QUANTITATIVE REASONING

Transcription

1 Guide For Exminees Inter-University Psychometric Entrnce Test QUNTITTIVE RESONING The Quntittive Resoning domin tests your bility to use numbers nd mthemticl concepts to solve mthemticl problems, s well s your bility to nlyze dt presented in vriety of wys, such s in tble or grph form. Only bsic knowledge of mthemtics is required (the mteril studied up to 9th or 0th grdes in most Isreli high schools). ll of the Quntittive Resoning problems tke the form of multiple-choice questions, tht is, question followed by four possible responses, only one of which is the correct nswer. The Quntittive Resoning sections consist of two ctegories of questions Questions nd Problems, nd Grph or Tble omprehension. Questions nd Problems cover vriety of subjects tken from lgebr nd geometry. Some of the questions re presented in mthemticl terms; others re word problems, which you must trnslte into mthemticl terms before solving. Grph or Tble omprehension questions relte to informtion ppering in grph or tble. grph presents dt in grphic form, such s br chrt, line grph or sctter plot. tble presents dt rrnged in columns nd rows. In generl, ll questions of given type re rrnged in scending order of difficulty. The esier questions, requiring reltively less time to solve, pper first, with the questions becoming progressively more difficult nd requiring more time to solve. The figures ccompnying some of the questions re not necessrily drwn to scle. Do not rely solely on the figure's ppernce to deduce line length, ngle size, nd so forth. However, if line in figure ppers to be stright, you my ssume tht it is, in fct, stright line. Formul Pge ppers t the beginning of ech Quntittive Resoning section. This pge contins instructions, generl comments nd mthemticl formuls, which you my refer to during the test. The Formul Pge lso ppers in the Guide (on the next pge) nd in the Quntittive Resoning sections of the prctice test. You should fmilirize yourself with its contents prior to tking the test. Pges 8-66 contin review of bsic mthemticl concepts, covering much of the mteril upon which the questions in the Quntittive Resoning sections re bsed. The ctul test my, however, include some questions involving mthemticl concepts nd theorems tht do not pper on these pges. Pges 67-8 contin exmples of different types of questions, ech followed by the nswer nd detiled explntion. 7

2 Quntittive Resoning FORMUL PGE This section contins 0 questions. The time llotted is 0 minutes. This section consists of questions nd problems involving Quntittive Resoning. Ech question is followed by four possible responses. hoose the correct nswer nd mrk its number in the pproprite plce on the nswer sheet. Note: The words ppering ginst gry bckground re trnslted into severl lnguges t the bottom of ech pge. Generl omments bout the Quntittive Resoning Section * The figures ccompnying some of the problems re provided to help solve the problems, but re not necessrily drwn to scle. Therefore, do not rely on the figures lone to deduce line length, ngle size, nd so forth. * If line in figure ppers to be stright, you my ssume tht it is in fct stright line. * When geometric term (side, rdius, re, volume, etc.) ppers in problem, it refers to term whose vlue is greter thn 0, unless stted otherwise. * When ( > 0) ppers in problem, it refers to the positive root of. * "0" is neither positive nor negtive number. * "0" is n even number. * "" is not prime number. Formuls. Percentges: % of x is equl to 00 $ x. Exponents: For every tht does not equl 0, nd for ny two integers n nd m - 0. The re of trpezoid with one bse length, the other bse length b, nd ltitude h is ] + b g h $ h b 8. n n b. m + n m n c. m n m _ i n (0 <, 0 < m) d. n m ( n ) m. ontrcted Multipliction Formuls: ( ± b) ± b + b ( + b)( b) b 4. Distnce Problems: distnce speed (rte) time 5. Work Problems: mount of work output (rte) time 6. Fctorils: n! n(n )(n ) Proportions: If D E F then nd DE DE EF DF 8. Tringles:. The re of tringle with bse of length nd ltitude to the bse of length h is h $ b. Pythgoren Theorem: In ny right tringle, s in the figure, the following lwys holds true: + c. In ny right tringle whose ngles mesure 0, 60, 90, the length of the leg opposite the 0 ngle is equl to hlf the length of the hypotenuse. 9. The re of rectngle of length nd width b is b b leg h D hypotenuse leg E F. The sum of the internl ngles of n n-sided polygon is (80n 60) degrees. In regulr n-sided polygon, ech internl ngle mesures 80 n 60 n k n k degrees.. ircle:. The re of circle with rdius r is πr (π.4...) b. The circumference of circle is πr c. The re of sector of circle x with centrl ngle of x is π r $ 60. ox (Rectngulr Prism), ube:. The volume of box of length, width b nd height c is b c b. The surfce re of the box is b + bc + c c. In cube, b c 4. ylinder:. The lterl surfce re of cylinder with bse rdius r nd height h is πr h b. The surfce re of the cylinder is πr + πr h πr(r + h) c. The volume of the cylinder is πr h 5. The volume of cone with bse rdius r πr $ h nd height h is 6. The volume of pyrmid with bse re S nd height h is S h $ r x r r h c r h b

3 Guide For Exminees Inter-University Psychometric Entrnce Test REVIEW OF SI MTHEMTIL ONEPTS SYMOLS Symbol b b «x y x y x < y x y < x, y x + x x : y Mening of the Symbol lines nd b re prllel line is perpendiculr to stright line b 90º ngle (right ngle) the ngle formed by line segments nd x equls y x does not equl y x is less thn y x is less thn or equl to y both x nd y re greter thn x my be equl to or to (-) the bsolute vlue of x the rtio of x to y TYPES OF NUMERS Integer: n integer, lso clled whole number, is number composed of whole units. n integer my be positive, negtive, or 0 (zero). Exmple:..., -4, -, -, -, 0,,,, 4,... Note: 0 (zero) is n integer tht is neither positive nor negtive. Non-integer: onsecutive numbers: number tht cnnot be expressed in whole units. Exmple:, -,, 7. Integers tht follow in sequence in differences of. For exmple, 4 nd 5 re consecutive numbers;,, nd 4 re consecutive numbers; (-) nd (-) re lso consecutive numbers. If n is n integer, then n nd (n + ) re consecutive numbers. This is sometimes expressed s: (n + ) is the next consecutive integer fter n. Even number: n integer which, when divided by, produces n integer (in other words, it is evenly divisible by ). If n is n integer, then n is n even number. Note: 0 is n even number. Odd number: n integer which, when divided by, produces non-integer (in other words, when it is divided by, reminder of one is obtined). If n is n integer, then n + is n odd number. 9

