regisss/math_qa huggingface dataset:
Problem: in a certain math department , students are required to enroll in either calculus or trigonometry , each of which is offered in beginner and advanced courses . the number of students enrolled in trigonometry is 50 % greater than the number of students enrolled in calculus , and 70 % of calculus students are enrolled in the beginner course . if 4 / 5 of students are in the beginner courses , and one student is selected at random , what is the probability that a beginner trigonometry student is selected ?
Options: a ) 40 % , b ) 44 % , c ) 48 % , d ) 52 % , e ) 56 %
Rationale: "let x be the number of students in calculus . then the number of students in trigonometry is 1.5 x the number of students on beginner calculus is 0.7 x the number of students in beginner trigonometry is 4 / 5 * ( 2.5 x ) - 0.7 x = 1.3 x the percentage in beginner trigonometry is 1.3 x / 2.5 x = 52 % the answer is d ."
Answer: d
Problem: a and b can do a work in 12 days , b and c in 30 days and c and a in 36 days . in how many days will the work be completed , if all three of them work together ?
Options: a ) 180 / 15 , b ) 180 / 10 , c ) 180 / 13 , d ) 180 / 15 , e ) 180 / 11
Rationale: "one day work of a and b = 1 / 12 one day work of b and c = 1 / 30 one day work of c and a = 1 / 36 2 ( a + b + c ) = 1 / 12 + 1 / 30 + 1 / 36 2 ( a + b + c ) = 13 / 90 ( a + b + c ) = 13 / 180 number of days required = 180 / 13 days . answer : c"
Answer: c
Problem: if the median of a list of numbers is m , the first quartile of the list is the median of the numbers in the list that are less than m . what is the first quartile of the list of numbers 42 , 24 , 30 , 22 , 28 , 19 , 33 and 35 ?
Options: a ) 33 , b ) 25 , c ) 27 , d ) 24 , e ) 23
Rationale: "it is given that a quartile is the middle number of all numbers less than median . . so lets arrange the number in ascending order - 42 , 24 , 30 , 22 , 28 , 19 , 33 and 35 19 , 22 , 24 , 28 , 30 , 33 , 35 , 42 . . . numbers less than median are 19 , 22 , 24 , 28 . . the median of these numbers = center of 22 and 24 = 23 e"
Answer: e
Problem: a and b can do a piece of work in 18 days ; band c can do it in 24 days a and c can do it in 36 days . in how many days will a , band c finish it together ?
Options: a ) 15 days , b ) 12 days , c ) 8 days , d ) 16 days , e ) 9 days
Rationale: sol . ( a + b ) ' s 1 day ' s work = ( 1 / 18 ) ( b + c ) ' s 1 day ' s work = ( 1 / 24 ) and ( a + c ) ' s 1 day ' s work = ( 1 / 36 ) adding , we get : 2 ( a + b + c ) ' s 1 day ' s work = ¬ ( 1 / 18 + 1 / 24 + 1 / 36 ) = 9 / 72 = 1 / 8 ( a + b + c ) ' s 1 day ' s work = 1 / 16 thus , a , band c together can finish the work in 16 days . ans : d
Answer: d
Problem: what is the sum of all possible solutions to | x - 3 | ^ 2 + | x - 3 | = 30 ?
Options: a ) 4 , b ) 5 , c ) 3 , d ) 2 , e ) 1
Rationale: "denote | x - 3 | as y : y ^ 2 + y = 30 - - > y = - 6 or y = 5 . discard the first solution since y = | x - 3 | , so it ' s an absolute value and thus can not be negative . y = | x - 3 | = 5 - - > x = 8 or x = - 3 . the sum = 5 . answer : b ."
Answer: b
Problem: let the number which when multiplied by 12 is increased by 198 .
Options: a ) 18 , b ) 20 , c ) 26 , d ) 28 , e ) 30
Rationale: "solution let the number be x . then , 12 x - x = 198 ‹ = › 11 x = 198 x ‹ = › 18 . answer a"
Answer: a
Problem: 200 college students were asked in a survey if they preferred windows or mac brand computers . 70 students claimed that they preferred mac to windows brand computers . one third as many of the students who preferred mac to windows , equally preferred both brands . 80 of the students had no preference . how many of the students in the survey preferred windows to mac brand computers ?
Options: a ) 25 , b ) 50 , c ) 57 , d ) 60 , e ) 75
Rationale: "200 = 70 ( mac ) + x ( window ) + 80 ( both ) = > x = 50 answer : b"
Answer: b
Problem: what is the smallest no . that should be added to 27452 to make it exactly divisible by 9 ?
Options: a ) 1 , b ) 3 , c ) 5 , d ) 6 , e ) 9
Rationale: if a number is divisible by 9 , the sum of its digits must be a multiple of 9 . here , 2 + 7 + 4 + 5 + 2 = 20 , the next multiple of 9 is 27 . 7 must be added to 27452 to make it divisible by 9 e
Answer: e
Problem: a circle graph shows how the megatech corporation allocates its research and development budget : 14 % microphotonics ; 25 % home electronics ; 15 % food additives ; 29 % genetically modified microorganisms ; 8 % industrial lubricants ; and the remainder for basic astrophysics . if the arc of each sector of the graph is proportional to the percentage of the budget it represents , how many degrees of the circle are used to represent basic astrophysics research ?
Options: a ) 8 ° , b ) 10 ° , c ) 18 ° , d ) 33 ° , e ) 52 °
Rationale: "here all percentage when summed we need to get 100 % . as per data 14 + 25 + 15 + 29 + 8 = 91 % . so remaining 9 % is the balance for the astrophysics . since this is a circle all percentage must be equal to 360 degrees . 100 % - - - - 360 degrees then 9 % will be 33 degrees . . imo option d ."
Answer: d
Problem: two trains 140 m and 160 m long run at the speed of 60 km / hr and 40 km / hr respectively in opposite directions on parallel tracks . the time ( in seconds ) which they take to cross each other , is :
Options: a ) 9 , b ) 9.6 , c ) 10 , d ) 10.8 , e ) 7
Rationale: "explanation : relative speed = ( 60 + 40 ) km / hr = [ 100 x ( 5 / 18 ) ] m / sec = ( 250 / 9 ) m / sec . distance covered in crossing each other = ( 140 + 160 ) m = 300 m . required time = [ 300 x ( 9 / 250 ) ] sec = ( 54 / 5 ) sec = 10.8 sec . answer : d"
Answer: d
Problem: a certain company that sells only cars and trucks reported that revenues from car sales in 1997 were down 11 percent from 1996 and revenues from truck sales were up 7 percent from 1996 . if total revenues from car sales and truck sales in 1997 were up 1 percent from 1996 , what is the ratio t of revenue from car sales in 1996 to revenue from truck sales in 1996 ?
Options: a ) 1 : 2 , b ) 4 : 5 , c ) 1 : 1 , d ) 3 : 2 , e ) 5 : 3
Rationale: a . . i have probably solved this question 3 - 4 times by now . . remember the answer . . 1 : 2
Answer: a
Problem: in a recent election , geoff received 0.5 percent of the 6,000 votes cast . to win the election , a candidate needed to receive more than x % of the vote . if geoff needed exactly 3,000 more votes to win the election , what is the value of x ?
