Cheukting/mathqa huggingface dataset:
Question (physics): how long does a train 150 m long traveling at 60 kmph takes to cross a bridge of 170 m in length ?
Options: a ) 16.5 , b ) 19.2 , c ) 16.4 , d ) 16.8 , e ) 16.1
Rationale: "d = 150 + 170 = 320 m s = 60 * 5 / 18 = 50 / 3 t = 320 * 3 / 50 = 19.2 sec answer : b"
Answer: b
Question (general): if a randomly selected non - negative single digit integer is added to { 2 , 3 , 4 , 7 } . what is the probability that the median of the set will increase but the range still remains the same ?
Options: a ) 0.2 , b ) 0.3 , c ) 0.4 , d ) 0.5 , e ) 0.6
Rationale: "we are selecting from non - negative single digit integers , so from { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . these 10 digits represent the total number of outcomes . hence , the total number of outcomes is 10 . we need to find the probability that the median of the set will increase but the range still remains the same . the median of the set is ( 3 + 4 ) / 2 = 3.5 , thus the number selected must be 4 or greater . for the range to remain the same , the number must be between 2 and 7 inclusive . to satisfy both conditions , the number selected must be 4 , 5 , 6 , or 7 . the probability is 4 / 10 = 0.4 the answer is c ."
Answer: c
Question (general): what is the remainder when the number w = 14 ^ 2 * 15 ^ 8 is divided by 5 ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 4 , e ) 5
Rationale: "14 ^ 2 has units digit 6 15 ^ 8 has units digit 5 thus w = 14 ^ 2 * 15 ^ 8 has units digit 0 and will be divisible by 5 . the remainder will be zero answer : ( a )"
Answer: a
Question (general): the product of two numbers is 2028 and their h . c . f is 13 . the number of such pairs is :
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: "let the numbers be 13 a and 13 b . then , 13 a * 13 b = 2028 = > ab = 12 . now , co - primes with product 12 are ( 1 , 12 ) and ( 3 , 4 ) . so , the required numbers are ( 13 * 1 , 13 * 12 ) and ( 13 * 3 , 13 * 4 ) . clearly , there are 2 such pairs . answer : b"
Answer: b
Question (general): a student chose a number , multiplied it by 6 , then subtracted 250 from the result and got 122 . what was the number he chose ?
Options: a ) 59 , b ) 60 , c ) 61 , d ) 62 , e ) 63
Rationale: "let x be the number he chose , then 6 ⋅ x − 250 = 122 6 x = 372 x = 62 correct answer d"
Answer: d
Question (general): find the average of all prime numbers between 30 and 50
Options: a ) 15 , b ) 42 , c ) 45 , d ) 34 , e ) 26
Rationale: "prime numbers between 30 and 50 are 37 , 41 , 43 , 47 required average = ( 37 + 41 + 43 + 47 ) / 4 = 168 / 4 = 42 answer is b"
Answer: b
Question (general): if the average marks of 3 batches of 55 , 60 and 45 students respectively is 40 , 62 , 58 , then the average marks of all the students is
Options: a ) 54.48 , b ) 53.31 , c ) 54.6 , d ) 54.58 , e ) none of these
Rationale: explanation : ( 55 ã — 40 ) + ( 60 ã — 62 ) + ( 45 ã — 58 ) / 55 + 60 + 45 8530 / 160 = 53.3 option b
Answer: b
Question (general): what is the sum of the integers from - 60 to 65 , inclusive ?
Options: a ) 315 , b ) 215 , c ) 115 , d ) 165 , e ) 765
Rationale: "in an arithmetic progression , the nth term is given by tn = a + ( n - 1 ) d here tn = 65 , a = - 60 , d = 1 hence , 65 = - 60 + ( n - 1 ) or n = 126 sum of n terms can be calculated by sn = n / 2 ( a + l ) a = first term , l = last term , n = no . of terms sn = 126 * ( - 60 + 65 ) / 2 sn = 126 * 5 / 2 = 315 answer : a"
Answer: a
Question (general): what is the sum of natural numbers between 60 and 100
Options: a ) 4800 , b ) 4860 , c ) 5000 , d ) 5500 , e ) 3280
Rationale: "a = first number l = last number sn = n / 2 [ a + l ] between 60 and 100 numbers = 41 = > 100 - 60 = 40 + 1 = 41 sn = 41 / 2 ã — 160 = 41 ã — 80 = 3280 answer : e"
Answer: e
Question (general): in an election contested by two parties , party d secured 12 % of the total votes more than party r . if party r got 132000 votes , by how many votes did it lose the election ?
