Cheukting/mathqa huggingface dataset:
Question (geometry): the points a ( 0 , 0 ) , b ( 0 , 4 a - 4 ) and c ( 2 a + 1 , 2 a + 6 ) form a triangle . if angle abc = 90 , what is the area of triangle abc ?
Options: a ) 88 , b ) 98 , c ) 108 , d ) 78 , e ) 90
Rationale: "1 / 2 bh = 1 / 2 ( 2 a + 1 ) ( 2 a + 6 ) now 4 a - 4 = 2 a + 6 2 a = 10 , a = 5 therefore , a ( 0,0 ) ; b ( 0,16 ) ; c ( 11,16 ) 1 / 2 * 11 * 16 = 88 answer : a"
Answer: a
Question (physics): machine a takes 100 hours to complete a certain job and starts that job at 7 am . after two hour of working alone , machine a is joined by machine b and together they complete the job at 6 pm . how long would it have taken machine b to complete the jobif it had worked alone for the entire job ?
Options: a ) 10.11 , b ) 9.99 , c ) 11.1 , d ) 44.11 , e ) 1.01
Rationale: "let us assume total job = 100 units a finishes 100 units in 100 hrs ( given ) hence a ( working rate ) = 1 units / hr now given that a works for 2 hr ( so 2 units done ) then a and b finish total work in 15 hours . hence a and b finish 98 units in 9 hours . of these 1 x 9 = 9 units were done by a . hence b did 89 units in 9 hours . hence b ( working rate ) = 89 / 9 units / hr hence b takes 100 x 9 / 89 = 10.11 hours to complete the job . answer a ."
Answer: a
Question (general): average age of students of an adult school is 46 years . 120 new students whose average age is 32 years joined the school . as a result the average age is decreased by 4 years . find the number of students of the school after joining of the new students .
Options: a ) 1200 , b ) 120 , c ) 360 , d ) 168 , e ) none of these
Rationale: "explanation : let the original no . of students be x . according to situation , 46 x + 120 * 32 = ( x + 120 ) 36 ⇒ x = 48 so , required no . of students after joining the new students = x + 120 = 168 . answer : d"
Answer: d
Question (physics): on a map , 3.5 inches represent 35 miles . how many miles approximately is the distance if you measured 127 centimeters assuming that 1 - inch is 2.54 centimeters ?
Options: a ) 515 , b ) 500 , c ) 525 , d ) 530 , e ) 535
Rationale: "1 inch = 2.54 cm 3.5 inch = 2.54 * 3.5 cm 8.89 cm = 35 miles 127 cms = 35 / 8.89 * 127 = 500 miles answer : b"
Answer: b
Question (physics): how long does a train 110 m long traveling at 60 kmph takes to cross a bridge of 140 m in length ?
Options: a ) 16.5 , b ) 16.0 , c ) 15 , d ) 16.8 , e ) 16.1
Rationale: "d = 110 + 140 = 250 m s = 60 * 5 / 18 = 50 / 3 t = 250 * 3 / 50 = 15 sec answer : c"
Answer: c
Question (physics): if the product of two numbers is 51216 and their h . c . f . is 33 , find their l . c . m .
Options: a ) 600 , b ) 680 , c ) 582 , d ) 560 , e ) none of these
Rationale: "explanation : hcf * lcm = 51216 , because we know product of two numbers = product of hcf and lcm lcm = 51216 / 88 = 582 option c"
Answer: c
Question (gain): at an elementary school , 70 % of the faculty members are women and 40 % of the faculty members are married . if 1 ⁄ 3 of the men are single , what fraction of the women are married ?
Options: a ) 5 ⁄ 7 , b ) 7 ⁄ 10 , c ) 2 / 7 , d ) 7 ⁄ 30 , e ) 5 ⁄ 70
Rationale: "- - - - - - - - - - - - - - - - - - - - m - - - - - - w - - - - - - - - total marrried - - - - - - - - - - 20 - - - - - 20 - - - - - - - - - 40 not married - - - - - 10 - - - - - 50 - - - - - - - - - 60 total - - - - - - - - - - - - - 30 - - - - - 70 - - - - - - - - 100 need married woman / total woman , so 20 / 70 = 2 / 7 c"
Answer: c
Question (physics): a train which has 300 m long , is running 80 kmph . in what time will it cross a person moving at 16 kmph in same direction ?