Options: a ) 50 , b ) 50.5 , c ) 56 , d ) 60 , e ) 63
Rationale: word problems are tricky in somehow more than other problem because you have the additional step to translate . breaking the problem : geoff how many votes he receives ? ? 30 votes he needs 3571 more votes so : 30 + 3000 = 3030 now what ' s the problem wants ? ? a x % . . . . . . . . 3030 is what % of total votes 6000 . . . . . . . . translating : 3030 = x / 100 * 6000 - - - x = 50.5 % . . . . . . . . . . b
Answer: b
Problem: if a ^ 3 is divisible by 240 what is the least possible value of integer a ?
Options: a ) 12 , b ) 30 , c ) 60 , d ) 90 , e ) 120
Rationale: 240 can be written as ( 2 ^ 4 ) * 3 * 5 . for a ^ 3 to be divisible by 240 it should contain atleast 2 ^ 4 and 3 and 5 in its factors . we can leave out option a because 12 doesnt have 5 as one of its factor . now if we check for option b , 30 can be written as 2 * 3 * 5 , hence 30 ^ 3 will have 3 as the maximum power of 2 , so we can leave out this option too . option c is the right answer if we follow the same method as we followed for other two previous options . 60 = ( 2 ^ 2 ) * 3 * 5 ; 60 ^ 3 = ( 2 ^ 6 ) * ( 3 ^ 3 ) * ( 5 ^ 3 ) . so it shows that 60 ^ 3 is divisible by 240 and hence the answer . answer : c
Answer: c
Problem: a goods train runs at a speed of 72 kmph and crosses a 250 m long platform in 26 seconds . what is the length of the goods train ?
Options: a ) 230 m , b ) 240 m , c ) 260 m , d ) 270 m , e ) 250 m
Rationale: "s = 250 + x / t 72 * 5 / 18 = 250 + x / 26 x = 270 answer : d"
Answer: d
Problem: a metallic sheet is of rectangular shape with dimensions 40 m x 30 m . from each of its corners , a square is cut off so as to make an open box . if the length of the square is 8 m , the volume of the box ( in m cube ) is :
Options: a ) 3220 m cube
b ) 4140 m cube
c ) 2688 m cube
d ) 4000 m cube
e ) none of these
Rationale: explanation : l = ( 40 - 16 ) m = 24 m , [ because 8 + 8 = 16 ] b = ( 30 - 16 ) m = 14 m , h = 8 m . volume of the box = ( 24 x 14 x 8 ) m cube = 2688 m cube . option c
Answer: c
Problem: the average of temperatures at noontime from monday to friday is 60 ; the lowest one is 50 , what is the possible maximum range of the temperatures ?
Options: a ) 50 , b ) 25 , c ) 40 , d ) 45 , e ) 75
Rationale: "average = 60 , sum of temperatures = 60 * 5 = 300 as the min temperature is 50 , max would be 300 - 4 * 50 = 100 - - > the range = 100 ( max ) - 50 ( min ) = 50 answer : a"
Answer: a
Problem: find the area of a rhombus one side of which measures 20 cm and one diagonal is 26 cm .
Options: a ) 600 cm 2 , b ) 500 cm 2 , c ) 516 cm 2 , d ) 416 cm 2 , e ) 100 cm 2
Rationale: "explanation : let other diagonal = 2 x cm . since diagonals of a rhombus bisect each other at right angles , we have : ( 20 ) 2 = ( 12 ) 2 + ( x ) 2 = > x = √ ( 20 ) 2 – ( 12 ) 2 = √ 256 = 16 cm . _ i so , other diagonal = 32 cm . area of rhombus = ( 1 / 2 ) x ( product of diagonals ) = ( 1 / 2 × 26 x 32 ) cm 2 = 416 cm 2 answer : option d"
Answer: d
Problem: what are the last two digits of ( 301 * 402 * 503 * 604 * 646 * 547 * 448 * 349 ) ^ 2
Options: a ) 96 , b ) 76 , c ) 56 , d ) 36 , e ) 16
Rationale: "( ( 301 * 402 * 503 * 604 * 646 ) * ( 547 * 448 * 349 ) ) ^ 2 if you observe above digits , last digit are : 1,2 , 3,4 , 6,7 , 8,9 ; 5 is missing ; so i have rearranged them so that multiplication will be easy for me as initial 4 digits have last two digits as 01 , 02,03 , 04,46 and final three as 47 * 48 * 49 . solving for only last two digits and multiplying them we get : ( ( 06 * 04 * 46 ) ( 56 * 49 ) ) ^ 2 = ( 44 * 44 ) ^ 2 = 36 ^ 2 = 76 hence answer is b"
Answer: b
Problem: if two of the 4 expressions x + y , x + 5 y , x - y , and 5 x - y are chosen at random , what is the probability t that their product will be of the form of x ^ 2 - ( by ) ^ 2 , where b is an integer ?
Options: a ) 1 / 2 , b ) 1 / 3 , c ) 1 / 4 , d ) 1 / 5 , e ) t = 1 / 6
Rationale: only ( x + y ) ( x - y ) pair will give the form x ^ 2 - ( by ) ^ 2 the probability of selecting these two pairs are 1 / 4 * 1 / 3 = 1 / 12 , assuming x + y is picked first , then x - y , but x - y can be picked first followed by x + y . so the probability t = 1 / 12 * 2 = 1 / 6 ans e
Answer: e
Problem: if n = 2.0767 and n * is the decimal obtained by rounding n to the nearest hundredth , what is the value of n * – n ?
Options: a ) - 0.0053 , b ) - 0.0003 , c ) 0.0007 , d ) 0.0047 , e ) 0.0033
Rationale: "n * = 2.08 n * - n = 2.08 - 2.0767 0.0033 answer : e"
Answer: e
Problem: list f consists of 12 consecutive integers . if - 4 is the least integer in list f , what is the range of positive integers in list f ?
Options: a ) 5 , b ) 6 , c ) 7 , d ) 11 , e ) 12
Rationale: since - 4 is the least integer in list f , then 7 is the largest integer in that list . thus the range of the positive integers in the list is 7 - 1 = 6 . answer : b .
Answer: b
Problem: a man is 32 years older than his son . in two years , his age will be twice the age of his son . the present age of the son is ?
Options: a ) 16 , b ) 19 , c ) 15 , d ) 22 , e ) 30
Rationale: "let the son ' s present age be x years . then , man ' s present age = ( x + 24 ) years . ( x + 32 ) + 2 = 2 ( x + 2 ) x + 34 = 2 x + 4 = > x = 30 . answer : e"
Answer: e
Problem: a man can row upstream at 8 kmph and downstream at 13 kmph . the speed of the stream is ?
Options: a ) 3 kmph , b ) 2.5 kmph , c ) 3.5 kmph , d ) 4.3 kmph , e ) 5 kmph
Rationale: "speed of stream = 1 / 2 ( 13 - 8 ) = 2.5 kmph answer is b"
Answer: b
Problem: a vessel of capacity 30 litres is fully filled with pure milk . nine litres of milk is removed from the vessel and replaced with water . nine litres of the solution thus formed is removed and replaced with water . find the quantity of pure milk in the final milk solution ?
