Cheukting/mathqa huggingface dataset:
Question (physics): 50 men do a work in 100 days . how many men are needed to finish the work in 20 days ?
Options: a ) 50 , b ) 100 , c ) 250 , d ) 300 , e ) 400
Rationale: men required to finish the work in 2 days = 50 * 100 / 20 = 250 answer is c
Answer: c
Question (gain): what is the rate percent when the simple interest on rs . 900 amount to rs . 160 in 4 years ?
Options: a ) 4.44 , b ) 5.44 , c ) 6.44 , d ) 7.44 , e ) 8.44
Rationale: "160 = ( 900 * 4 * r ) / 100 r = 4.44 % answer : a"
Answer: a
Question (physics): a car takes 6 hours to cover a distance of 432 km . how much should the speed in kmph be maintained to cover the same direction in 3 / 2 th of the previous time ?
Options: a ) 48 kmph , b ) 50 kmph , c ) 52 kmph , d ) 56 kmph , e ) 60 kmph
Rationale: "time = 6 distence = 432 3 / 2 of 6 hours = 6 * 3 / 2 = 9 hours required speed = 432 / 9 = 48 kmph a"
Answer: a
Question (general): in a recent election , james received 5 percent of the 2,000 votes cast . to win the election , a candidate needed to receive more than 50 percent of the vote . how many additional votes would james have needed to win the election ?
Options: a ) 901 , b ) 989 , c ) 990 , d ) 991 , e ) 1,001
Rationale: "james = ( 5 / 100 ) * 2000 = 100 votes to win = ( 50 / 100 ) * total votes + 1 = ( 50 / 100 ) * 2000 + 1 = 1001 remaining voted needed to win election = 1001 - 100 = 901 answer : option a"
Answer: a
Question (general): the scoring system in a certain football competition goes as follows : 3 points for victory , 1 point for a draw , and 0 points for defeat . each team plays 20 matches . if a team scored 14 points after 5 games , what is the least number of the remaining matches it has to win to reach the 40 - point mark by the end of the tournament ?
Options: a ) 6 , b ) 7 , c ) 8 , d ) 9 , e ) 10
Rationale: "to get 40 points as end of season we need another 26 points or more from remaining 15 matches : option a = 6 * 3 + 9 * 1 = 27 hence option a - 6"
Answer: a
Question (general): find large number from below question the difference of two numbers is 1390 . on dividing the larger number by the smaller , we get 6 as quotient and the 15 as remainder
Options: a ) 1245 , b ) 1345 , c ) 1455 , d ) 1577 , e ) 1665
Rationale: "let the smaller number be x . then larger number = ( x + 1390 ) . x + 1390 = 6 x + 15 5 x = 1375 x = 275 large number = 275 + 1390 = 1665 e"
Answer: e
Question (general): for the past n days , the average ( arithmetic mean ) daily production at a company was 50 units . if today ' s production of 105 units raises the average to 55 units per day , what is the value of n ?
Options: a ) 8 , b ) 10 , c ) 12 , d ) 14 , e ) 16
Rationale: "the daily production was raised by 5 units for n days , which has a weighting of 5 n . 5 n = 105 - 55 = 50 n = 10 the answer is b ."
Answer: b
Question (other): a company sells pens and pencils . the revenue from pens in 2007 was up 5 % from 2006 . the revenue from pencils declined 13 % over the same period . overall revenue was down 1 % from 06 to 07 . what was the ratio of pencil revenue to pen revenue in 2006
Options: a ) 1 / 2 , b ) 1 / 3 , c ) 1 / 4 , d ) 1 / 5 , e ) 1 / 6
Rationale: let the revenue of pens be ' x ' let the revenue of pencils be ' y ' the revenue of pens up from 2006 by 5 % = > 105 % the revenue of pencils down from 2006 by 13 % = > 87 % overall revenue down by 1 % = > 99 % hence the ratio of pencils to pens = 6 / 12 = 1 / 2 answer : a
Answer: a
Question (gain): a family made a down payment of $ 300 and borrowed the balance on a set of encyclopedias that cost $ 750 . the balance with interest was paid in 9 monthly payments of $ 57 each and a final payment of $ 21 . the amount of interest paid was what percent of the amount borrowed ?
