Cheukting/mathqa huggingface dataset:
Question (general): x and y are positive integers . when x is divided by 12 , the remainder is 5 , and when x is divided by 13 , the remainder is 3 . when y is divided by 14 , the remainder is 5 , and when y is divided by 7 , the remainder is 5 . what is the least possible value of x + y ?
Options: a ) 24 , b ) 12 , c ) 64 , d ) 48 , e ) 25
Rationale: "when x is divided by 12 , the remainder is 5 : so , the possible values of x are : 5 , 17 , 29 , 41 , etc . when x is divided by 13 , the remainder is 3 : so , the possible values of x are : 3 , 16,29 . . . stop . since both lists include 29 , the smallest possible value of x is 29 . when y is divided by 14 , the remainder is 5 : so , the possible values of y are : 5 , 19 , 33,47 etc . when y is divided by 7 , the remainder is 5 : so , the possible values of y are : 5 , 19 . . . stop . since both lists include 19 , the smallest possible value of y is 19 since the smallest possible values of x and y are 29 and 19 respectively , the smallest possible value of x + y is 48 . so , d is the correct answer to the original question ."
Answer: d
Question (physics): a train running at the speed of 180 km / hr crosses a pole in 7 seconds . what is the length of the train ?
Options: a ) 530 , b ) 305 , c ) 503 , d ) 430 , e ) 350
Rationale: "speed = ( 180 * 5 / 18 ) m / sec = ( 50 ) m / sec length of the train = ( speed x time ) = ( 50 * 7 ) m = 350 m . answer : e"
Answer: e
Question (geometry): the length of a rectangular plot is thrice its breadth . if the area of the rectangular plot is 432 sq m , then what is the breadth of the rectangular plot ?
Options: a ) 12 , b ) 17 , c ) 18 , d ) 19 , e ) 14
Rationale: "let the breadth of the plot be b m . length of the plot = 3 b m ( 3 b ) ( b ) = 432 3 b 2 = 432 b 2 = 144 b = 12 m . answer : option a"
Answer: a
Question (geometry): one side of a rectangle is 3 cm shorter than the other side . if we increase the length of each side by 1 cm , then the area of the rectangle will increase by 16 cm 2 . find the lengths of all sides .
Options: a ) 10 and 3 , b ) 9 and 6 , c ) 10 and 7 , d ) 3 and 10 , e ) 10 and 10
Rationale: "let x be the length of the longer side x > 3 , then the other side ' s length is x − 3 cm . then the area is s 1 = x ( x - 3 ) cm 2 . after we increase the lengths of the sides they will become ( x + 1 ) and ( x − 3 + 1 ) = ( x − 2 ) cm long . hence the area of the new rectangle will be a 2 = ( x + 1 ) ⋅ ( x − 2 ) cm 2 , which is 16 cm 2 more than the first area . therefore a 1 + 16 = a 2 x ( x − 3 ) + 16 = ( x + 1 ) ( x − 2 ) x 2 − 3 x + 16 = x 2 + x − 2 x − 2 2 x = 18 x = 9 . so , the sides of the rectangle are 9 cm and ( 9 − 3 ) = 6 cm long . so answer is b ."
Answer: b
Question (gain): the difference between simple interest and compound interest on rs . 1200 for one year at 10 % per annum reckoned half - yearly is :
Options: a ) rs . 2.50 , b ) rs . 3 , c ) rs . 3.75 , d ) rs . 4 , e ) rs . 5
Rationale: s . i = ( 1000 * 10 * 4 ) / 100 = rs . 400 c . i = 1200 * 1 + 5 / 100 ) 2 – 1200 = 123 . difference = rs . ( 123 - 120 ) = rs . 3 answer : b
Answer: b
Question (geometry): a rectangular tank needs to be coated with insulation . the tank has dimensions of 3 feet , 6 feet , and 2 feet . each square foot of insulation costs $ 20 . how much will it cost to cover the surface of the tank with insulation ?
Options: a ) $ 980
b ) $ 1060
c ) $ 1220
d ) $ 1440
e ) $ 1650
Rationale: the total surface area is 2 ( 2 * 3 + 3 * 6 + 2 * 6 ) = 72 square feet the total cost is 72 * $ 20 = $ 1440 the answer is d .
