Cheukting/mathqa huggingface dataset:
Question (physics): two goods train each 500 m long , are running in opposite directions on parallel tracks . their speeds are 45 km / hr and 30 km / hr respectively . find the time taken by the slower train to pass the driver of the faster one .
Options: a ) 12 sec , b ) 24 sec , c ) 48 sec , d ) 60 sec , e ) 65 sec
Rationale: "explanation : relative speed = = ( 45 + 30 ) km / hr = 75 x 5 / 18 m / sec = 125 / 6 m / sec we have to find the time taken by the slower train to pass the driver of the faster train and not the complete train . so , distance covered = length of the slower train . therefore , distance covered = 500 m . required time = 500 x 6 / 125 = 24 sec . answer is b"
Answer: b
Question (general): the age of somu is one - third his father ' s . 5 years back he was one - fifth of his father ' s age . what is his persent age ?
Options: a ) 11 , b ) 13 , c ) 14 , d ) 12 , e ) 10
Rationale: "explanation : let somu ' s age be x and that of his father be 3 x . so , x - 5 = 3 x - 5 / 5 = x = 10 answer : option e"
Answer: e
Question (general): a charitable association sold an average of 66 raffle tickets per member . among the female members , the average was 70 raffle tickets . the male to female ratio of the association is 1 : 2 . what was the average number w of tickets sold by the male members of the association
Options: a ) 50 , b ) 56 , c ) 58 , d ) 62 , e ) 66
Rationale: "given that , total average w sold is 66 , male / female = 1 / 2 and female average is 70 . average of male members isx . ( 70 * f + x * m ) / ( m + f ) = 66 - > solving this equation after substituting 2 m = f , x = 58 . ans c ."
Answer: c
Question (physics): amit and ananthu can do a work in 15 days and 90 days respectively . amit started the work and left after 3 days . ananthu took over and completed the work . in how many days was the total work completed ?
Options: a ) 68 days , b ) 70 days , c ) 73 days , d ) 75 days , e ) 77 days
Rationale: "amit ’ s one day ’ s work = 1 / 15 amit ’ s 3 day ’ s work = 1 / 15 * 3 = 1 / 5 work left = 1 - 1 / 5 = 4 / 5 ananthu ’ s one day ’ s work = 1 / 90 ananthu can do work in = 4 / 5 * 90 = 72 days so total days = 72 + 3 = 75 days answer : d"
Answer: d
Question (general): city x has a population 6 times as great as the population of city y , which has a population twice as great as the population of city z . what is the ratio of the population of city x to the population of city z ?
Options: a ) 1 : 8 , b ) 1 : 4 , c ) 2 : 1 , d ) 4 : 1 , e ) 12 : 1
Rationale: "x = 6 y , y = 2 * z x : y , y : z 6 : 1 , 2 : 1 12 : 2 , 2 : 1 so , x : z = 12 : 1 ( e )"
Answer: e
Question (physics): a and b can do a work in 2 days , b and c in 4 days and c and a in 6 days . in how many days will the work be completed , if all three of them work together ?
Options: a ) 10 / 12 , b ) 16 / 28 , c ) 24 / 11 , d ) 42 / 16 , e ) 18 / 12
Rationale: one day work of a and b = 1 / 2 one day work of b and c = 1 / 4 one day work of c and a = 1 / 6 2 ( a + b + c ) = 1 / 2 + 1 / 4 + 1 / 6 . 2 ( a + b + c ) = 11 / 12 ( a + b + c ) = 11 / 24 number of days required = 24 / 11 days . answer : c
Answer: c
Question (general): what is the sum of all the multiples of 7 between 30 and 100 ?
Options: a ) 692 , b ) 700 , c ) 677 , d ) 654 , e ) 665
Rationale: "you first have to know all the multiples of 7 between 30 and 100 . they are 35 , 42 , 49 , 56 , 63 , 70 , 77 , 84 , 91 , and 98 . if you add all these numbers together , you get 665 . final answer : e"
Answer: e
Question (general): if { x } is the product of all even integers from 1 to x inclusive , what is the greatest prime factor of { 20 } + { 24 } ?