Options: a ) 6 % , b ) 12 % , c ) 14 % , d ) 18.66 % , e ) 20 %
Rationale: interest paid = 57 * 9 + 21 - 450 = 84 % off the balance = ( 84 / 450 ) * 100 = 18.66 % answer is d .
Answer: d
Question (general): there are 10 teams in a certain league and each team plays each of the other teams exactly once . what is the total number of games played ?
Options: a ) 30 , b ) 36 , c ) 42 , d ) 45 , e ) 50
Rationale: "10 c 2 = 45 the answer is d ."
Answer: d
Question (geometry): consider a quarter of a circle of radius 20 . let r be the radius of the circle inscribed in this quarter of a circle . find r .
Options: a ) 20 * ( sqr 2 - 1 )
b ) 8 * ( sqr 3 - 1 )
c ) 4 * ( sqr 7 - 1 )
d ) 12 * ( sqr 7 - 1 )
e ) none of these
Rationale: i got 20 / ( sqr 2 + 1 ) and just forgot to multiply by ( sqr 2 - 1 ) . answer a
Answer: a
Question (general): what two - digit number is less than the sum of the square of its digits by 12 and exceeds their doubled product by 5 ?
Options: a ) 95 , b ) 99 , c ) 26 , d ) 73 , e ) none of the above
Rationale: "let the digits be x and y . the number would be 10 x + y . we are given that 2 xy + 5 = 10 x + y = x ^ 2 y ^ 2 - 12 thus 2 xy + 5 = x ^ 2 + y ^ 2 - 12 x ^ 2 + y ^ 2 - 2 xy = 16 ( x - y ) ^ 2 = 16 ( x - y ) = 4 or - 4 substituting the values of ( x - y ) in the equation 2 xy + 5 = 10 x + y x comes out to be 1 or 9 . . . thus the two numbers can be 15 or 99 thus the answer is b"
Answer: b
Question (general): if x is to be chosen at random from the integers between 1 to 5 , inclusive , and y is to be chosen at random from the integers between 7 and 10 , inclusive , what is the probability that x + y will be even ?
Options: a ) 1 / 2 , b ) 2 / 3 , c ) 3 / 4 , d ) 4 / 5 , e ) 5 / 6
Rationale: x + y will be even if x and y are both even or both odd . p ( x and y are both even ) = 2 / 5 * 2 / 4 = 1 / 5 p ( x and y are both odd ) = 3 / 5 * 2 / 4 = 3 / 10 p ( x + y is even ) = 1 / 5 + 3 / 10 = 1 / 2 the answer is a .
Answer: a
Question (general): if x ^ 2 + ( 1 / x ^ 2 ) = 8 , x ^ 4 + ( 1 / x ^ 4 ) = ?
Options: a ) 10 , b ) 11 , c ) 62 , d ) 14 , e ) 15
Rationale: "- > x ^ 4 + ( 1 / x ^ 4 ) = ( x ^ 2 ) ^ 2 + ( 1 / x ^ 2 ) ^ 2 = ( x ^ 2 + 1 / x ^ 2 ) ^ 2 - 2 x ^ 2 ( 1 / x ^ 2 ) = 8 ^ 2 - 2 = 62 . thus , the answer is c ."
Answer: c
Question (physics): the perimeter of a rhombus is 52 units . one of its diagonal is 24 units . what is its second diagonals length ?
Options: a ) 5
b ) 10
c ) 15
d ) 20
e ) 25
Rationale: when the perimeter of a triangle is 52 units each side is 13 units next , the diagonal of a rhombus bisect each other at 90 degree thus the half of the diagonal is 12 units now in a right angle triangle so formed , the perpendicular is 12 and the hypptenuse is 13 ( that is the side of the triangle ) finding the base ( which is half of the other diagonal ) : 13 ^ 2 - 12 ^ 2 = 25 thus half of the diagonal is 5 and the diagonal is 10 units answer : b
Answer: b
Question (general): how much is 60 % of 40 is greater than 4 / 5 of 25 ?
