Cheukting/mathqa huggingface dataset:
Question (physics): a man can do a piece of work in 5 days , but with the help of his son he can do it in 3 days . in what time can the son do it alone ?
Options: a ) 7 1 / 2 days , b ) 6 1 / 2 days , c ) 5 1 / 2 days , d ) 4 1 / 2 days , e ) 3 1 / 2 days
Rationale: "explanation : in this type of question , where we have one person work and together work done . then we can easily get the other person work just by subtracting them . as , son ' s one day work = ( 1 / 3 − 1 / 5 ) = ( 5 − 3 / 15 ) = 2 / 15 so son will do whole work in 15 / 2 days which is = 7 1 / 2 days option a"
Answer: a
Question (general): a luxury liner , queen marry ii , is transporting several cats as well as the crew ( sailors , a cook , and one - legged captain ) to a nearby port . altogether , these passengers have 15 heads and 43 legs . how many cats does the ship host ?
Options: a ) 4 , b ) 5 , c ) 6 , d ) 7 , e ) 8
Rationale: "sa ' s + co + ca + cats = 15 . sa ' s + 1 + 1 + cats = 15 or sa ' s + cats = 13 . sa ' s ( 2 ) + 2 + 1 + cats * 4 = 43 sa ' s * 2 + cats * 4 = 40 or sa ' s + cats * 2 = 20 or 13 - cats + cat * 2 = 20 then cats = 7 d"
Answer: d
Question (gain): find the amount on rs . 5000 in 2 years , the rate of interest being 20 % per first year and 25 % for the second year ?
Options: a ) 3377 , b ) 2678 , c ) 5460 , d ) 7500 , e ) 1671
Rationale: "5000 * 120 / 100 * 125 / 100 = > 7500 answer : d"
Answer: d
Question (physics): two trains are moving in opposite directions at 60 km / hr and 90 km / hr . their lengths are 1.10 km and 0.9 km respectively . the time taken by the slower train to cross the faster train in seconds is ?
Options: a ) 26 sec , b ) 76 sec , c ) 48 sec , d ) 27 sec , e ) 22 sec
Rationale: "relative speed = 60 + 90 = 150 km / hr . = 150 * 5 / 18 = 125 / 3 m / sec . distance covered = 1.10 + 0.9 = 2 km = 2000 m . required time = 2000 * 3 / 125 = 48 sec . answer : c"
Answer: c
Question (general): a , b and c are partners . a receives 2 / 3 of profits , b and c dividing the remainder equally . a ' s income is increased by rs . 300 when the rate to profit rises from 5 to 7 percent . find the capital of c ?
Options: a ) 3377 , b ) 2899 , c ) 3750 , d ) 2778 , e ) 1991
Rationale: "a : b : c = 2 / 3 : 1 / 6 : 1 / 6 = 4 : 1 : 1 x * 2 / 100 * 2 / 3 = 300 c ' s capital = 22500 * 1 / 6 = 3750 answer : c"
Answer: c
Question (general): define a * by the equation a * = π - x . then ( ( − π ) * ) * =
Options: a ) − 2 π , b ) - 1 , c ) − π , d ) 2 π , e ) 4 π
Rationale: for a * f ( f ( − π ) ) = f ( π − ( − π ) ) = f ( π + π ) = f ( 2 π ) = π − 2 π = − π = c
Answer: c
Question (general): the average age of a group of n people is 15 years old . one more person aged 35 joins the group and the new average is 17 years old . what is the value of n ?
Options: a ) 8 , b ) 9 , c ) 10 , d ) 11 , e ) 12
Rationale: 15 n + 35 = 17 ( n + 1 ) 2 n = 18 n = 9 the answer is b .
Answer: b
Question (geometry): what is the measure of the radius of the circle inscribed in a triangle whose sides measure 4 , 11 and 12 units ?