Options: a ) 14.7 , b ) 2.9 , c ) 38.3 , d ) 78.3 , e ) 79.3
Rationale: "explanation : let the initial quantity of milk in vessel be t litres . let us say y litres of the mixture is taken out and replaced by water for n times , alternatively . quantity of milk finally in the vessel is then given by [ ( t - y ) / t ] ^ n * t for the given problem , t = 30 , y = 9 and n = 2 . hence , quantity of milk finally in the vessel = [ ( 30 - 9 ) / 30 ] ^ 2 ( 30 ) = 14.7 litres . answer : option a"
Answer: a
Problem: the tax on a commodity is diminished by 20 % but its consumption is increased by 5 % . find the decrease percent in the revenue derived from it ?
Options: a ) 12 % , b ) 14 % , c ) 16 % , d ) 20 % , e ) 22 %
Rationale: "explanation : 100 * 100 = 10000 80 * 105 = 8400 10000 - - - - - - - 1600 100 - - - - - - - ? = 16 % c )"
Answer: c
Problem: if 15 ! / 3 ^ x is an integer , what is the greatest possible value of x ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "15 - 5 * 3 12 - 4 * 3 9 - 3 * 3 6 - 2 * 3 3 - 1 * 3 hence max of 3 ^ 6 is allowed . imo d ."
Answer: d
Problem: how many cubes of 50 cm edge can be put in a cubical box of 1 m edge .
Options: a ) 17 cm , b ) 10 cm , c ) 8 cm , d ) 6 cm , e ) 7 cm
Rationale: "number of cubes = 100 â ˆ — 100 â ˆ — 100 / 50 * 50 * 50 = 8 number of cubes = 100 â ˆ — 100 â ˆ — 10010 â ˆ — 10 â ˆ — 10 = 1000 note : 1 m = 100 cm answer : c"
Answer: c
Problem: we have boxes colored either red or blue . in each blue box there is a fixed number of blueberries . in each red box there is a fixed number of strawberries . if we dispose of one blue box for one additional red box , the total number of berries would increase by 20 , and the difference between the total number of strawberries and the total number of blueberries would increase by 80 . each blue box contains how many blueberries ?
Options: a ) 25 , b ) 30 , c ) 35 , d ) 40 , e ) 45
Rationale: let x be the number of blueberries in each blue box . then there are x + 20 strawberries in each red box . x + ( x + 20 ) = 80 x = 30 the answer is b .
Answer: b
Problem: a portion of the 90 % solution of chemicals was replaced with an equal amount of 20 % solution of chemicals . as a result , 40 % solution of chemicals resulted . what part of the original solution was replaced ?
Options: a ) 5 / 7 , b ) 6 / 7 , c ) 9 / 5 , d ) 13 / 9 , e ) 13 / 6
Rationale: this is a weighted average question . say x % of the solution was replaced - - > equate the amount of chemicals : 0.90 ( 1 - x ) + 0.2 * x = 0.4 - - > x = 5 / 7 . answer : a .
Answer: a
Problem: in smithtown , the ratio of right - handed people to left - handed people is 3 to 1 and the ratio of men to women is 3 to 2 . if the number of right - handed men is maximized , then what s percent of all the people in smithtown are left - handed women ?
Options: a ) 50 % , b ) 40 % , c ) 25 % , d ) 20 % , e ) 10 %
Rationale: "looking at the ratio we can take total number of people = 20 . . ans 5 / 20 or 25 % c"
Answer: c
Problem: a man is 24 years older than his son . in two years , his age will be twice the age of his son . the present age of this son is
Options: a ) 21 years , b ) 22 years , c ) 23 years , d ) 24 years , e ) 26 years
Rationale: "explanation : let ' s son age is x , then father age is x + 24 . = > 2 ( x + 2 ) = ( x + 24 + 2 ) = > 2 x + 4 = x + 26 = > x = 22 years option b"
Answer: b
Problem: how many cubes of 10 cm edge can be put in a cubical box of 1 m edge
Options: a ) 10 , b ) 100 , c ) 1000 , d ) 10000 , e ) none of these
Rationale: "explanation : number of cubes = ( 100 x 100 x 100 ) / ( 10 x 10 x 10 ) = 1000 answer : c"
Answer: c
Problem: each of the integers from 1 to 14 is written on the a seperate index card and placed in a box . if the cards are drawn from the box at random without replecement , how many cards must be drawn to ensure that the product of all the integers drawn is even ?
Options: a ) 19 , b ) 12 , c ) 11 , d ) 10 , e ) 8
Rationale: "out of the 14 integers : 7 are odd and 7 are even . if we need to make sure that the product of all the integers withdrawn is even then we need to make sure that we have at least one even number . in the worst case : 1 . we will end up picking odd numbers one by one , so we will pick all 7 odd numbers first 2 . 8 th number will be the first even number so we need to withdraw at least 8 numbers to make sure that we get one even number and the product of all the integers picked is even . so , answer will be 8 . ( e )"
Answer: e
Problem: two trains of length 200 m and 300 m are 200 m apart . they start moving towards each other on parallel tracks , at speeds 36 kmph and 72 kmph . in how much time will the trains cross each other ?
Options: a ) 70 / 8 , b ) 70 / 7 , c ) 70 / 5 , d ) 70 / 3 , e ) 70 / 1
Rationale: "relative speed = ( 36 + 72 ) * 5 / 18 = 6 * 5 = 30 mps . the time required = d / s = ( 200 + 200 + 300 ) / 30 = 700 / 30 = 70 / 3 sec . answer : d"
Answer: d
Problem: a salesman ’ s terms were changed from a flat commission of 5 % on all his sales to a fixed salary of rs . 1200 plus 2.5 % commission on all sales exceeding rs . 4,000 . if his remuneration as per new scheme was rs . 1200 more than that by the previous schema , his sales were worth ?
Options: a ) 12028 , b ) 12000 , c ) 12019 , d ) 16000 , e ) 12012
Rationale: "[ 1200 + ( x - 4000 ) * ( 2.5 / 100 ) ] - x * ( 5 / 100 ) = 600 x = 16000 answer : d"
Answer: d
Problem: i chose a number and divide it by 6 . then i subtracted 15 from the result and got 5 . what was the number i chose ?
Options: a ) 600 , b ) 700 , c ) 120 , d ) 900 , e ) none
Rationale: "solution : let xx be the number i chose , then x / 6 â ˆ ’ 15 = 5 x / 6 = 20 x = 120 answer c"
Answer: c
Problem: veena ranks 44 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 ) 88 , b ) 108 , c ) 110 , d ) 90 , e ) 117
Rationale: "total student = 182 failed = 22 paasd student = 182 - 22 = 160 from bottom her rank is = 160 - 44 + 1 = 117 answer : e"
Answer: e
Problem: an alloy of copper and zinc contains copper and zinc in the ratio 5 : 4 . another alloy of copper and zinc contains copper and zinc in the ratio 3 : 6 . in what ratio should the two alloys be mixed so that the resultant alloy contains equal proportions of copper and zinc ?