Options: a ) 23 , b ) 25 , c ) 11 , d ) 5 , e ) 2
Rationale: "soln : { 24 } + { 20 } = 24 * { 20 } + { 20 } = 25 * { 20 } answer : b"
Answer: b
Question (general): the maximum number of students among them 1001 pens and 910 pencils can be distributed in such a way that each student get the same number of pens and same number of pencils ?
Options: a ) 87 , b ) 91 , c ) 100 , d ) 96 , e ) 101
Rationale: "number of pens = 1001 number of pencils = 910 required number of students = h . c . f . of 1001 and 910 = 91 answer is b"
Answer: b
Question (general): how many even integers n , such that 20 < = n < = 250 are of the form 3 k + 4 , where k is any natural number ?
Options: a ) 33 , b ) 36 , c ) 39 , d ) 42 , e ) 45
Rationale: "the first number is 22 = 16 + 6 ( 1 ) . we can continue adding 6 to make a list : 22 , 28 , 34 , . . . the last number is 250 = 16 + 6 ( 39 ) there are 39 numbers in the list . the answer is c ."
Answer: c
Question (physics): a trader sells 85 meters of cloth for $ 8925 at the profit of $ 15 per metre of cloth . what is the cost price of one metre of cloth ?
Options: a ) $ 69 , b ) $ 78 , c ) $ 90 , d ) $ 86 , e ) $ 77
Rationale: c $ 90 sp of 1 m of cloth = 8925 / 85 = $ 105 cp of 1 m of cloth = sp of 1 m of cloth - profit on 1 m of cloth = $ 105 - $ 15 = $ 90 .
Answer: c
Question (general): a fruit - salad mixture consists of apples , peaches , and grapes in the ratio 12 : 8 : 7 , respectively , by weight . if 54 pounds of the mixture is prepared , the mixture includes how many more pounds of apples than grapes ?
Options: a ) 15 , b ) 10 , c ) 9 , d ) 6 , e ) 4
Rationale: "we can first set up our ratio using variable multipliers . we are given that a fruit - salad mixture consists of apples , peaches , and grapes , in the ratio of 6 : 5 : 2 , respectively , by weight . thus , we can say : apples : peaches : grapes = 12 x : 8 x : 7 x we are given that 54 pounds of the mixture is prepared so we can set up the following question and determine a value for x : 12 x + 8 x + 7 x = 54 27 x = 54 x = 2 now we can determine the number of pounds of apples and of grapes . pounds of grapes = ( 7 ) ( 2 ) = 14 pounds of apples = ( 12 ) ( 2 ) = 24 thus we know that there are 24 - 14 = 10 more pounds of apples than grapes . answer is b ."
Answer: b
Question (general): of the six - digit positive integers that have no digits equal to zero , how many have two digits that are equal to each other and the remaining digit different from the other two ?
Options: a ) 2443200 , b ) 3643200 , c ) 5443200 , d ) 1443200 , e ) 2163200
Rationale: "of the six - digit positive integers that have no digits equal to zero , how many have two digits that are equal to each other and the remaining digit different from the other two ? a . 24 b . 36 c . 72 d . 144 e . 216 choosing the digit for p - 9 ways ; choosing the digit for q - 8 ways ; choosing the digit for r - 7 ways ; choosing the digit for s - 6 ways ; choosing the digit for t - 5 ways ; # of permutations of 3 digits in ppqrst - 6 ! / 2 ! total : 9 * 8 * 7 * 6 * 5 * 6 ! / 2 ! = 5443200 . answer : c ."