Options: a ) 1.6 units
b ) 6 units
c ) 3 units
d ) 5 units
e ) 12 units
Rationale: sides are 4 , 11 and 12 . . . thus it is right angle triangle since 12 ^ 2 = 4 ^ 2 + 11 ^ 2 therefore , area = 1 / 2 * 11 * 4 = 22 we have to find in - radius therefore , area of triangle = s * r . . . . where s = semi - perimeter and r = in - radius now s = semi - perimeter = 12 + 11 + 4 / 2 = 13,5 thus , 22 = 13,5 * r and hence r = in - radius = 1.6 option a
Answer: a
Question (general): what is x if x + 2 y = 10 and y = 4 ?
Options: a ) a ) 10 , b ) b ) 8 , c ) c ) 6 , d ) d ) 4 , e ) e ) 2
Rationale: "x = 10 - 2 y x = 10 - 8 . x = 2 answer : e"
Answer: e
Question (general): we bought a total of 90 books at the store . math books cost $ 4 and history books cost $ 5 . the total price was $ 396 . how many math books did we buy ?
Options: a ) 47 , b ) 54 , c ) 56 , d ) 61 , e ) 64
Rationale: "m + h = 90 h = 90 - m 4 m + 5 h = 396 4 m + 5 * ( 90 - m ) = 396 m = 54 the answer is b ."
Answer: b
Question (geometry): what is the smallest positive perfect square that is divisible by 12 , 15 , and 18 ?
Options: a ) 900 , b ) 1,600 , c ) 2,500 , d ) 3,600 , e ) 4,900
Rationale: "take the lcm of 12 , 15,18 that will come 180 . the smallest positive perfect square given in the option that can be divided with 180 is 900 . so the ans is ( a )"
Answer: a
Question (probability): find the probability that a leap year selected at random will have 53 mondays
Options: a ) 6 / 7 , b ) 5 / 7 , c ) 4 / 7 , d ) 3 / 7 , e ) 2 / 7
Rationale: there are 366 days in a leap year : 52 weeks and 2 more days . so , 52 mondays and 2 days . these 2 days can be : { mon , tue } , { tue , wed } , { wed , thu } , { thu , fri } , { fri , sat } , { sat , sun } and { sun , mon } ( 7 cases ) . in order to have 53 mondays we should have either { mon , tuesday } or { sun , mon } case . the probability of that is 2 / 7 . answer : e .
Answer: e
Question (physics): a train passes a station platform in 36 sec and a man standing on the platform in 20 sec . if the speed of the train is 81 km / hr . what is the length of the platform ?
Options: a ) 240 , b ) 288 , c ) 277 , d ) 360 , e ) 422
Rationale: "speed = 81 * 5 / 18 = 22.5 m / sec . length of the train = 22.5 * 20 = 450 m . let the length of the platform be x m . then , ( x + 450 ) / 36 = 22.5 = > x = 360 m . answer : d"
Answer: d
Question (gain): solution x is 30 % chemical a and 70 % chemical b by volume . solution y is 40 % chemical a and 60 % chemical b by volume . if a mixture of x and y is 32 % chemical a , what percent of the mixture is solution x ?
Options: a ) 85 % , b ) 80 % , c ) 75 % , d ) 70 % , e ) 65 %
Rationale: the volume of the mixture be x + y . 0.3 x + 0.4 y = 0.32 ( x + y ) x = 4 y x / ( x + y ) = 4 / 5 = 80 % . the answer is b .
Answer: b
Question (general): the wages earned by robin is 30 % more than that earned by erica . the wages earned by charles is 50 % more than that earned by erica . how much percent is the wages earned by charles more than that earned by robin ?
Options: a ) 18.75 % , b ) 23 % , c ) 15 % , d ) 50 % , e ) 100 %
Rationale: "let wage of erica = 10 wage of robin = 1.3 * 10 = 13 wage of charles = 1.5 * 10 = 15 percentage by which wage earned by charles is more than that earned by robin = ( 15 - 13 ) / 13 * 100 % = 2 / 13 * 100 % = 15 % answer c"
Answer: c
Question (general): 8 x 5.4 - 0.6 x 10 / 1.2 = ?
Options: a ) 30 , b ) 45 , c ) 50 , d ) 31 , e ) 21
Rationale: "given expression = ( 43.2 - 6 ) / 1.2 = 37.2 / 1.2 = 31 answer is d ."