Options: a ) 25 sec , b ) 35 sec , c ) 15 sec , d ) 45 sec , e ) 20 sec
Rationale: time taken to cross a moving person = length of train / relative speed time taken = 300 / ( ( 80 - 16 ) ( 5 / 18 ) = 300 / 64 * ( 5 / 18 ) = 300 / 20 = 15 sec answer : c
Answer: c
Question (other): what is the number of integers from 1 to 500 ( inclusive ) that are divisible by neither 10 nor by 15 ?
Options: a ) 170 , b ) 250 , c ) 300 , d ) 560 , e ) 420
Rationale: in 500 consecutive numbers , number of multiples of 10 = 500 / 10 = 50 ( ignore decimals ) in 500 consecutive numbers , number of multiples of 15 = 500 / 15 = 33 number of multiples of 10 * 15 i . e . 150 = 500 / 150 = 3 number of integers from 1 to 500 that are divisible by neither 10 nor by 15 = 500 - ( 50 + 33 - 3 ) { using the concept of sets here ) = 420 answer is e
Answer: e
Question (general): 495 x 495 = ?
Options: a ) 250000 , b ) 250025 , c ) 5000 , d ) 245025 , e ) 240000
Rationale: "495 x 495 = ( 495 ) 2 = ( 500 - 5 ) 2 = ( 500 ) 2 + ( 5 ) 2 - ( 2 x 500 x 5 ) = 250000 + 25 - 5000 = 250025 - 5000 = 245025 . d )"
Answer: d
Question (general): the average weight of a group of boys is 30 kg . after a boy of weight 33 kg joins the group , the average weight of the group goes up by 1 kg . find the number of boys in the group originally ?
Options: a ) 4 , b ) 5 , c ) 2 , d ) 7 , e ) 8
Rationale: "let the number off boys in the group originally be x . total weight of the boys = 30 x after the boy weighing 33 kg joins the group , total weight of boys = 30 x + 33 so 30 x + 33 = 31 ( x + 1 ) = > x = 2 . answer : c"
Answer: c
Question (other): what is the number of integers from 11 to 1010 ( inclusive ) that are divisible by neither 12 nor by 15 ?
Options: a ) 565 , b ) 856 , c ) 783 , d ) 590 , e ) 300
Rationale: "in 1000 consecutive numbers , number of multiples of 12 = 1000 / 12 = 83 ( ignore decimals ) in 1000 consecutive numbers , number of multiples of 15 = 1000 / 15 = 66 number of multiples of 12 * 15 i . e . 180 = 1000 / 180 = 5 number of integers from 11 to 1010 that are divisible by neither 12 nor by 15 = 1000 - ( 83 + 66 - 5 ) { using the concept of sets here ) = 856 answer is b"
Answer: b
Question (general): the ages of ashley and mary are in the ratio 4 : 7 . the sum of their ages is 22 . find the age of mary .
Options: a ) 11 yrs , b ) 12 yrs , c ) 13 yrs , d ) 14 yrs , e ) 15 yrs
Rationale: "a : m = 4 : 7 let the ages of a & m be x & 22 - x x / 22 - x = 4 / 7 x = 8 a = 8 yrs m = 14 yrs answer : d"
Answer: d
Question (geometry): a rectangular field has a length 10 meters more than it is width . if the area of the field is 171 , what is the length ( in meters ) of the rectangular field ?
Options: a ) 17
b ) 19
c ) 21
d ) 23
e ) 25
Rationale: area = l * w = ( l ) * ( l - 10 ) = 171 trial and error : 20 * 10 = 200 ( too high ) 19 * 9 = 171 the length is 19 meters . the answer is b .