4 Quntittive Resoning Prime number: positive integer tht is evenly divisible by only two numbers itself nd the number. For exmple, is prime number becuse it is evenly divisible by only nd. Note: is not defined s prime number. Opposite numbers (dditive inverse): pir of numbers whose sum equls zero. For exmple, 4 nd (-4) re opposite numbers. In generl, nd (-) re opposite numbers ( + (-) 0). In other words, (-) is the opposite number of. Reciprocls (multiplictive inverse:) pir of numbers whose product is equl to. For exmple, nd re reciprocls, s re 7 nd 7. In generl, for, b 0: nd re reciprocls $ k. We cn lso sy tht is the reciprocl of. b nd b re reciprocls b b $ b l, or in other words, b is the reciprocl of b. bsolute vlue: If x > 0, then x x. If 0 > x, then x -x RITHMETIL OPERTIONS WITH EVEN ND ODD NUMERS even + even even odd + odd even odd + even odd even even even odd odd even even odd odd odd even odd even even even odd odd odd odd even even 40

5 Guide For Exminees Inter-University Psychometric Entrnce Test There re no similr rules for division. The quotient of two even numbers my be odd b6 l, even 4 k, or non-integer b6 4 l. FTORS (DIVISORS) ND MULTIPLES fctor (divisor) of positive integer is ny positive integer tht divides it evenly (tht is, without reminder). For exmple, the fctors of 4 re,,, 4, 6, 8,, nd 4. common fctor of x nd y is number tht is fctor of x nd lso fctor of y. For exmple, 6 is common fctor of 4 nd 0. prime fctor is fctor tht is lso prime number. For exmple, the prime fctors of 4 re nd. ny positive integer (greter thn ) cn be written s the product of prime fctors. For exmple, 4 multiple of n integer x is ny integer tht is evenly divisible by x. For exmple, 6,, nd 88 re multiples of 8. When the word "divisible" ppers in question, it mens "evenly divisible" or "divisible without reminder." MTHEMTIL OPERTIONS WITH FRTIONS Reduction When the numertor nd denomintor of frction hve common fctor, ech of them cn be divided by tht common fctor. The resulting frction, which hs smller numertor nd denomintor, equls the originl frction. For exmple, if we divide the numertor nd the denomintor of 6 by 4, the result is 4, b 6 4 l. Multipliction To multiply two frctions, multiply the numertors by ech other nd the denomintors by ech other. Exmple: $ $ $ 7 Division To divide number by frction, multiply the number by the reciprocl of the divisor. Exmple: $ 8 $ To multiply or divide n integer by frction, the integer cn be regrded s frction whose denomintor is. Exmple: 4

6 Quntittive Resoning ddition nd Subtrction To dd or subtrct frctions, they must be converted into frctions tht hve common denomintor. common denomintor is number tht is evenly divisible by the denomintors of ll of the originl frctions. (Tht is, the denomintors of the originl frctions re ll fctors of the common denomintor.) fter finding suitble common denomintor, ech of the frctions must be converted into frction tht hs this common denomintor. To do so, multiply the numertor nd denomintor of ech of the frctions by the sme integer, so tht the number obtined in the denomintor will be the number tht ws chosen to be the common denomintor. Multiplying the numertor nd denomintor by the sme number is the sme s multiplying by, nd its vlue remins unchnged. Once the frctions hve common denomintor, dd or subtrct the new numertors nd reduce the resulting frctions to lowest terms where possible. EXMPLE ? We know tht 4, 6, nd 8 re ll fctors of 4. Therefore, we cn use 4 s common denomintor: 4 6, 4 4, We will now convert ech of the frctions into frctions with this common denomintor: 4 8 4, 6 4 4, Thus, PERENTGES Percentges re wy of expressing hundredths: % of x is hundredths of x, or 00 $ x. In questions contining percentges, convert the percentges to hundredths, nd solve s in ordinry frction problems. EXMPLE Wht is 60 percent of 80? Insted of 60 percent, substitute 60 hundredths nd solve it s you would for ny multipliction of frctions: $ $ $ Thus, 60% of 80 is 48. Questions deling with chnge expressed s percentge refer to the percentge of the originl vlue, unless otherwise specified. EXMPLE The price of n item tht cost 80 shekels ws rised by 5%. Wht is the new price? Since 5% ws dded to the old price, the new price is 5% of the old price (00% + 5%). Therefore, you must clculte 5% of 80. Substitute hundredths for percent nd solve: 5 00 $ Thus, the new price is 00 shekels. 4

7 Guide For Exminees Inter-University Psychometric Entrnce Test EXMPLE The price of n item dropped from 5 shekels to shekels. y wht percentge did the price drop? In this exmple, the chnge in the price of n item is given, nd you re sked to clculte this chnge s percentge. The difference in the price is shekels out of 5 shekels. You hve to clculte wht percent of 5 is. onvert the question into mthemticl expression: 00 $ 5. Solve the eqution for : $ Thus, the price dropped by 0%. RTIO The rtio of x to y is written s x : y. EXMPLE The rtio between the number of pirs of socks nd the number of shirts tht Eli hs is :. In other words, for every pirs of socks, Eli hs shirts. Stting it differently, the number of socks tht Eli hs is greter thn the number of shirts tht he hs. MEN (VERGE) The rithmetic men (verge) of set of numericl vlues is the sum of the vlues divided by the number of vlues in the set. When the word "verge" ppers in question, it refers to the rithmetic men. For exmple, the verge of the set of vlues,, 5, 0, nd is 8 becuse If the verge of set of numericl vlues is given, their sum cn be clculted by multiplying the verge by the number of vlues. EXMPLE Dnny bought 5 items whose verge price is 0 shekels. How much did Dnny py for ll of the items? In this question we re sked to find the sum bsed on the verge. If we multiply the verge by the number of items, we will obtin Thus, Dnny pid totl of 50 shekels for ll of the items which he bought. weighted verge is n verge tht tkes into ccount the reltive weight of ech of the vlues in set. 4