Answer: d
Question (gain): if a trader sold two cars each at rs . 325475 and gains 13 % on the first and loses 13 % on the second , then his profit or loss percent on the whole is ?
Options: a ) 1.69 % , b ) 1.74 % , c ) 1.84 % , d ) 1.47 % , e ) 1.24 %
Rationale: "sp of each car is rs . 325475 , he gains 13 % on first car and losses 13 % on second car . in this case , there will be loss and percentage of loss is given by = [ ( profit % ) ( loss % ) ] / 100 = ( 13 ) ( 13 ) / 100 % = 1.69 % answer : a"
Answer: a
Question (general): p , q and r have rs . 5000 among themselves . r has two - thirds of the total amount with p and q . find the amount with r ?
Options: a ) rs . 3000 , b ) rs . 3600 , c ) rs . 2000 , d ) rs . 4000 , e ) none of these
Rationale: "let the amount with r be rs . r r = 2 / 3 ( total amount with p and q ) r = 2 / 3 ( 5000 - r ) = > 3 r = 10000 - 2 r = > 5 r = 10000 = > r = 2000 . answer : c"
Answer: c
Question (general): a cab driver 5 days income was $ 250 , $ 400 , $ 750 , $ 400 , $ 500 . then his average income is ?
Options: a ) $ 440 , b ) $ 450 , c ) $ 460 , d ) $ 470 , e ) $ 480
Rationale: "avg = sum of observations / number of observations avg income = ( 250 + 400 + 750 + 400 + 500 ) / 5 = 460 answer is c"
Answer: c
Question (general): 1903 x 1903 = ?
Options: a ) a ) 3623216 , b ) b ) 3624216 , c ) c ) 3624316 , d ) d ) 3621409 , e ) e ) 3625216
Rationale: "1903 x 1903 = ( 1903 ) 2 = ( 1900 + 3 ) 2 = ( 1900 ) 2 + ( 3 ) 2 + ( 2 x 1900 x 3 ) = 3610000 + 9 + 11400 . = 3621409 . d )"
Answer: d
Question (physics): a man , a woman and a boy can together complete a piece of work in 3 days . if a man alone can do it in 6 days and a boy alone in 9 days , how long will a woman take to complete the work ?
Options: a ) 12 days , b ) 15 days , c ) 16 days , d ) 24 days , e ) 18 days
Rationale: explanation : ( 1 man + 1 woman + 1 boy ) ’ s 1 day ’ s work = 1 / 3 1 man ’ s 1 day work = 1 / 6 1 boy ’ s 1 day ’ s work = 1 / 9 ( 1 man + 1 boy ) ‘ s 1 day ’ s work = 1 / 6 + 1 / 9 = 5 / 18 therefore , 1 woman ’ s 1 day ’ s work = 1 / 3 – 5 / 18 = 1 / 18 therefore , the woman alone can finish the work in 18 days . answer : option e
Answer: e
Question (general): a grocer has a sale of rs . 6435 , rs . 6927 , rs . 6855 , rs . 7230 and rs . 6562 for 5 consecutive months . how much sale must he have in the sixth month so that he gets an average sale of rs . 6700 ?
Options: a ) s . 4991 , b ) s . 5991 , c ) s . 6001 , d ) s . 6191 , e ) s . 7991
Rationale: "total fr 5 mnths = ( 6435 + 6927 + 6855 + 7230 + 6562 ) = rs 34009 . reqd . sale = rs . [ ( 6700 * 6 ) - 34009 ] = rs . ( 40200 - 34009 ) = rs . 6191 . answer : d"
Answer: d
Question (geometry): find the cube root of 804357 ?
Options: a ) 22 , b ) 88 , c ) 93 , d ) 19 , e ) 18
Rationale: "unit digit of the answer is 3 . ( cube of 3 = 27 ) . now omit 357 and find after which cube 804 lies . it is 729 . so our tenth place is 9 . answer = c"
Answer: c
Question (general): if p and q are prime numbers , how many divisors does the product p ^ 3 * q ^ 7 have ?