Options: a ) 4 : 1 , b ) 6 : 1 , c ) 1 : 3 , d ) 2 : 1 , e ) 4 : 1
Rationale: let alloy _ 1 be x units , and let alloy _ 2 be y units . so , fraction of copper in alloy _ 1 = 5 x / 9 , and fraction of zinc in alloy _ 1 = 4 x / 9 . similarly , fraction of copper in alloy _ 2 = 3 y / 9 , and fraction of zinc in alloy _ 2 = 6 y / 9 . mixing them , we get copper = 5 x / 9 + 3 y / 9 ; zinc = 4 x / 9 + 6 y / 9 so , 5 x + 3 y = 4 x + 6 y - > x = 3 y - > x / y = 1 / 3 so , they must be mixed in the ratio 1 : 3 answer : c
Answer: c
Problem: find the area of a parallelogram with base 26 cm and height 14 cm ?
Options: a ) 287 cm 2 , b ) 887 cm 2 , c ) 364 cm 2 , d ) 368 cm 2 , e ) 668 cm 2
Rationale: "area of a parallelogram = base * height = 26 * 14 = 364 cm 2 answer : c"
Answer: c
Problem: after decreasing 24 % in the price of an article costs rs . 532 . find the actual cost of an article ?
Options: a ) 118 , b ) 677 , c ) 700 , d ) 2688 , e ) 1991
Rationale: "cp * ( 76 / 100 ) = 532 cp = 7 * 100 = > cp = 700 answer : c"
Answer: c
Problem: a motorcyclist started riding at highway marker a , drove 120 miles to highway marker b , and then , without pausing , continued to highway marker c , where she stopped . the average speed of the motorcyclist , over the course of the entire trip , was 25 miles per hour . if the ride from marker a to marker b lasted 3 times as many hours as the rest of the ride , and the distance from marker b to marker c was half of the distance from marker a to marker b , what was the average speed , in miles per hour , of the motorcyclist while driving from marker b to marker c ?
Options: a ) 40 , b ) 45 , c ) 108 , d ) 55 , e ) 60
Rationale: "a - b = 120 miles b - c = 60 miles avg speed = 25 miles time taken for a - b 3 t and b - c be t avg speed = ( 120 + 60 ) / total time 25 = 180 / 4 t t = 108 b - c = 108 mph answer c"
Answer: c
Problem: an engineer designed a ball so that when it was dropped , it rose with each bounce exactly one - half as high as it had fallen . the engineer dropped the ball from a 16 - meter platform and caught it after it had traveled 44.5 meters . how many times did the ball bounce ?
Options: a ) 4 , b ) 6 , c ) 7 , d ) 8 , e ) 9
Rationale: "division of total diatance travelled will be 16 + 16 + 8 + 4 + 0.5 ans : 4"
Answer: a
Problem: of 70 players on a cricket team , 37 are throwers . the rest of the team is divided so one third are left - handed and the rest are right handed . assuming that all throwers are right handed , how many right - handed players are there total ?
Options: a ) 54 , b ) 55 , c ) 59 , d ) 71 , e ) 92
Rationale: "total = 90 thrower = 37 rest = 90 - 37 = 33 left handed = 33 / 3 = 11 right handed = 22 if all thrower are right handed then total right handed is 37 + 22 = 59 so c . 59 is the right answer"
Answer: c
Problem: a rectangular grass field is 75 m * 55 m , it has a path of 2.5 m wide all round it on the outside . find the area of the path and the cost of constructing it at rs . 5 per sq m ?
Options: a ) 675 , rs . 3375 , b ) 575 , rs . 1350 , c ) 1350 , rs . 675 , d ) 1450 , rs . 775 , e ) 1550 , rs . 875
Rationale: "area = ( l + b + 2 d ) 2 d = ( 75 + 55 + 2.5 * 2 ) 2 * 2.5 = > 675 675 * 5 = rs . 3375 answer : a"
Answer: a
Problem: an integer n between 1 and 100 , inclusive , is to be chosen at random . what is the probability that n ( n + 1 ) will be divisible by 4 ?
Options: a ) 1 / 9 , b ) 1 / 3 , c ) 1 / 2 , d ) 2 / 3 , e ) 5 / 6
Rationale: "n ( n + 1 ) to be divisible by 4 either n or n + 1 must be a multiples of 4 . in each following group of numbers : { 1 , 2 , 3 , 4 } , { 5 , 6 , 7 , 8 } , . . . , { 97 , 98 , 99 , 100 } there are exactly 2 numbers out of 3 satisfying the above condition . for example in { 1 , 2 , 3 , 4 } n can be : 3 , or 4 . thus , the overall probability is 2 / 4 = 1 / 2 . answer : c ."
Answer: c
Problem: the product of two numbers is 120 and the sum of their squares is 289 . the sum of the number is ?
Options: a ) 23 , b ) 25 , c ) 27 , d ) 31 , e ) 35
Rationale: let the numbers be x and y . then , xy = 120 and x 2 + y 2 = 289 . ( x + y ) 2 = x 2 + y 2 + 2 xy = 289 + ( 2 x 120 ) = 529 x + y = 529 = 23 . option a
Answer: a
Problem: an army ’ s recruitment process included n rounds of selection tasks . for the first a rounds , the rejection percentage was 60 percent per round . for the next b rounds , the rejection percentage was 50 percent per round and for the remaining rounds , the selection percentage was 70 percent per round . if there were 100000 people who applied for the army and 1400 were finally selected , what was the value of n ?
Options: a ) 4 , b ) 5 , c ) 6 , d ) 8 , e ) 10
Rationale: "since 70 % is accepted and figure will be 1400 therefore in last stage tested / interviewed no . candidates should be 2000 so from here i will work for those two steps of decrements of 100000 candidates . fastly i reduce 60 % till it gets closer to our required 2000 candidates step ( 1 ) 40000 accepted . step ( 2 ) another 40 % of 40000 = 16000 accepted . here it is quiet observable that if we further deduct candidate by 60 % it would change our probablity of easy going 2000 candidate . so i would get to second stage of recruitment where 50 % is accepted step ( 3 ) 50 % of 16000 = 8000 step ( 4 ) 50 % of 8000 = 4000 step ( 5 ) 50 % of 4000 = 2000 . . . here we are done with total 5 steps and last step of accepting 70 % of 2000 = 1400 ( our target ) total 6 steps required . answer : c"
Answer: c
Problem: express a speed of 342 kmph in meters per second ?
Options: a ) 10 mps , b ) 59 mps , c ) 79 mps , d ) 90 mps , e ) 19 mps
Rationale: "342 * 5 / 18 = 90 mps answer : d"
Answer: d
Problem: how many positive integers less than 50 are multiples of 3 but not multiples of 5 ?
