Cheukting/mathqa huggingface dataset:
Question (general): if a ^ 2 + b ^ 2 = 15 and ab = 25 , what is the value of the expression ( a - b ) ^ 2 + ( a + b ) ^ 2 ?
Options: a ) 10 , b ) 20 , c ) 30 , d ) 60 , e ) 50
Rationale: "( a - b ) ^ 2 = a ^ 2 + b ^ 2 - 2 ab = 15 - 25 = - 10 ( a + b ) ^ 2 = a ^ 2 + b ^ 2 + 2 ab = 15 + 25 = 40 so ( a + b ) ^ 2 + ( a - b ) ^ 2 = 40 - 10 = 50 e"
Answer: e
Question (general): equal amount of water were poured into two empty jars of different capacities , which made one jar 1 / 3 full and other jar 1 / 2 full . if the water in the jar with lesser capacity is then poured into the jar with greater capacity , what fraction of the larger jar will be filled with water ?
Options: a ) 2 / 3 , b ) 2 / 7 , c ) 1 / 2 , d ) 7 / 12 , e ) 1 / 7
Rationale: same amount of water made bigger jar 1 / 3 full , thenthe same amount of water ( stored for a while in smaller jar ) were added to bigger jar , so bigger jar is 1 / 3 + 1 / 3 = 2 / 3 full . answer : a .
Answer: a
Question (gain): a certain candy manufacturer reduced the weight of candy bar m by 5 percent buy left the price unchanged . what was the resulting percent increase in the price per ounce of candy bar m ?
Options: a ) 5.5 , b ) 5.4 , c ) 5.3 , d ) 5.2 , e ) 5.1
Rationale: assume 1 oz candy cost $ 1 before . now price remain same $ 1 but weight of candy reduces to 0.95 oz new price of candy = 1 / 0.95 = 1.052 price increase 5.2 % d
Answer: d
Question (general): if | x - 9 | = 3 x , then x = ?
Options: a ) 1.5 , b ) 3.1 , c ) 4.5 , d ) - 2.25 , e ) - 4.5
Rationale: "| x - 9 | = 3 x . . . ( given ) x ^ 2 - 18 x + 81 = 9 x ^ 2 8 * x ^ 2 + 18 * x - 81 = 0 . . . . ( by solving above eq . we get ) , x = - 4.5 or 2.25 = = = > ans - e"
Answer: e
Question (general): mary ' s income is 60 percent more than tim ' s income , and tim ' s income is 60 percent less than juan ' s income . what percent of juan ' s income is mary ' s income ?
Options: a ) 64 % , b ) 120 % , c ) 96 % , d ) 80 % , e ) 64 %
Rationale: "juan ' s income = 100 ( assume ) ; tim ' s income = 40 ( 60 percent less than juan ' s income ) ; mary ' s income = 64 ( 60 percent more than tim ' s income ) . thus , mary ' s income ( 64 ) is 64 % of juan ' s income ( 100 ) . answer : a ."
Answer: a
Question (general): 12 . what is the dividend . divisor 17 , the quotient is 9 and the remainder is 7 .
Options: a ) 150 , b ) 152 , c ) 154 , d ) 158 , e ) 160
Rationale: d = d * q + r d = 17 * 9 + 7 d = 153 + 7 d = 160 answer e
Answer: e
Question (gain): if the simple interest on a certain amount in at 4 % rate 5 years amounted to rs 1920 less than the principal . what was the principal ?
Options: a ) 1500 , b ) 2500 , c ) 2507 , d ) 3200 , e ) 2400
Rationale: "p - 1920 = ( p * 5 * 4 ) / 100 p = 2400 answer : e"
Answer: e
Question (geometry): the diagonals of a rhombus are 15 cm and 20 cm . find its area ?
Options: a ) 158 , b ) 129 , c ) 150 , d ) 123 , e ) 117
Rationale: "1 / 2 * 15 * 20 = 150 answer : c"
Answer: c
Question (general): how many different positive integers exist between 10 ^ 3 and 10 ^ 4 , the sum of whose digits is equal to 2 ?