Answer: b
Question (general): a man is 26 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 ) 24 , e ) 10
Rationale: "let the son ' s present age be x years . then , man ' s present age = ( x + 26 ) years . ( x + 26 ) + 2 = 2 ( x + 2 ) x + 28 = 2 x + 4 = > x = 24 . answer : d"
Answer: d
Question (general): if a 2 + b 2 + c 2 = 250 and ab + bc + ca = 3 , then a + b + c is
Options: a ) 4 , b ) 8 , c ) 16 , d ) 32 , e ) 64
Rationale: "by formula , ( a + b + c ) ^ 2 = a ^ 2 + b ^ 2 + c ^ 2 + 2 ( ab + bc + ca ) , since , a ^ 2 + b ^ 2 + c ^ 2 = 250 and ab + bc + ca = 3 , ( a + b + c ) ^ 2 = 250 + 2 ( 3 ) = 256 = 16 ^ 2 therefore : a + b + c = 16 answer : c"
Answer: c
Question (general): if x is a positive number and 5 the square root of x is equal to x / 25 , then x =
Options: a ) 10 , b ) 25 , c ) 5 , d ) 2 , e ) 1
Rationale: "5 of sqrt ( x ) = x / 25 , which means that sqrt ( x ) = x / 5 or x = x ^ 2 / 25 - > divide by x 1 = x / 25 x = 25 b ."
Answer: b
Question (general): if sales tax is reduced from 5 % to 4 % , then what difference it will make if you purchase an item of rs . 1000
Options: a ) 10 , b ) 20 , c ) 30 , d ) 40 , e ) 50
Rationale: explanation : clue : answer will be 5 % of 1000 - 4 % of 1000 answer : option a
Answer: a
Question (general): of all the homes on gotham street , 1 / 3 are termite - ridden , and 1 / 4 of these are collapsing . what fraction of the homes are termite - ridden , but not collapsing ?
Options: a ) 1 / 3 , b ) 2 / 5 , c ) 1 / 4 , d ) 1 / 6 , e ) 3 / 4
Rationale: "the fraction of homes which are termite - infested but not collapsing is 3 / 4 * 1 / 3 = 1 / 4 the answer is c ."
Answer: c
Question (physics): convert 100 miles into yards ?
Options: a ) 25630 yards , b ) 35200 yards , c ) 39520 yards , d ) 42560 yards , e ) 176000 yards
Rationale: "1 mile = 1760 yards 100 miles = 100 * 1760 = 176000 yards answer is e"
Answer: e
Question (general): take number 1000 and then add 20 to it . now add 1000 one more time . now add 30 . now add 1000 one more time . now add 40 . now add 1000 one more time . now add 10 . what is the total ?
Options: a ) 4100 , b ) 3590 , c ) 2800 , d ) 3700 , e ) 3600
Rationale: a 4100
Answer: a
Question (general): elena ’ s bread recipe calls for 2 ounces of butter for each 5 cups of flour used . she needs to make 4 times the original recipe . if 12 ounces of butter is used , then how many cups of flour are needed ?
Options: a ) 1 , b ) 4 , c ) 9 , d ) 13 , e ) 20
Rationale: "solving through algebra route : 2 b + 5 f = x amount if we multiply this equation with 4 we get : 8 b + 20 f = 4 x therefore , we got 8 ounces of butter and 4 x amount of quantity when we use 20 ounces of floor . ans : e"
Answer: e
Question (geometry): three walls have wallpaper covering a combined area of 300 square meters . by overlapping the wallpaper to cover a wall with an area of 180 square meters , the area that is covered by exactly two layers of wallpaper is 38 square meters . what is the area that is covered with three layers of wallpaper ?
Options: a ) 5 square meters
b ) 36 square meters
c ) 42 square meters
d ) 83.3 square meters
e ) 41 square meters
Rationale: 300 - 180 = 120 sq m of the wallpaper overlaps ( in either two layers or three layers ) if 36 sq m has two layers , 120 - 38 = 82 sq m of the wallpaper overlaps in three layers . 82 sq m makes two extra layers hence the area over which it makes two extra layers is 41 sq m . answer ( e ) .
Answer: e
Question (general): a dog is tied to a tree by a long nylon cord . if the dog runs from the due north side of the tree to the due south side of the tree with the cord extended to its full length at all items , and the dog ran approximately 30 feet , what was the approximate length of the nylon cord l in feet ?