Options: a ) 4 , b ) 6 , c ) 10 , d ) 8 , e ) 13
Rationale: "60 / 100 ) * 40 â € “ ( 4 / 5 ) * 25 24 - 20 = 4 answer : a"
Answer: a
Question (general): if x and y are integers , what is the least positive number of 24 x + 18 y ?
Options: a ) 3 , b ) 6 , c ) 9 , d ) 12 , e ) 15
Rationale: "24 x + 21 y = 6 ( 4 x + 3 y ) which will be a minimum positive number when 4 x + 3 y = 1 . 4 ( 1 ) + 3 ( - 1 ) = 1 then 6 ( 4 x + 3 y ) can have a minimum positive value of 6 . the answer is b ."
Answer: b
Question (general): the jogging track in a sports complex is 726 m in circumference . suresh and his wife start from the same point and walk in opposite direction at 4.5 km / hr and 3.75 km / hr respectively . they will meet for the first time in ?
Options: a ) 5.2 minutes , b ) 5.21 minutes , c ) 5.25 minutes , d ) 5.28 minutes , e ) 6 minutes
Rationale: let both of them meet after tt min 4500 m are covered by suresh in 60 m . in tt min he will cover 4500 t 604500 t 60 likewise , in tt min suresh ' s wife will cover 3750 t 603750 t 60 given , 4500 t 60 + 3750 t 60 = 7264500 t 60 + 3750 t 60 = 726 tt = 5.28 minutes option ( d ) is correct
Answer: d
Question (physics): a boatman goes 3 km against the current of the stream in 3 hour and goes 1 km along the current in 30 minutes . how long will it take to go 3 km in stationary water ?
Options: a ) 5 hours , b ) 4 hours , c ) 3 hours , d ) 2 hours , e ) 1 hours
Rationale: "explanation : speed upstream = 3 / 3 = 1 km / hr speed downstream = 1 / ( 30 / 60 ) = 2 km / hr speed in still water = 1 / 2 ( 2 + 1 ) = 3 / 2 km / hr time taken to travel 3 km in still water = 3 / ( 3 / 2 ) = 6 / 3 = 2 hours answer : option d"
Answer: d
Question (general): the average age of 8 men increases by 2 years when two women are included in place of two men of ages 20 and 24 years . find the average age of the women ?
Options: a ) 39 , b ) 27 , c ) 30 , d ) 26 , e ) 29
Rationale: "20 + 24 + 8 * 2 = 60 / 2 = 30 answer : c"
Answer: c
Question (general): how many paying stones , each measuring 2 1 / 2 m * 2 m are required to pave a rectangular court yard 20 m long and 16 1 / 2 m board ?
Options: a ) 66 , b ) 18 , c ) 16 , d ) 10 , e ) 15
Rationale: "20 * 33 / 2 = 5 / 2 * 2 * x = > x = 66 answer : a"
Answer: a
Question (gain): a corporation paid $ 5 million in federal taxes on its first $ 50 million of gross profits and then $ 30 million in federal taxes on the next $ 150 million in gross profits . by approximately what percent did the ratio of federal taxes to gross profits increase from the first $ 50 million in profits to the next $ 150 million in profits ?