Answer: c
Question (general): two employees x and y are paid a total of rs . 330 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 . 150 , b ) s . 200 , c ) s . 250 , d ) s . 350 , e ) s . 400
Rationale: "let the amount paid to x per week = x and the amount paid to y per week = y then x + y = 330 but x = 120 % of y = 120 y / 100 = 12 y / 10 â ˆ ´ 12 y / 10 + y = 330 â ‡ ’ y [ 12 / 10 + 1 ] = 330 â ‡ ’ 22 y / 10 = 330 â ‡ ’ 22 y = 3300 â ‡ ’ y = 3300 / 22 = 300 / 2 = rs . 150 a )"
Answer: a
Question (general): if the price of a certain computer increased 30 percent from d dollars to 338 dollars , then 2 d =
Options: a ) 540 , b ) 520 , c ) 619 , d ) 649 , e ) 700
Rationale: "before price increase price = d after 30 % price increase price = d + ( 30 / 100 ) * d = 1.3 d = 338 ( given ) i . e . d = 338 / 1.3 = $ 260 i . e . 2 d = 2 * 260 = 520 answer : option b"
Answer: b
Question (gain): simple interest on a certain sum of money for 2 years at 16 % per annum is one fourth the compound interest on rs . 4000 for 2 years at 8 % per annum . the sum placed on simple interest is
Options: a ) rs . 4000 , b ) rs . 665.6 , c ) rs . 520 , d ) rs . 166.4 , e ) none
Rationale: "solution c . i . = rs [ 4000 x ( 1 + 8 / 100 ) â ² - 4000 ] rs . ( 4000 x 108 / 100 x 108 / 100 - 4000 ) = rs . 665.6 . sum = rs . [ 166.4 x 100 / 2 x 16 ] = rs . 520 . answer c"
Answer: c
Question (general): each digit 1 through 5 is used exactly once to create a 5 - digit integer . if the 1 and the 5 can not be adjacent digits in the integer , how many 5 - digit integers are possible ?
Options: a ) 48 , b ) 66 , c ) 76 , d ) 78 , e ) none of these
Rationale: "number of arrangements using 5 distinct digits = 5 ! number of arrangements in which 1 and 5 are adjacent - consider 1 and 5 together as one group . now you have 4 numbers / groups to arrange which can be done in 4 ! ways . in each of these arrangements , 1 and 5 can be arranged as 15 or 51 . number of arrangements in which 3 and 4 are not adjacent = 5 ! - 2 * 4 ! = 72 answer : e"
Answer: e
Question (general): a pupil ' s marks were wrongly entered as 79 instead of 45 . 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 ) 80 , c ) 20 , d ) 68 , e ) 26
Rationale: "let there be x pupils in the class . total increase in marks = ( x * 1 / 2 ) = x / 2 . x / 2 = ( 79 - 45 ) = > x / 2 = 34 = > x = 68 . answer : d"
Answer: d
Question (general): in a can , there is a mixture of milk and water in the ratio 2 : 5 . if it is filled with an additional 8 litres of milk the can would be full and ratio of milk and water would become 4 : 5 . find the capacity of the can ?
Options: a ) 40 , b ) 44 , c ) 48 , d ) 50 , e ) 8
Rationale: "let the capacity of the can be t litres . quantity of milk in the mixture before adding milk = 2 / 7 ( t - 8 ) after adding milk , quantity of milk in the mixture = 4 / 9 t . 4 t / 9 - 8 = 2 / 7 ( t - 8 ) 2 t = 72 - 56 = > t = 8 . answer : e"
Answer: e
Question (general): 90 % of the population of a village is 45000 . the total population of the village is ?
Options: a ) 26799 , b ) 24000 , c ) 50000 , d ) 29973 , e ) 12312
Rationale: "x * ( 90 / 100 ) = 45000 x = 500 * 100 x = 50000 answer : c"
Answer: c
Question (general): total 52 matches are conducted in knockout match type . how many players will be participated in that tournament ?
Options: a ) 60 , b ) 51 , c ) 53 , d ) 56 , e ) 50
Rationale: "51 players answer : b"
Answer: b
Question (other): the ratio between the number of sheep and the number of horses at the stewart farm is 1 to 7 , if each horse is fed 230 ounces of horse food per day and the farm needs a total 12,880 ounces of horse food per day , what is the number of sheep in the farm ?