Options: a ) 4 , b ) 6 , c ) 8 , d ) 10 , e ) 12
Rationale: "imo the answer is c ( 8 numbers ) the lcm of 3 and 5 is 15 . if x < 50 and x is divisible by 3 not by 5 - - > x is not divisible by 15 . from 1 - - > 50 , we have 3 numbers which is divisible by 15 : 15 , 30 , 45 . from 1 - - > 50 , we have ( 45 - 3 ) / 3 + 1 = 15 numbers divisible by 3 . therefore , our answer is 15 - 3 = 12 numbers . e"
Answer: e
Problem: out of 9 persons , 8 persons spent rs . 30 each for their meals . the ninth one spent rs . 20 more than the average expenditure of all the 9 . the total money spent by all of them was
Options: a ) rs . 292.50 , b ) rs . 297.50 , c ) rs . 298 , d ) rs . 298.50 , e ) none of these
Rationale: explanation : let the average expenditure be rs . x then , 9 x = [ 8 × 30 + ( x + 20 ) ] ⇔ 9 x = x + 260 ⇔ x = 32.50 total money spent = 9 x = rs . ( 9 x 32.5 o ) = rs 292 . 50 answer : a
Answer: a
Problem: a driver just filled the car ' s gas tank with 36 liters of gasohol , a mixture consisting of 5 % ethanol and 95 % gasoline . if the car runs best on a mixture consisting of 10 % ethanol and 90 % gasoline , how many liters of ethanol must be added into the gas tank for the car to achieve optimum performance ?
Options: a ) 1 , b ) 1.5 , c ) 2 , d ) 2.5 , e ) 3
Rationale: "let x be the number of liters of ethanol added to the gas tank . 0.05 ( 36 ) + x = 0.1 ( 36 + x ) 0.9 x = 3.6 - 1.8 = 1.8 x = 2 liters the answer is c ."
Answer: c
Problem: a cylinder and a cone have the same height and same radius of base . the ratio between the volumes of the cylinder and the cone is :
Options: a ) 4 : 1
b ) 2 : 1
c ) 7 : 1
d ) 5 : 1
e ) 3 : 1
Rationale: volume of the cylinder = π r ( power 2 ) h volume of the cone = 1 / 3 π r ( power 2 ) h ( π r ( power 2 ) h ) / ( 1 / 3 π r ( power 2 ) h ) = 3 / 1 = 3 : 1 answer is e .
Answer: e
Problem: when working alone , painter w can paint a room in 2 hours , and working alone , painter x can paint the same room in f hours . when the two painters work together and independently , they can paint the room in 3 / 4 of an hour . what is the value of f ?
Options: a ) 3 / 4 , b ) 1 [ 1 / 5 ] , c ) 1 [ 2 / 5 ] , d ) 1 [ 3 / 4 ] , e ) 2
Rationale: rate * time = work let painter w ' s rate be w and painter x ' s rate be x r * t = work w * 2 = 1 ( if the work done is same throughout the question then the work done can be taken as 1 ) = > w = 1 / 2 x * f = 1 = > x = 1 / f when they both work together then their rates get added up combined rate = ( w + x ) r * t = work ( w + x ) * 3 / 4 = 1 = > w + x = 4 / 3 = > 1 / 2 + 1 / f = 4 / 3 = > 1 / f = ( 8 - 3 ) / 6 = 5 / 6 = > f = 6 / 5 = 1 [ 1 / 5 ] answer b
Answer: b
Problem: a woman invested $ 1,000 , part at 5 % and the rest at 6 % . her total investment with interest at the end of the year was $ 1,054 . how much did she invest at 5 % ?
Options: a ) $ 500 , b ) $ 600 , c ) $ 700 , d ) $ 900 , e ) $ 950
Rationale: et x be the portion invested at 5 % and let ( 1 - x ) be the rest which is invested at 6 % the question states that the return after 1 year is ( 1054 / 1000 ) - 1 = 0.054 = 5.4 % we want to find the dollar amount invested in x using our defined variables , put together the equation and solve for x ( the percentage of 1000 invested at 5 % ) 0.05 x + 0.06 ( 1 - x ) = 0.054 ( 0.05 ) x + 0.06 - ( 0.06 ) x = 0.054 - 0.01 x = - 0.006 x = - 0.006 / - 0.01 = 6 / 10 = 60 % so x = 60 % of the 1000 which is 600 answer : b
Answer: b
Problem: what is the rate percent when the simple interest on rs . 800 amount to rs . 192 in 4 years ?
Options: a ) 5 % , b ) 6 % , c ) 3 % , d ) 9 % , e ) 1 %
Rationale: "192 = ( 800 * 4 * r ) / 100 r = 6 % answer : b"
Answer: b
Problem: the average of 10 numbers is calculated as 16 . it is discovered later on that while calculating the average , the number 45 was incorrectly read as 25 , and this incorrect number was used in the calculation . what is the correct average ?
Options: a ) 17 , b ) 18 , c ) 22 , d ) 26 , e ) 36
Rationale: "the total sum of the numbers should be increased by 20 . then the average will increase by 20 / 10 = 2 . the correct average is 18 . the answer is b ."
Answer: b
Problem: what percent of 15 is 15 percent of 1 ?
Options: a ) 0.001 , b ) 0.01 , c ) 0.1 , d ) 1 , e ) 101
Rationale: "15 % of 1 = ( 15 / 100 ) * 1 = 15 / 100 to determine what percentage of 15 this is : [ 15 ] [ / 100 * 15 ] * 100 = 1 % ans : d"
Answer: d
Problem: two trains a and b starting from two points and travelling in opposite directions , reach their destinations 9 hours and 4 hours respectively after meeting each other . if the train a travels at 70 kmph , find the rate at which the train b runs .
Options: a ) 40 , b ) 60 , c ) 120 , d ) 80 , e ) 105
Rationale: "if two objects a and b start simultaneously from opposite points and , after meeting , reach their destinations in ‘ a ’ and ‘ b ’ hours respectively ( i . e . a takes ‘ a hrs ’ to travel from the meeting point to his destination and b takes ‘ b hrs ’ to travel from the meeting point to his destination ) , then the ratio of their speeds is given by : sa / sb = √ ( b / a ) i . e . ratio of speeds is given by the square root of the inverse ratio of time taken . sa / sb = √ ( 4 / 9 ) = 2 / 3 this gives us that the ratio of the speed of a : speed of b as 2 : 3 . since speed of a is 80 kmph , speed of b must be 70 * ( 3 / 2 ) = 105 kmph e"
Answer: e
Problem: 7528 : 5306 : : 4673 : ?
Options: a ) 2351 , b ) 2451 , c ) 2551 , d ) 2651 , e ) 2751
Rationale: "as there is a difference of 2222 . 7528 - 2222 = 5306 . so 4673 - 2222 = 2451 answer : b"
Answer: b
Problem: in a urban village of india named ` ` owlna ' ' , 70 % people have refrigerator , 75 % people have television , 90 % people got computers and 85 % got air - conditionor . how many people ( minimum ) got all these luxury .
Options: a ) 23 % , b ) 17 % , c ) 11 % , d ) 10 % , e ) 20 %
Rationale: "e 10 % 100 - [ ( 100 - 85 ) + ( 100 - 90 ) + ( 100 - 75 ) + ( 100 - 70 ) ] = 100 - ( 30 + 25 + 10 + 15 ) = 100 - 80"
Answer: e
Problem: a merchant has selected two items to be placed on sale , one of which currently sells for 20 percent less than the other . if he wishes to raise the price of the cheaper item so that the two items are equally priced , by what percentage must he raise the price of the less expensive item ?
Options: a ) 23 % , b ) 24 % , c ) 25 % , d ) 26 % , e ) 27 %
Rationale: expensive item = $ 100 ; cheap item = $ 80 ; we must increase $ 80 to $ 100 , so by $ 20 , which is 25 % increase : ( 100 - 80 ) / 80 = 1 / 4 = 0.25 answer : c .
