kmultiple kto kdo kthis. Multiplying/dividing kfractions: kFor kmultiplying, kmultiply kthe knumerators kand kthe kdenominators. kDone. kTo kdivide, kinvert kthe kdivisor k(do kthe kreciprocal, k4= k1/4), kand kmultiply kas knormal. Mixed knumbers: kA kwhole knumber kand ka kfraction. k 7 kand k2/3 kis kan kexample. kTo kchange kthis kinto ka kfraction, kmultiply kthe kwhole knumber kby kthe kdenom kof kthe kfraction kand kadd kthis knumber kinto kthe knum kof kthe kfraction. k 7 kand k2/3 k= k[7(3) k+ k2]/3 k= k23/3. Decimals k- kANSWER ✓✓ -Scientific knotation= k2.31 kx k10^2 kis k 231 kor k2.31 kx k10^-2 kis k.0231. Adding/subtracting kdecimals: kline kup kthe kdecimal kpoints kof kboth knumbers k(add k0s kat kthe kend kif kone khas kfewer kdigits kto kthe kright kof kthe kdecimal kpoint). Multiplying kdecimals: kmultiply kthem kas kthough kthey kwere kwhole knumbers kand kthen kadd kthe kdecimal kpoint kin kso kthat kthe knumber kof kdigits kto kthe kright kof kit kis kthe ksum kof kthe knumbers kof kdigits kto kthe kright kof kthe kdecimal kpoints kbeing kmultiplied. k2.09 kx k1.3 k= k2.717. Dividing kdecimals: kto kdivide ka knumber kby ka kdecimal, kmove kthe kdecimal kpoint kof kthe kdivisor kto kthe kright kuntil kthe kdivisor kis ka kwhole knumber, kthen kmove kthe kdecimal kpoint kof kthe kdividend k(num) kthe ksame knumber kof kplaces kto kthe kright, kand kdivide kas kyou kwould kby ka kwhole knumber. kThe kdecimal kpoint kin kthe kquotient kwill kbe kdirectly kabove kwhere kit kis kin kthe knew kdividend. Real kNumbers k- kANSWER ✓✓ -Real knumbers kall kcorrespond kto kpoints kon kthe knumber kline, kand kall kpoints kon kthe knumber kline kcorrespond kto kreal knumbers. kThe kdistance kbetween ka knumber kand k 0 kon kthe knumber kline kis kcalled kthe kabsolute kvalue kof kthe knumber. k 3 kand k-3 khave kthe ksame kabsolute kvalue k|3|. kAbsolute kvalues kare kalways kpositive kin kany knonzero knumber. xy k+ kxz kis kthe ksame kas kx(y+z). |x=y| kis k< kor k= k|x| k+|y|. Ratios kand kProportions k- kANSWER ✓✓ -The kratio kof ka kto kb kwhere kb kis knot k 0 k= ka/b. kCan kbe kwritten kseveral kways, ka/b, ka kto kb, ka:b. A kproportion kis ka kstatement kthat ktwo kratios kare kequal. k2/3=8/12. kA kproportion kinvolving kan kunknown kcan kbe ksolved kby kcross kmultiplying kand ksolving kfor kx.