Options: a ) 240000 , b ) 300000 , c ) 168000 , d ) 36000 , e ) 24,000
Rationale: explanatory answer let the percentage of the total votes secured by party d be x % then the percentage of total votes secured by party r = ( x - 12 ) % as there are only two parties contesting in the election , the sum total of the votes secured by the two parties should total up to 100 % i . e . , x + x - 12 = 100 2 x - 12 = 100 or 2 x = 112 or x = 56 % . if party d got 56 % of the votes , then party got ( 56 - 12 ) = 44 % of the total votes . 44 % of the total votes = 132,000 i . e . , 44 / 100 * t = 132,000 = > t = 132000 * 100 / 44 = 300,000 votes . the margin by which party r lost the election = 12 % of the total votes = 12 % of 300,000 = 36,000 . the correct choice is ( d )
Answer: d
Question (geometry): in measuring the sides of a rectangle , one side is taken 4 % in excess , and the other 3 % in deficit . find the error percent in the area calculated from these measurements .
Options: a ) 0.11 % , b ) 0.7 % , c ) 0.4 % , d ) 0.6 % , e ) 0.88 %
Rationale: "let x and y be the sides of the rectangle . then , correct area = xy . calculated area = ( 26 / 25 ) x ( 32 / 33 ) y = ( 344 / 341 ) ( xy ) error in measurement = ( 344 / 341 ) xy - xy = ( 3 / 341 ) xy error percentage = [ ( 3 / 341 ) xy ( 1 / xy ) 100 ] % = ( 22 / 25 ) % = 0.88 % . answer is e ."
Answer: e
Question (general): a multiple choice test consists of 4 questions , and each question has 5 answer choices . in how many e ways can the test be completed if every question is unanswered ?
Options: a ) 24 , b ) 120 , c ) 625 , d ) 720 , e ) 1024
Rationale: "5 choices for each of the 4 questions , thus total e of 5 * 5 * 5 * 5 = 5 ^ 4 = 625 ways to answer all of them . answer : c ."
Answer: c
Question (physics): a and b undertake to do a piece of work for $ 600 . a alone can do it in 6 days while b alone can do it in 8 days . with the help of c , they finish it in 3 days . find the share of a ?
Options: a ) $ 100 , b ) $ 150 , c ) $ 300 , d ) $ 250 , e ) $ 350
Rationale: c ' s 1 day work = ( 1 / 3 ) - ( 1 / 6 + 1 / 8 ) = 1 / 24 a : b : c = 1 / 6 : 1 / 8 : 1 / 24 = 4 : 3 : 1 a ' s share = 600 * 4 / 8 = $ 300 answer is c
Answer: c
Question (general): two employees x and y are paid a total of rs . 660 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 . 300
Rationale: "let the amount paid to x per week = x and the amount paid to y per week = y then x + y = 660 but x = 120 % of y = 120 y / 100 = 12 y / 10 â ˆ ´ 12 y / 10 + y = 660 â ‡ ’ y [ 12 / 10 + 1 ] = 660 â ‡ ’ 22 y / 10 = 660 â ‡ ’ 22 y = 6600 â ‡ ’ y = 6600 / 22 = 600 / 2 = rs . 300 e"
Answer: e
Question (general): 54671 - 10456 - 33466 = ?
Options: a ) 2449 , b ) 10449 , c ) 10749 , d ) 10449 , e ) 6468
Rationale: "c if we calculate we will get 10749"
Answer: c
Question (physics): a 250 metres long train running at the speed of 120 kmph crosses another train running in opposite direction at the speed of 80 kmph in 9 seconds . what is the length of the other train ?