Answer: c
Problem: what is the ratio between perimeters of two squares one having 5 times the diagonal then the other ?
Options: a ) 5 : 1 , b ) 3 : 1 , c ) 5 : 8 , d ) 3 : 2 , e ) 8 : 3
Rationale: "d = 5 d d = d a √ 2 = 5 d a √ 2 = d a = 5 d / √ 2 a = d / √ 2 = > 5 : 1 answer : a"
Answer: a
Problem: find the area of a parallelogram with base 48 cm and height 36 cm ?
Options: a ) 1730 cm 2 , b ) 1728 cm 2 , c ) 1870 cm 2 , d ) 1890 cm 2 , e ) 668 cm 2
Rationale: "area of a parallelogram = base * height = 48 * 36 = 1728 cm 2 answer : b"
Answer: b
Problem: a and b complete a work in 3 days . a alone can do it in 9 days . if both together can do the work in how many days ?
Options: a ) 3.75 days , b ) 4.75 days , c ) 3.25 days , d ) 3.15 days , e ) 2.75 days
Rationale: "1 / 3 + 1 / 9 = 4 / 9 9 / 4 = 4.75 days answer : b"
Answer: b
Problem: a question paper has 2 parts , a & b , each containing 10 questions . if a student has to choose 8 from part a & 5 from part b , in how many ways can he choose the questions ?
Options: a ) 1100 , b ) 1200 , c ) 1235 , d ) 1354 , e ) 1140
Rationale: there 10 questions in part a out of which 8 question can be chosen as = 10 c 8 . similarly , 5 questions can be chosen from 10 questions of part b as = 10 c 5 . hence , total number of ways , = 10 c 8 * 10 c 5 = [ 10 ! / ( 2 ! 8 ! ) ] * [ 10 ! / ( 5 ! * 5 ) ] = { 10 * 9 / 2 } * { 10 * 9 * 8 * 7 * 6 / ( 5 * 4 * 3 * 2 * 1 ) } = 1140 . e
Answer: e
Problem: what must be added to each term of the ratio 4 : 15 , so that it may become equal to 3 : 4 ?
Options: a ) 18 , b ) 29 , c ) 39 , d ) 49 , e ) 59
Rationale: "let x be added to each term of the ratio 4 : 15 . new ratio becomes ( 4 + x ) / ( 15 + x ) , which is given to be equal to 3 : 4 . this means , ( 4 + x ) / ( 15 + x ) = 3 / 4 solving we get , x = 29 . ans b"
Answer: b
Problem: what is the greatest prime factor of ( 11 ! × 10 !   + 10 ! × 9 ! ) / 111 ?
Options: a ) 2 , b ) 3 , c ) 5 , d ) 7 , e ) 11
Rationale: ( 11 ! × 10 !   + 10 ! × 9 ! ) / 111 in numerator take out 10 ! * 9 ! as common term 10 ! ∗ 9 ! ( 11 × 10   + 1 ) / 111 10 ! ∗ 9 ! ∗ 111 / 111 10 ! ∗ 9 ! 10 ! has 7 as the greatest prime factor , so ans is 7 answer : d
Answer: d
Problem: a sum of money is to be distributed among a , b , c , d in the proportion of 5 : 2 : 4 : 3 . if c gets rs . 1000 more than d , what is b ' s share ?
Options: a ) 4000 , b ) 5000 , c ) 2000 , d ) 1000 , e ) 6000
Rationale: "let the shares of a , b , c and d be rs . 5 x , rs . 2 x , rs . 4 x and rs . 3 x respectively . then , 4 x - 3 x = 1000 x = 1000 . b ' s share = rs . 2 x = rs . ( 2 x 1000 ) = rs . 2000 . answer is c ."
Answer: c
Problem: there are 11 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 ) 40 , b ) 45 , c ) 50 , d ) 55 , e ) 60
Rationale: "11 c 2 = 55 the answer is d ."
Answer: d
Problem: the average of 10 numbers was calculated as 17 . it is discovered later on that while calculating the average , one number , namely 56 , was incorrectly read as 26 . what is the correct average ?
Options: a ) 19 , b ) 20 , c ) 21 , d ) 22 , e ) 23
Rationale: 10 * 17 - 26 + 56 = 200 200 / 10 = 20 the answer is b .
Answer: b
Problem: the average of 6 observations is 15 . a new observation is included and the new average is decreased by 1 . the seventh observation is ?
Options: a ) 1 , b ) 8 , c ) 5 , d ) 6 , e ) 7
Rationale: "let seventh observation = x . then , according to the question we have = > ( 90 + x ) / 7 = 14 = > x = 8 hence , the seventh observation is 8 answer : b"
Answer: b
Problem: a bag holds 4 red marbles and 4 green marbles . if you removed two randomly selected marbles from the bag , without replacement , what is the probability that both would be red ?
Options: a ) 1 / 7 , b ) 1 / 5 , c ) 3 / 14 , d ) 2 / 7 , e ) 1 / 2
Rationale: "probability of selecting first red marble = 4 / 8 probability of selecting second red marble without replacement = 3 / 7 final probability = 4 / 8 * 3 / 7 = 3 / 14 the correct answer is c ."
Answer: c
Problem: a motorcyclist goes from bombay to pune , a distance of 288 kms at an average of 32 kmph speed . another man starts from bombay by car 2 ½ hours after the first , and reaches pune ½ hour earlier . what is the ratio of the speed of the motorcycle and the car ?
Options: a ) 1 : 2 , b ) 1 : 5 , c ) 1 : 4 , d ) 2 : 3 , e ) 1 : 8
Rationale: "t = 288 / 32 = 9 h t = 9 - 3 = 6 time ratio = 9 : 6 = 3 : 2 speed ratio = 2 : 3 answer : d"
Answer: d
Problem: what is the area of square field whose side of length 5 m ?
Options: a ) 80 sq m , b ) 26 sq m , c ) 25 sq m , d ) 45 sq m , e ) 96 sq m
Rationale: "5 * 5 = 25 sq m answer : c"
Answer: c
Problem: if 25 % of x is 30 less than 15 % of 1500 , then x is ?
Options: a ) 872 , b ) 738 , c ) 837 , d ) 780 , e ) 83
Rationale: "25 % of x = x / 4 ; 15 % of 1500 = 15 / 100 * 1500 = 225 given that , x / 4 = 225 - 30 = > x / 4 = 195 = > x = 780 . answer : d"
Answer: d
Problem: by selling an article at rs . 1000 , a shopkeeper makes a profit of 25 % . at what price should he sell the article so as to make a loss of 25 % ?
Options: a ) s . 600 , b ) s . 480 , c ) s . 500 , d ) s . 450 , e ) s . 550
Rationale: "sp = 1000 profit = 25 % cp = ( sp ) * [ 100 / ( 100 + p ) ] = 1000 * [ 100 / 125 ] = 800 loss = 25 % = 25 % of 800 = rs . 200 sp = cp - loss = 800 - 200 = rs . 600 answer : a"
Answer: a
Problem: if the population of a certain country increases at the rate of one person every 20 seconds , by how many persons does the population increase in 1 hour ?