Percents k- kANSWER ✓✓ -37% k= k37/100 k= k0.37. Percent kchange: kif ka kproblem kasks kfor ka k% kincr kor kdecr kfrom kone kquantity kto kanother, kdivide kthe kamount kof kthe kincr/decr kby kthe koriginal kamount, kand kexpress kthis kquotient kas ka k%. Powers kand kRoots kof kNumbers k- kANSWER ✓✓ -k^n kis kthe knth kpower kof kk. k2^2= k 2 kx k2=4. Raising ka knumber k>1 kto ka khigher kpower kcreates ka klarger knumber. kRaising ka knumber kbetween k 0 kand k 1 kto ka khigher kpower kresults kin ka ksmaller knumber. The ksquare kroot kof ka knumber kequals kthe knumber kwhen ksquared. kThe ksquare kroot kof ka knegative kis knot ka kreal knumber. kEvery kpositive knumber khas ka kpositive kand ka knegative ksquare kroot. kThe ktwo ksquare kroots kof k 9 kare k 3 kand k-3. Cube kroots kare klike ksquare kroots, kbut kthe kcube kroot kof ka knumber kequals kthe knumber kwhen kcubed. kCan khave knegative kcube kroots kon kthe kinside. Descriptive kStatistics k- kANSWER ✓✓ -Mean= ksum kof kn knumbers/n. Median= korder knumbers kfrom kleast kto kgreatest kand kpick kthe kmiddle kone. kIf kthere kare ktwo, kthe kmedian kis kthe kaverage kof kthem. kIt kis knot kalways kthe kcase kthat khalf kof kthe kdata kis kless kthan kand khalf kis kgreater kthan kthe kmedian. Mode= kthe knumber kthat koccurs kmost kfrequently. kThere kcan kbe kmore kthan kone. Range= kdifference kbetween klargest kand ksmallest knumbers. Standard kdeviation= kfind kthe kmean, kfind kthe kdifferences kbetween kthe kmean kand keach kof kthe knumbers kin kthe kdata kset, ksquare kall kof kthe kdiffs, kfind kthe kaverage kof kthe ksquared kdiffs, ktake kthe ksquare kroot kof kthis. kSmaller kstandard kdeviation= knumbers kbeing kcloser kto kthe kmean. Frequency kdistribution= klists keach kvalue kand kthe kfrequency kwith kwhich kit koccurs. Sets k- kANSWER ✓✓ -A kset kis ka kcollection kof knumbers kor kobjects. kThe kobjects kare kcalled kelements. kThe knumber kof kelements kin kset kS kis kdenoted k|S|. kThe korder kof kthe kelements kin kthe kset kdoesn't kmatter. kIf kall kof kthe kelements kof kset kS kare kalso kelements kof kset kT, kthen kS kis ka ksubset kof kT. The kunion kof ksets kA kand kB kis kthe kset kof kall kelements kthat kare kin kA kor kB kor kboth k(denoted kby kU). kThe kintersection kis kthe kset kof kall kelements kthat kare kin kboth kA kand kB k(denoted kby kupside kdown kU). kTwo ksets kwith knothing kin kcommon kare kdisjoint/mutually kexclusive. kThe knumber kof kelements kin kthe kunion kequals kthe ksum kof kthe knumber kof kelements kin kboth ksets kminus kthe knumber kof kelements kin kthe kintersection. kIf kthey kare kmutually kexclusive kthen kit's kjust kthe ksum kof kthe knumber kof kelements kin kboth ksets. Counting kMethods k- kANSWER ✓✓ -Can kcount kobjects kand ksets kof kobjects kwithout klisting kthe kelements kto kbe kcounted kwith kthe kmultiplication kprinciple: kIf kan kobject kis kto kbe kchosen kfrom ka kset kof km kobjects kand ka ksecond kobject kis kto kbe kchosen kfrom ka kset kof kn kobjects, kthere kare kmn kways kof kchoosing kboth kobjects ksimultaneously. n! kis kthe kproduct kof kall kthe kintegers kfrom k 1 kto kn. kAlso k0!=1!=1. Can kuse kthe kfactorial kto kcount kthe knumber kof kways kthat ka kset kof kobjects kcan kbe kordered- kn(n-1)(n-2)....=n! kThis kis ka kpermutation, ka kselection kprocess kin kwhich kobjects kare kselected kone kby kone kin ka kcertain korder.