Options: a ) 30 , b ) 25 , c ) 15 , d ) 10 , e ) 5
Rationale: "because the cord was extended to its full length at all items , the dog ran along a semi - circular path , from north to south . the circumference of a full circle is 2 * pi * r , but since we only care about the length of half the circle , the semi - circle path is pi * r . l = pi * r = 30 . round pi = 3 , then r = 10 . chord is about 10 feet long . d"
Answer: d
Question (general): all factors of a positive integer x are multiplied and the product obtained is x ^ 3 . if x is greater than 1 , how many factors does x have ?
Options: a ) 2 , b ) 3 , c ) 4 , d ) 5 , e ) 6
Rationale: the question can be done in seconds if you understand the properties of factors of a number . note that every distinct factor of a number which is not a perfect square has a complementary factor . for example , factors of 6 : 1 , 2 , 3 , 6 1 has a complementary factor 6 such that 1 * 6 = 6 ( the original number ) 2 has a complementary factor 3 such that 2 * 3 = 6 ( the original number ) so if you have a number , n , which is not a perfect square , and start multiplying its factors , you will get n to a certain power . the power will half of the number of factors . for example , 6 has 4 factors and when you multiply all 4 factors , you get 6 ^ 2 ( 2 is half of 4 ) . so if you know that the product of all factors is x ^ 3 , it means there must have been twice the number of factors i . e . 3 * 2 = 6 factors . e )
Answer: e
Question (gain): mike needs 30 % to pass . if he scored 212 marks and falls short by 19 marks , what was the maximum marks he could have got ?
Options: a ) 543 , b ) 677 , c ) 770 , d ) 867 , e ) 832
Rationale: "if mike had scored 19 marks more , he could have scored 30 % therefore , mike required 212 + 19 = 231 marks let the maximum marks be m . then 30 % of m = 231 ( 30 / 100 ) × m = 231 m = ( 231 × 100 ) / 30 m = 23100 / 30 m = 770 answer : c"
Answer: c
Question (gain): mary works in a restaurant a maximum of 40 hours . for the first 20 hours , she is paid $ 8 per hour . for each overtime hour , she is paid at a rate which is 25 % higher than her regular rate . how much mary can earn in a week ?
Options: a ) 300 , b ) 420 , c ) 360 , d ) 320 , e ) 400
Rationale: "mary receives $ 8 ( 20 ) = $ 160 for the first 20 hours . for the 20 overtime hours , she receives $ 8 ( 0.25 ) + $ 8 = $ 10 per hour , that is $ 10 ( 20 ) = $ 200 . the total amount is $ 160 + $ 200 = $ 360 answer c 360 ."
Answer: c
Question (gain): sonika deposited rs . 13000 which amounted to rs . 10200 after 4 years at simple interest . had the interest been 5 % more . she would get how much ?
Options: a ) 18990 , b ) 13002 , c ) 12800 , d ) 12300 , e ) 14500
Rationale: "( 13000 * 4 * 5 ) / 100 = 2600 10200 - - - - - - - - 12800 answer : c"
Answer: c
Question (general): a is two years older than b who is twice as old as c . if the total of the ages of a , b and c be 47 , the how old is b ?
Options: a ) 10 years , b ) 12 years , c ) 14 years , d ) 18 years , e ) 16 years
Rationale: "let c ' s age be x years . then , b ' s age = 2 x years . a ' s age = ( 2 x + 2 ) years . ( 2 x + 2 ) + 2 x + x = 47 5 x = 45 x = 9 . hence , b ' s age = 2 x = 19 years . d )"
Answer: d
Question (gain): a shopkeeper sells 400 metres of cloth for rs . 18000 at a loss of rs . 5 per metre . find his cost price for one metre of cloth ?
Options: a ) rs . 59 , b ) rs . 58 , c ) rs . 09 , d ) rs . 50 , e ) rs . 13
Rationale: "sp per metre = 18000 / 400 = rs . 45 loss per metre = rs . 5 cp per metre = 45 + 5 = rs . 50 answer : d"
Answer: d
Question (physics): the positive integers m and n leave remainders of 2 and 3 , respectively , when divided by 6 . m > n . what is the remainder when m – n is divided by 6 ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: m is 2 more than a multiple of 6 ; n is 3 more than a multiple of 6 . m - n will be 1 less than a multiple of 6 ( 5 , 11 , 17 , . . . ) , therefore m - n will yield a remainder of 5 when divided by 6 . answer : e .