Options: a ) 230 m , b ) 240 m , c ) 250 m , d ) 260 m , e ) 270 m
Rationale: "speed = ( 120 + 80 ) km / h ( because direction is opposite hence relative velocity is added ) = 500 / 9 m / s time = 9 sec let the lenght of second train is x total distance covered = 250 + x therefore , d = speed * time thus 250 + x = 500 / 9 * 9 x = 500 - 250 = 250 m answer : c"
Answer: c
Question (general): the value of x + x ( xx ) when x = 7
Options: a ) a ) 350 , b ) b ) 346 , c ) c ) 358 , d ) d ) 336 , e ) e ) 364
Rationale: x + x ( xx ) put the value of x = 7 in the above expression we get , 7 + 7 ( 77 ) = 7 + 7 ( 7 ã — 7 ) = 7 + 7 ( 49 ) = 7 + 343 = 350 the answer is ( a )
Answer: a
Question (general): in a class of 40 students , 2 students did not borrow any books from the library , 12 students each borrowed 1 book , 11 students each borrowed 2 books , and the rest borrowed at least 3 books . if the average number of books per student was 2 , what is the maximum number of books any single student could have borrowed ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "the class borrowed a total of 40 * 2 = 80 books . the 25 students who borrowed 0 , 1 , or 2 books borrowed a total of 12 + 11 * 2 = 34 . to maximize the number of books borrowed by 1 student , let ' s assume that 14 students borrowed 3 books and 1 student borrowed the rest . 80 - 34 - 3 * 14 = 4 the maximum number of books borrowed by any student is 4 . the answer is b ."
Answer: b
Question (general): if ( t - 8 ) is a factor of t ^ 2 - kt - 45 , then k =
Options: a ) 16 , b ) 12 , c ) 2 , d ) 6 , e ) 14
Rationale: t ^ 2 - kt - 48 = ( t - 8 ) ( t + m ) where m is any positive integer . if 48 / 8 = 6 , then we know as a matter of fact that : m = + 6 and thus k = 8 - 6 = 12 t ^ 2 - kt - m = ( t - a ) ( t + m ) where a > m t ^ 2 + kt - m = ( t - a ) ( t + m ) where a < m t ^ 2 - kt + m = ( t - a ) ( t - m ) t ^ 2 + kt + m = ( t + a ) ( t + m ) b
Answer: b
Question (general): set a contains all the even numbers between 12 and 50 inclusive . set b contains all the even numbers between 112 and 150 inclusive . what is the difference between the sum of elements of set b and the sum of the elements of set a ?
Options: a ) 2000 , b ) 2550 , c ) 5050 , d ) 6275 , e ) 11325
Rationale: set a contains 12,14 , 16 . . . 50 set b contains 112 , 114 , 116 . . . 150 number of terms in each set = 20 difference between corresponding terms in set a and b = 100 difference between sum of set b and set a = 100 * 20 = 2000 answer a
Answer: a
Question (geometry): how much greater is the combined area in square inches of the front and back of a rectangular sheet of paper measuring 11 inches by 15 inches than that of a rectangular sheet of paper measuring 7.5 inches by 11 inches ?
Options: a ) 50 % , b ) 87 % , c ) 100 % , d ) 187 % , e ) 200 %
Rationale: "let ' s just look at the dimensions ( no calculation needed ) . with dimension 11 the same , the other dimension 15 is twice 7.5 then the area will be double which means 100 % greater . the answer is c ."
Answer: c
Question (geometry): the number 150 can be written as the sum of the squares of 3 different positive integers . what is the sum of these 3 integers ?
Options: a ) 17 , b ) 18 , c ) 15 , d ) 14 , e ) 13
Rationale: "10 ^ 2 + 7 ^ 2 + 1 ^ 2 = 150 = = = > sum of these 3 integers = 10 + 7 + 1 = 18 b"
Answer: b
Question (geometry): find the area of a rhombus one side of which measures 20 cm and one diagonal is 24 cm ?
