Cheukting/mathqa huggingface dataset:
Question (general): in a river flowing at 2 km / hr , a boat travels 64 km upstream and then returns downstream to the starting point . if its speed in still water be 6 km / hr , find the total journey time .
Options: a ) 10 hours , b ) 12 hours , c ) 14 hours , d ) 24 hours , e ) none of these
Rationale: "explanation : speed of the boat = 6 km / hr speed downstream = ( 6 + 2 ) = 8 km / hr speed upstream = ( 6 - 2 ) = 4 km / hr distance travelled downstream = distance travelled upstream = 64 km total time taken = time taken downstream + time taken upstream = ( 64 / 8 ) + ( 64 / 4 ) = = 24 hr . answer : option d"
Answer: d
Question (general): ( 128.5 x 32 ) + ( 13.8 x 30 ) = ? x 25
Options: a ) 524.48 , b ) 556.02 , c ) 574.36 , d ) 181.04 , e ) 595.46
Rationale: "explanation : ? = ( 128.5 x 32 ) + ( 13.8 x 30 ) / 25 = 4112 + 414 / 25 = 181.04 answer : option d"
Answer: d
Question (general): if ( m - 8 ) is a factor of m ^ 2 - pm - 24 , then p =
Options: a ) 3 , b ) 5 , c ) 6 , d ) 11 , e ) 16
Rationale: "( m - 8 ) ( m - a ) = m ^ 2 - pm - 24 a = - 3 p = 8 + a = 5 = b"
Answer: b
Question (general): you and your friend spent a total of $ 15 for lunch . your friend spent $ 1 more than you . how much did your friend spend on their lunch ?
Options: a ) $ 9 , b ) $ 8 , c ) $ 4 , d ) $ 6 , e ) $ 5
Rationale: "my lunch = l , my friends lunch = l + 1 ( l ) + ( l + 1 ) = 15 l + l + 1 - 1 = 15 - 1 2 l = 14 l = 7 my friends lunch l + 1 = 7 + 1 = $ 8 , the answer is b"
Answer: b
Question (general): how many multiples of 4 are there between 8 and 160 ?
Options: a ) 28 , b ) 37 , c ) 32 , d ) 35 , e ) 39
Rationale: "it should be mentioned whether 8 and 160 are inclusive . if 8 and 160 are inclusive , then the answer is ( 160 - 8 ) / 4 + 1 = 39 . if 8 and 160 are not inclusive , then the answer is ( 156 - 12 ) / 4 + 1 = 37 . since oa is b , then we have not inclusive case ."
Answer: b
Question (gain): √ 7 percent of 7 √ 7 =
Options: a ) 0.55 , b ) 0.96 , c ) 0.1 , d ) 0.45 , e ) 0.6
Rationale: "the other solutions are great , but here ' s one more approach . first , we scan the answer choices to see that they are quite spread apart . this means we can also use someapproximation all test - takers must memorize the following approximations : √ 2 ≈ 1.4 √ 3 ≈ 1.7 √ 5 ≈ 2.2 √ 6 ≈ 2.4 √ 7 ≈ 2.6 so , √ 7 percent of 7 √ 7 ≈ 2.6 percent of ( 7 ) ( 2.6 ) ≈ ( 2.6 / 100 ) ( 18.2 ) ≈ 47.32 / 100 ≈ 0.47 check the answer choices to see that d is the only answer that ' s close . answer : d"
Answer: d
Question (general): harold works at a resort from the beginning of march to the end of september . during the month of august this past year , he made 7 times the average ( arithmetic mean ) of his monthly totals in tips for the other months . his total tips for august were what fraction of his total tips for all of the months he worked ?