Answer: d
Question (other): veena ranks 65 rd from the top in a class of 182 . what is her rank from the bottom if 22 students have failed the examination ?
Options: a ) 96 , b ) 108 , c ) 110 , d ) 90 , e ) 93
Rationale: "total student = 182 failed = 22 paasd student = 182 - 22 = 160 from bottom her rank is = 160 - 65 + 1 = 96 answer : a"
Answer: a
Question (general): there were 36000 hardback copies of a certain novel sold before the paperback version was issued . from the time the first paperback copy was sold until the last copy of the novel was sold 9 times as many paperback copies as hardback copies were sold . if a total of 440000 copies of the novel were sold in all , how many paperback copies were sold ?
Options: a ) 45,000 , b ) 360,000 , c ) 364,500 , d ) 363,600 , e ) 396,900
Rationale: say x was the # of hardback copies sold from the time the first paperback copy was sold . then the total # of paperback copies sold was 9 x ; hence the total # of copies sold was ( hardback ) + ( paperback ) = ( 36 + x ) + ( 9 x ) = 440 - - > x = 40.4 . so , the total # of paperback copies sold was 9 x = 9 * 40.4 = 363.6 or 363,600 . answer : d .
Answer: d
Question (gain): a certain car ' s price decreased by 2.5 % ( from the original price ) each year from 1996 to 2002 , during that time the owner of the car invested in a new carburetor and a new audio system for the car , which increased car ' s price by $ 1,500 . if the price of the car in 1996 was $ 22,000 , what is the car ' s price in 2002 ?
Options: a ) $ 18,400 , b ) $ 19,500 , c ) $ 20,200 , d ) $ 20,400 , e ) $ 21,100
Rationale: "important point to notice - 2.5 % decrease from the original price 2.5 % of 22,000 = 550 total reduction in 6 years = 550 * 6 = 3300 final price = 22,000 + 1500 - 3300 = 20,200 c is the answer"
Answer: c
Question (physics): in a covering a certain distance , the speeds of a and b are in the ratio of 3 : 4 . a takes 30 minutes more than b to reach the destination . the time taken by a to reach the destination is ?
Options: a ) 8 hrs , b ) 9 hrs , c ) 2 hrs , d ) 2 hrs , e ) 9 hrs
Rationale: ratio of speeds = 3 : 4 ratio of times taken = 4 : 3 suppose a takes 4 x hrs and b takes 3 x hrs to reach the destination . then , 4 x - 3 x = 30 / 60 = > x = 1 / 2 time taken by a = 4 x hrs = 4 * 1 / 2 = 2 hrs . answer : c
Answer: c
Question (general): in a group of ducks and cows , the total number of legs are 24 more than twice the no . of heads . find the total no . of buffaloes .
Options: a ) 10 , b ) 12 , c ) 14 , d ) 16 , e ) 19
Rationale: "let the number of buffaloes be x and the number of ducks be y = > 4 x + 2 y = 2 ( x + y ) + 24 = > 2 x = 24 = > x = 12 b"
Answer: b
Question (gain): rohan spends 40 % of his salary on food , 20 % on house rent , 10 % on entertainment and 10 % on conveyance . if his savings at the end of a month are rs . 2500 . then his monthly salary is
Options: a ) rs . 12500 , b ) rs . 1000 , c ) rs . 8000 , d ) rs . 6000 , e ) rs . 15000
Rationale: "sol . saving = [ 100 - ( 40 + 20 + 10 + 10 ] % = 20 % . let the monthly salary be rs . x . then , 20 % of x = 2500 â ‡ ” 20 / 100 x = 2500 â ‡ ” x = 2500 ã — 5 = 12500 . answer a"
Answer: a
Question (physics): a man complete a journey in 10 hours . he travels first half of the journey at the rate of 20 km / hr and second half at the rate of 20 km / hr . find the total journey in km .
Options: a ) 220 km , b ) 224 km , c ) 230 km , d ) 232 km , e ) 200 km
Rationale: "0.5 x / 20 + 0.5 x / 20 = 10 - - > x / 20 + x / 20 = 20 - - > 2 x = 20 x 20 - - > x = ( 20 x 20 ) / 2 = 200 km . answer : e ."