8 Quntittive Resoning EXMPLE Robert's score on the midterm exm ws 75, nd his score on the finl exm ws 90. If the weight of the finl exm is twice tht of the midterm exm, wht is Robert's finl grde in the course? The set of vlues in this cse is 75 nd 90, but ech hs different weight in Robert's finl grde for the course. The score of 75 hs weight of, while the score of 90 hs weight of. To clculte the weighted verge, multiply ech score by the weight ssigned to it, nd divide by the sum of the weights: $ 75 $ Thus, Robert's finl grde in the course is 85. This clcultion is identicl to the clcultion of simple verge of the three numbers 75, 90 nd 90. POWERS ND ROOTS Rising number to the nth power (when n is positive integer) mens multiplying it by itself n times: n $... $ $ n times For exmple: (-) (-)(-)(-) -7. The expression n is clled power; n is clled the exponent; nd is clled the bse. ny number other thn zero, rised to the 0th power, equls. Thus, for ny 0, 0. power with negtive exponent is defined s the power obtined by rising the reciprocl of the bse to the opposite power: - n n k. Exmple: - k $ $ 8 The n th root of positive number, expressed s n, is the positive number b, which if rised to the n th n power, will give. In other words, b becuse b n 4. For exmple, 8 becuse 4 8. When the root is not specified, nd-order root is intended. nd-order root is lso clled squre root. For exmple, root cn lso be expressed s power in which the exponent is frction. This frction is the n reciprocl of the order of the root: n ] 0 < g. Note: When (0 < ) ppers in question, it refers to the positive root of. sic rules for opertions involving powers (for ny m nd n): Multipliction: To multiply powers with the sme bse, dd the exponents: m n (m+n). Division: To divide power by nother power with the sme bse, subtrct the exponent in m the denomintor from the exponent in the numertor: m n n ^ h. Note: When the powers do not hve the sme bse, the exponents cnnot be dded or subtrcted. 44

9 Guide For Exminees Inter-University Psychometric Entrnce Test Rising to power: To rise power to power, multiply the exponents: ^ h ( m n $ ). m n Rising product or quotient to power: ( b) m m b m, b m m k m. b Since roots cn lso be expressed s powers, the rules for powers cn lso be pplied to roots. m n For exmple, to clculte the product $ ] 0 < g, first express the roots s powers: m n m n $ $. m n m + n Then multiply the powers, tht is, dd the exponents: $ b l. Inequlities involving powers: If 0 < b < nd 0 < n then b n < n If 0 < b < nd n < 0 then n < b n If < nd m < n then m < n If 0 < < nd m < n then n < m ONTRTED MULTIPLITION FORMULS To multiply two expressions enclosed in prentheses, ech of which is the sum of two terms, multiply ech of the terms in the first expression by ech of the terms in the second expression, then dd the products. For exmple, ( + b) (c + d) c + d + bc + bd. This generl formul cn be used for finding the product of ny two expressions, but to sve time, you might wnt to memorize some common formuls: ( + b) ( + b) ( + b) + b + b ( b) ( b) ( b) b + b ( b) ( + b) b OMINTORIL NLYSIS Multi-Stge Experiment EXMPLE If we toss die nd then toss coin, how mny different results re possible? This experiment hs two stges tossing the die nd tossing the coin. The number of possible results of tossing die is 6. The number of possible results of tossing coin is. Thus, the number of possible results of the entire experiment is 6. One of the possible results is the number on the die nd tils on the coin. In effect, it mkes no difference whether we first toss the die nd only then toss the coin, or first toss the coin nd then toss the die, or toss both t the sme time. In ech cse, there re possible results. 45

10 Quntittive Resoning Now let us consider multi-stge experiment with set of n items, out of which n item is selected t rndom r times. Ech selection of n item from the set constitutes stge in the experiment, so tht the experiment hs totl of r stges. The number of possible results in ech of the r stges depends on the smpling method by which n item is selected. The totl number of possible results of the entire experiment is the product of the number of possible results obtined in ech of the r stges. Ech possible result in the experiment is referred to s smple. There re four bsic types of multistge experiments. They re designted by the smpling method used: whether or not the order of smpling mtters (clled ordered nd unordered) nd whether or not the smpled item is returned to the originl set (clled with replcement nd without replcement). Ordered Smples with Replcement Smpling method: The smpled item is immeditely returned to the set nd the order in which the items re smpled mtters. Note: In this smpling method, n item my be smpled more thn once. Number of possible results: The number of possible results in ech stge is n. Thus, the number of possible results of ll r stges tht is, of the entire experiment is n n... n n r. The number of ordered smples with replcement is n r. EXMPLE box contins 9 blls, numbered through 9. One bll is removed t rndom from the box nd replced, nd this process is repeted two more times. The numbers on the blls tht re removed from the box re written down in the order in which they re removed, forming threedigit number. How mny different three-digit numbers cn be obtined in this wy? In this question, the order in which the results re obtined is importnt. For exmple, if blls numbered, 8, nd re removed, in tht order, the number 8 is obtined; but if the order in which they re removed is,, nd 8, the result is different number 8. There re stges in this experiment, nd the number of possible results in ech stge is 9. Thus, the number of possible results of the entire experiment is In other words, 79 different three-digit numbers cn be obtined. Ordered Smples Without Replcement Smpling method: The smpled item is not returned to the set fter being smpled, nd the order in which the items re smpled mtters. Number of possible results: The number of possible results in the first stge is n; the number of possible results in the second stge is n (becuse the item tht ws smpled in the first stge is not returned, nd only n items remin to be smpled); nd so on, until the lst stge, stge r, in which the number of possible results is n r +. Thus, the number of possible results of the entire experiment is n (n )... (n r + ). The number of ordered smples without replcement is n (n )... (n r + ). 46