Options: a ) 25 , b ) 30 , c ) 32 , d ) 35 , e ) 40
Rationale: "when a number n = a ^ x * b ^ y , where a and b are prime numbers , and x , y are positive integers , the number of divisors of n = ( x + 1 ) ( y + 1 ) therefore , the answer is c . 4 * 8 = 32"
Answer: c
Question (gain): a person spends 40 % of his salary on food , 25 % on house rent , 15 % on entertainment and 15 % on conveyance . if his savings at the end of the month is rs . 1200 , then his salary per month in rupees is :
Options: a ) 4000 , b ) 24000 , c ) 8000 , d ) 10000 , e ) none of these
Rationale: "total expenditure = 40 + 25 + 15 + 15 = 95 % saving = ( 100 - 95 ) = 5 % 5 / 100 × salary = 1200 , salary = 24000 rs . answer : b"
Answer: b
Question (physics): a train , 135 meters long travels at a speed of 45 km / hr crosses a bridge in 30 seconds . the length of the bridge is
Options: a ) 270 m , b ) 245 m , c ) 235 m , d ) 220 m , e ) 240 m
Rationale: "explanation : assume the length of the bridge = x meter total distance covered = 135 + x meter total time taken = 30 s speed = total distance covered / total time taken = ( 135 + x ) / 30 m / s = > 45 ã — ( 10 / 36 ) = ( 135 + x ) / 30 = > 45 ã — 10 ã — 30 / 36 = 135 + x = > 45 ã — 10 ã — 10 / 12 = 135 + x = > 15 ã — 10 ã — 10 / 4 = 135 + x = > 15 ã — 25 = 135 + x = 375 = > x = 375 - 135 = 240 answer : option e"
Answer: e
Question (other): find the ratio between 1 hr 30 minutes and 2 hrs 40 minutes .
Options: a ) 9 : 16 , b ) 5 : 9 , c ) 12 : 17 , d ) 13 : 24 , e ) none of these
Rationale: solution : the ratio = 1 hr 30 min : 2 hr 40 min . 90 min : 160 min = 9 : 16 . answer is a
Answer: a
Question (general): a batsman makes a score of 87 runs in the 17 th inning and thus increases his averages by 3 . what is his average after 17 th inning ?
Options: a ) 39 , b ) 35 , c ) 42 , d ) 40.5 , e ) 41
Rationale: "explanation : let the average after 17 innings = x total runs scored in 17 innings = 17 x average after 16 innings = ( x - 3 ) total runs scored in 16 innings = 16 ( x - 3 ) total runs scored in 16 innings + 87 = total runs scored in 17 innings = > 16 ( x - 3 ) + 87 = 17 x = > 16 x - 48 + 87 = 17 x = > x = 39 answer : option a"
Answer: a
Question (other): mrs . evans gave a test to her freshmen economics class , which has 40 students enrolled and 30 of them answered question 1 correctly . if 29 answered question 2 correctly and 10 did not take the test then how many answered both questions correctly ?
Options: a ) 24 , b ) 27 , c ) 29 , d ) 31 , e ) 33
Rationale: "total number of enrolled students = 40 number of students who did not take test = 10 hence , number of students who took test = 40 - 10 = 30 number of students who answered q 2 correctly = 29 , therefore , all students who took test answered q 2 correctly . so , number of students who answered q 1 correctly , also answered q 2 correctly = 29 . number of students who answered both q 1 & q 2 correctly = 29 . answer : c"
Answer: c
Question (general): nina has exactly enough money to purchase 5 widgets . if the cost of each widget were reduced by $ 1.25 , then nina would have exactly enough money to purchase 8 widgets . how much money does nina have ?
Options: a ) $ 16.7 , b ) $ 24 , c ) $ 30 , d ) $ 36 , e ) $ 40
Rationale: "a its is . let price = x ( x - 1.25 ) 8 = 5 x x = 10 / 3 hence total money = 5 * 10 / 3 = 16.7"
Answer: a
Question (general): if the sales tax reduced from 3 1 / 2 % to 3 1 / 3 % , then what difference does it make to a person who purchases an article with market price of rs . 9000 ?
Options: a ) 11 , b ) 13 , c ) 15 , d ) 18 , e ) 10
Rationale: "required difference = [ 3 ½ % of rs . 9000 ] – [ 3 1 / 3 % of rs . 9000 ] = [ ( 7 / 20 - ( 10 / 3 ) ] % of rs . 9000 = 1 / 6 % of rs . 9000 = rs . [ ( 1 / 6 ) 8 ( 1 / 100 ) * 9000 ] = rs . 15 . answer is c ."