Options: a ) 100 , b ) 120 , c ) 150 , d ) 180 , e ) 160
Rationale: "answer = 3 * 60 = 180 answer is d"
Answer: d
Problem: working alone , printers x , y , and z can do a certain printing job , consisting of a large number of pages , in 12 , 40 , and 50 hours , respectively . what is the ratio of the time it takes printer x to do the job , working alone at its rate , to the time it takes printers y and z to do the job , working together at their individual rates ?
Options: a ) 4 / 11 , b ) 1 / 2 , c ) 15 / 22 , d ) 200 / 9 , e ) 11 / 4
Rationale: "p 1 takes 12 hrs rate for p 2 p 3 together = 1 / 40 + 1 / 50 = 9 / 200 therefore they take 200 / 9 ratio = 200 / 9 = d"
Answer: d
Problem: jane and thomas are among the 9 people from which a committee of 4 people is to be selected . how many different possible committees of 4 people can be selected from these 9 people if at least one of either jane or thomas is to be selected ?
Options: a ) 67 , b ) 73 , c ) 79 , d ) 85 , e ) 91
Rationale: "the total number of ways to choose 4 people from 9 is 9 c 4 = 126 . the number of committees without jane or thomas is 7 c 4 = 35 . there are 126 - 35 = 91 possible committees which include jane and / or thomas . the answer is e ."
Answer: e
Problem: albert is 2 times mary ’ s age and 4 times as old as betty . mary is 22 years younger than albert . how old is betty ?
Options: a ) 8 , b ) 9 , c ) 11 , d ) 15 , e ) 18
Rationale: "a = 2 m = m + 22 m = 22 a = 44 a = 4 b , and so b = 11 the answer is c ."
Answer: c
Problem: a train speeds past a pole in 10 seconds and a platform 150 m long in 25 seconds . its length is :
Options: a ) 300 m . , b ) 250 m . , c ) 100 m . , d ) 200 m . , e ) 150 m .
Rationale: "let the length of the train be x meters and its speed be y m / sec . they , x / y = 10 = > y = x / 10 x + 150 / 25 = x / 10 x = 100 m . answer : option c"
Answer: c
Problem: the perimeter of a triangle is 32 cm and the inradius of the triangle is 2.5 cm . what is the area of the triangle
Options: a ) a ) 40 , b ) b ) 828 , c ) c ) 729 , d ) d ) 34 , e ) e ) 35
Rationale: "explanation : area of a triangle = r * s where r is the inradius and s is the semi perimeter of the triangle . area of triangle = 2.5 * 32 / 2 = 40 cm 2 answer : option a"
Answer: a
Problem: two brother x and y appeared for an exam . the probability of selection of x is 1 / 5 and that of b is 2 / 3 . find the probability that both of them are selected .
Options: a ) 2 / 69 , b ) 2 / 20 , c ) 2 / 18 , d ) 2 / 29 , e ) 2 / 15
Rationale: "explanation : let a be the event that x is selected and b is the event that y is selected . p ( a ) = 1 / 5 , p ( b ) = 2 / 3 . let c be the event that both are selected . p ( c ) = p ( a ) ã — p ( b ) as a and b are independent events : = ( 1 / 5 ) ã — ( 2 / 3 ) = 2 / 15 answer : e ) 2 / 15"
Answer: e
Problem: what is the smallest number which when increased by 7 is divisible by 25 , 49 , 15 , and 21 ?
Options: a ) 3,551 , b ) 3,668 , c ) 3,733 , d ) 3,822 , e ) 3,957
Rationale: "first factor each of 25 , 49 , 15 , and 21 into primes : 25 = 5 * 5 ; 49 = 7 * 7 ; 15 = 3 * 5 ; 21 = 3 * 7 the smallest number divisible by each of the numbers is 3 * 5 * 5 * 7 * 7 = 3,675 3,675 - 7 = 3,668 thus 3,668 + 7 is the smallest number divisible by 25 , 49 , 15 , and 21 b"
Answer: b
Problem: at the opening of a trading day at a certain stock exchange , the price per share of stock k was $ 10 . if the price per share of stock k was $ 15 at the closing of the day , what was the percent increase in the price per share of stock k for that day ?
Options: a ) 1.4 % , b ) 50 % , c ) 11.1 % , d ) 12.5 % , e ) 23.6 %
Rationale: opening = 10 closing = 15 rise in price = 5 so , percent increase = 5 / 10 * 100 = 50 % answer : b
Answer: b
Problem: find the least number which when divided by 5 , 6,7 , and 8 leaves a remainder 3 , but when divided by 9 leaves no remainder .
Options: a ) 1638 , b ) 1863 , c ) 1683 , d ) 1836 , e ) none of them
Rationale: "l . c . m . of 5,6 , 7,8 = 840 . required number is of the form 840 k + 3 least value of k for which ( 840 k + 3 ) is divisible by 9 is k = 2 . required number = ( 840 x 2 + 3 ) = 1683 answer is c ."
Answer: c
Problem: a can give b 100 meters start and c 150 meters start in a kilometer race . how much start can b give c in a kilometer race ?
Options: a ) 10.55 meters , b ) 11.55 meters , c ) 33.55 meters , d ) 55.55 meters , e ) none of these
Rationale: "explanation : a runs 1000 meters while b runs 900 meters and c runs 850 meters . therefore , b runs 900 meters while c runs 850 meters . so , the number of meters that c runs when b runs 1000 meters = ( 1000 x 850 ) / 900 = 944.44 meters thus , b can give c ( 1000 - 944.44 ) = 55.55 meters start answer : d"
Answer: d
Problem: if 213 × 16 = 3408 , then 1.6 × 2.13 is equal to :
Options: a ) 0.3408 , b ) 3.408 , c ) 34.08 , d ) 340.8 , e ) none of these
Rationale: "solution 1.6 × 2.13 = ( 16 / 10 x 213 / 100 ) = ( 16 x 213 / 1000 ) = 3408 / 100 = 3.408 . answer b"
Answer: b
Problem: in how many no . between 100 and 1000 exactly one of the digits is 3 ?
Options: a ) 648 , b ) 512 , c ) 252 , d ) 225 , e ) 26
Rationale: "3 scenarios possible 3 xy = 1 * 9 * 9 = 81 x 3 y ( with x ≠ ≠ 3 or 0 ) = 8 * 1 * 9 = 72 y 3 x ( with y ≠ ≠ 3 or 0 ) = 8 * 1 * 9 = 72 total cases possible = 144 + 81 = 225 d is the correct answer"
Answer: d
Problem: the owner of a furniture shop charges his customer 25 % more than the cost price . if a customer paid rs . 8340 for a computer table , then what was the cost price of the computer table ?
Options: a ) rs . 6672 , b ) rs . 6727 , c ) rs . 6908 , d ) rs . 6725 , e ) rs . 6728
Rationale: "cp = sp * ( 100 / ( 100 + profit % ) ) = 8340 ( 100 / 125 ) = rs . 6672 . answer : a"
Answer: a
Problem: everyone shakes hands with everyone else in a room . total number of handshakes is 78 . number of persons = ?