Options: a ) 320 cm 2 , b ) 280 cm 2 , c ) 384 cm 2 , d ) 290 cm 2 , e ) 350 cm 2
Rationale: "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 . so , other diagonal = 32 cm . area of rhombus = ( 1 / 2 ) x ( product of diagonals ) = ( 1 / 2 × 24 x 32 ) cm 2 = 384 cm 2 hence c"
Answer: c
Question (general): what is the least value of x . so that 22 x 57 is divisible by 3 .
Options: a ) 2 , b ) 0 , c ) 1 , d ) 3 , e ) 4
Rationale: "explanation : the sum of the digits of the number is divisible by 3 , then the number is divisible by 3 . 2 + 2 + x + 5 + 7 = 16 + x least value of x may be 2 therefore 16 + 2 = 18 is divisible by 3 . answer : option a"
Answer: a
Question (general): in a forest 120 deer were caught , tagged with electronic markers , then released . a week later , 50 deer were captured in the same forest . of these 50 deer , it was found that 5 had been tagged with the electronic markers . if the percentage of tagged deer in the second sample approximates the percentage of tagged deer in the forest , and if no deer had either left or entered the forest over the preceding week , what is the approximate number of deer in the forest ?
Options: a ) 1,200 , b ) 750 , c ) 1,250 , d ) 1,500 , e ) 2,500
Rationale: "the percentage of tagged deer in the second sample = 5 / 50 * 100 = 10 % . so , 120 tagged deers comprise 10 % of total # of deers - - > total # of deers = 120 * 10 = 1,200 answer : a"
Answer: a
Question (gain): if the selling price of 7 articles is same as the cost price of 10 articles . find the gain or loss percentage ?
Options: a ) 93 % , b ) 23 % , c ) 33 % , d ) 43 % , e ) 53 %
Rationale: "let the c . p of each article be re 1 . then , s . p of 7 articles = c . p of 10 articles = rs . 10 now , c . p of 7 articles = rs . 7 , s . p of 7 articles = rs 10 gain = rs ( 10 - 7 ) = rs 3 . gain % = ( 3 / 7 × 100 ) % = 43 % answer : d"
Answer: d
Question (other): the ratio of number of boys and girls in a school is 2 : 7 . if there are 360 students in the school , find the number of girls in the school ?
Options: a ) 150 , b ) 250 , c ) 300 , d ) 370 , e ) 280
Rationale: "let the number of boys and girls be 2 x and 7 x total students = 360 number of girls in the school = 7 * 360 / 9 = 280 answer is e"
Answer: e
Question (geometry): a spirit and water solution is sold in a market . the cost per liter of the solution is directly proportional to the part ( fraction ) of spirit ( by volume ) the solution has . a solution of 1 liter of spirit and 1 liter of water costs 30 cents . how many cents does a solution of 1 liter of spirit and 2 liters of water cost ?
Options: a ) 13 , b ) 30 , c ) 50 , d ) 51 , e ) 52
Rationale: "yes , ensure that you understand the relation thoroughly ! cost per liter = k * fraction of spirit 30 cents is the cost of 2 liters of solution ( 1 part water , 1 part spirit ) . so cost per liter is 15 cents . fraction of spirit is 1 / 2 . 15 = k * ( 1 / 2 ) k = 30 cost per liter = 30 * ( 1 / 3 ) ( 1 part spirit , 2 parts water ) cost for 3 liters = 30 * ( 1 / 3 ) * 3 = 50 cents b . 30 cents"
Answer: b
Question (other): you need to print a document of the area 216 sq cm . condition is 3 cm margin is to be left at both top & bottom and 2 cm at the sides . what is the optimized size of your paper ?
Options: a ) 60 sq . cm
b ) 70 sq . cm
c ) 95 sq . cm
d ) 80 sq . cm
e ) 90 sq . cm
Rationale: let us consider it is a rectangle . so area = 24 * 9 = 216 sq . cm now ( 24 - 3 * 2 ) * ( 9 - 2 * 2 ) = 18 * 5 = 90 sq . cm answer : e
Answer: e
Question (general): find the average of all numbers between 1 and 13 which are divisible by 2
Options: a ) 15 , b ) 21 , c ) 25 , d ) 30 , e ) 35
Rationale: "explanation : average = ( 2 + 4 + 6 + 8 + 10 + 12 ) / 2 = 42 / 2 = 21 option b"
Answer: b
Question (general): if x / y = 5 / 3 , then ( x + y ) / ( x - y ) = ?