Answer: c
Question (physics): in a kilometer race , a beats b by 40 meters or 15 seconds . what time does a take to complete the race ?
Options: a ) 277 sec , b ) 285 sec , c ) 267 sec , d ) 167 sec , e ) 276 sec
Rationale: "time taken by b run 1000 meters = ( 1000 * 15 ) / 50 = 300 sec . time taken by a = 300 - 15 = 285 sec . answer : b"
Answer: b
Question (gain): a reduction of 40 % in the price of bananas would enable a man to obtain 67 more for rs . 40 , what is reduced price per dozen ?
Options: a ) 1.86 , b ) 2.86 , c ) 4.86 , d ) 3.86 , e ) 5.86
Rationale: "40 * ( 40 / 100 ) = 16 - - - 67 ? - - - 12 = > rs . 2.86 answer : b"
Answer: b
Question (general): a waitress ' s income consists of her salary and tips . during one week , her tips were 7 / 4 of her salary . what fraction of her income for the week came from tips ?
Options: a ) 1 / 9 , b ) 1 / 6 , c ) 2 / 3 , d ) 4 / 9 , e ) 7 / 11
Rationale: "her tips were 7 / 4 of her salary . let ' s say her salary = $ 4 this mean her tips = ( 7 / 4 ) ( $ 4 ) = $ 7 so , her total income = $ 4 + $ 7 = $ 11 what fraction of her income for the week came from tips $ 7 / $ 11 = 7 / 11 = e"
Answer: e
Question (physics): if one cat eats 1.5 rats in 1.5 days , then in howmany days 3 cats will eat 3 rats ? ?
Options: a ) 1 day , b ) 2 days , c ) 3 days , d ) 4 days , e ) 5 days
Rationale: 1 cat - - - - - - 3 / 2 rat - - - - - - - - - 3 / 2 days 1 cat - - - - - - ( 3 / 2 ) * 2 rat - - - - - ( 3 / 2 ) * 2 days ( rat doubled , days doubled ) 1 * 3 cat - - - - - 3 rat - - - - - - - - - - - 3 / 3 = 1 day ( cat * 3 , same rat , days / 3 ) answer : a
Answer: a
Question (physics): the distance between two cities a and b is 465 km . a train starts from a at 8 a . m . and travel towards b at 60 km / hr . another train starts from b at 9 a . m and travels towards a at 75 km / hr . at what time do they meet ?
Options: a ) 09 , b ) 07 , c ) 12 , d ) 05 , e ) 03
Rationale: "explanation : suppose they meet x hrs after 8 a . m then , [ distance moved by first in x hrs ] + [ distance moved by second in ( x - 1 ) hrs ] = 465 therefore , 60 x + 75 ( x - 1 ) = 465 = > x = 3 . so , they meet at ( 8 + 4 ) i . e , 12 noon answer : c )"
Answer: c
Question (physics): a train 100 m long crosses a platform 125 m long in 10 sec ; find the speed of the train ?
Options: a ) 81 , b ) 77 , c ) 54 , d ) 33 , e ) 99
Rationale: "d = 100 + 125 = 225 t = 10 s = 225 / 10 * 18 / 5 = 81 kmph answer : a"
Answer: a
Question (general): calculate the largest 3 digit number which is exactly divisible by 89 ?
Options: a ) 911 , b ) 969 , c ) 979 , d ) 977 , e ) 971
Rationale: "largest 4 digit number is 999 after doing 999 ÷ 89 we get remainder 20 hence largest 3 digit number exactly divisible by 89 = 999 - 20 = 979 c"
Answer: c
Question (general): on dividing 83 by a number , the quotient is 9 and the remainder is 2 . find the divisor ?
Options: a ) 7 , b ) 8 , c ) 9 , d ) 10 , e ) 11
Rationale: "d = ( d - r ) / q = ( 83 - 2 ) / 9 = 81 / 9 = 9 c )"
Answer: c
Question (gain): albert invested rs . 1000 in a scheme for 3 years at compound interest rate 10 % p . a . how much amount will albert get on maturity of the fixed deposit ?