Options: a ) 6 , b ) 7 , c ) 5 , d ) 4 , e ) 18
Rationale: "so , the numbers should be from 1,000 to 10,000 the following two cases are possible for the sum of the digits to be 2 : 1 . two 1 ' s and the rest are 0 ' s : 1,001 1,010 1,100 3 numbers . 2 . one 2 and the rest are 0 ' s : 2,000 1 number . total = 4 numbers . answer : d"
Answer: d
Question (general): if 1 / ( x + 4 ) + 1 / ( x - 4 ) = 1 / ( x - 4 ) , what is the value of x ?
Options: a ) 5 , b ) 4 , c ) 1 , d ) 2 , e ) 3
Rationale: "if we solve the question , we get x = 4 . option : b"
Answer: b
Question (general): 8 : 18 : : 24 : ?
Options: a ) 38 , b ) 32 , c ) 44 , d ) 43 , e ) 42
Rationale: "3 ^ 2 - 1 = 8 4 ^ 2 + 2 = 18 5 ^ 2 - 1 = 24 there for 6 ^ 2 + 2 = 38 answer : a"
Answer: a
Question (general): in n is a positive integer less than 200 , and 2 n / 108 is an integer , then n has how many different positive prime factors ?
Options: a ) 2 , b ) 3 , c ) 5 , d ) 6 , e ) 8
Rationale: "( b ) . 2 n / 108 must be an integer . = > n / 54 must be an integer . hence n must be a multiple of 2 * 3 * 9 . = > n has 3 different prime integers ."
Answer: b
Question (gain): what percent is 100 gm of 1 kg ?
Options: a ) 25 % , b ) 40 % , c ) 10 % , d ) 8 % , e ) 12 %
Rationale: "1 kg = 1000 gm 100 / 1000 × 100 = 10000 / 1000 = 10 % answer is c"
Answer: c
Question (general): the average of 9 observations was 8 , that of the 1 st of 5 being 10 and that of the last 5 being 8 . what was the 5 th observation ?
Options: a ) 18 , b ) 12 , c ) 15 , d ) 17 , e ) 19
Rationale: explanation : 1 to 9 = 8 * 9 = 72 1 to 5 = 5 * 10 = 50 5 to 9 = 5 * 8 = 40 5 th = 50 + 40 = 90 – 72 = 18 option a
Answer: a
Question (general): sophia finished 2 / 3 of a book . she calculated that she finished 30 more pages than she has yet to read . how long is her book ?
Options: a ) a ) 90 , b ) b ) 150 , c ) c ) 130 , d ) d ) 190 , e ) e ) 210
Rationale: "let x be the total number of pages in the book , then she finished 2 / 3 * x pages . then she has x − 2 / 3 * x = 1 / 3 * x pages left . 2 / 3 * x − 1 / 3 * x = 30 1 / 3 * x = 30 x = 90 so the book is 90 pages long . answer is a ."
Answer: a
Question (gain): in a recent election , geoff received 0.5 percent of the 6,000 votes cast . to win the election , a candidate needed to receive more than x % of the vote . if geoff needed exactly 3,571 more votes to win the election , what is the value of x ?
Options: a ) 50 , b ) 54 , c ) 56 , d ) 60 , e ) 63
Rationale: "we are told that a candidate needs to receive more than x % of the vote . we got that a candidate needs to receive more than 3,600 votes ( 3,601 ) . 3,600 is 60 % of total votes . thus x = 60 % . answer : d"
Answer: d
Question (general): which expression is the greatest ?