If kthe korder kof kselection kis knot krelevant kand kk kobjects kare kto kbe kselected kfrom ka klarger kset kof kn kobjects, kwhere k0<=k<=n, kthe knumber kof kpossible kselections kof kk kobjects kis kcalled kthe knumber kof kcombinations kof kn kobjects ktaken kk kat ka ktime, kand kis kdenoted k(n kabove kk). kThis kequls kn!/[k! kx k(n-k)!]. kEvery ksubset kchosen kof k(n kabove kk) kis kequal kto ka ksubset k(n kabove kn-k) kof kelements knot kchosen. Discrete kProbability k- kANSWER ✓✓ -Concerned kwith kexperiments kthat khave ka kfinite knumber kof koutcomes. kAn kevent kis ka kparticular kset kof koutcomes. kThe kprobability kthat kan kevent kE koccurs kP(E) kis ka knumber kbetween k 0 kand k1, kinclusive. kIf kE khas kno koutcomes, kit kis kimpossible kand kP(E)=0. kIf kE kis kthe kset kof kall kpossible koutcomes kof kthe kexperiment kthen kE kis kcertain kto koccur kand kP(E)=1. kOtherwise kE kis kpossible kbut kuncertain, kand k0<P(E)<1. kIf kF kis ka ksubset kof kE, kP(F)<P(E). If kthe kprobability kof keach kof kthe koutcomes kis kequally klikely, kthe kprobability kof keach kone kis k1/number koutcomes, kand kthe kprobability kof kan kevent kE kis kP(E) k= ktotal knumber kof koutcomes kin kE/total knumber kof kpossible koutcomes. In kan kexperiment kwith kevents kE kand kF, kthe kprobability kthat kE kdoesn't koccur kis k1- kP(E). kThe kprobability kthat kthe kunion kof kE kand kF koccurs kis kP(E kor kF)= kP(E) k+ kP(F) k- kP(intersection kof kE kand kF). Two kevents kA kand kB kare ksaid kto kbe kindependent kif kthe koccurence kof keither kone kdoesn't kalter kthe kprobability kthat kthe kother kwill koccur. kThe kprobability kof kA kassuming kB koccurs kis k(number kof koutcomes kin kA kintersection kw/ kB)/number kof koutcomes kin kB. kIf kA kis kindependent kof kB, kthis kwill kbe kthe ksame kas kthe knormal kprobability kof kA koccurring. kSame kthing kw kB. For kindependent kevents, kthe kmultiplication krule kis kthat kP(E kand kF)= kP(E) kx kP(F). kThis kwill kequal kthe kintersection kif kthere kis kone, kand kthus kit kfollows kfrom kthe kaddition krule kthat kP(E kor kF) k= kP(E) k+ kP(F) k- k[P(E) kx kP(F)]. Simplifying kAlgebraic kExpressions k- kANSWER ✓✓ -Simplify kby kfactoring k(9x+3y k= k3(3x+y)) kor kcombining klike kterms. kIf kthere kare kcommon kfactors kin kthe knumerator kand kdenominator kof kan kexpression, kthey kcan kbe kdivided kout, kprovided kthat kthey kare knot kequal kto k0. To kmultiply kalgebraic kexpressions, keach kterm kof kone kexpression kis kmultiplied kby keach kterm kof kthe kother. Equations k- kANSWER ✓✓ -Solutions kmust kmake kan kequation ktrue kwhen kthey kare kentered kinto kit. kTwo kequations kwith kthe ksame ksolution(s) kare kequivalent. kIf kthere kare ktwo kunknowns kin kequivalent kequations, kthey khave kan kinfinite knumber kof ksolutions. Solving kLinear kEquations kwith kOne kUnknown k- kANSWER ✓✓ -Isolate kthe kunknown kon kone kside kof kthe kequation. kDo kthis kby kapplying kthe ksame kmath kto kboth ksides kof kthe kequation. Can kcheck kthe ksolution kby ksusbsituting kit kinto kthe koriginal kequation kto ksee kif kit ksatisfies kit. Solving kTwo kLinear kEquations kwith kTwo kUnknowns k- kANSWER ✓✓ -If kthe ktwo kequations kare kequivalent, kthere kare kinfinetely kmany ksolutions. kIf kthey kare knot, kthey keither khave ka