11 Guide For Exminees Inter-University Psychometric Entrnce Test EXMPLE box contins 9 blls, numbered through 9. Three blls re removed t rndom from the box, one fter nother, nd re not replced. The numbers on the blls removed from the box re written down in the order in which they re removed, forming three-digit number. How mny different three-digit numbers cn be obtined in this wy? In this experiment, too, the order in which the blls re removed is importnt, but unlike the previous exmple, in this experiment bll tht is removed from the box is not returned. Thus, the number of possible results in the first stge is 9, in the second stge, 8, nd in the third stge, 7. The number of possible results of the entire experiment is In other words, 504 different three-digit numbers cn be obtined. rrngements (Permuttions) of n Ordered Smple When creting n ordered smple without replcement out of ll n items in set (tht is, if r n), ech possible result describes the rrngement of the items which item is first, which is second, nd so on. The question is: How mny possible rrngements re there? If we substitute r n in the formul for obtining the number of ordered smples without replcement, we obtin n (n ).... This number is clled "n fctoril" nd is written s n!. The number of possible rrngements of n items is n!. EXMPLE grndmother, mother, nd dughter wish to rrnge themselves in row in order to be photogrphed. How mny different wys cn they rrnge themselves? Let us regrd the person on the right s the first in the set, the person in the middle s the second, nd the person on the left s the third. The question is then: How mny possible rrngements re there of the grndmother, mother nd dughter? The grndmother, mother, nd dughter cn be considered set of items. Thus, the number of possible rrngements for this set is! 6. The possible rrngements re: grndmother-mother-dughter, grndmotherdughter-mother, mother-grndmother-dughter, mother-dughter-grndmother, dughtergrndmother-mother, dughter-mother-grndmother. Unordered Smples Smpling method: The smpled item is not returned to the set fter being smpled, nd the order in which the items re smpled does not mtter. When the order does not mtter, ll smples contining the sme r items (only the smpling order is different in ech smple) re regrded s the sme result. The number of smples of this type is ctully the number of rrngements of the r items, tht is, r!. To clculte the possible number of results in n unordered smple, clculte the number of possible results s if the order mtters, nd divide it by the number of rrngements of the r items. The number of unordered smples the number of ordered smples without replcement the number of rrngements in the smple n (n )... (n r + ) r! 47

12 Quntittive Resoning EXMPLE box contins 9 blls, numbered through 9. Three blls re removed t rndom from the box, one fter nother, nd re not replced. The blls tht were removed re plced in ht. How mny possible different combintions re there of the blls in the ht? In this question, the composition of the blls in the ht is importnt nd not the order in which they re removed from the box. For exmple, if the blls re removed in the order of 5,, nd 4, the composition of the blls in the ht is, 4, nd 5. This would be the composition of the blls in the ht even if they re removed from the box in the order of 4, 5, nd or in ny other of the! possible orders: -4-5, -5-4, 4--5, 4-5-, 5--4, nd (ctully, the fct tht the blls re removed one fter nother is irrelevnt; they could be removed t the sme time without ffecting the result.) Thus, the number of possible combintions is 9$ 8$ 7!, which equls 84 different possible combintions of the blls in the ht. PROILITY Probbility theory is mthemticl model for phenomen (experiments) whose occurrence is not certin. Ech possible outcome of n experiment is clled simple event, nd the collection of outcomes is clled n event. (For the ske of brevity, we will use the term "event" to lso denote simple event.) Ech event is ssigned number from 0 to, which represents the probbility (likelihood) tht the event will occur. The higher the probbility, the greter the chnce tht the event will occur. n event tht is certin to occur hs probbility of, nd n event tht hs no possibility of occurring hs probbility of 0. The sum of the probbilities of ll the simple events in n experiment is. When ech of the n-possible results of prticulr experiment hs n equl likelihood of occurring, we sy tht they hve equl probbilities. In such cse, ech result hs probbility of n. EXMPLE Experiment: The tossing of coin. Possible results: Either side of the coin. They re mrked or 0 (heds or tils). If we re tossing fir coin, both results re eqully likely. In other words, the probbility of "" coming up is equl to the probbility of "0" coming up. Thus, the probbility of ech possible result is. EXMPLE Experiment: The tossing of fir die. Possible results: The numbers,,, 4, 5, nd 6 which re mrked on the fces of the die. If we re tossing fir die, the probbility of obtining ech of the possible results is 6. When ll possible results re eqully likely, the probbility tht n event will occur is: the number of possible results of prticulr event the totl number of possible results of the experiment 48

13 Guide For Exminees Inter-University Psychometric Entrnce Test EXMPLE Experiment: The tossing of fir die. Event: The result is less thn 4. Results of this event: the numbers,, nd. Probbility of the event: 6. EXMPLE Experiment: The removl of bll from box contining 5 white blls nd 5 blck blls. Event: The removl of blck bll. Probbility of the event: the number of blck blls 5 0. the totl number of blls in the box The probbility tht two events will occur When two events occur t the sme time or one fter nother, there re two possible scenrios:. The events re independent, tht is, the probbility of one event occurring is not ffected by the occurrence of the other event.. The events re dependent, tht is, the probbility of one event is ffected by the occurrence of nother event. In other words, the probbility of prticulr event occurring fter (or given tht) nother event hs occurred, is different from the probbility of tht prticulr event occurring independent of the other event. EXMPLE There re 0 blls in box, 5 white nd 5 blck. Two blls re removed from the box, one fter nother. The first bll tht is removed is blck. Wht is the probbility tht the second bll tht is removed is lso blck? () The first bll is returned to the box. Since the first bll is returned to the box, there is no chnge either in the totl number of blls in the box or in the number of blck blls. The probbility of removing second blck bll is 5 0 the first bll tht ws removed is blck. nd is equl to the probbility tht The fct tht blck bll ws removed the first time does not chnge the probbility of removing blck bll the second time. In other words, the two events re independent. (b) The first bll is not returned to the box. fter removing blck bll, totl of 9 blls remin in the box, of which 4 re blck. Thus, the probbility of removing second blck bll is 9 4. This probbility is different from the probbility of removing blck bll the first time. Thus, the second event is dependent on the first event. 49