Options: a ) 3252 / 3257 , b ) 3456 / 3461 , c ) 3591 / 3596 , d ) 3346 / 3351 , e ) 3453 / 3458
Rationale: let 3252 / 3257 be the original fraction . notice that it is a positive proper fraction . hence adding the same constant to both the numerator and the denominator will increase the fraction as the fraction will move closer to one . hence we just have to pick the number that adds the greatest value to both the numerator and the denominator . options can be re - written as ( x - 5 ) x = > 1 - ( 5 / x ) a ) 1 - ( 5 / 3257 ) b ) 1 - ( 5 / 3461 ) c ) 1 - ( 5 / 3596 ) d ) 1 - ( 5 / 3351 ) e ) 1 - ( 5 / 3458 ) answer : c
Answer: c
Question (physics): the end of a blade on an airplane propeller is 10 feet from the center . if the propeller spins at the rate of 1,320 revolutions per second , how many miles will the tip of the blade travel in one minute ? ( 1 mile = 5,280 feet )
Options: a ) 200 π , b ) 240 π , c ) 300 π , d ) 480 π , e ) 1,200 π
Rationale: "distance traveled in 1 revolution = 2 π r = 2 π 10 / 5280 revolutions in one second = 1320 revolutions in 60 seconds ( one minute ) = 1320 * 60 total distance traveled = total revolutions * distance traveled in one revolution 1320 * 60 * 2 π 10 / 5280 = 300 π c is the answer"
Answer: c
Question (general): if there are 200 questions in a 3 hr examination . among these questions are 100 type a problems , which requires twice as much as time be spent than the rest of the type b problems . how many minutes should be spent on type a problems ?
Options: a ) 72 min , b ) 62 min , c ) 70 min , d ) 74 min , e ) 120 min
Rationale: "x = time for type b prolems 2 x = time for type a problem total time = 3 hrs = 180 min 100 x + 100 * 2 x = 180 x = 180 / 300 x = 0.6 time taken for type a problem = 100 * 2 * 0.6 = 120 min answer : e"
Answer: e
Question (physics): a and b are 90 km apart they start to move each other simultaneously a at speed 10 and b at 5 km / hr if every hour they double their speed what is distance that a pass until he meet b ?
Options: a ) 30 km , b ) 40 km , c ) 50 km , d ) 60 km , e ) 70 km
Rationale: in 2 hours a covers ( 10 + 20 ) = 30 km b covers ( 5 + 10 ) = 15 km , total covered = 45 in the next hour they have to cover ( 90 - 45 ) km with the speeds 40 km / hr and 20 km / hr respectively let them take another t hrs to meet so sum of the distance to be covered by both is 45 km ( 40 * t ) + ( 20 * t ) = 45 km gives t = ( 3 / 4 ) hrs we need to fid out the distance covered by a which equals to = 30 + ( 40 * ( 3 / 4 ) ) = 60 km answer : d
Answer: d
Question (general): a card board of size 34 * 14 has to be attached to a wooden box and a total of 35 pins are to be used on the each side of the card box . find the total number of pins used .
Options: a ) 135 , b ) 136 , c ) 137 , d ) 138 , e ) 139
Rationale: total 35 pins are there and 4 sides of card board . . . so 35 x 4 = 140 now in the rectangle 4 vertices have 4 pins which is common to the sides . . . . so 140 - 4 = 136 answer : b
Answer: b
Question (physics): a cistern can be filled by a tap in 2 hours while it can be emptied by another tap in 4 hours . if both the taps are opened simultaneously then after how much time will the cistern get filled ?
Options: a ) 2 hr , b ) 5 hr , c ) 7 hr , d ) 1 hr , e ) 4 hr
Rationale: "net part filled in 1 hour 1 / 2 - 1 / 4 = 1 / 4 the cistern will be filled in 4 hr answer is e"
Answer: e
Question (general): evaluate combination 98 c 96 = 98 ! / ( 96 ) ! ( 2 ) !
Options: a ) 1617 , b ) 4753 , c ) 1417 , d ) 4353 , e ) none of these
Rationale: explanation : ncr = n ! / ( r ) ! ( n − r ) ! 98 c 96 = 98 ! / ( 96 ) ! ( 2 ) ! = 98 ∗ 97 * 96 ! / ( 96 ) ! ( 2 ) ! = 98 * 97 / 2 ∗ 1 = 4753 option b
Answer: b
Question (general): a ranch has both horses and ponies . exactly 3 / 10 of the ponies have horseshoes , and exactly 5 / 8 of the ponies with horseshoes are from iceland . if there are 4 more horses than ponies , what is the minimum possible combined number of horses and ponies on the ranch ?