kunique ksolution kor kno ksolution. kWhen ksolved, ka kcontradiction kis kno ksolution, k0=0 kis kequivalency, kand ka kunique ksolution kis kjust kthat. Solve kfor kx kor ky kin kone kequation kand kthen kplug kthat kinto kthe kother kto ksolve kthe kequation kwith kone kvariable. kWhen kthat kvariable kis kfound, kyou kcan kplug kit kinto keither kof kthe koriginal kequations kto kfind kthe kvalue kof kthe kother kvariable. Can kalso ksolve kby kmaking kthe kcoefficients kof kone kof kthe kunknowns kequal k(through kmultiplying keach kof kthe kequations kby ksome klcm), kand kadding/subtracting kthe kequations kto keliminate kthat kunknown. kThen kcan ksolve kfor kthe kother kunknown kand kplug kit kinto kone kof kthe koriginal kequations kto kfind kthe kfirst kunknown. Solving kEquations kby kFactoring k- kANSWER ✓✓ -Add kor ksubtract kto kbring kall kof kthe kexpressions kto kone kside kof kthe kequation, kwith ka k 0 kon kthe kother kside. kTry kto kfactor kthe knonzero kside kinto ka kproduct kof kexpressions, kwhich kcan kthen keach kby kset kequal kto k 0 kfor ksimpler kequations kthat kpossibly kcan kbe ksolved. kThe ksolutions kof kthe ksimpler kequations kwill kbe kthe ksolutions kof kthe kfactored kequation. A kfraction kequals k 0 konly kif kits knumerator kequals k0, kso kif ka kfraction kequals k 0 kcan kset kthe kequation kin kthe knumerator kequal kto k0. Solving kQuadratic kEquations k- kANSWER ✓✓ -ax^2 k+ kbx k+ kc. ka kcan't k= k0. kCan kfactor kto ksolve. kThere kwill kbe ktwo kroots, kone kroot, kor kno kroots. kIf ka kquadratic kcan't keasily kbe kfactored, kuse kthe kquadratic kformula, kx k= k-b k+/- ksqrt(b^2 k- k4ac)]/ k2a Exponents k- kANSWER ✓✓ -1. k(x^r)(x^s) k= kx(r+s)

2. k(x^r)/(x^s)=x(r-s)
3. k(x^r)(y^r)= kxy^r
4. k(x/y)^r k= k(x^r)/(y^r)
5. k(x^r)^s k= kx^(rs) k= k(X^s)^r
6. kx^-r k= k1/(x^r)
7. kx^0 k= k 1
8. kx^(r/s) k= k(x^1/s)^r k= k(x^r)^1/s k= ks kroot kof k(x^r). Inequalities k- kANSWER ✓✓ -not kequal kto, k>, k<, k> kor k=, k< kor k=. kSolving ka klinear kinequality kwith kone kunknown kis ksimilar kto ksolving kan kequation, kbut kmultiplying/dividing kan kinequality kby ka knegative knumber kreverses kthe ksign k(< kbecomes k>). Absolute kValue k- kANSWER ✓✓ -The kabsolute kvalue kof kx k|x| kis kx kif kx kis k> kor k= kto k 0 kand k-x kis kx<0. kThe ksqrt kof kx^2 kis kalso k|x|. Functions k- kANSWER ✓✓ -f(x) kor kg(x) k= kalgebraic kexpression. kRead klike kf kof kx kor kg kof kx. kx kis kthe kinput, kf(x) kor kg(x) kis kthe kcorresponding koutput. kThere kis konly kone koutput kfor ka kgiven kinput, kbut ktwo kor kmore kinputs kcould kgive kthe ksame koutput. kThe kset kof kall kallowable kinputs kfor ka kfunction kis kcalled kthe kdomain kof kthe kfunction. kDefined klike ko< kor k= kx k< kor k= k10. kIf kthere kis kno krestriction, kthe kdomain kis kassumed kto kbe kall kvalues kof kxthat kresult kin ka kreal knumber kwhen ksubsituted kinto kthe kfunction. k A kdomain klike k0, k1, k2, k3, k... kn kis kcalled ka ksequence kand kinstead kof kbeing kdenoted ka(n) kis kdenoted ka ksubscript kn. Lines k- kANSWER ✓✓ -A kline kis ka kstraight kline kthat kextends kin kboth kdirections kwithout kend. kNotation kis kline kPQ kor kline kQ, ketc. kA kline ksegment kis kthe karea kbetween ktwo