14 Quntittive Resoning The probbility of two independent events occurring (in prllel or one fter nother) is equl to the product of the probbilities of ech individul event occurring. EXMPLE Experiment: The tossing of two fir dice, one red nd the other yellow. Event : obtining number tht is less thn on the red die. The probbility of event is 6. Event : obtining n even number on the yellow die. The probbility of event is 6. Since the result of tossing one die does not ffect the probbility of the result obtined by tossing the other die, event nd event re independent events. The probbility of both event nd event occurring (together) is thus, the probbility the probbility of event of event, tht is, $ 6. Let us define two dependent events, nd (in ny given experiment). The probbility of event occurring given tht event hs occurred is: the number of results common to both nd the number of results of EXMPLE Experiment: The tossing of die. Wht is the probbility of obtining result tht is less thn 4 if we know tht the result obtined ws n even number? Event : n even result is obtined. Event : result tht is less thn 4 is obtined. We will rephrse the question in terms of the events: Wht is the probbility of, given tht we know tht occurred? Event hs results:, 4, nd 6. Event hs results:,, nd. ut, if we know tht event occurred, there is only one possible result for. In other words, the result is the only result tht is common to both nd. Thus, the probbility of, given tht we know tht occurred is. This probbility is different from the probbility of occurring (without our knowing nything bout ), which is equl to. 50 Distnce Problems: Distnce, Speed (Rte), Time The speed (rte) t which n object moves is the distnce tht the object covers in unit of time. The formul for the reltionship between the speed, the distnce the object covers, nd the mount of time it requires to cover tht distnce is: v s t where v speed (rte) s distnce t time

15 Guide For Exminees Inter-University Psychometric Entrnce Test ll possible reltionships between distnce, speed nd time cn be derived from this formul: t v s nd s v t. EXMPLE trin trveled 40 kilometers (km) t speed of 80 kilometers per hour (kph). How long did the journey tke? We re given v (80 kph) nd s (40 km), nd we hve to clculte t. Since the speed is given in kilometers per hour, the trveling time must be clculted in hours. Substituting the given informtion into the formul t v s, we get t Thus, the journey took hours. The unit of mesurement of two of the vribles determines the unit of mesurement of the third vrible. For exmple, if the distnce is expressed in kilometers (km) nd the time in hours, then the speed will be expressed in kilometers per hour (kph). If the distnce is expressed in meters nd the time in seconds, then the speed will be expressed in meters per second. Meters cn be converted to kilometers nd seconds to hours, nd vice vers. There re,000 meters in every kilometer ( meter, 000 kilometer). In every hour, there re,600 seconds, which equl 60 minutes ( second, 600 hour). speed of meter per hour is equl to speed of 5, meters per second c, m. speed of meter per second is equl to speed of.6 kilometers per hour Work Problems: Output (Rte), Work, Time Output is the mount of work per unit of time., 000, 600, , 000 The formul for the reltionship between output, mount of work nd the time needed to do the work is p w t, where p output (rte ) w mount of work t time ll possible reltionships between output, mount of work nd time cn be derived from this formul: t w p nd w p t. 5

16 Quntittive Resoning EXMPLE builder cn finish building one wll in hours. How mny hours would be needed for two builders working t the sme rte to finish building 5 wlls? We re given the mount of work of one builder ( wll), nd the mount of time he spent working ( hours). Therefore his output is of wll in n hour. Since the question involves two builders, the output of the two of them together is $ wlls per hour. We re lso given the mount of work which the two builders must do 5 wlls. We cn therefore clculte the mount of time they will need: t 5 5$ 5 7. Thus, they will need 7 hours to build the wlls. 5

17 Guide For Exminees Inter-University Psychometric Entrnce Test PRLLEL (STRIGHT) LINES Prllel lines tht intersect ny two lines divide those lines into segments tht re proportionl in length. Thus, in the figure, c b, d b d c nd c + b c+ d. Other reltionships between the segments cn be deduced bsed on the bove reltionships. b c d NGLES right ngle is 90 ngle. In ll of the figures, right ngles re mrked by. n cute ngle is n ngle tht is less thn 90. n obtuse ngle is n ngle tht is greter thn 90. stright ngle is 80 ngle. djcent Supplementry ngles The two ngles tht re formed between stright line nd ry tht extends from point on the stright line re clled djcent supplementry ngles. Together they form stright ngle nd their sum therefore equls 80. For exmple, in the figure, x nd y re djcent supplementry ngles; thus, x + y 80. y x Verticl ngles When two stright lines intersect, they form four ngles. Ech pir of non-djcent ngles re clled verticl ngles nd they re equl in size. For exmple, in the figure, x nd z re verticl ngles, s re w nd y. Therefore, x z nd w y. x y w z When two prllel lines re intersected by nother line (clled trnsversl), eight ngles re formed. In the figure, these ngles re designted, b, c, d, e, f, g, nd h. orresponding ngles ngles locted on the sme side of the trnsversl nd on the sme side of the prllel lines re referred to s corresponding ngles. orresponding ngles re equl. Thus, in the figure, e, b f, c g, d h. b c d e f g h lternte ngles ngles locted on opposite sides of the trnsversl nd on opposite sides of the prllel lines re clled lternte ngles. lternte ngles re equl. Thus, in the figure, h, b g, c f, d e. 5