Options: a ) 116 , b ) 132 , c ) 148 , d ) 164 , e ) 180
Rationale: 3 / 10 * p are ponies with horseshoes , so p is a multiple of 10 . 5 / 8 * 3 / 10 * p = 3 / 16 * p are icelandic ponies with horseshoes , so p is a multiple of 16 . the minimum value of p is 80 . then h = p + 4 = 84 . the minimum number of horses and ponies is 164 . the answer is d .
Answer: d
Question (general): there are 5 chess amateurs playing in villa ' s chess club tournament . if each chess amateur plays with exactly 8 other amateurs , what is the total number of chess games possible to be played in the tournament ?
Options: a ) 10 , b ) 20 , c ) 40 , d ) 60 , e ) 120
Rationale: "method 2 : each person is one participant of 4 games . so there are in all 8 * 5 = 40 instances of one participant games . but each game has 2 participants so total number of games = 40 / 2 = 20 b"
Answer: b
Question (physics): the speed of a train is 90 kmph . what is the distance covered by it in 12 minutes ?
Options: a ) 15 , b ) 18 , c ) 77 , d ) 52 , e ) 42
Rationale: 90 * 12 / 60 = 18 kmph answer : b
Answer: b
Question (other): in a chess competition involving some men and women , every player needs to play exactly one game with every other player . it was found that in 45 games , both the players were women and in 190 games , both players were men . what is the number of games in which one person was a man and other person was a woman ?
Options: a ) 40 , b ) 120 , c ) 180 , d ) 200 , e ) 220
Rationale: this is similar to handshake problem , if there are nn people , then number of handshakes = 1 + 2 + . . . n = n ∗ ( n − 1 ) / 2 no . of women be xx and given x ∗ ( x − 1 ) / 2 = 45 this gives x = 10 | | | y no . of men : solve y ∗ ( y − 1 ) / 2 = 190 . this gives y = 20 so total men vs . women = 10 * 20 = 200 . answer is d
Answer: d
Question (gain): 11 is subtracted from 40 % of a number and results is 23 . what is the number ?
Options: a ) 160 , b ) 92 , c ) 88 , d ) 85 , e ) 34
Rationale: "( 40 / 100 ) * x â € “ 11 = 23 4 x = 340 x = 85 answer : d"
Answer: d
Question (general): 0.1 + 0.1 ^ 2 + 0.1 ^ 3 =
Options: a ) 0.1 , b ) 0.111 , c ) 0.1221 , d ) 0.2341 , e ) 0.3
Rationale: "you have a mistake in your question . choice b should be ( . 111 ) and not ( . 11 ) . the equation in the solution can be simplified to ( . 1 + . 01 + . 001 ) . adding those together you get : . 100 . 010 . 001 - - - - - - . 111 - which is answer choice ( b )"
Answer: b
Question (gain): an article is bought for rs . 675 and sold for rs . 900 , find the gain percent ?
Options: a ) 33 1 / 8 % , b ) 33 9 / 3 % , c ) 33 1 / 3 % , d ) 93 1 / 3 % , e ) 33 2 / 3 %
Rationale: "675 - - - - 225 100 - - - - ? = > 33 1 / 3 % answer : c"
Answer: c
Question (physics): in a garment industry , 12 men working 8 hours per day complete a piece of work in 10 days . to complete the same work in 8 days , working 15 hours a day , the number of men required is :
Options: a ) 4 , b ) 5 , c ) 6 , d ) 8 , e ) 9
Rationale: "explanation : let the required number of men be x . less days , more men ( indirect proportion ) more working hrs per day , less men ( indirect proportion ) days 8 : 10 working hrs 15 : 8 : : 12 : x = > 8 x 15 x x = 10 x 8 x 12 = > x = 10 x 8 x 12 / ( 8 x 15 ) = > x = 8 answer : d"
Answer: d
Question (physics): what is the maximum number of pieces of birthday cake of size 4 ” by 4 ” that can be cut from a cake 20 ” by 20 ” ?