Answer: e
Question (general): the mean of 50 observations was 36 . it was found later that an observation 44 was wrongly taken as 23 . the corrected new mean is
Options: a ) 35.22 , b ) 36.12 , c ) 36.22 , d ) 36.42 , e ) none
Rationale: "solution correct sum = ( 36 x 50 + 44 - 23 ) = 1821 . â ˆ ´ correct mean = 1821 / 50 = 36.42 . answer d"
Answer: d
Question (geometry): the ratio of the length and the width of a rectangle is 4 : 3 and the area of the rectangle is 5292 sq cm . what is the ratio of the width and the area of the rectangle ?
Options: a ) 1 : 72 , b ) 1 : 76 , c ) 1 : 80 , d ) 1 : 84 , e ) 1 : 88
Rationale: "let the length and the width be 4 x and 3 x respectively . area = ( 4 x ) ( 3 x ) = 5292 12 x ^ 2 = 5292 x ^ 2 = 441 x = 21 the ratio of the width and the area is 3 x : 12 x ^ 2 = 1 : 4 x = 1 : 84 the answer is d ."
Answer: d
Question (general): is two years older than b who is twice as old as c . if the total of the ages of a , b and c be 27 , then how old is b ?
Options: a ) 7 , b ) 8 , c ) 9 , d ) 10 , e ) 11
Rationale: "let c ' s age be x years . then , b ' s age = 2 x years . a ' s age = ( 2 x + 2 ) years . ( 2 x + 2 ) + 2 x + x = 27 5 x = 25 = > x = 5 hence , b ' s age = 2 x = 10 years . answer : d"
Answer: d
Question (general): if a * b = 2 a – 3 b + ab , then 5 * 7 + 7 * 5 is equal to :
Options: a ) 34 , b ) 18 , c ) 72 , d ) 78 , e ) 82
Rationale: 5 * 7 + 7 * 5 = ( 2 * 5 – 5 * 7 + 5 * 7 ) + ( 2 * 7 – 5 * 5 + 7 * 5 ) = ( 10 + 14 – 25 + 35 ) = 34 . answer : a
Answer: a
Question (other): two family reunions are happening at the park avenue hotel , the oates reunion and the hall reunion . all 100 guests at the hotel attend at least one of the reunions . if 40 people attend the oates reunion and 70 people attend the hall reunion , how many people attend both reunions
Options: a ) 2 , b ) 5 , c ) 10 , d ) 16 , e ) 22
Rationale: no of people in oates reunion = 40 no of people in hall reunion = 70 attending both = x all guests attend at least one . therefore , 100 = 40 + 70 - ( both ) both = 10 answer c
Answer: c
Question (general): a whale goes on a feeding frenzy that lasts for 9 hours . for the first hour he catches and eats x kilos of plankton . in every hour after the first , it consumes 4 kilos of plankton more than it consumed in the previous hour . if by the end of the frenzy the whale will have consumed a whopping accumulated total 450 kilos of plankton , how many kilos did he consume on the sixth hour ?
Options: a ) 38 , b ) 47 , c ) 50 , d ) 53 , e ) 54
Rationale: if you list the amount eaten each hour , you ' ll get an equally spaced list , increasing by 4 each hour . in any equally spaced list , the median equals the mean . here , the mean is 450 / 9 = 50 , so the median is also 50 , and that is the amount eaten in the 5 th hour . we need to add 4 to find the total eaten in the next hour , so the answer is 54 option e .
Answer: e
Question (general): a waiter ' s salary consists of his salary and tips . during one week his tips were 9 / 4 of his salary . what fraction of his income came from tips ?