Options: a ) s . 1300 , b ) s . 1005 , c ) s . 1031 , d ) s 1331 , e ) s . 1120
Rationale: "amount = [ 1000 * ( 1 + 10 / 100 ) 3 ] = 1000 * 11 / 10 * 11 / 10 * 11 / 10 = rs . 1331 answer : d"
Answer: d
Question (general): if a ' s height is 35 % less than that of b , how much percent b ' s height is more than that of a ?
Options: a ) 66.66 % , b ) 66.68 % , c ) 53.84 % , d ) 86.66 % , e ) 66.65 %
Rationale: "excess of b ' s height over a ' s = [ ( 35 / ( 100 - 35 ) ] x 100 % = 53.84 % answer : c )"
Answer: c
Question (gain): what is 5 / 8 as a percent
Options: a ) 62.5 % , b ) 63 % , c ) 63.5 % , d ) 64.5 % , e ) 65 %
Rationale: "5 / 8 as a percent = ( 5 / 8 ) x 100 = 0.625 x 100 = 62.5 % answer : a i . e . 62.5 %"
Answer: a
Question (general): a group of students decided to collect as many paise from each member of group as is the number of members . if the total collection amounts to rs . 77.44 , the number of the member is the group is :
Options: a ) 57 , b ) 67 , c ) 77 , d ) 88 , e ) 97
Rationale: "money collected = ( 77.44 x 100 ) paise = 7744 paise . number of members = square root of 7744 = 88 . answer : option d"
Answer: d
Question (general): a library has an average of 510 visitors on sunday and 240 on other days . the average number of visitors per day in a month of 30 days beginning with a sunday is :
Options: a ) 288 , b ) 278 , c ) 267 , d ) 285 , e ) 231
Rationale: "since the month begins with a sunday , so there will be five sundays in the month . required average = [ ( 510 * 5 ) + ( 240 * 25 ) ] / 30 = 8550 / 30 = 285 . answer : d"
Answer: d
Question (gain): some of the 50 % solution of acid was removed and this was replaced with an equal amount of 10 % solution of acid . as a result , a 20 % solution of acid was obtained . what fraction of the original solution was replaced ?
Options: a ) 1 / 2 , b ) 2 / 3 , c ) 3 / 4 , d ) 3 / 5 , e ) 4 / 5
Rationale: "let x be the fraction of the original solution that was replaced . 0.5 * ( 1 - x ) + 0.1 ( x ) = 0.2 0.4 x = 0.3 x = 3 / 4 the answer is c ."
Answer: c
Question (gain): in a park there are two ponds with both brown ducks and green ducks . in the smaller pond there are 30 ducks and in the larger pond there are 50 ducks . if 20 % of the ducks in the smaller pond are green and 12 % of the ducks in the larger pond are green , then what percentage of ducks are green ?
Options: a ) 13 % , b ) 14 % , c ) 15 % , d ) 16 % , e ) 17 %
Rationale: number of ducks in small pond = 30 green ducks in small pond = 20 % of 30 = 6 ducks number of ducks in large pond = 50 green ducks in large pond = 12 % of 50 = 6 ducks total number of ducks = 30 + 50 = 80 total number of green ducks = 6 + 6 = 12 ducks percentage of green ducks = 12 / 80 * 100 = 15 % answer : c
Answer: c
Question (physics): the h . c . f of two numbers is 17 and their l . c . m is 5134 . if one of the numbers is 289 , then the other is ?
Options: a ) 302 , b ) 304 , c ) 306 , d ) 308 , e ) 310
Rationale: "other number = ( 17 * 5134 ) / 289 = 302 . answer : a"
Answer: a
Question (general): a fruit drink is made of orange , watermelon , and grape juice , where 15 percent of the drink is orange juice and 60 percent is watermelon juice . if the drink is made with 35 ounces of grape juice , how many ounces is the drink total ?
Options: a ) 120 , b ) 140 , c ) 160 , d ) 180 , e ) 200
Rationale: "let the total number of ounces in the drink be x . % of orange = 15 % % of watermelon = 60 % % of grape = 100 % - 75 % = 25 % 0.25 x = 35 x = 140 therefore there are a total of 140 ounces in the drink . the answer is b ."
Answer: b
Question (geometry): the dimensions of a room are 25 feet * 15 feet * 12 feet . what is the cost of white washing the four walls of the room at rs . 8 per square feet if there is one door of dimensions 6 feet * 3 feet and three windows of dimensions 4 feet * 3 feet each ?