Answer: e
Question (general): the probability of a student possessing a ball point pen in exam is 3 / 5 & possessing an ink pen is 2 / 3 . find his probability of possessing at least one of them
Options: a ) 10 / 15 , b ) 11 / 15 , c ) 12 / 15 , d ) 13 / 15 , e ) 14 / 15
Rationale: the probability of a student possessing a ball point pen in exam is 3 / 5 . the probability of a student not possessing a ball point pen in exam is 2 / 5 & possessing an ink pen is 2 / 3 . & not possessing an ink pen is 1 / 3 . his probability of possessing none of them = 2 / 5 * 1 / 3 = 2 / 15 his probability of possessing one of them = 1 - 2 / 15 = 13 / 15 answer : d
Answer: d
Question (general): if 200 ! / 10 ^ n is an integer , what is the largest possible value of n ?
Options: a ) 40 , b ) 42 , c ) 44 , d ) 48 , e ) 49
Rationale: "we have to basically count the number of 5 s withing 200 200 / 5 = 40 200 / 5 ^ 2 = 8 200 / 5 ^ 3 = 1 we know for sure that there will be at least 49 even numbers ( multiples of 2 ) within 200 so there are 49 10 s in 200 ! answer : e"
Answer: e
Question (general): if you write down all the numbers from 1 to 50 , then how many times do you write 3 ?
Options: a ) a ) 11 , b ) b ) 15 , c ) c ) 20 , d ) d ) 21 , e ) e ) 22
Rationale: "explanation : explanation : clearly , from 1 to 50 , there are ten numbers with 3 as the unit ' s digit - 3 , 13 , 23 , 33 , 43 , and ten numbers with 3 as the ten ' s digit - 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 . so , required number = 5 + 10 = 15 . answer : b"
Answer: b
Question (general): evaluate 248 + 64 − − √ − − − − − − − − − √
Options: a ) 14 , b ) 26 , c ) 16 , d ) 36 , e ) 46
Rationale: explanation : = 248 + 64 − − √ − − − − − − − − − √ = 248 + 8 − − − − − − √ = 256 − − − √ = 16 answer : c
Answer: c
Question (general): two employees x and y are paid a total of rs . 638 per week by their employer . if x is paid 120 percent of the sum paid to y , how much is y paid per week ?
Options: a ) s . 250 , b ) s . 280 , c ) s . 290 , d ) s . 299 , e ) s . 300
Rationale: "let the amount paid to x per week = x and the amount paid to y per week = y then x + y = 638 but x = 120 % of y = 120 y / 100 = 12 y / 10 ∴ 12 y / 10 + y = 638 ⇒ y [ 12 / 10 + 1 ] = 638 ⇒ 22 y / 10 = 638 ⇒ 22 y = 6380 ⇒ y = 6380 / 22 = 580 / 2 = rs . 290 c"
Answer: c
Question (general): for any integer k > 1 , the term “ length of an integer ” refers to the number of positive prime factors , not necessarily distinct , whose product is equal to k . for example , if k = 24 , the length of k is equal to 4 , since 24 = 2 × 2 × 2 × 3 . if x and y are positive integers such that x > 1 , y > 1 , and x + 3 y < 920 , what is the maximum possible sum of the length of x and the length of y ?
Options: a ) 14 , b ) 12 , c ) 10 , d ) 16 , e ) 18
Rationale: "we know that : x > 1 , y > 1 , and x + 3 y < 1000 , and it is given that length means no of factors . for any value of x and y , the max no of factors can be obtained only if factor is smallest noall factors are equal . hence , lets start with smallest no 2 . 2 ^ 1 = 2 2 ^ 2 = 4 2 ^ 3 = 8 2 ^ 4 = 16 2 ^ 5 = 32 2 ^ 6 = 64 2 ^ 7 = 128 2 ^ 8 = 256 2 ^ 9 = 512 2 ^ 10 = 1024 ( it exceeds 920 , so , x ca n ' t be 2 ^ 10 ) so , max value that x can take is 2 ^ 9 , for which has length of integer is 9 . now , since x = 512 , x + 3 y < 920 so , 3 y < 408 = = > y < 136 so , y can take any value which is less than 136 . and to get the maximum no of factors of smallest integer , we can say y = 2 ^ 7 for 2 ^ 7 has length of integer is 7 . so , combined together : 9 + 7 = 16 . d"
Answer: d
Question (geometry): a circular grass lawn of 35 metres in radius has a path 7 metres wide running around it on the outside . find the area of path .