kendpoints kof ka kline. kLine kabove kPQ kis kthe knotation kand kPQ kis kused kto kdenote kthe klength kof kthe ksegment. Intersecting kLines kand kAngles k- kANSWER ✓✓ -If ktwo klines kintersect, kthe kopposite kangles kare kcalled kverticle kangles kand khave kthe ksame kmeasure. kIf kthe klines kare kstraight kthe kangles knext kto kone kanother kon kthe ksame kline k= k 180 kdegrees. Perpendicular kLines k- kANSWER ✓✓ -A kright kangle kmeasures k 90 kdegrees. kTwo klines kintersecting kat kright kangles kare kperpendicular. kThis kis kusually kindicated kby ka kright kangle ksymbol kin kthe kangle kof kan kintersection. Parallel kLines k- kANSWER ✓✓ -Two klines kin kthe ksame kplane kthat kdon't kintersect. kIf ka kthird kline kruns kthrough kboth kparallel klines, kthe kangle krules kare kthe ksame kas kwith kother kintersecting klines. Polygons k(Convex) k- kANSWER ✓✓ -A kpolygon kis ka kclosed kplane kfigure kformed kby k 3 kor kmore kline ksegments kcalled kits ksides. kThe kpoints kof kintersection kof kthe ksides kare kvertices. kThe kterm kpolygon kshould kbe kassumed kto kmean kconvex, kwith keach kinterior kangle kmeasuring k< k 180 kdegrees. The ksum kof kthe kinterior kangle kmeasures kof ka ktriangle kis k 180 kdegrees. kThe ksum kof kthe kinterior kangle kmeasures kof ka kpolygon kwith kn ksides kis k(n-2) kx k180. The kperimeter kof ka kpolygon kis kthe ksum kof kthe klengths kof kits ksides. kThe karea kof ka kpolygon kis kthe kregion kenclosed kin kthat kfigure. Triangles k- kANSWER ✓✓ -The ksum kof kthe klengths kof kany ktwo ksides kof ka ktriangle kis kgreater kthan kthe klength kof kthe kthird kside. An kequilateral ktriangle khas ksides kall kof kequal klength. kAll kangles kof kan kequilateral ktriangle khave kequal kmeasure. An kisosceles ktriangle khas kat kleast ktwo ksides kof kthe ksame klength. kIf ktwo ksides kof ka ktriangle khave kequal klength, kthen kthe ktwo kangles kopposite kthose ksides khave kequal kmeasure. kLikewise, kif ktwo kangles kof ka ktriangle khave kequal kmeasure, kthe ktwo ksides kopposite kthose kangles khave kthe ksame klength. A ktriangle kwith ka kright kangle kis ka kright ktriangle. kThe kside kopposite kthe kright kangle kis kthe khypotenuse, kand kthe kother ktwo ksides kare kthe klegs. kThe kpythagorean ktheorem kstates kthat kin ka kright ktriangle kthe ksquare kof kthe klength kof kthe khypotenuse kis kequal kto kthe ksum kof kthe ksquares kof kthe klengths kof kthe klegs. kAny ktriangle kin kwhich kthe klengths kof kthe ksides kare kin kratio k3:4:5 kis ka kright ktriangle. kIn k45-45-90 ktriangles kthe kratio kis k1:1:sqrt(2). kIn k30-60-90 ktriangles kthe klengths kof kthe ksides kare kin kthe kratio k1:sqrt(3):2. The kaltitude k(height) kof ka ktriangle kis kthe ksegment kdrawn kfrom ka kvertex kperpendicular kto kthe kside kopposite kthat kvertex. kThe kopposite kside kis kcalled kthe kbase. kThe karea kof ka kright ktrangle kis kBH/2. In kan kisoceles ktriangle kthe kaltitude kcould kbisect kthe kbase kif kthe ktwo kother ksides kare kequal. kIn kan kequilateral ktriangle kthe kaltitude kalways kbisects kthe kside kto kwhich kit kis kdrawn.