Options: a ) 6 % , b ) 14 % , c ) 20 % , d ) 140 % , e ) 43 %
Rationale: difference in ratios = ( 30 / 150 ) - ( 5 / 50 ) = ( 5 / 50 ) % change = ( change ( 5 / 50 ) / original ratio ( 7 / 50 ) ) * 100 = 140 % answer - d
Answer: d
Question (general): a textile manufacturing firm employees 70 looms . it makes fabrics for a branded company . the aggregate sales value of the output of the 70 looms is rs 5 , 00,000 and the monthly manufacturing expenses is rs 1 , 50,000 . assume that each loom contributes equally to the sales and manufacturing expenses are evenly spread over the number of looms . monthly establishment charges are rs 75000 . if one loom breaks down and remains idle for one month , the decrease in profit is :
Options: a ) 13000 , b ) 7000 , c ) 10000 , d ) 5000 , e ) none of these
Rationale: explanation : profit = 5 , 00,000 â ˆ ’ ( 1 , 50,000 + 75,000 ) = rs . 2 , 75,000 . since , such loom contributes equally to sales and manufacturing expenses . but the monthly charges are fixed at rs 75,000 . if one loan breaks down sales and expenses will decrease . new profit : - = > 500000 ã — ( 69 / 70 ) â ˆ ’ 150000 ã — ( 69 / 70 ) â ˆ ’ 75000 . = > rs 2 , 70,000 . decrease in profit = > 2 , 75,000 â ˆ ’ 2 , 70,000 = > rs . 5,000 . answer : d
Answer: d
Question (general): there are 5 thieves , each loot a bakery one after the other such that the first one takes 1 / 2 of the total no . of the breads plus 1 / 2 of a bread . similarly 2 nd , 3 rd , 4 th and 5 fth also did the same . after the fifth one no . of breads remained are 3 . initially how many breads were there ?
Options: a ) 124 , b ) 125 , c ) 126 , d ) 127 , e ) 128
Rationale: let the number of breads initial be x you should go in reverse order as left bread is 3 i . e x = ? after 1 st person ate a = x - ( x / 2 ) - ( 1 / 2 ) i . e a = ( 2 x - x ) / 2 - ( 1 / 2 ) i . e a = ( x / 2 ) - ( 1 / 2 ) i . e a = ( 2 x - 2 ) / 4 i . e a = ( x - 1 ) / 2 like wise - - - - after 2 nd person ate b = ( a - 1 ) / 2 like wise - - - - after 3 rd person ate c = ( b - 1 ) / 2 like wise - - - - after 4 th person ate d = ( c - 1 ) / 2 like wise - - - - after 5 th person ate e = ( d - 1 ) / 2 now as after 5 th person ate the value of remaining bread is - - - 3 therefore the value of e = 3 ; now go reverse from botom to top to get the value of x after calcuating value of d = 7 value of c = 15 value of b = 31 value of a = 63 value of x = 127 so the final answer is 127 breads were there answer : d
Answer: d
Question (gain): in an examination 35 % of the students passed and 260 failed . how many students appeared for the examination ?
Options: a ) a ) 540 , b ) b ) 400 , c ) c ) 700 , d ) d ) 650 , e ) e ) 840
Rationale: "let the number of students appeared be x then , 65 % of x = 260 65 x / 100 = 260 x = 260 * 100 / 65 = 400 answer is b"
Answer: b
Question (physics): if 8 women can color 400 m long cloth in 5 days , then 10 women can color 500 m long cloth in ?
Options: a ) 6 days , b ) 4 days , c ) 5 days , d ) 15 days , e ) 10 days
Rationale: "the length of cloth painted by one woman in one day = 400 / 8 × 5 = 10 m no . of days required to paint 500 m cloth by 10 women = 500 / 10 × 10 = 5 days answer : c"
Answer: c
Question (general): average of 6 numbers comes out to be 3.95 . average of two of the numbers is 3.4 while the average of other two is 3.85 . determine the average of the two numbers left .
Options: a ) 4.7 , b ) 4.8 , c ) 4.3 , d ) 4.6 , e ) 4.9
Rationale: explanation : in the given question we have taken average of 6 numbers taken 2 at a time which makes 3 numbers . = > 3.95 ( average of 1 st and 2 nd number ) = > 3.85 ( average of 3 rd and 4 th number ) = > 3.4 ( average of 5 th and 6 th number ) 3.95 = ( 3.4 + 3.85 + x ) / 3 x = 4.6 answer d
Answer: d
Question (general): the average weight of 8 person ' s increases by 4 kg when a new person comes in place of one of them weighing 55 kg . what might be the weight of the new person ?