Options: a ) s . 4529
b ) s . 4586
c ) s . 7248
d ) s . 4530
e ) s . 4528
Rationale: area of the four walls = 2 h ( l + b ) since there are doors and windows , area of the walls = 2 * 12 ( 15 + 25 ) - ( 6 * 3 ) - 3 ( 4 * 3 ) = 906 sq . ft . total cost = 906 * 8 = rs . 7248 answer : c
Answer: c
Question (gain): a money lender finds that due to a fall in the annual rate of interest from 8 % to 7 3 / 4 % his yearly income diminishes by rs . 61.50 , his capital is ?
Options: a ) 24,607 , b ) 24,609 , c ) 24,600 , d ) 24,602 , e ) 24,604
Rationale: "let the capital be rs . x . then , ( x * 8 * 1 ) / 100 - ( x * 31 / 4 * 1 / 100 ) = 61.50 32 x - 31 x = 6150 * 4 x = 24,600 . answer : c"
Answer: c
Question (physics): a 210 m long train running at the speed of 120 km / hr crosses another train running in opposite direction at the speed of 80 km / hr in 9 sec . what is the length of the other train ?
Options: a ) 290 , b ) 280 , c ) 270 , d ) 260 , e ) 250
Rationale: "relative speed = 120 + 80 = 200 km / hr . = 200 * 5 / 18 = 500 / 9 m / sec . let the length of the other train be x m . then , ( x + 210 ) / 9 = 500 / 9 = > x = 290 . answer : option a"
Answer: a
Question (physics): walking with 4 / 5 of my usual speed , i miss the bus by 6 minutes . what is my usual time ?
Options: a ) 16 min , b ) 26 min , c ) 34 min , d ) 24 min , e ) 12 min
Rationale: "speed ratio = 1 : 4 / 5 = 5 : 4 time ratio = 4 : 5 1 - - - - - - - - 6 4 - - - - - - - - - ? è 24 answer : d"
Answer: d
Question (general): the age of father 8 years ago was thrice the age of his son . 5 years hence , father ' s age will be twice that of his son . the ratio of their present ages is :
Options: a ) 47 : 11 , b ) 37 : 21 , c ) 27 : 21 , d ) 47 : 21 , e ) 47 : 31
Rationale: let the ages of father and son 8 years ago be 3 x and x years respectively . then , ( 3 x + 8 ) + 5 = 2 [ ( x + 8 ) + 5 ] 3 x + 13 = 2 x + 26 x = 13 . required ratio = ( 3 x + 8 ) : ( x + 8 ) = 47 : 21 . answer : option d
Answer: d
Question (general): if the average ( arithmetic mean ) of ( 2 a + 16 ) and ( 3 a - 8 ) is 69 , what is the value of a ?
Options: a ) 26 , b ) 30 , c ) 28 , d ) 36 , e ) 42
Rationale: ( ( 2 a + 16 ) + ( 3 a - 8 ) ) / 2 = ( 5 a + 8 ) / 2 = 69 a = 26 the answer is a .
Answer: a
Question (general): what is the remainder when 43 ^ 89 is divided by 5 ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: "the units digit of the exponents of 3 cycle in a group of 4 : { 3 , 9 , 7 , 1 } 89 has the form 4 k + 1 so the units digit of 43 ^ 89 is 3 . the remainder when dividing by 5 is 3 . the answer is d ."
Answer: d
Question (general): the value of 99 ^ ( 91 / 99 ) x 99 is :
Options: a ) 9989 , b ) 9896 , c ) 9892 , d ) 9809 , e ) 9810
Rationale: ( 100 â € “ 8 / 99 ) x 99 = 9900 â € “ 8 = 9892 . answer : c
Answer: c
Question (general): for any integer k greater than 1 , the symbol k * denotes the product of all the fractions of the form 1 / t , where t is an integer between 1 and k , inclusive . what is the value of 3 * / 5 * ?