Options: a ) 5 , b ) 10 , c ) 16 , d ) 20 , e ) 25
Rationale: "each length = 20 ' ' each piece = 4 ' ' cake could be divided to maximum of 25 pieces answer : e"
Answer: e
Question (gain): the present worth of rs . 845 due in 2 years at 4 % per annum compound interest is
Options: a ) rs . 150.50 , b ) rs . 154.75 , c ) rs . 781.25 , d ) rs . 158 , e ) none
Rationale: "solution present worth = rs . [ 845 / ( 1 + 4 / 100 ) ² ] = rs . ( 845 x 25 / 26 x 25 / 26 ) = rs . 781.25 answer c"
Answer: c
Question (gain): on the first of the year , james invested x dollars at proudstar bank in an account that yields 4 % in interest every quarter year . at the end of the year , during which he made no additional deposits or withdrawals , he had y dollars in the account . if james had invested the same amount in an account which pays interest on a yearly basis , what must the interest rate be for james to have y dollars at the end of the year ?
Options: a ) 22.04 % , b ) 26.12 % , c ) 28 % , d ) 16.25 % , e ) 30 %
Rationale: per quarter , interest = 4 % so for a year , interest = 16 % due to quarter cumulation , effective yield ( ytm ) would be slight higher than 16.25 % answer = 16.25 % = answer : d
Answer: d
Question (general): if a store owner increases a product ’ s price by 20 percent and then increases this price by another 15 percent , what percent of the original price is the total price increase ?
Options: a ) 20 % , b ) 35 % , c ) 38 % , d ) 65 % , e ) 135 %
Rationale: let the initial price be 100 1 st increase = 20 % the price now is 120 2 nd increase of 15 % on 120 = 18 final price = 138 , total increase = 38 % answer : c
Answer: c
Question (general): if 5 students are made to sit in a row , then 6 do n ' t have seats . if 12 students are made to sit in a row , then 3 rows are empty . how many students have to sit a row such that each row has equal number of students and all students are seated ?
Options: a ) 0 , b ) 1 , c ) 4 , d ) 5 , e ) 6
Rationale: let there be a rows . then 5 a + 6 = 12 ( a - 3 ) [ equating the number of students ] = > a = 6 and total number of students = 36 therefore 6 students much sit in each row such that each row has an equal number of students and all students are seated . option ( e )
Answer: e
Question (general): a pupil ' s marks were wrongly entered as 85 instead of 33 . due to that the average marks for the class got increased by half . the number of pupils in the class is :
Options: a ) 30 , b ) 104 , c ) 20 , d ) 25 , e ) 26
Rationale: "let there be x pupils in the class . total increase in marks = ( x * 1 / 2 ) = x / 2 . x / 2 = ( 85 - 33 ) = > x / 2 = 52 = > x = 104 . answer : b"
Answer: b
Question (other): if the ratio of two number is 6 : 7 and lcm of the number is 450 then what is the number .
Options: a ) 11 , b ) 20 , c ) 25 , d ) 30 , e ) 35
Rationale: "product of two no = lcm * hcf 6 x * 7 x = 450 * x x = 11 answer : a"
Answer: a
Question (physics): a person travels equal distances with speeds of 3 km / hr , 4 km / hr and 5 km / hr and takes a total time of 54 minutes . the total distance is ?
Options: a ) 1 km , b ) 2 km , c ) 3 km , d ) 3.4 km , e ) 5 km
Rationale: "d 3 km let the total distance be 3 x km . then , x / 3 + x / 4 + x / 5 = 54 / 60 47 x / 60 = 54 / 60 = > x = 1.14 total distance = 3 * 1.14 = 3.4 km ."
Answer: d
Question (gain): jim is able to sell a hand - carved statue for $ 620 which was a 25 % profit over his cost . how much did the statue originally cost him ?
Options: a ) $ 496.00 , b ) $ 512.40 , c ) $ 555.40 , d ) $ 574.90 , e ) $ 588.20
Rationale: "620 = 1.25 * x x = 620 / 1.25 = 496.00 which rounds to $ 496.00 , which is ( a ) ."
Answer: a
Question (other): praveen starts business with rs . 3780 and after 5 months , hari joins with praveen as his partner . after a year , the profit is divided in the ratio 2 : 3 . what is hari ’ s contribution in the capital ?