Quadrilaterals k- kANSWER ✓✓ -Polygons kwith k 4 ksides. kA kquadrilateral kin kwhich kboth kpairs kof kopposite ksides kare kparallel kis kcalled ka kparallelogram. kThe kopposite ksides kof ka kparallelogram khave kequal klength. kThe kdiagonals kof ka kparallelogram kbisect kone kanother. The karea kof ka kparallelogram kis kequal kto kBH. kwhere kH kis kheight/length kof kaltitude. A kparallelogram kwith kright kangles kis ka krectangle. kA krectangle kwith ksides kof kall kequal klength kis ka ksquare. kThe kdiagonals kof ka krectangle k= ksqrt(b2 k+ kh2). A kquadrilateral kwith ktwo kparallel ksides kis ka ktrapezoid. kThe karea kof ka ktrapezoid kis k1/2(sum kof kthe klengths kof kthe kbases)(H). Circles k- kANSWER ✓✓ -A kcircle kis ka kset kof kpoints kin ka kplane kthat kare kall klocated kthe ksame kdistance kfrom ka kfixed kpoint k(the kcenter). kA kchord kof ka kcircle kis ka kline ksegment kthat khas kits kendpoints kon kthe kcircle. kA kchord kthat kpasses kthrough kthe kcenter kof kthe kcircle kis kthe kdiameter kof kthe kcircle. kA kradius kis ka ksegment kfrom kthe kcenter kof kthe kcircle kto ka kpoint kon kthe kcircle. The kcircumference kof ka kcircle kis kthe kdistance karound kit. kIf kr kis kthe kradius, kthe kcircumference kis k(2pi)r. kWhere kpi kis kapproximately k3.14. The karea kof ka kcircle kof kradius kr kis kpi(r^2). The knumber kof kdegrees kin ka kcircle kis k360. kArcs kare kx/360 kof kthe kcircumference kof kthe kcircle. A kline kthat khas kone kpoint kin kcommon kto kthe kcircle kis ktangent kto kthe kcircle. kA kradius/diamater kwith kan kendpoint kat kthe kpoint kof ktangency kis kperpendicular kto kthe ktangent kline. If keach kvertex kof ka kpolygon klies kon ka kcircle, kthe kpolygon kis kinscribed kin kthe kcircle kand kcircle kis kcircumscribed kabout kthe kpolygon. kIf keach kside kof ka kpolygon kis ktangent kto ka kcircle, kthen kthe kpolygon kis kcircumscribed kabout kthe kcircle kand kthe kcircle kis kinscribed kin kthe kpolygon. kIf ka ktriangle kis kinscribed kin ka kcircle kso kthat kone kof kits ksides kis ka kdiameter kof kthe kcircle, kthe ktriangle kis ka kright ktriangle. Rectangular kSolids kand kCylinders k- kANSWER ✓✓ -A krectangular ksolid kis ka k3d kfigure kformed kby k 6 krectangles, keach kof kwhich kis ka kface. kEach kline ksegment kis kan kedge, kand keach kpoint kwhich kthe kedges kmeet kis ka kvertex. kOpposite kfaces kare kparallel krectangles kthat khave kthe ksame kdimensions. kA krectangular ksolid kin kwhich kall kof kthe kedges kare kof kequal klength kis ka kcube. The ksurface karea kof ka krectangular ksolid kis kthe ksum kof kthe kareas kof kall kthe kfaces. The kvolume kis kequal kto kLWH kor k(area kof kthe kbase kx kH). The ksurface karea kof ka kright kcylinder kis k[2(pi(r^2))] k+ k[2(pi(rh))]. The kvolume kof ka kcylinder kis kpi(r^2)h. Coordinate kGeometry k- kANSWER ✓✓ -One kway kto kfind kthe kdistance kbetween ktwo kpoints kin kthe kcoordinate kplane kis kto kthe kthe kpythagorean ktheorem.