Options: a ) 80 kg , b ) 87 kg , c ) 90 kg , d ) 100 kg , e ) 110 kg
Rationale: "total weight increased = ( 8 x 4 ) kg = 32 kg . weight of new person = ( 55 + 32 ) kg = 87 kg . b )"
Answer: b
Question (physics): a 900 m long train crosses a platform in 39 sec while it crosses a signal pole in 18 sec . what is the length of the platform ?
Options: a ) 1000 , b ) 1050 , c ) 1060 , d ) 370 , e ) 380
Rationale: "speed = 900 / 18 = 50 / 1 m / sec . let the length of the platform be x meters . then , ( x + 300 ) / 39 = 50 / 1 = > x = 1950 m . l = 1950 - 900 = 1050 answer : option b"
Answer: b
Question (geometry): a horse is tethered to one corner of a rectangular grassy field 40 m by 24 m with a rope 14 m long . over how much area of the field can it graze ?
Options: a ) 154 cm 2
b ) 308 m 2
c ) 150 m 2
d ) 407 m 2
e ) none of these
Rationale: area of the shaded portion = 1 ⁄ 4 × π × ( 14 ) 2 = 154 m 2 answer a
Answer: a
Question (general): reeya obtained 55 , 67 , 76 , 82 and 55 out of 100 in different subjects , what will be the average
Options: a ) 70 , b ) 75 , c ) 67 , d ) 85 , e ) 90
Rationale: "explanation : ( 55 + 67 + 76 + 82 + 55 / 5 ) = 67 option c"
Answer: c
Question (geometry): a circular garden is surrounded by a fence of negligible width along the boundary . if the length of the fence is 1 / 8 of th area of the garden . what is the radius of the circular garden ?
Options: a ) 1
b ) 2
c ) 4
d ) 8
e ) 16
Rationale: as per the question - - width is negligible now , let l be the length of the fence = 2 pir l = 1 / 8 ( pir ^ 2 ) pir ^ 2 = 16 pir r = 16 answer : e
Answer: e
Question (gain): at the end of the first quarter , the share price of a certain mutual fund was 20 percent higher than it was at the beginning of the year . at the end of the second quarter , the share price was 60 percent higher than it was at the beginning of the year . what was the percent increase in the share price from the end of the first quarter to the end of the second quarter ?
Options: a ) 20 % , b ) 25 % , c ) 30 % , d ) 33.33 % , e ) 40 %
Rationale: "another method is to use the formula for 2 successive percentage changes : total = a + b + ab / 100 60 = 20 + b + 20 b / 100 b = 33.33 answer ( d )"
Answer: d
Question (general): the class mean score on a test was 50 , and the standard deviation was 15 . if jack ' s score was within 2 standard deviations of the mean , what is the lowest score he could have received ?
Options: a ) 20 , b ) 31 , c ) 45 , d ) 90 , e ) 89
Rationale: "1 sd from the mean ranges from 35 to 65 , where 65 is within sd above the mean and 35 within 1 sd below the mean 2 sd = 15 twice = 30 from the the mean , which is 80 to 20 , where 80 is within 2 sd above the mean and 20 is within 2 sd below the mean . answer = a"
Answer: a
Question (physics): find the area of a cuboid of length 4 cm , breadth 6 cm . and height 5 cm .
Options: a ) 120 cm cube
b ) 110 cm cube
c ) 220 cm cube
d ) 430 cm cube
e ) 480 cm cube
Rationale: area of a cuboid = lxbxh = 4 cm x 6 cm x 5 cm = 120 cm cube answer : a
Answer: a
Question (physics): a man traveled a total distance of 1800 km . he traveled one - third of the whole trip by plane and the distance traveled by train is two - thirds of the distance traveled by bus . if he traveled by train , plane and bus , how many kilometers did he travel by bus ?
Options: a ) 660 , b ) 680 , c ) 720 , d ) 760 , e ) 800
Rationale: "total distance traveled = 1800 km . distance traveled by plane = 600 km . distance traveled by bus = x distance traveled by train = 2 x / 3 x + 2 x / 3 + 600 = 1800 5 x / 3 = 1200 x = 720 km the answer is c ."