Options: a ) 18 , b ) 28 , c ) 32 , d ) 56 , e ) 8
Rationale: "let the number of sheeps and horses be 1 x and 7 x . now total number of horses = total consumption of horse food / consumption per horse = 12880 / 230 = 56 , which is equal to 7 x . = > x = 8 sheeps = 1 x = 1 * 8 = 8 . hence e"
Answer: e
Question (general): there are 12 teams in a soccer league and each team plays each of the other teams once . how many games will be played in total ?
Options: a ) 58 , b ) 62 , c ) 66 , d ) 70 , e ) 74
Rationale: 12 c 2 = 66 the answer is c .
Answer: c
Question (geometry): the length of a rectangle is increased by 45 % and its breadth is decreased by 20 % . what is the effect on its area ?
Options: a ) 1288 , b ) 1299 , c ) 1000 , d ) 10000 , e ) 11600
Rationale: "100 * 100 = 10000 145 * 80 = 11600 answer : e"
Answer: e
Question (general): michael cashed a check for $ 1270 and received only $ 10 and $ 50 bills in return . during the course of a day , he used 15 bills and then lost the rest of the money . if the number of $ 10 bills used was either one more or one less than the number of $ 50 bills used , what is the minimum possible amount of money that was lost ?
Options: a ) $ 830 , b ) $ 800 , c ) $ 770 , d ) $ 730 , e ) $ 700
Rationale: 1 . 15 bills spent 2 . number of $ 10 bills is one more or one less than number of $ 50 bills . so , 2 choices 1 . 7 * 10 + 8 * 50 = $ 470 2 . 8 * 10 + 7 * 50 = $ 430 least money lost implies most money spent . i . e $ 470 therefore , lost money = 1270 - 470 = 800 . answer b
Answer: b
Question (physics): a man ' s regular pay is $ 3 per hour up to 40 hours . overtime is twice the payment for regular time . if he was paid $ 180 , how many hours overtime did he work ?
Options: a ) 8 , b ) 5 , c ) 9 , d ) 6 , e ) 10
Rationale: "at $ 3 per hour up to 40 hours , regular pay = $ 3 x 40 = $ 120 if total pay = $ 168 , overtime pay = $ 180 - $ 120 = $ 60 overtime rate ( twice regular ) = 2 x $ 3 = $ 6 per hour = > number of overtime hours = $ 60 / $ 6 = 10 ans is e"
Answer: e
Question (general): if a farmer sells 5 of his goats , his stock of feed will last for 4 more days than planned , but if he buys 10 more goats , he will run out of feed 3 days earlier than planned . if no goats are sold or bought , the farmer will be exactly on schedule . how many goats does the farmer have ?
Options: a ) 12 , b ) 20 , c ) 48 , d ) 55 , e ) 60
Rationale: say farmer has n goat and he is good for d days . : - we have 3 equations given in question : - ( n - 5 ) * d + 4 = ( n + 10 ) * ( d - 3 ) = n * d solving these : ( you can solve 1 st and 3 rd and 2 nd and 3 rd together ) we get : 10 d - 3 n = 30 4 n - 5 d = 20 = > n = 20 ans b it is !
Answer: b
Question (physics): cole drove from home to work at an average speed of 30 kmh . he then returned home at an average speed of 90 kmh . if the round trip took a total of 2 hours , how many minutes did it take cole to drive to work ?
Options: a ) 66 , b ) 70 , c ) 72 , d ) 75 , e ) 90
Rationale: "let the distance one way be x time from home to work = x / 30 time from work to home = x / 90 total time = 2 hrs ( x / 30 ) + ( x / 90 ) = 2 solving for x , we get x = 45 time from home to work in minutes = ( 45 ) * 60 / 30 = 90 minutes ans = e"
Answer: e
Question (physics): when tom works alone he chops 3 lb . salad in 2 minutes , and when tammy works alone she chops 2 lb . salad in 3 minutes . they start working together , and after some time finish chopping 65 lb . of salad . of those 80 lb . , the salad quantity chopped by tammy is what percent lesser than the quantifying chopped by tom ?