Options: a ) 4 , b ) 1 / 5 , c ) - 1 / 6 , d ) - 1 / 5 , e ) - 5
Rationale: "any x and y satisfying x / y = 5 / 3 should give the same value for ( x + y ) / ( x - y ) . say x = 5 and y = 3 , then ( x + y ) / ( x - y ) = ( 5 + 3 ) / ( 5 - 3 ) = 4 . answer : a ."
Answer: a
Question (general): a number , when 35 is subtracted from it , reduces to its 80 percent . what is 4 - fifth of that number ?
Options: a ) 130 , b ) 155 , c ) 490 , d ) 140 , e ) 160
Rationale: explanation : x - 35 = 80 x / 100 = > x = 175 = > 4 x / 5 = 4 x 175 / 5 = 140 . answer d
Answer: d
Question (gain): a broker invested her own money in the stock market . during the first year , she increased her stock market wealth by 40 percent . in the second year , largely as a result of a slump in the stock market , she suffered a 30 percent decrease in the value of her stock investments . what was the net increase or decrease on her overall stock investment wealth by the end of the second year ?
Options: a ) − 5 % , b ) − 2 % , c ) 15 % , d ) 20 % , e ) 80 %
Rationale: "the actual answer is obtained by multiplying 140 % by 70 % and subtracting 100 % from this total . that is : 140 % × 70 % = 98 % ; 98 % − 100 % = - 2 % . answer : b"
Answer: b
Question (general): a library has an average of 425 visitors on sundays and 325 on other days . the average number of visitors per day in a month of 30 days beginning with a sunday is :
Options: a ) 140 , b ) 240 , c ) 260 , d ) 280 , e ) 340
Rationale: explanation : since the month begins with a sunday , so there will be five sundays in the month , required average = ( 415 * 5 + 325 * 25 ) / 30 = 10200 / 30 = 340 answer : e ) 340
Answer: e
Question (probability): from a pack of 52 cards , two cards are drawn together at random . what is the probability that the one is heart and other two is diamond ?
Options: a ) 238 / 5525 , b ) 176 / 5534 , c ) 253 / 5523 , d ) 234 / 5525 , e ) 1 / 5525
Rationale: "solution let s be the sample space . then , n ( s ) = 52 c 3 = 22100 let e = event of getting 1 face card . n ( e ) = number of ways of choosing 1 face card out of 26 = 13 c 1 * 13 c 2 = 13 * 72 = 936 p ( e ) = n ( e ) / n ( s ) = 936 / 22100 = 234 / 5525 = 234 / 5525 . answer d"
Answer: d
Question (physics): the sector of a circle has radius of 14 cm and its perimeter 50 cm . find its central angel ?
Options: a ) 180 o , b ) 225 o , c ) 270 o , d ) 150 o , e ) 90 o
Rationale: lte central angle = x perimeter of the sector = length of the arc + 2 ( radius ) 50 = ( x / 360 * 2 * 22 / 7 * 14 ) + 2 ( 14 ) 50 = 88 x / 360 + 28 88 x / 360 = 22 88 x = 7920 x = 90 answer : e
Answer: e
Question (general): a certain list consists of 21 different numbers . if n is in the list and n is 4 times the average ( arithmetic mean ) of the other 20 numbers in the list , then n is what fraction q of the sum of the 21 numbers in the list ?
Options: a ) 1 / 20 , b ) 1 / 6 , c ) 1 / 5 , d ) 4 / 21 , e ) 5 / 21
Rationale: this is how i used to calculate which i think works pretty well : if you let the average of the 20 other numbers equal a , can you write this equation for sum of the list ( s ) n + 20 a = s the question tells us that n = 4 a plug this back into the first equation and you get that the sum is 24 a 4 a + 20 a = 24 a therefore fraction q of n to the total would be 4 a / 24 a or 1 / 6 answer b
Answer: b
Question (physics): if 12 men can reap 120 acres of land in 16 days , how many acres of land can 36 men reap in 32 days ?