Options: a ) 1694 m 2
b ) 1700 m 2
c ) 1598 m 2
d ) 1500 m 2
e ) none of thes
Rationale: radius of a circular grass lawn ( without path ) = 35 m ∴ area = π r 2 = π ( 35 ) 2 radius of a circular grass lawn ( with path ) = 35 + 7 = 42 m ∴ area = π r 2 = π ( 42 ) 2 ∴ area of path = π ( 42 ) 2 – π ( 35 ) 2 = π ( 422 – 352 ) = π ( 42 + 35 ) ( 42 – 35 ) answer a
Answer: a
Question (gain): sonika deposited rs . 6000 which amounted to rs . 9200 after 3 years at simple interest . had the interest been 2 % more . she would get how much ?
Options: a ) 9660 , b ) 6560 , c ) 7560 , d ) 9560 , e ) 8560
Rationale: "( 6000 * 3 * 2 ) / 100 = 360 9200 - - - - - - - - 9560 answer : d"
Answer: d
Question (physics): excluding stoppages , the speed of a train is 42 kmph and including stoppages it is 27 kmph . of how many minutes does the train stop per hour ?
Options: a ) a ) 19.42 , b ) b ) 20.42 , c ) c ) 21.42 , d ) d ) 22.42 , e ) e ) 23.42
Rationale: "explanation : t = 15 / 42 * 60 = 21.42 answer : option c"
Answer: c
Question (general): the prices of tea and coffee per kg were the same in june . in july the price of coffee shot up by 20 % and that of tea dropped by 20 % . if in july , a mixture containing equal quantities of tea and coffee costs 80 / kg . how much did a kg of coffee cost in june ?
Options: a ) 50 , b ) 60 , c ) 80 , d ) 100 , e ) 120
Rationale: "let the price of tea and coffee be x per kg in june . price of tea in july = 1.2 x price of coffee in july = 0.8 x . in july the price of 1 / 2 kg ( 800 gm ) of tea and 1 / 2 kg ( 800 gm ) of coffee ( equal quantities ) = 80 1.2 x ( 1 / 2 ) + 0.8 x ( 1 / 2 ) = 80 = > x = 80 thus proved . . . option c ."
Answer: c
Question (gain): working alone , mary can pave a driveway in 6 hours and hillary can pave the same driveway in 6 hours . when they work together , mary thrives on teamwork so her rate increases by 25 % , but hillary becomes distracted and her rate decreases by 20 % . if they both work together , how many hours will it take to pave the driveway ?
Options: a ) 3 hours , b ) 4 hours , c ) 5 hours , d ) 6 hours , e ) 7 hours
Rationale: "initial working rates : mary = 1 / 6 per hour hillary = 1 / 6 per hour rate when working together : mary = 1 / 6 + ( 1 / 4 * 1 / 6 ) = 1 / 5 per hour hillary = 1 / 6 - ( 1 / 5 * 1 / 6 ) = 2 / 15 per hour together they work 1 / 5 + 2 / 15 = 1 / 3 per hour so they will need 3 hours to complete the driveway . the correct answer is a ."
Answer: a
Question (gain): two boys starts from the same place walking at the rate of 4.5 kmph and 5.5 kmph respectively in the same direction . what time will they take to be 9.5 km apart ?
Options: a ) 9.5 , b ) 5.5 , c ) 8.5 , d ) 9.6 , e ) 9.7
Rationale: explanation : relative speed = 5.5 - 4.5 = 1 kmph ( because they walk in the same direction ) distance = 9.5 km time = distance / speed = 9.5 / 1 = 9.5 hr answer : a
Answer: a
Question (physics): a train is moving at 4 / 5 of its usual speed . the train is 45 minutes too late . what is the usual time ( in hours ) for the train to complete the journey ?