18 Quntittive Resoning EXMPLE Given: Stright lines p nd q re prllel. d + f? c nd d re djcent supplementry ngles. Therefore, d + c 80. c nd f re lternte ngles. Therefore, c f. Thus, d + f d + c 80, nd the nswer is 80. p q b c d e f g h TRINGLES ngles of Tringle The sum of the interior ngles of ny tringle is 80. For exmple, in the figure, + b + g 80. n ngle formed by the extension of one side of tringle nd the djcent side is clled n exterior ngle, nd it equls the sum of the other two ngles of the tringle. For exmple, in the figure, d is the ngle djcent to b, nd therefore δ α + γ. In ny tringle, the longer side lies opposite the lrger ngle. For exmple, in the figure, if γ < α < β, it follows tht side (which is opposite ngle β) is longer thn side (which is opposite ngle ), nd side is longer thn side (which is opposite ngle g). α δ β γ The medin of tringle is line segment tht joins vertex of tringle to the midpoint of the opposite side. For exmple, in the tringle in the figure, D is the medin to side (D D). D ltitude of Tringle The ltitude to side of tringle is line tht is drwn from vertex of the tringle to the opposite side (or its extension) nd is perpendiculr to tht side. For exmple, in the tringles in the figures, h is the ltitude to side. h re of Tringle The re of tringle equls hlf the length of one of the sides multiplied by the ltitude to tht side. For exmple, the re of ech tringle in the bove figures is h $. h Inequlity in Tringle In ny tringle, the sum of the lengths of ny two sides is greter thn the length of the third side. For exmple, in the tringles in the figures, + >. 54

19 Guide For Exminees Inter-University Psychometric Entrnce Test ongruent Tringles Two geometric figures re congruent if one of them cn be plced on the other in such wy tht the two coincide. ongruent tringles re one exmple of congruent geometric figures. In congruent tringles, the corresponding sides nd ngles re equl. α For exmple, if tringles nd DEF in the figure re congruent, then their corresponding sides re equl: DE, EF, nd DF, nd their corresponding ngles re equl: α δ, β τ, nd γ ε. Ech of the following four theorems enbles us to deduce tht two tringles re congruent: E β τ ε γ F () Two tringles re congruent if two sides of one tringle equl the two corresponding sides of the other tringle nd the ngle between these sides in one tringle equls the corresponding ngle in the other tringle (Side-ngle-Side SS). δ D For exmple, if DE, DF, nd α δ, then the two tringles in the figure re congruent. (b) Two tringles re congruent if two ngles of one tringle equl the two corresponding ngles of nother tringle, nd the length of the side between these ngles in one tringle equls the length of the corresponding side in the other tringle (ngle-side-ngle S). For exmple, if α δ, β τ, nd DE, then the two tringles in the figure re congruent. (c) Two tringles re congruent if the three sides of one tringle equl the three sides of the other tringle (Side-Side-Side SSS). (d) Two tringles re congruent if two sides of one tringle equl the corresponding two sides of the other tringle, nd the ngle opposite the longer of the two sides of one tringle is equl to the corresponding ngle in the other tringle (Side-Side-ngle SS). For exmple, the tringles in the figure bove re congruent if > nd DE > DF; nd DE, DF, nd γ ε. Similr Tringles Two tringles re similr if the three ngles of one tringle re equl to the three ngles of the other tringle. In similr tringles, the rtio between ny two sides of one tringle is the sme s the rtio between the corresponding two sides of the other tringle. For exmple, in the figure, tringles nd DEF re similr. DE Therefore,. DF It lso follows tht DE. DF EF E D F ongruent tringles re necessrily lso similr tringles. 55

20 Quntittive Resoning TYPES OF TRINGLES n equilterl tringle is tringle whose sides re ll of equl length. For exmple, in the figure,. In tringle of this type, ll of the ngles re lso equl (60 ). 60 If the length of the side of such tringle is, then its ltitude is $ nd its re is $ n isosceles tringle is tringle with two sides of equl length. For exmple, in the figure,. The third side of n isosceles tringle is clled the bse. The two ngles opposite the equl sides re equl. For exmple, in the figure, β γ. β γ n cute tringle is tringle in which ll the ngles re cute. n obtuse tringle is tringle with one obtuse ngle. right tringle is tringle with one right ngle (90 ). The side opposite the right ngle (side in the figure) is clled the hypotenuse, nd the other two sides re clled legs (sides nd in the figure). ccording to the Pythgoren theorem, in right tringle the squre of the hypotenuse is equl to the sum of the squres of the legs. For exmple, in the figure, +. leg leg hypotenuse This formul cn be used to find the length of ny side if the lengths of the other two sides re given. In right tringle whose ngles mesure 0, 60 nd 90, the length of the leg opposite the 0 ngle equls hlf the length of the hypotenuse. 0 For exmple, in the figure, the length of the hypotenuse is. Therefore, the length of the leg opposite the 0 ngle is. It follows from the Pythgoren theorem tht the length of the leg opposite the 60 ngle is. 60 In n isosceles right tringle, the ngles mesure 45, 45, nd 90, the two legs re of equl length, nd the length of the hypotenuse is times greter thn the length of the legs (bsed on the Pythgoren theorem) For exmple, in the figure, the length of ech leg is, nd therefore the length of the hypotenuse is. 56

21 Guide For Exminees Inter-University Psychometric Entrnce Test QUDRILTERLS qudrilterl is ny four-sided polygon. For exmple: Rectngles nd Squres rectngle is qudrilterl whose ngles re ll right ngles. In rectngle, ech pir of opposite sides is equl in length. The perimeter of the rectngle in the figure is + b ( + b). The length of digonl of rectngle in the figure is + b (bsed on the Pythgoren theorem). The re of rectngle is the product of the lengths of two djcent sides. The re of the rectngle in the figure is b. b +b b D squre is rectngle whose sides re ll of equl length. The perimeter of the squre in the figure is 4. D The length of digonl of the squre in the figure is +. The re of squre equls the squre of the length of its side. The re of the squre in the figure is. Prllelogrms nd Rhombuses prllelogrm is qudrilterl in which ech pir of opposite sides is prllel nd of equl length. For exmple, in the prllelogrm in the figure: D, D D, D D The digonls of prllelogrm bisect ech other. b h b The perimeter of the prllelogrm in the figure is + b. The ltitude of prllelogrm is line tht connects two opposite sides (or their extensions) nd is perpendiculr to them. The re of prllelogrm equls the product of side multiplied by the ltitude to tht side. For exmple, the re of the prllelogrm in the figure is h. rhombus is qudrilterl whose four sides re ll equl. Ech pir of opposite sides in rhombus is prllel, nd it cn therefore be regrded s prllelogrm with equl sides. Digonls of rhombus Since rhombus is type of prllelogrm, its digonls lso bisect ech other. In rhombus, the digonls re lso perpendiculr to ech other. h D 57