Answer: c
Question (gain): a , b and c enter into a partnership . a invests 3 times as much as b invests and 2 / 3 of what c invests . at the end of the year , the profit earned is rs . 55000 . what is the share of c ?
Options: a ) rs . 12250 , b ) rs . 13375 , c ) rs . 16750 , d ) rs . 15000 , e ) none of these
Rationale: explanation : let the investment of c be rs . x . the inverstment of b = rs . ( 2 x / 3 ) the inverstment of a = rs . ( 3 × ( 2 / 3 ) x ) = rs . ( 2 x ) ratio of capitals of a , b and c = 2 x : 2 x / 3 : x = 6 : 2 : 3 c ' s share = rs . [ ( 3 / 11 ) × 55000 ] = rs . 15000 answer : option d
Answer: d
Question (general): the average runs of a cricket player of 5 innings was 20 . how many runs must he make in his next innings so as to increase his average of runs by 3 ?
Options: a ) 12 , b ) 42 , c ) 44 , d ) 76 , e ) 84
Rationale: "explanation : average after 6 innings = 23 required number of runs = ( 24 * 6 ) – ( 20 * 5 ) = 144 – 100 = 44 answer c"
Answer: c
Question (general): the sum of three consecutive even numbers is 20 more than the first of these numbers . what is the middle number ?
Options: a ) 7 , b ) 9 , c ) 11 , d ) can not be determined , e ) none of these
Rationale: "solution let the numbers be x , x + 2 and x + 4 . then , x + ( x + 2 ) + ( x + 4 ) = x + 20 ⇔ 2 x = 14 ⇔ x = 7 . ∴ middle number = x + 2 = 9 . answer b"
Answer: b
Question (gain): a certain sum is invested at simple interest at 18 % p . a . for two years instead of investing at 12 % p . a . for the same time period . therefore the interest received is more by rs . 600 . find the sum ?
Options: a ) 7000 , b ) 7029 , c ) 2778 , d ) 2800 , e ) 5000
Rationale: "let the sum be rs . x . ( x * 18 * 2 ) / 100 - ( x * 12 * 2 ) / 100 = 600 = > 36 x / 100 - 24 x / 100 = 600 = > 12 x / 100 = 600 = > x = 5000 . answer : e"
Answer: e
Question (general): there are 100 freshmen at a particular college , all of whom must take at least one of the 3 core classes : art , biology , and calculus . of these freshmen , 17 take only biology , 10 take only calculus , 9 take all 3 classes , and 20 take art and exactly one of the other two core classes . if the number of freshmen who take only art is 3 times the number of freshmen who take every core class except art , how many freshmen take art ?
Options: a ) 25 , b ) 32 , c ) 36 , d ) 48 , e ) 62
Rationale: make a venn diagram to get a clear picture . look at the diagram : each letter represents only one color . b represents the people who take only art . d represents people who take only art and bio etc . d + f = 20 ( people who take art and one other class ) b = 3 e ( people who take only art is 3 times the people who take bio and calculus ) 17 + 10 + 9 + b + d + e + f = 100 ( total people ) b + b / 3 = 44 b = 33 number of freshmen who take art = 33 + 20 + 9 = 62 answer e
Answer: e
Question (general): in how many ways can an answer key for a quiz be written if the quiz contains 4 true - false questions followed by 3 multiple - choice questions with 4 answer choices each , if the correct answers to all true - false questions can not be the same ?
Options: a ) 512 , b ) 656 , c ) 740 , d ) 896 , e ) 972
Rationale: "there are 2 ^ 4 = 16 possibilities for the true - false answers . however we need to remove two cases for tttt and ffff . there are 4 * 4 * 4 = 64 possibilities for the multiple choice questions . the total number of possibilities is 14 * 64 = 896 . the answer is d ."
Answer: d
Question (physics): a goods train runs at the speed of 72 km / hr and crosses a 250 m long platform in 26 sec . what is the length of the goods train ?