Options: a ) 269 , b ) 512 , c ) 369 , d ) 720 , e ) 450
Rationale: 12 men 120 acres 16 days 36 men ? 32 days 120 * 36 / 12 * 32 / 16 120 * 3 * 2 120 * 6 = 720 answer : d
Answer: d
Question (gain): in a certain pond , 50 fish were caught , tagged , and returned to the pond . a few days later , 50 fish were caught again , of which 4 were found to have been tagged . if the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond , what is the approximate number of fish in the pond ?
Options: a ) 200 , b ) 325 , c ) 565 , d ) 625 , e ) 700
Rationale: "total fish = x percentage of second catch = ( 4 / 50 ) * 100 = 8 % so , x * 8 % = 50 x = 625 ans d ."
Answer: d
Question (general): a , b and c play a cricket match . the ratio of the runs scored by them in the match is a : b = 2 : 3 and b : c = 2 : 5 . if the total runs scored by all of them are 75 , the runs scored by b are ?
Options: a ) 11 , b ) 18 , c ) 99 , d ) 77 , e ) 51
Rationale: "a : b = 2 : 3 b : c = 2 : 5 a : b : c = 4 : 6 : 15 6 / 25 * 75 = 18 answer : b"
Answer: b
Question (general): the hcf and lcm of two numbers m and n are respectively 8 and 200 . if m + n = 84 , then 1 / m + 1 / n is equal to
Options: a ) 1 / 35 , b ) 3 / 50 , c ) 5 / 37 , d ) 2 / 35 , e ) none
Rationale: "answer we have , m x n = 8 x 200 = 1600 â ˆ ´ 1 / m + 1 / n = ( m + n ) / mn = 84 / 1600 = 3 / 50 correct option : b"
Answer: b
Question (gain): an inspector rejects 0.04 % of the meters as defective . how many will he examine to reject 2 ?
Options: a ) a ) 1500 , b ) b ) 2000 , c ) c ) 2500 , d ) d ) 5000 , e ) e ) 3100
Rationale: "let the number of meters to be examined be x then , 0.04 % of x = 2 ( 4 / 100 ) * ( ( 1 / 100 ) * x = 2 x = 5000 answer is d"
Answer: d
Question (gain): john makes $ 50 a week from his job . he earns a raise andnow makes $ 90 a week . what is the % increase ?
Options: a ) 15 % , b ) 16.66 % , c ) 17.8 % , d ) 80 % , e ) 21 %
Rationale: "increase = ( 40 / 50 ) * 100 = ( 4 / 5 ) * 100 = 80 % . d"
Answer: d
Question (general): a father said his son , ` ` i was as old as you are at present at the time of your birth . ` ` if the father age is 48 now , the son age 5 years back was
Options: a ) 14 , b ) 17 , c ) 11 , d ) 19 , e ) 99
Rationale: "let the son ' s present age be x years . then , ( 48 - x ) = x x = 24 . son ' s age 5 years back = ( 24 - 5 ) = 19 years answer : d"
Answer: d
Question (physics): a man can row upstream at 25 kmph and downstream at 43 kmph , and then find the speed of the man in still water ?
Options: a ) 86 , b ) 34 , c ) 30 , d ) 15 , e ) 17
Rationale: "us = 25 ds = 43 m = ( 43 + 25 ) / 2 = 34 answer : b"
Answer: b
Question (general): if | x - 20 | = 40 what is the sum of all the values of x .
Options: a ) 0 , b ) 60 , c ) - 80 , d ) 40 , e ) 80
Rationale: "there will be two cases x - 20 = 40 and x - 20 = - 40 solve for x = > x = 40 + 20 = > x = 60 or x = - 40 + 20 = > x = - 20 the sum of both values will be 60 + - 20 = 40 answer is d"
Answer: d
Question (general): a certain car traveled twice as many miles from town a to town b as it did from town b to town c . from town a to town b , the car averaged 12 miles per gallon , and from town b to town c , the car averaged 14 miles per gallon . what is the average miles per gallon that the car achieved on its trip from town a through town b to town c ?