Options: a ) 44 % , b ) 100 % , c ) 15 % , d ) 125 % , e ) 400 %
Rationale: "tom chops 4 lbs in 6 minutes tammy chops 9 lbs in 6 minutes so in the same amount of time , tammy chops 125 % more than tom , since 9 is 125 % greater than 4 . so 125 % is the answer . note that the actual time does n ' t matter . if you multiply the time each work by x , you ' ll multiply the work each does by x , and 9 x is still 125 % greater than 4 x . 125 % lesser than tom ans : d"
Answer: d
Question (general): what is the smallest integer y for which 27 ^ y > 3 ^ 24 ?
Options: a ) 7 , b ) 8 , c ) 9 , d ) 10 , e ) 12
Rationale: "27 ^ y > 3 ^ 24 converting into the same bases : 27 ^ y > 27 ^ 8 therefore for the equation to hold true , y > 8 or y = 9 option c"
Answer: c
Question (physics): a and b together can do a piece of work in 4 days . if a alone can do the same work in 20 days , then b alone can do the same work in ?
Options: a ) 0.35 days , b ) 0.45 days , c ) 0.55 days , d ) 0.25 days , e ) 0.2 days
Rationale: "b = 1 / 4 – 1 / 20 = 0.2 days answer : e"
Answer: e
Question (general): a bell curve ( normal distribution ) has a mean of − 1 and a standard deviation of 1 / 8 . how many integer values z are within 3 standard deviations of the mean ?
Options: a ) 0 , b ) 1 , c ) 3 , d ) 6 , e ) 7
Rationale: got the question correctly - - the second item - - [ highlight ] bthe list of elements in the set is required . [ / highlight ] is not required . with the new information , there is only one integer value ( - 1 ) that is between ( - 1.375 , - 0.625 ) i . e . , falls within the three 3 sd range . b
Answer: b
Question (gain): a man ' s speed with the current is 15 km / hr and the speed of the current is 2.8 km / hr . the man ' s speed against the current is ?
Options: a ) 9.4 , b ) 20 , c ) 50 , d ) 30 , e ) 40
Rationale: "man ' s speed with the current = 15 km / hr = > speed of the man + speed of the current = 15 km / hr speed of the current is 2.8 km / hr hence , speed of the man = 15 - 2.8 = 12.2 km / hr man ' s speed against the current = speed of the man - speed of the current = 12.2 - 2.8 = 9.4 km / hr answer is a ."
Answer: a
Question (general): 1 / 3 + 1 / 2 - 5 / 6 + 1 / 5 + 1 / 4 - 9 / 20 - 2 / 15 =
Options: a ) 0 , b ) 2 / 15 , c ) 2 / 5 , d ) 9 / 20 , e ) 5 / 6
Rationale: "we need to determine the result of 1 / 3 + 1 / 2 - 5 / 6 + 1 / 5 + 1 / 4 - 9 / 20 let ’ s add the given fractions in two groups . in the group of the first three fractions , notice that 1 / 3 and 1 / 2 share a common denominator of 6 with 5 / 6 . 1 / 2 + 1 / 3 = 3 / 6 + 2 / 6 = 5 / 6 thus , 5 / 6 – 5 / 6 = 0 looking at the 2 nd group of the fractions ( 1 / 5 , 1 / 4 , and 9 / 20 ) , notice that 1 / 5 and 1 / 4 share a common denominator of 20 with 9 / 20 . 1 / 5 + 1 / 4 = 4 / 20 + 5 / 20 = 9 / 20 thus , 9 / 20 – 9 / 20 = 0 . thus , the result of 1 / 3 + 1 / 2 – 5 / 6 + 1 / 5 + 1 / 4 – 9 / 20 is 2 / 15 . answer : b"
Answer: b
Question (physics): a river 2 m deep and 45 m wide is flowing at the rate of 3 kmph the amount of water that runs into the sea per minute is ?
Options: a ) 4500 , b ) 2678 , c ) 2689 , d ) 2761 , e ) 2882
Rationale: "( 3000 * 2 * 5 ) / 60 = 4500 m 3 answer : a"
Answer: a
Question (general): in a coconut grove , ( x + 2 ) trees yield 60 nuts per year , x trees yield 120 nuts per year and ( x – 2 ) trees yield 180 nuts per year . if the average yield per year per tree be 100 , find x .