Options: a ) 228 , b ) 2000 , c ) 267 , d ) 270 , e ) 274
Rationale: "speed = 72 * 5 / 18 = 20 m / sec . time = 26 sec . let the length of the train be x meters . then , ( x + 250 ) / 26 = 20 x = 270 m . answer : d"
Answer: d
Question (general): when 52416 is divided by 312 , the quotient is 168 . what will be the quotient when 52.416 is divided by 0.0168 ?
Options: a ) 5 , b ) 6 , c ) 4 , d ) 3 , e ) 8
Rationale: "for the 1 st no . there are 2 digits after decimal for the 2 nd no . there are 5 digits after decimal total no . of decimals = 7 req . no . of digits = ( n - 1 ) = ( 7 - 1 ) = 6 answer : b"
Answer: b
Question (physics): a goods train runs at the speed of 72 km / hr and crosses a 150 m long platform in 26 sec . what is the length of the goods train ?
Options: a ) 299 , b ) 277 , c ) 276 , d ) 370 , e ) 281
Rationale: "speed = 72 * 5 / 18 = 20 m / sec . time = 26 sec . let the length of the train be x meters . then , ( x + 150 ) / 26 = 20 x = 370 m . answer : d"
Answer: d
Question (other): if w is the set of all the integers between 49 and 76 , inclusive , that are either multiples of 3 or multiples of 2 or multiples of both , then w contains how many numbers ?
Options: a ) 19 , b ) 91 , c ) 41 , d ) 18 , e ) 14
Rationale: "official solution : number of multiples of 3 step 1 . subtract the extreme multiples of 3 within the range ( the greatest is 75 , the smallest is 51 ) : 75 - 51 = 24 step 2 . divide by 3 : 24 / 3 = 8 step 3 . add 1 : 8 + 1 = 9 . so there are 9 multiples of 3 within the range : examples are 51 , 54 , 57 , 60 , etc . number of multiples of 2 step 1 . subtract the extreme multiples of 2 within the range ( the greatest is 76 , the smallest is 50 ) : 76 - 50 = 26 step 2 . divide by 2 : 26 / 2 = 13 step 3 . add 1 : 13 + 1 = 14 . so there are 14 multiples of 2 within the range : examples are 50 , 52 , 54 , 56 , 58 , 60 etc . add the 9 multiples of 3 and the 14 multiples of 2 : 9 + 14 = 23 . however , by adding the multiples of 2 and the multiples of 3 , we are effectively counting several numbers twice : for example , 54 and 60 are parts of both the lists above . so we ca n ' t just take 9 + 14 = 23 . find the number of multiples of 6 ( which are the double counted , as 6 is divisible by both 2 and 3 ) , and subtract it from 23 : step 1 . subtract the extreme multiples of 6 within the range ( the greatest is 72 , the smallest is 54 ) : 72 - 54 = 18 step 2 . divide by 6 : 18 / 6 = 3 step 3 . add 1 : 3 + 1 = 4 . so there are 4 multiples of 6 within the range : we counted 4 numbers twice . subtract the 4 multiples of 6 from the sum of the multiples of 2 and 3 : = 9 + 14 - 4 = 23 - 4 = 19 therefore , the final number of multiples of 2 , 3 or 6 is 19 . hence , this is the correct answer . ( a )"
Answer: a
Question (other): the ratio between the sale price and the cost price of an article is 10 : 7 . what is the ratio between the profit and the cost price of that article ?
Options: a ) 2 : 9 , b ) 2 : 5 , c ) 3 : 7 , d ) 2 : 0 , e ) 2 : 1
Rationale: "let c . p . = rs . 7 x and s . p . = rs . 10 x . then , gain = rs . 3 x required ratio = 3 x : 7 x = 3 : 7 answer : c"
Answer: c
Question (physics): two cyclist start on a circular track from a given point but in opposite direction with speeds of 7 m / s and 8 m / s . if the circumference of the circle is 300 meters , after what time will they meet at the starting point ?
Options: a ) 20 sec , b ) 15 sec , c ) 30 sec , d ) 50 sec , e ) 1 min
Rationale: they meet every 300 / 7 + 8 = 20 sec answer is a
Answer: a