Options: a ) 5 , b ) 5 / 4 , c ) 4 / 5 , d ) 1 / 4 , e ) 1 / 5
Rationale: "when dealing with ' symbolism ' questions , it often helps to ' play with ' the symbol for a few moments before you attempt to answer the question that ' s asked . by understanding how the symbol ' works ' , you should be able to do the latter calculations faster . here , we ' re told that k * is the product of all the fractions of the form 1 / t , where t is an integer between 1 and k , inclusive . based on this definition . . . . if . . . . k = 2 k * = ( 1 / 1 ) ( 1 / 2 ) = 1 / 2 if . . . . k = 3 k * = ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) = 1 / 6 we ' re asked to find the value of 5 * / 4 * now that we know how the symbol ' works ' , solving this problem should n ' t be too difficult . you can actually choose to do the math in a couple of different ways . . . . 5 * = ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) ( 1 / 5 ) do n ' t calculate this just yet though . . . . since we ' re dividing by 4 * , many of those fractions will ' cancel out . ' 4 * = ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) we ' re looking for the value of : ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) ( 1 / 5 ) / ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) since the first four fraction in the numerator and denominator cancel out , we ' re left with just one fraction : 5 / 3 b"
Answer: b
Question (physics): the length of the bridge , which a train 130 m long and traveling at 45 km / hr can cross in 30 sec is ?
Options: a ) 288 , b ) 269 , c ) 245 , d ) 888 , e ) 121
Rationale: "speed = 45 * 5 / 18 = 25 / 2 m / sec . time = 30 sec let the length of bridge be x meters . then , ( 130 + x ) / 30 = 25 / 2 x = 245 m . answer : c"
Answer: c
Question (general): if the average of 55 , 48 , 507 , 2 , 684 and y is 223 , then what is y ?
Options: a ) 701 , b ) 120 , c ) 42 , d ) 868 , e ) 1
Rationale: y = 42 , because : ( 55 + 48 + 507 + 2 + 684 + y ) / 6 = 223 ( 55 + 48 + 507 + 2 + 684 + y ) / 6 * 6 = 223 * 6 ( 55 + 48 + 507 + 2 + 684 + y ) - 1296 = 1338 - 1296 y = 42 so the correct answer is c , 42 .
Answer: c
Question (general): in the xy - coordinate system , if ( m , n ) and ( m + 5 , n + k ) are two points on the line with the equation x = 2 y + 5 , then k =
Options: a ) 1 / 2 , b ) 1 , c ) 2 , d ) 5 / 2 , e ) 4
Rationale: "since ( m , n ) and ( m + 2 , n + k ) are two points on the line with the equation x = 2 y + 5 they should satisfy m = 2 n + 5 and m + 5 = 2 * ( n + k ) + 5 . by 1 st equation we have m - 2 n = 5 and by 2 nd equation m - 2 n = 2 k - - - > 5 = 2 k - - - > k = 5 / 2 . the answer is , therefore , ( d ) ."
Answer: d
Question (geometry): the surface area of a sphere is 4 π r 2 , where r is the radius of the sphere . if the area of the base of a hemisphere is 3 , what is the surface area r of that hemisphere ?
Options: a ) 6 / π , b ) 9 / π , c ) 6 , d ) 9 , e ) 12
Rationale: "given area of the base of a hemisphere is 3 = pi * r ^ 2 thus r = sqrt ( 3 / pi ) . surface area of whole sphere = 4 * pi * r ^ 2 . = 4 * pi * 3 / pi = 12 . since the hemisphere is half of a sphere the surface area of the hemisphere = 12 / 2 = 6 ( curved part , not including the flat rounded base ) . but the total surface area = 6 + area of the base of a hemisphere . = 6 + 3 = 9 . answer is d ! !"
Answer: d
Question (general): the sum of the first 70 positive even integers is 4970 . what is the sum of the first 70 odd integers ?
Options: a ) 4500 , b ) 4600 , c ) 4300 , d ) 4900 , e ) 5000
Rationale: sum of first n even numbers = n ( n + 1 ) = 4970 sum of first n odd numbers = n ^ 2 = 70 * 70 = 4900 ( here n = 70 ) answer : d
Answer: d
Question (physics): a train crosses a platform of 150 m in 10 sec , same train crosses another platform of length 250 m in 15 sec . then find the length of the train ?
Options: a ) 150 m , b ) 180 m , c ) 159 m , d ) 250 m , e ) 50 m
Rationale: "length of the train be ‘ x ’ x + 150 / 10 = x + 250 / 15 3 x + 450 = 2 x + 500 x = 50 m answer : e"
Answer: e