Options: a ) 1 , b ) 1.5 , c ) 2 , d ) 2.5 , e ) 3
Rationale: new time = d / ( 4 v / 5 ) = 5 / 4 * usual time 45 minutes represents 1 / 4 of the usual time . the usual time is 4 * 45 minutes = 3 hours . the answer is e .
Answer: e
Question (other): in a class of 52 students , 12 enrolled for both english and german . 22 enrolled for german . if the students of the class enrolled for at least one of the two subjects , then how many students enrolled for only english and not german ?
Options: a ) 30 , b ) 10 , c ) 18 , d ) 28 , e ) 32
Rationale: total = english + german - both + neither - - > 52 = english + 22 - 12 + 0 - - > english = 42 - - > only english = english - both = 42 - 12 = 30 answer : a .
Answer: a
Question (other): if the ratio of apples to bananas is 5 to 2 and the ratio of bananas to cucumbers is 1 to 4 , what is the ratio of apples to cucumbers ?
Options: a ) 1 : 3 , b ) 2 : 5 , c ) 3 : 5 , d ) 5 : 8 , e ) 4 : 7
Rationale: the ratio of bananas to cucumbers is 1 to 4 which equals 2 to 8 . the ratio of apples to bananas to cucumbers is 5 to 2 to 8 . the ratio of apples to cucumbers is 5 to 8 . the answer is d .
Answer: d
Question (probability): a box contains 100 balls , numbered from 1 to 100 . if 3 balls are selected at random and with replacement from the box . if the 3 numbers on the balls selected contain two odd and one even . what is the probability l that the first ball picked up is odd numbered ?
Options: a ) 0 , b ) 1 / 3 , c ) 1 / 2 , d ) 2 / 3 , e ) 1
Rationale: answer - d selecting the balls either even or odd is having probability 50 / 100 = 1 / 2 we have already selected 3 balls with 2 odd numbers and 1 even number . so we have 3 combinations ooe , oeo , eoo . we have 3 outcomes and 2 are favourable as in 2 cases 1 st number is odd . so probability l is 2 / 3 . d
Answer: d
Question (general): the average of marks obtained by 120 candidates was 35 . if the avg of marks of passed candidates was 39 and that of failed candidates was 39 and that of failed candidates was 15 , the no . of candidates who passed the examination is ?
Options: a ) 80 , b ) 90 , c ) 100 , d ) 120 , e ) 130
Rationale: let the number of candidate who passed = y then , 39 y + 15 ( 120 - y ) = 120 x 35 ⇒ 24 y = 4200 - 1800 ∴ y = 2400 / 24 = 100 c
Answer: c
Question (geometry): in may , the groundskeeper at spring lake golf club built a circular green with an area of 90 π square feet . in august , the groundskeeper doubled the distance from the center of the green to the edge of the green . what is the total area of the renovated green ?
Options: a ) 360 π , b ) 400 π , c ) 450 π , d ) 500 π , e ) 600 π
Rationale: "area = π r ^ 2 , so doubling the radius results in an area that is 4 times the original area . 4 ( 90 π ) = 360 π the answer is a ."
Answer: a
Question (general): the average of first six prime numbers which are between 40 and 80 is
Options: a ) 35.4 , b ) 50.7 , c ) 45.7 , d ) 57 , e ) 67
Rationale: "explanation : first six prime numbers which are between 40 and 80 = 41 , 43 , 47 , 53 , 59 , 61 average = ( 41 + 43 + 47 + 53 + 59 + 61 ) / 6 = 50.7 answer : b"
Answer: b
Question (gain): boy sells a book for rs . 720 he gets a loss of 10 % , to gain 10 % , what should be the sp ?
Options: a ) 430 , b ) 880 , c ) 550 , d ) 590 , e ) 600
Rationale: "cost price = 720 / 90 x 100 = 800 to gain 10 % = 800 x 10 / 100 = 80 sp = cp + gain = 800 + 80 = 880 answer : b"
Answer: b
Question (physics): renu can do a piece of work in 5 days , but with the help of her friend suma , she can do it in 4 days . in what time suma can do it alone ?