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "( x + 2 ) × 60 + x × 120 + ( x − 2 ) × 180 / ( x + 2 ) + x + ( x − 2 ) = 100 ⇒ 360 x − 240 / 3 x = 100 ⇒ 60 x = 240 ⇒ x = 4 answer e"
Answer: e
Question (general): tourist purchased a total of 30 travelers checks in $ 50 and $ 100 denominations . the total worth of the travelers checks is $ 1800 . how many checks of $ 50 denominations can he spend so that average amount ( arithmetic mean ) of the remaining travelers checks is $ 70 ?
Options: a ) 4 , b ) 12 , c ) 15 , d ) 20 , e ) 24
Rationale: "you could set - up a quick table and brute force the answer . a 4 * 50 200 1800 - 200 1600 26 61.54 b 12 * 50 600 1800 - 600 1200 18 66.67 c 15 * 50 750 1800 - 750 1050 15 70.00 d 20 * 50 1000 1800 - 1000 800 10 80.00 e 24 * 50 1200 1800 - 1200 600 6 100.00 answer is c"
Answer: c
Question (general): in a graduating class , 40 percent of the students are male . in this class , 50 percent of the male students and 40 percent of the female students are 25 years old or older . if one student in the class is randomly selected , approximately what is the probability that he or she will be less than 25 years old ?
Options: a ) 0.44 , b ) 0.5 , c ) 0.56 , d ) 0.62 , e ) 0.68
Rationale: "let x be the total number of students . the number students who are younger than 25 is 0.5 * 0.4 x + 0.6 * 0.6 x = 0.56 x the answer is c ."
Answer: c
Question (general): what is the remainder when 14,451 × 15,654 × 16,783 is divided by 5 ?
Options: a ) 1 , b ) 3 , c ) 2 , d ) 4 , e ) 5
Rationale: "only the unit ' s digit of the product will decide the remainder when divided by 5 . hence , 1 * 4 * 3 = will give units digit as 2 so , whatever be the number , if it ends in 2 , the remainder after dividing with 5 will be 2 . option c"
Answer: c
Question (other): up to 20 , how many no . are odd and prime both ?
Options: a ) 9 , b ) 5 , c ) 6 , d ) 7 , e ) 8
Rationale: we have to calculate it manually . odd 1,3 , 5,7 , 9,11 , 13,15 , 17,19 prime 2,3 , 5,7 , 11,13 , 17,19 common 3 , 5,7 , 11,13 , 17,19 total no . 7 answer d
Answer: d
Question (geometry): the area of sector of a circle whose radius is 15 metro and whose angle at the center is 42 â ° is ?
Options: a ) 52.6 , b ) 52.9 , c ) 52.8 , d ) 52.1 , e ) 82.5
Rationale: "42 / 360 * 22 / 7 * 15 * 15 = 82.5 m 2 answer : e"
Answer: e
Question (physics): 9 . on level farmland , two runners leave at the same time from the intersection of two country roads . one runner jogs due north at a constant rate of 8 miles per hour while the second runner jogs due east at a constant rate that is 42 miles per hour faster than the first runner ' s rate . how far apart , to the nearest mile , will they be after 1 / 2 hour ?
Options: a ) 6 , b ) 7 , c ) 8 , d ) 12 , e ) 14
Rationale: "if runner 1 is going north and runner 2 is going east they are like two sides of a 90 degree triangle . side 1 = 8 m / h - - > 4 m in 1 / 2 hr side 2 = 10 m / h - - > 5 m in 1 / 2 hr to complete this right angle triangle d ^ 2 = 4 ^ 2 + 5 ^ 2 d ^ 2 = 41 = ~ 6 answer option a"
Answer: a
Question (general): if x , y , and z are positive integers and 2 x = 4 y = 5 z , then the least possible value of x + y + z is
Options: a ) 40 , b ) 50 , c ) 55 , d ) 60 , e ) 19
Rationale: "given 2 x = 4 y = 5 z x + y + z in terms of x = x + ( 2 x / 4 ) + ( 2 x / 5 ) = 38 x / 20 = 19 x / 10 now checking with each of the answers and see which value gives a minimum integer value . a x = 10 / 19 * 40 , not an integer b , c , e can be ruled out similarly . d is minimum value as x = 19 * 10 / 19 = 10 answer is e"
Answer: e
Question (physics): a boat can travel with a speed of 13 km / hr in still water . if the speed of the stream is 4 km / hr , find the time taken by the boat to go 68 km downstream .