Options: a ) s . 7500 , b ) s . 8000 , c ) s . 8500 , d ) s . 9720 , e ) s . 6000
Rationale: "let hari ’ s capital be rs . x . then , 3780 * 12 / 7 x = 2 / 3 = > 14 x = 136080 = > x = 9720 . answer : d"
Answer: d
Question (general): find the greatest number that will divide 48 , 210 and 136 leaving 4 , 12 and 4 as remainders respectively
Options: a ) 19 , b ) 17 , c ) 13 , d ) 22 , e ) 21
Rationale: "explanation : answer will be hcf of ( 48 - 4 , 210 - 12 , 136 - 4 ) hcf of ( 44 , 198 , 132 ) = 22 option d"
Answer: d
Question (general): a student has to obtain 33 % of the total marks to pass . he got 125 marks and failed by 40 marks . the maximum marks are ?
Options: a ) 450 , b ) 300 , c ) 500 , d ) 610 , e ) 175
Rationale: "let the maximum marks be x then , 33 % of x = 125 + 40 33 x / 100 = 165 x = 500 answer is c"
Answer: c
Question (general): the telephone company wants to add an area code composed of 2 letters to every phone number . in order to do so , the company chose a special sign language containing 524 different signs . if the company used 522 of the signs fully and two remained unused , how many additional area codes can be created if the company uses all 524 signs ?
Options: a ) 46 , b ) 248 , c ) 2092 , d ) 15,128 , e ) 30,256
Rationale: # of 2 - letter codes possible from 524 different signs = 524 * 524 . # of 2 - letter codes possible from 522 different signs = 522 * 522 . the difference = 524 ^ 2 - 522 ^ 2 = ( 524 - 522 ) ( 524 + 522 ) = 2092 . answer : c .
Answer: c
Question (general): a case contains c cartons . each carton contains b boxes , and each box contains 200 paper clips . how many paper clips are contained in 2 cases ?
Options: a ) 200 bc , b ) 200 b / c , c ) 400 bc , d ) 400 b / c , e ) 400 / bc
Rationale: "2 cases * c cartons / case * b boxes / carton * 200 clips / box = 400 bc paper clips the answer is c ."
Answer: c
Question (general): what is the value of x ^ 2 yz − xyz ^ 2 , if x = − 2 , y = 1 , and z = 1 ?
Options: a ) 6 , b ) 24 , c ) 30 , d ) 32 , e ) 48
Rationale: 4 * 1 * 1 - ( - 2 * 1 * 1 ) = 4 + 2 = 6 ans : a
Answer: a
Question (general): 2.205 divided by 0.3 gives
Options: a ) 7.35 , b ) 6.25 , c ) 8.15 , d ) 6.95 , e ) none of these
Rationale: "explanation : 2.205 / 0.3 = 22.05 / 3 = 7.35 . answer : option a"
Answer: a
Question (general): having received his weekly allowance , john spent 3 / 5 of his allowance at the arcade . the next day he spent one third of his remaining allowance at the toy store , and then spent his last $ 0.88 at the candy store . what is john ’ s weekly allowance ?
Options: a ) $ 2.70 , b ) $ 3.00 , c ) $ 3.30 , d ) $ 3.60 , e ) $ 3.90
Rationale: "x = 3 x / 5 + 1 / 3 * 2 x / 5 + 88 4 x / 15 = 88 x = 330 = $ 3.30 the answer is c ."
Answer: c
Question (general): a doctor prescribed 18 cubic centimeters of a certain drug to a patient whose body weight was 120 pounds . if the typical dosage is 4 cubic centimeters per 15 pounds of the body weight , by what percent was the prescribed dosage lesser than the typical dosage ?
Options: a ) 8 % , b ) 9 % , c ) 11 % , d ) 12.5 % , e ) - 87.5 %
Rationale: "typical dosage per 15 pound of the body weight = 4 c . c typical dosage per 120 pound of the body weight = 4 * ( 120 / 15 ) = 4 * 8 = 32 c . c dosage prescribed by doctor for 120 pound patient = 18 c . c % prescribed dosage greater than the typical dosage = ( 18 - 32 / 16 ) * 100 % = ( - 14 / 16 ) * 100 % = - 87 . . 5 % answer e"
Answer: e
Question (physics): calculate how much time could be saved if the train would run at its usual speed , given that it ran at 2 / 3 of its own speed and got to its destination in 20 hours ?