Options: a ) 4 / 9 , b ) 5 / 4 , c ) 9 / 13 , d ) 5 / 9 , e ) 6 / 9
Rationale: "income = salary ( s ) + tips = s + s * 9 / 4 = s * 13 / 4 tips = s * 9 / 4 fraction of his income came from tips = ( s * 9 / 4 ) / ( s * 13 / 4 ) = 9 / 13 answer : c"
Answer: c
Question (general): the largest 5 digit number exactly divisible by 91 is :
Options: a ) 25698 , b ) 581396 , c ) 369741 , d ) 99918 , e ) 1258
Rationale: "explanation : largest 5 - digit number = 99999 91 ) 99999 ( 1098 91 - - - 899 819 - - - - 809 728 - - - 81 - - - required number = ( 99999 - 81 ) = 99918 . answer : d"
Answer: d
Question (general): the average age of a husband and his wife was 23 years at the time of their marriage . after 5 years they have a one - year old child . the average age of the family now is :
Options: a ) 25 , b ) 23 , c ) 19 , d ) 18 , e ) 17
Rationale: explanation : sum of the present ages of husband , wife and child = ( 23 * 2 + 5 * 2 ) + 1 = 57 years . required average = ( 57 / 3 ) = 19 years . answer : c
Answer: c
Question (general): what sum of money put at c . i amounts in 2 years to rs . 8820 and in 3 years to rs . 9261 ?
Options: a ) rs . 8000 , b ) rs . 8400 , c ) rs . 7500 , d ) rs . 7800 , e ) none of these
Rationale: "explanation : 8820 - - - - 441 100 - - - - ? = > 5 % x * 105 / 100 * 105 / 100 = 8820 x * 1.1025 = 8820 x = 8820 / 1.1025 = > 8000 answer is a"
Answer: a
Question (geometry): if the length of a rectangular field is 30 metres more than its breadth and the perimeter of the field is 540 metres , what is the area of the field in square metres ?
Options: a ) 18000
b ) 40000
c ) 31000
d ) 19000
e ) 16000
Rationale: let breadth is x then length is 30 + x now the perimeter = 2 ( x + 30 + x ) = 540 solving this we get x = 120 now the area is 120 * 150 = 18000 answer : a
Answer: a
Question (general): what is the greater of the two numbers whose product is 2560 , given that the sum of the two numbers exceeds their difference by 32 ?
Options: a ) a ) 90 , b ) b ) 100 , c ) c ) 160 , d ) d ) 70 , e ) of these
Rationale: "let the greater and the smaller number be g and s respectively . gs = 2560 g + s exceeds g - s by 32 i . e . , g + s - ( g - s ) = 32 i . e . , 2 s = 32 = > s = 16 g = 2560 / s = 160 . answer : c"
Answer: c
Question (gain): the profits of qrs company rose 10 % from march to april , then dropped 20 % from april to may , then rose 50 % from may to june . what was the percent increase for the whole quarter , from march to june ?
Options: a ) 15 % , b ) 32 % , c ) 40 % , d ) 62 % , e ) 80 %
Rationale: "we need multipliers . multiplier for a 10 % increases = 1 + 0.10 = 1.1 multiplier for a 20 % decreases = 1 – 0.20 = 0.8 multiplier for a 50 % increases = 1 + 0.50 = 1.5 now , multiply these . first , multiply ( 0.8 ) and ( 1.5 ) , using the doubling & halving trick . half of 0.80 is 0.40 , and twice 1.5 is 3 ( 0.8 ) * ( 1.5 ) = ( 0.4 ) * ( 3 ) = 1.2 now , multiply this by 1.1 1.2 * 1.1 = 1.32 thus , the three percent changes combined produce a 32 % increase . answer = ( b ) ."
Answer: b
Question (geometry): what will be the cost of building a fence around a circular plot with area equal to 289 sq ft , if the price per foot of building the fence is rs . 58 ?