Options: a ) 2 , b ) 4 , c ) 6 , d ) 5 , e ) 3
Rationale: "speed downstream = ( 13 + 4 ) km / hr = 17 km / hr . time taken to travel 68 km downstream = 68 / 17 = 4 hrs ans - b"
Answer: b
Question (general): the events a and b are independent , the probability that event a occurs is greater than 0 , and the probability that event a occurs is twice the probability that event b occurs . the probability that at least one of events a and b occurs is 5 times the probability that both events a and b occur . what is the probability that event a occurs ?
Options: a ) 1 / 45 , b ) 1 / 2 , c ) 1 / 8 , d ) 2 / 3 , e ) 2 / 11
Rationale: let us say probability of a occuring is a . let us say probability of b occuring is b . a = 2 b probability ( either a or b or both ) = 5 times probability ( a and b ) a * ( 1 - b ) + b * ( 1 - a ) + ab = 5 * ab substituting a = 2 b in the second equation : 2 b * ( 1 - b ) + b * ( 1 - 2 b ) + 2 b * b = 5 * 2 b * b 3 b - 2 b ^ 2 = 110 b ^ 2 3 b = 12 b ^ 2 b = 3 / 12 = 1 / 4 so , a = 2 b = 1 / 2
Answer: b
Question (general): a , b , and c are integers and a < b < c . s is the set of all integers from a to b , inclusive . q is the set of all integers from b to c , inclusive . the median of set s is ( 3 / 4 ) * b . the median of set q is ( 7.5 / 8 ) * c . if r is the set of all integers from a to c , inclusive , what fraction of c is the median of set r ?
Options: a ) 3 / 8 , b ) 1 / 2 , c ) 11 / 16 , d ) 23 / 32 , e ) 3 / 4
Rationale: "the answer isc : 11 / 16 . the key to this problem is remembering that the median for a consecutive set of numbers is equivalent to its mean . for example , the mean and median of a set consisting of x , x + 1 , x + 2 , . . . , y will always be ( x + y ) / 2 . for set s , consisting of numbers ( a , a + 1 , . . . , b ) , the median is given to be 3 / 4 * b : ( a + b ) / 2 = ( 3 / 4 ) * b a = b / 2 for set q , consisting of numbers ( b , b + 1 , . . . , c ) , the median is given to be 7.5 / 8 * c : ( b + c ) / 2 = ( 7.5 / 8 ) * c b = ( 7 / 8 ) * c for set r , consisting of numbers ( a , a + 1 , . . . c ) , the median needs to be found : a = b / 2 = ( 7 / 8 * c ) / 2 = ( 7 / 16 ) * c median = ( a + c ) / 2 = ( 7 / 16 * c + c ) / 2 = ( 23 / 16 ) * c / 2 = ( 23 / 32 ) * c ( answer d )"
Answer: d
Question (general): barbata invests $ 1400 in the national bank at 5 % . how much additional money must she invest at 8 % so that the total annual income will be equal to 6 % of her entire investment ?
Options: a ) 700 , b ) 300 , c ) 1000 , d ) 360 , e ) 2400
Rationale: let the additional invested amount for 8 % interest be x ; equation will be ; 1400 + 0.05 * 2400 + x + 0.08 x = 1400 + x + 0.06 ( 2400 + x ) 0.05 * 1400 + 0.08 x = 0.06 x + 0.06 * 1400 0.02 x = 1400 ( 0.06 - 0.05 ) x = 1400 * 0.01 / 0.02 = 700 ans : a
Answer: a