Options: a ) 5.67 hrs , b ) 9.67 hrs , c ) 8.95 hrs , d ) 8.67 hrs , e ) 6.67 hrs
Rationale: "new speed = 2 / 3 of usual speed new time = 2 / 3 of usual time 2 / 3 of usual time = 20 hrs usual time = 20 * 2 / 3 = 13.33 hrs time saved = 22 - 13.33 = 8.67 hrs answer is d"
Answer: d
Question (general): exactly 3 / 7 of the people in the room are under the age of 21 , and exactly 5 / 10 of the people in the room are over the age of 65 . if the total number of the people in the room is greater than 50 and less than 100 , how many people in the room are under the age of 21 ?
Options: a ) 21 , b ) 35 , c ) 39 , d ) 30 , e ) 65
Rationale: "the total number of the people in the room must be a multiple of both 7 and 10 ( in order 3 / 7 and 5 / 10 of the number to be an integer ) , thus the total number of the people must be a multiple of lcm of 7 and 10 , which is 70 . since , the total number of the people in the room is greater than 50 and less than 100 , then there are 70 people in the room . therefore there are 3 / 7 * 70 = 30 people in the room under the age of 21 . answer : d ."
Answer: d
Question (general): a 5 - digit number is formed by the digits 2 , 45 , 68 ( each digit used exactly once ) . what is the probability that the number formed is divisible by 4 ?
Options: a ) 2 / 5 , b ) 3 / 5 , c ) 4 / 5 , d ) 6 / 5 , e ) 7 / 5
Rationale: a no . is divisible by 4 if last two digit is divisible by 4 so numbers ending with 2428 , 4852 , 5664 , 6884 are divisible by 4 last two places are fixed by 2428 , 4852 , 5664 , 6884 so remaining 3 places can be filled in 3 ! ways for each total = 8 * 3 ! = 48 = n ( e ) n ( s ) = 5 c 4 = 120 p = 48 / 120 = 2 / 5 answer : a
Answer: a
Question (general): if x is 20 percent more than y and y is 60 percent less than z , then x is what percent of z ?
Options: a ) 500 % , b ) 48 % , c ) 500 / 3 % , d ) 125 % , e ) 60 %
Rationale: "z = 100 ; y = 40 so x = 48 x as % of z = 48 / 100 * 100 = > 48 % answer will be ( b )"
Answer: b
Question (general): the area of one square is x ^ 2 + 4 x + 4 and the area of another square is 4 x ^ 2 − 12 x + 9 . if the sum of the perimeters of both squares is 32 , what is the value of x ?
Options: a ) 0 , b ) 2 , c ) 2.5 , d ) 4.67 , e ) 3
Rationale: "spotting the pattern of equations both are in form of ( x + c ) ^ 2 so a 1 = ( x + 2 ) ^ 2 a 2 = ( 2 x - 3 ) ^ 2 l 1 = x + 2 l 2 = 2 x - 3 p 1 = 4 ( x + 2 ) p 2 = 4 ( 2 x - 3 ) p 1 + p 2 = 32 4 ( x + 2 ) + 4 ( 2 x - 3 ) = 32 . . . . . . . . . . . . . . > x = 3 answer : e"
Answer: e
Question (physics): from mumbai railway station , two trains a and b start simultaneously from stations 400 miles apart , and travel the same route toward each other on adjacent parallel tracks . if train a and train b travel at a constant rate of 40 miles per hour and 40 miles per hour , respectively , how many miles will train a have traveled when the trains pass each other , to the nearest mile ?
Options: a ) 112 , b ) 133 , c ) 150 , d ) 167 , e ) 180
Rationale: since we know the distance ( 400 ) and the combined rate ( 80 ) , we plug it into the formula : distance = rate * time 400 = 80 * time we can solve for the time they will meet cause we added the rate of train a and train b together . so the time will be 400 / 80 from dividing 80 on both sides to isolate time in the equation above . time will be 20 hours so now you can plug that in for train a ’ s distance . distance = rate * time distance = 40 * 20 distance = 180 according to answer choice e .
Answer: e