Options: a ) 13 , b ) 12 , c ) 11 , d ) 15 , e ) 16
Rationale: in a room of n people , the number of possible handshakes is c ( n , 2 ) or n ( n - 1 ) / 2 so n ( n - 1 ) / 2 = 78 or n ( n - 1 ) = 156 or n = 13 answer is ( a )
Answer: a
Problem: if n is a positive integer and n ^ 2 is divisible by 134 , then the largest positive integer that must divide n is
Options: a ) 6 , b ) 12 , c ) 24 , d ) 36 , e ) 48
Rationale: "the question asks aboutthe largest positive integer that must divide n , not could divide n . since the least value of n for which n ^ 2 is a multiple of 72 is 12 then the largest positive integer that must divide n is 12 . complete solution of this question is given above . please ask if anything remains unclear . i spent a few hours on this one alone and i ' m still not clear . i chose 12 at first , but then changed to 48 . i ' m not a native speaker , so here is how i interpreted this question : the largest positive integer that must divide n = the largest positive factor of n . since n is a variable ( i . e . n is moving ) , so is its largest factor . please correct if i ' m wrong here . i know that if n = 12 , n ^ 2 = 144 = 2 * 72 ( satisfy the condition ) . when n = 12 , the largest factor of n is n itself , which is 12 . check : 12 is the largest positive number that must divide 12 - - > true however if n = 48 , n ^ 2 = 48 * 48 = 32 * 72 ( satisfy the condition too ) . when n = 48 , the largest factor of n is n itself , which is 48 . check : 48 is the largest positive number that must divide 48 - - > true so , i also notice that the keyword ismust , notcould . the question is , why is 48 notmust divide 48 , but instead onlycould divide 48 ? i ' m not clear right here . why is 12 must divide 12 ? what ' s the difference between them ? only restriction we have on positive integer n is that n ^ 2 is divisible by 72 . the least value of n for which n ^ 2 is divisible by 72 is 12 , thus nmustbe divisible by 12 ( n is in any case divisible by 12 ) . for all other values of n , for which n ^ 2 is divisible by 72 , n will still be divisible by 12 . this means that n is always divisible by 12 if n ^ 2 is divisible by 72 . now , ask yourself : if n = 24 , is n divisible by 48 ? no . so , n is not always divisible by 48 . c"
Answer: c
Problem: in a group of 400 players some are outdoor players , some are indoor players and some plays both . 350 plays outdoors and 110 plays indoor . how many plays indoor as well as outdoor games .
Options: a ) 70 , b ) 60 , c ) 90 , d ) 100 , e ) 110
Rationale: consider total number of players n ( o u i ) = 400 players who plays outdoors n ( o ) = 350 players who plays indoors n ( i ) = 110 players who plays both n ( o â ˆ © i ) = ? n ( o u i ) = n ( o ) + n ( i ) - n ( o â ˆ © i ) 400 = 350 + 110 - n ( o â ˆ © i ) n ( o â ˆ © i ) = 460 - 400 n ( o â ˆ © i ) = 60 so people read both science fiction and literacy works are 60 answer : b
Answer: b
Problem: if 50 % of x equals the sum of y and 20 , then what is the value of x – 2 y ?
Options: a ) 20 , b ) 40 , c ) 60 , d ) 80 , e ) 90
Rationale: explanation : given , 50 % of x equals the sum of y and 20 . = > ( 50 / 100 ) × x = y + 20 . = > x / 2 = y + 20 . = > x = 2 y + 40 . = > x – 2 y = 40 . answer : b
Answer: b
Problem: weights of two friends ram and shyam are in the ratio of 4 : 5 . ram ’ s weight increases by 10 % and the total weight of ram and shyam together becomes 82.8 kg , with an increase of 15 % . by what per cent did the weight of shyam increase ?
Options: a ) 12.5 % , b ) 17.5 % , c ) 19 % , d ) 21 % , e ) none of these
Rationale: let the weights of ram and shyam be 4 x and 5 x . now , according to question , 4 x × 110 / 100 + shyam ? snewwt 4 x × 110100 + shyam ? s new wt = 82.8 . . . . . . . . ( i ) and 4 x + 5 x = 9 x + 115 / 100 = 82.8 . . . . . . . . ( ii ) from ( ii ) , x = 8 putting in ( i ) , we get shyam ’ s new wt = ( 82.8 – 35.2 ) = 47.6 % increase in shyam ’ s wt = ( 47.6 − 40 / 40 × 100 ) = 19 answer c
Answer: c
Problem: 1397 x 1397
Options: a ) 1951609 , b ) 1981709 , c ) 18362619 , d ) 2031719 , e ) none of these
Rationale: "1397 x 1397 = ( 1397 ) 2 = ( 1400 - 3 ) 2 = ( 1400 ) 2 + ( 3 ) 2 - ( 2 x 1400 x 3 ) = 1960000 + 9 - 8400 = 1960009 - 8400 = 1951609 . answer : option a"
Answer: a
Problem: if n divided by 9 has a remainder of 8 , what is the remainder when 2 times n is divided by 9 ?
Options: a ) 1 , b ) 7 , c ) 3 , d ) 5 , e ) 6
Rationale: "as per question = > n = 9 p + 8 for some integer p hence 2 n = > 18 q + 16 but again , 16 can be divided by 9 to get remainder 7 for some integer q hence b"
Answer: b
Problem: barbata invests $ 2400 in the national bank at 4 % . 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 ) 1200 , b ) 3000 , c ) 2400 , d ) 3600 , e ) 2450
Rationale: "let the additional invested amount for 8 % interest be x ; equation will be ; 2400 + 0.04 * 2400 + x + 0.08 x = 2400 + x + 0.06 ( 2400 + x ) 0.04 * 2400 + 0.08 x = 0.06 x + 0.06 * 2400 0.02 x = 2400 ( 0.06 - 0.04 ) x = 2400 * 0.02 / 0.02 = 2400 ans : c"
Answer: c
Problem: what is the value of ( 4 ) ^ - 5 ?
Options: a ) 1 / 602 , b ) 1 / 120 , c ) 1 / 650 , d ) 1 / 1024 , e ) 1 / 1000
Rationale: "4 ^ - 5 = 1 / ( 4 ) ^ 5 = 1 / 1024 answer : d"
Answer: d
Problem: if m is an integer such that ( - 2 ) ^ 2 m = 2 ^ ( 12 - m ) then m = ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: "2 m = 12 - m 3 m = 12 m = 4 the answer is d ."
Answer: d
Problem: p and q are the only two applicants qualified for a short - term research project that pays 540 dollars in total . candidate p has more experience and , if hired , would be paid 50 percent more per hour than candidate q would be paid . candidate q , if hired , would require 10 hours more than candidate p to do the job . candidate p ’ s hourly wage is how many dollars greater than candidate q ’ s hourly wage ?
Options: a ) $ 9 , b ) $ 15 , c ) $ 18 , d ) $ 21 , e ) $ 27
Rationale: "let q ' s hourly wage be x , then p ' s hourly wage is 1.5 x let t be the number of hours that q needs , then p needs t - 10 hours to do the job . since they both are paid an equal total amount of $ 540 : x * t = 1.5 x * ( t - 10 ) t = 30 hours and q ' s hourly wage is 540 / 30 = $ 18 p ' s hourly wage is 540 / ( t - 10 ) = $ 27 which is $ 9 per hour more . the answer is a ."
Answer: a