22 Quntittive Resoning The perimeter of the rhombus in the figure is 4. re of rhombus Since rhombus is type of prllelogrm, its re, too, cn be clculted by multiplying side by the ltitude to tht side. For exmple, the re of the rhombus in the figure is h. The re of rhombus cn lso be clculted s hlf the product of the length of its digonls. For exmple, the re of the rhombus in the figure is D $. Trpezoid trpezoid is qudrilterl with only one pir of prllel sides. The prllel sides re clled bses, nd the other two sides re clled legs. The bses of trpezoid re not equl, nd re therefore referred to s the long bse nd the short bse. The ltitude of trpezoid is line tht joins the bses of the trpezoid nd is perpendiculr to them. D h b The re of trpezoid equls hlf the sum of the bse lengths multiplied by the ltitude. For exmple, the re of the trpezoid in the figure is h $ ] + b g. n isosceles trpezoid is trpezoid whose legs re of equl length. For exmple, in the figure, D. The ngles of the long bse of n isosceles trpezoid re equl, s re the ngles of the short bse. For exmple, in the figure, «D «D α, nd ««D β. In n isosceles trpezoid, if two ltitudes re drwn from the ends of the short bse to the long bse, rectngle nd two congruent right tringles re obtined (P nd DQ). D α α β β P Q right trpezoid is trpezoid in which one of the legs forms right ngle with ech of the bses. D Kite (Deltoid) kite is qudrilterl formed by two isosceles tringles joined t their bses. For exmple, kite D in the figure is composed of tringles D nd D ( D, D). The digonl joining the vertices of the two isosceles tringles bisects the digonl tht is the bse of these two tringles nd is perpendiculr to it. For exmple, in the figure, bisects D (P PD) nd D. The perimeter of the kite in the figure is + b. The re of kite equls hlf the product of the lengths of the digonls. b P b D For exmple, the re of the kite in the figure is D $. 58

23 Guide For Exminees Inter-University Psychometric Entrnce Test REGULR POLYGON regulr polygon is polygon whose sides re ll of equl length nd whose interior ngles re ll equl. Exmples: regulr octgon is regulr polygon with 8 sides. regulr pentgon is regulr polygon with 5 sides. squre is regulr polygon with 4 sides. n equilterl tringle is regulr polygon with sides. The size of the interior ngle of regulr polygon with n sides cn be clculted using the formul α 80c 60c 80c n k n 60c n k. α For exmple, in regulr hexgon, such s in the figure, ech of the interior ngles is 0 : α 80c 60c 6 0c. IRLE rdius is line segment tht joins the center of circle to point on its circumference. chord of circle is line segment tht psses though the circle nd joins two different points on its circumference. dimeter of circle is chord tht psses through its center. The length of circle's dimeter is twice the length of its rdius. If the rdius of circle is r, the dimeter of the circle is r. The circumference of circle with rdius r is pr. (The vlue of π is pproximtely.4.) The re of circle with rdius r is pr. n rc is prt of the circle's circumference bounded by two points. sector is prt of the re of circle bounded by two rdii nd n rc. Inscribed ngle n inscribed ngle is n ngle whose vertex lies on the circumference of circle nd whose sides re chords of the circle. Inscribed ngles intercepting the sme rc re equl. β For exmple, in the figure, ngles nd b re inscribed ngles, both of which intercept rc ; therefore, α β. α n inscribed ngle tht lies on the dimeter of circle (tht is, it intercepts n rc whose length equls hlf the circle's circumference) is right ngle. entrl ngle centrl ngle is n ngle whose vertex is the center of the circle nd whose sides re rdii of the circle. centrl ngle is twice the size of ny inscribed ngle tht intercepts the sme rc. For exmple, in the figure, is centrl ngle nd β is n inscribed ngle, nd both intercept the sme rc. Therefore, α β. β α 59

24 Quntittive Resoning rc Length Two points on the circumference of circle define two rcs. For exmple, in the figure, points nd define two rcs one corresponding to centrl ngle α nd the other corresponding to centrl ngle b. The minor rc corresponds to α, the smller of the two ngles. If r is the rdius of the circle, then the length of this rc is πr $ α 60. α β re of Sector The ngle formed between the two rdii tht bound sector is centrl ngle. For exmple, the shded region in the figure is the sector of circle with centrl ngle x. The re of the sector of the circle is π r $ x. 60 Tngent to ircle tngent to circle is line tht touches the circumference of circle t only one point, clled the point of tngency. The ngle formed by the rdius nd the tngent t the point of tngency is right ngle. For exmple, in the figure, line segment is tngent to the circle whose rdius is r. r xº r r Two tngents to the sme circle tht intersect t prticulr point re sid to originte t the sme point. The length of ech tngent is the length of the segment tht joins the tngents' point of intersection nd the point of tngency. Tngents to circle tht originte t the sme point re equl in length. For exmple, in the figure, is the point of intersection, nd re the points of tngency, nd therefore. Polygon ircumscribing ircle polygon tht circumscribes circle is polygon whose sides re ll tngent to the circle. Polygon Inscribed in ircle polygon inscribed in circle is polygon whose vertices ll lie on the circumference of the circle. Inscribed Tringle ny tringle cn be inscribed in circle. Every tringle hs only one circle tht circumscribes it. If n inscribed tringle is right tringle, the center of the circle tht circumscribes it is the midpoint of the tringle's hypotenuse. Qudrilterl Inscribed in ircle Not every qudrilterl cn be inscribed in circle. The sum of the opposite ngles of qudrilterl inscribed in circle lwys equls 80. For exmple, in the qudrilterl in the figure, α + γ 80 β + δ 80 60