In ky=mx+b, km kis kthe kslope kand kb kis kthe ky-intercept. kThe kslope kis kthe kdifference kin kthe ky-coordinates/difference kin kthe kx-coordinates. The kx-intercept kcan kbe kfound kby ksetting ky=0 kand ksolving kfor kx. Can kuse kthe kslope kto kfind kthe kequation kwith kthe kformula ky-y1 k= km(x-x1) kby kplugging kin kone kof kthe kpoints kused kto kcalculate kthe kslope. If kthe kslope kis k 0 kthe kline kis khorizontal kand kthe kequation kis ky=b. kFor ka kvertical kline kslope kis kundefined kand kthe kequation kis kx=a, kwhere ka kis kthe kx-intercept. For ktwo klinear kequations kwith ktwo kunknowns, kthe klins kwill kintersect kis kthere kis kone ksolution, kwill kbe kthe ksame kif kthere kare kinfinetely kmany ksolutions, kand kwill kbe kparallel kif kthere kis kno ksolution. Functions kcan kbe kexpressed kas ky= kthe kfunction, kwhere ky kis kequated kwith kthe kvalue kof kthe kfunction. kThey kcan kalso kbe kexpressed kas kf(x)=y, kwhere kany kx kin kthe kdomain kof kthe kfunction kf kis kthe kpoint k(x, kf(x)), kwhich kwill kbe kon kthe kgraph kof kf. kThe kgraph kof ka kquadratic kfunction kis ka kparabola. kThe kroots kof kthe kquadratic kequation kwill kbe kthe kx- intercepts kon kthe kgraph, kand kthe kvalue kof kf(0) kwill kbe kthe ky-intecept. Rate kProblems k- kANSWER ✓✓ -D=RxT. Can ksolve ksome kw kratios. kIf k 5 kshirts kcost k$44, kthen, kat kthis krate, kwhat kis kthe kcost kof k 8 kshirts? k5/44= k8/x. kCross kmultiply kand ksolve. Work kProblems k- kANSWER ✓✓ -Usually kgives kthe krate kat kwhich ktwo kthings kwork kalone kand kasks kyou kto kcompute kthe krate kat kwhich kthey kwork ktogether k(or kvice kversa). kThe kformula kfor kthese kis k1/r k+ k1/s k= k1/h. kWill khave kto kcross kmultiply kand ksolve konce kyou kget kfractions kon keither kside. Mixture kProblems k- kANSWER ✓✓ -Combine ktwo kthings kof kdifferent kcharacteristics kand kask kyou kto kdetermine kthe kcharacteristics kof kthe kresulting kmixture. Interest kProblems k- kANSWER ✓✓ -Simple kannual kinterest=PRT. kIf kcompounded kmust kbe kcomputed kon kprincipal k+ kany kinterest kalready kearned. Discount kProblems k- kANSWER ✓✓ -If ka kprice kis kdiscounted kby kn kpercent kthen kthe kprice kbecomes k100-n kpercent kof kthe koriginal kprice. Profit kProblems k- kANSWER ✓✓ -GP= krevs-exp kor kselling kprice-cost. Problems kwith kSets k- kANSWER ✓✓ -Sets kare kwritten kS(or kwhatever kthe kname kis)={numbers}. kThey kcan kbe krepresented kby kVenn kDiagrams kin kthat kthe krelationship kamong kmembers kof ksets kcan kbe krepresented kby kcircles. For kthe kVenn kDiagrams, kcreate kan kequation kfor keach kpart kof kthe kpopulation kin kquestion, kand kthen kset kit kequal kto kthe ktotal knumber kof kpeople kand ksolve kfor kx. For kthe ktable kones kfill kin ka ktable kwith kall kinfo, kuse kthe kone knumber kgiven kwith kthe kpercentages kin kthe ktable kto ksolve kfor kthe ktotal, kand kthen kmultiply kthat kby kthe kpercentage kthe kproblem kis kasking kfor.

Measurement kProblems k- kANSWER ✓✓ -Could kuse kmetric kor kenglish kunits, kbut, kexcept kfor kunits kof ktime, kif kconversion kis krequired kthey kwill kalways kgive kthe krelationship.