Options: a ) a ) rs . 3944
b ) b ) rs . 3495.28
c ) c ) rs . 3988
d ) d ) rs . 3949
e ) e ) rs . 3923
Rationale: let the side of the circular plot be a ft . pi * ( r ) ^ 2 = 289 = > r = 9.59 length of the fence = perimeter of the plot = 2 * pi * r = 60.26 ft . cost of building the fence = 60.26 * 58 = rs . 3495.28 answer : b
Answer: b
Question (gain): the amount of principal rs . 4000 at compound interest at the ratio of 10 % p . a . for 3 years is
Options: a ) rs . 5342 , b ) rs . 4325 , c ) rs . 5324 , d ) rs . 3425 , e ) rs . 3525
Rationale: 5324 answer : c
Answer: c
Question (geometry): twenty eight meters of wire is available to fence off a flower bed in the form of a circular sector . what must the radius of the circle in meters be , if we wish to have a flower bed with the greatest possible surface area ?
Options: a ) 2 √ 2
b ) 2 √ 5
c ) 5
d ) 7
e ) none of these
Rationale: area of sector , a = x / 360 * pi * r ^ 2 circumference of the sector = 28 = > x / 360 * 2 * pi * r + 2 r = 28 = > 2 a / r + 2 r = 28 = > a = r 14 - r ^ 2 = r 14 - r ^ 2 we will now max using derivations max value of a will found at a = 0 i . e 14 - 2 r = 0 r = 7 d
Answer: d
Question (general): if 2 x = 3 y = 20 , then 12 xy = ?
Options: a ) 1,200 , b ) 800 , c ) 120 , d ) 40 , e ) 20
Rationale: "2 x = 20 ; x = 10 3 y = 20 ; y = 20 / 3 multiply : 12 xy = 12 * 10 * 20 / 3 = 800 . answer : b ."
Answer: b
Question (general): when a number is divided by 6 & then multiply by 12 the answer is 12 what is the no . ?
Options: a ) 4.5 , b ) 5 , c ) 5.5 , d ) 5.8 , e ) 6
Rationale: "if $ x $ is the number , x / 6 * 12 = 12 = > 2 x = 12 = > x = 6 e"
Answer: e
Question (other): divide $ 1162 among a , b , c in the ratio 35 : 28 : 20 .
Options: a ) 490 , 490,490 , b ) 360 , 392,460 , c ) 490 , 392,280 , d ) 260 , 160,60 , e ) 492 , 390,280
Rationale: "sum of ratio terms = ( 35 + 28 + 20 ) = 83 . a ' s share = $ ( 1162 x ( 35 / 83 ) ) = $ 490 ; b ' s share = $ ( 1162 x ( 28 / 83 ) ) = $ . 392 ; c ' s share = $ ( 1162 x ( 20 / 83 ) ) = $ 280 . answer c 490 , 392,280"
Answer: c
Question (general): running at their respective constant rate , machine x takes 2 days longer to produce w widgets than machines y . at these rates , if the two machines together produce 5 w / 4 widgets in 3 days , how many days would it take machine x alone to produce 2 w widgets .
Options: a ) 4 , b ) 6 , c ) 8 , d ) 10 , e ) 12
Rationale: "rx * ( t + 2 ) = w ry * ( t ) = w . also , w / ( t + 2 ) + w / t = ( 5 w / 4 ) * ( 1 / 3 ) or 1 / t + 1 / ( t + 2 ) = 5 / 12 - - > ( 3 + 2 ) / 12 = 1 / 4 + 1 / 6 thus , t = 4 . replacing , rx = w / 6 , w / 6 * t = 2 w or t = 12 days . answer : e"
Answer: e
Question (gain): how long will a boy take to run round a square field of side 60 meters , if he runs at the rate of 9 km / hr ?
Options: a ) 96 sec , b ) 45 sec , c ) 1 min , d ) 32 sec , e ) 25 sec
Rationale: "speed = 9 km / hr = 9 * 5 / 18 = 5 / 2 m / sec distance = 60 * 4 = 240 m time taken = 240 * 2 / 5 = 96 sec answer is a"
Answer: a
Question (physics): a 60 cm long wire is to be cut into two pieces so that one piece will be 2 / 4 th of the other , how many centimeters will the shorter piece be ?
Options: a ) a ) 73 , b ) b ) 20 , c ) c ) 83 , d ) d ) 21 , e ) e ) 52
Rationale: "explanation : 1 : 2 / 4 = 4 : 2 2 / 6 * 60 = 20 answer : option b"
Answer: b