Options: a ) 10 , b ) 12 , c ) 14 , d ) 15 , e ) 20
Rationale: "renu â € ™ s one day â € ™ s work = 1 / 5 suma â € ™ s one day â € ™ s work = 1 / 4 - 1 / 5 = 1 / 20 suma can do it alone in 20 days . answer : e"
Answer: e
Question (general): when positive integer n is divided by 5 , the remainder is 1 . when n is divided by 7 , the remainder is 5 . what is the smallest positive integer p , such that ( n + p ) is a multiple of 30 ?
Options: a ) 1 , b ) 2 , c ) 4 , d ) 19 , e ) 20
Rationale: "when positive integer n is divided by 5 , the remainder is 1 i . e . , n = 5 x + 1 values of n can be one of { 1 , 6 , 11 , 16 , 21 , 26 , 31 . . . . . . . . . . . . . 46 , 51 , 56,61 . . . . . . . . . . . . . . . . . . } similarly , when n is divided by 7 , the remainder is 5 . . i . e . , n = 7 y + 5 values of n can be one of { 5 , 12 , 19 , 26 , 33 , 40 , 47 , 54 , 61 . . . . . . . . } combining both the sets we get n = { 19 , 52 , . . . . . . . . . . . } what is the smallest positive integer p , such that ( n + p ) is a multiple of 30 or 30 x in case of n = 26 p = 4 c is the answer ."
Answer: c
Question (other): the compounded ratio of ( 2 : 3 ) , ( 6 : 11 ) and ( 11 : 2 ) is :
Options: a ) 1 : 2 , b ) 5 : 9 , c ) 2 : 1 , d ) 11 : 24 , e ) none
Rationale: answer : option c 2 / 3 : 6 / 11 : 11 / 2 = 2 : 1
Answer: c
Question (other): in a graduating class of 232 students , 144 took geometry and 119 took biology . what is the difference between the greatest possible number p and the smallest possible number of students that could have taken both geometry and biology ?
Options: a ) 144 , b ) 119 , c ) 113 , d ) 88 , e ) 31
Rationale: "official solution : first of all , notice that since 144 took geometry and 119 took biology , then the number of students who took both geometry and biology can not be greater than 119 . { total } = { geometry } + { biology } - { both } + { neither } ; 232 = 144 + 119 - { both } + { neither } ; { both } = 31 + { neither } . { both } is minimized when { neither } is 0 . in this case { both } = 31 . the greatest possible number of students p that could have taken both geometry and biology , is 119 . thus , the answer is 119 - 31 = 88 . answer : d ."
Answer: d
Question (general): what is the remainder when 30 ^ 72 ^ 87 is divided by 11 .
Options: a ) 5 , b ) 7 , c ) 1 , d ) 3 , e ) 4
Rationale: "let us take n = 72 ^ 87 eq is . . 30 ^ n / 11 = 8 ^ n / 11 here cyclicity of 8 is 10 then n / 10 = 72 ^ 87 / 10 = 2 ^ 87 / 10 here cyclicity of 2 is then 87 % 4 = 3 ( 3 rd cycle of 2 ) . . . . i . e 2 ^ 3 % 10 = 8 ( 8 th cycle of 8 ) therefore final ans is 8 ^ 8 % 11 = 5 answer : a"
Answer: a
Question (other): the number of people who purchased book a is twice the number of people who purchased book b . the number of people who purchased both books a and b is 500 , which is twice the number of people who purchased only book b . what is the number of people v who purchased only book a ?
Options: a ) 250 , b ) 500 , c ) 750 , d ) 1000 , e ) 1500
Rationale: "this is best solved using overlapping sets or a venn diagram . we know that a = 2 b , and that 500 people purchased both a and b . further , those purchasing both was double those purchasing b only . this gives us 250 people purchasing b only . with the 500 that pruchased both , we have a total of 750 that purchased b and this is 1 / 2 of those that purchased a . so , 1500 purchased a . less the 500 that purchased both , v = 1000 purchased a only . ( this is much simpler to solve using the venn diagram ) . correct answer is d . 1000"
Answer: d