Options: a ) 13 , b ) 13.5 , c ) 14 , d ) 14.5 , e ) 15
Rationale: "ans is e given d _ ab = 2 * d _ bc let d _ ab = d and d _ bc = x so d = 2 x for average miles per gallon = ( d + x ) / ( ( d / 12 ) + ( x / 14 ) ) = 15 ( formula avg speed = total distance / total time )"
Answer: e
Question (general): what is the units digit of 222 ^ ( 333 ) * 333 ^ ( 222 ) ?
Options: a ) 0 , b ) 2 , c ) 4 , d ) 6 , e ) 8
Rationale: "each of the other explanations to this question has properly explained that you need to break down the calculation into pieces and figure out the repeatingpatternof the units digits . here ' s another way to organize the information . we ' re given [ ( 2222 ) ^ 333 ] [ ( 3333 ) ^ 222 ] we can ' combine ' some of the pieces and rewrite this product as . . . . ( [ ( 2222 ) ( 3333 ) ] ^ 222 ) [ ( 2222 ) ^ 111 ] ( 2222 ) ( 3333 ) = a big number that ends in a 6 taking a number that ends in a 6 and raising it to a power creates a nice pattern : 6 ^ 1 = 6 6 ^ 2 = 36 6 ^ 3 = 216 etc . thus , we know that ( [ ( 2222 ) ( 3333 ) ] ^ 222 ) will be a gigantic number that ends in a 6 . 2 ^ 111 requires us to figure out thecycleof the units digit . . . 2 ^ 1 = 2 2 ^ 2 = 4 2 ^ 3 = 8 2 ^ 4 = 16 2 ^ 5 = 32 2 ^ 6 = 64 2 ^ 7 = 128 2 ^ 8 = 256 so , every 4 powers , the pattern of the units digits repeats ( 2 , 4 , 8 , 6 . . . . . 2 , 4 , 8 , 6 . . . . ) . 111 = 27 sets of 4 with a remainder of 3 . . . . this means that 2 ^ 111 = a big number that ends in an 8 so we have to multiply a big number that ends in a 6 and a big number that ends in an 8 . ( 6 ) ( 8 ) = 48 , so the final product will be a gigantic number that ends in an 4 . final answer : c"
Answer: c
Question (physics): the length and breadth of a rectangular courtyard is 75 m and 32 m . find the cost of leveling it at the rate of $ 3 per m 2 . also , find the distance covered by a boy to take 4 rounds of the courtyard .
Options: a ) 3573 , b ) 3455 , c ) 8600 , d ) 7000 , e ) 7200
Rationale: length of the courtyard = 75 m breadth of the courtyard = 32 m perimeter of the courtyard = 2 ( 75 + 32 ) m = 2 × 107 m = 214 m distance covered by the boy in taking 4 rounds = 4 × perimeter of courtyard = 4 × 214 = 856 m we know that area of the courtyard = length × breadth = 75 × 32 m 2 = 2400 m 2 for 1 m 2 , the cost of levelling = $ 3 for 2400 m 2 , the cost of levelling = $ 3 × 2400 = $ 7200 answer : e
Answer: e
Question (physics): city a and city b are 140 miles apart . train c departs city a , heading towards city b , at 4 : 00 and travels at 40 miles per hour . train d departs city b , heading towards city a , at 4 : 40 and travels at 20 miles per hour . the trains travel on parallel tracks . at what time do the two trains meet ?
Options: a ) 5 : 00 , b ) 5 : 30 , c ) 6 : 40 , d ) 6 : 30 , e ) 7 : 00
Rationale: "train c has traveled 20 mi in the half hour before train d has started its journey . 140 - 20 = 120 40 + 20 = 60 mph 120 mi / 60 mph = 2 hrs 4 : 40 pm + 2 hrs = 6 : 40 pm answer : c . 6 : 40"
Answer: c