25 Guide For Exminees Inter-University Psychometric Entrnce Test Qudrilterl ircumscribing ircle Not every qudrilterl cn circumscribe circle. When qudrilterl circumscribes circle, the sums of the lengths of ech pir of opposite sides is equl. For exmple, in the qudrilterl in the figure, + c b + d. d c b When squre circumscribes circle, the length of the side of the squre equls the length of the dimeter of the circle. 6

26 Quntittive Resoning SOLIDS ox (Rectngulr Prism) nd ube box is three-dimensionl figure with six rectngulr fces. The box's three dimensions re its length, width nd height (, b nd c respectively, in the figure). Every fce of box is perpendiculr to the fces djcent to it. c b The surfce re of box is the sum of the res of its fces. The surfce re of the box in the figure is b + c + bc + b + c + bc b + c + bc. The volume of box is the product of its length, width nd height. The volume of the box in the figure is b c. cube is box whose length, width nd height re ll equl. ll of the fces of cube re congruent squres. The re of ech fce of the cube in the figure is d. Therefore, the surfce re of the cube is 6d. The volume of the cube in the figure is d. d d d ylinder cylinder is three-dimensionl figure whose two bses re congruent circles on prllel plnes joined by lterl surfce. The line joining the centers of the circles is perpendiculr to ech of the bses. The lterl surfce re of cylinder with bse rdius of length r nd height h equls the circumference of the bse multiplied by the height, tht is, pr h. The totl surfce re of cylinder is the sum of the res of the bses nd the lterl surfce. The re of ech bse is pr nd the lterl surfce re is pr h. Thus, the totl surfce re is pr h + pr pr (h + r). The volume of cylinder equls the re of one of the bses multiplied by the height, tht is, pr h. r h one right cone is three-dimensionl figure formed by joining the points on the circumference of circle to point lying outside the plne of the circle. This point is clled the cone's vertex nd it lies on line tht psses through the center of the circle nd is perpendiculr to the plne of the circle (see figure). The volume of cone with bse rdius r nd height h is π r h. $ h r 6 Prism right prism is three-dimensionl figure whose two bses re congruent polygons on prllel plnes nd whose lterl fces re rectngles. prism is referred to by the number of sides of its bse. For exmple, tringulr prism hs three-sided bses, qudrngulr prism hs four-sided bses, nd so on (see figures).

27 Guide For Exminees Inter-University Psychometric Entrnce Test The height of prism is the length of the segment tht joins the bses of the prism nd is perpendiculr to them. This is the distnce between the bses of the prism. The lterl surfce re of the prism is the sum of the res of ll the lterl fces. The lterl surfce cn lso be clculted by multiplying the perimeter of the prism's bse by the height of the prism. The totl surfce re of prism is the sum of the lterl surfce re nd the res of the two bses. The volume of prism equls the re of one of the bses multiplied by the height of the prism. Pyrmid right pyrmid is three-dimensionl figure formed by joining the vertices of ny regulr polygon to point outside the plne of the polygon. The polygon is clled the bse of the pyrmid nd the point is the pyrmid's vertex or pex. The lterl fces of pyrmid re tringles. pyrmid is referred to by the number of sides of its bse. For exmple, tringulr pyrmid hs three-sided bse, qudrngulr pyrmid hs four-sided bse, nd so on (see figure). The height of pyrmid is the length of the line segment extending perpendiculrly from the pyrmid's vertex to its bse. This is the distnce between the pyrmid's vertex nd bse (see figure). If S is the re of the pyrmid's bse nd h is the pyrmid's height, then the pyrmid's volume is S $ h h Edge The edge of three-dimensionl figure is the stright line formed where two fces meet. The bold line in the pyrmid in the bove figure is one of its edges. box hs edges. NUMER LINE (XIS) number line is geometric representtion of the reltionships between numbers The numbers long the xis increse to the right. The distnce between points on the xis is proportionl to the difference between the numericl vlues corresponding to the points. For exmple, the distnce between the points corresponding to the vlues (-4) nd (-) is equl to the distnce between the points corresponding to the vlues nd 5. 6

28 Quntittive Resoning RTESIN OORDINTE SYSTEM In crtesin coordinte system in plne, there re two number lines (xes) tht re perpendiculr to ech other. The horizontl line is clled the x-xis nd the verticl line is clled the y-xis. The numbers long the x-xis increse to the right. The numbers long the y-xis increse upwrds. The xes divide the plne into four qudrnts, designted in the figure by Romn numerls I, II, III, nd IV. Ech point in the coordinte plne corresponds to pir of x nd y vlues which describe their loction reltive to the xes. For exmple, in the figure, the x-vlue of point is 4, nd its y-vlue is. The x-vlue of point in the figure is (-) nd its y-vlue is. It is customry to write the x- nd y-vlues of the points in prentheses, with the x-vlue to the left of the y-vlue: (x, y). Sometimes, the vlues of point re written next to the letter representing the point, for exmple, (4, ) nd (-, ). The x- nd y-vlues of point re sometimes clled the coordintes of tht point. The point in the plne corresponding to (0, 0) is the point of intersection of the two xes nd is clled the origin. ll points on line prllel to the x-xis hve the sme y-coordinte, nd ll points on line prllel to the y-xis hve the sme x-coordinte. For exmple, in the figure, line k is prllel to the y-xis. Thus, ll of the points on line k hve the sme x-coordinte (in the figure, x.5). Line m is prllel to the x-xis. Thus, ll of the points on line m hve the sme y-coordinte (in the figure, y.5). Only one line cn be drwn through ny two points on plne. The prt of the line tht is locted between the two points is clled line segment. II m III y 4 I IV y k x x If line segment is prllel to the y-xis, its length is the difference (in bsolute vlue) between the y-coordintes of the two points. For exmple, in the figure, line segment is prllel to the y-xis. The y-coordinte of point is 4 nd the y-coordinte of point is (-). The difference between the vlues of the y-coordintes of the two points is 4 (-) 7. Therefore, the length of line segment is 7. The length of line segment prllel to the x-xis is clculted in the sme wy y x 64