Options: a ) 1 / 3 , b ) 2 / 5 , c ) 3 / 7 , d ) 1 / 2 , e ) 7 / 13
Rationale: "the time from beginning of march to the end of september is 7 months . if x is the average monthly tip for all months other than august then his august month tip will be 7 * x his total tip for the 7 months = 6 * ( average tip for the months other than august ) + 7 x = 13 x august tips as a fraction of total tips = 7 x / 13 x = 7 / 13 . so e"
Answer: e
Question (physics): maxwell leaves his home and walks toward brad ' s house . one hour later , brad leaves his home and runs toward maxwell ' s house . if the distance between their homes is 54 kilometers , maxwell ' s walking speed is 4 km / h , and brad ' s running speed is 6 km / h . what is the total time it takes maxwell before he meets up with brad ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 8
Rationale: "total distance = 54 kms maxwell speed = 4 kms / hr maxwell travelled for 1 hour before brad started , therefore maxwell traveled for 4 kms in 1 hour . time taken = total distance / relative speed total distance after brad started = 50 kms relative speed ( opposite side ) ( as they are moving towards each other speed would be added ) = 6 + 4 = 10 kms / hr time taken to meet brad after brad started = 50 / 10 = 5 hrs distance traveled by maxwell = maxwell ' s speed * time taken = 4 * 5 = 20 + 4 = 24 kms . . . therefore total time taken by maxwell to meet brad = distance travelled by maxwell / maxwell ' s speed = 24 / 4 = 6 hrs . . . answer d alternatively . . . total time taken by maxwell to meet brad = maxwell started 1 hour earlier to brad + total time taken to meet brad after brad started = 1 + 5 = 6 hrs . . . answer d"
Answer: d
Question (general): tom read a book containing 480 pages by reading the same number of pages each day . if he would have finished the book 5 days extra by reading 10 pages a day less , how many days did tom spend reading the book ?
Options: a ) 13 , b ) 15 , c ) 14 , d ) 13.19 , e ) 16
Rationale: "actually u can set up 2 equation p - - stands for the pages d - - stands for the days 1 ) p * d = 480 ( we want to find the days , sop = 480 / d ) 2 ) ( p - 10 ) ( d + 5 ) = 480 = > pd + 5 p - 10 d - 50 = 480 as the 1 ) stated u can put 1 ) into 2 ) = > 480 + 5 p - 10 d - 50 = 480 = > 5 p - 10 d = 50 put the bold one into it = > 5 * 480 / d - 10 d = 50 the we get the final equation 2400 / d - 10 d = 50 = > d ^ 2 + 5 d - 240 = 0 ( d - 15 ) ( d + 10 ) = 0 so d = 13.19 days . ans : d"
Answer: d
Question (general): what are the total number of divisors of 600 ( including 1 and 600 ) ?
Options: a ) 4 , b ) 24 , c ) 40 , d ) 16 , e ) 20
Rationale: write the prime factorization . 600 = ( 2 ^ 3 ) ( 3 ^ 1 ) ( 5 ^ 2 ) the number of divisors ( including 1 and the number itself ) is the product of one plus the exponents . ( 3 + 1 ) ( 1 + 1 ) ( 2 + 1 ) = 24 answer : b
Answer: b
Question (other): the overall ages of kumar , hathim and rahul is 93 years . 1 decade ago , the ratio of their ages was 2 : 3 : 4 . what is the current age of rahul ?
Options: a ) 38 , b ) 37 , c ) 34 , d ) 35 , e ) 33
Rationale: a 38 if the ages of kumar , hathim and rahul 1 decade ago be 2 a , 3 a and 4 a years correspondingly then , ( 2 a + 10 ) + ( 3 a + 10 ) + ( 4 a + 10 ) = 93 9 a = 93 - 30 9 a = 63 a = 7 rahul ’ s current age = ( 4 a + 10 ) = 38 years .
Answer: a
Question (general): a gardener wants to plant trees in his garden in such a way that the number of trees in each row should be the same . if there are 2 rows or 7 rows or 9 rows , then no tree will be left . find the least number of trees required
Options: a ) 22 , b ) 60 , c ) 126 , d ) 76 , e ) 21
Rationale: "explanation : the least number of trees that are required = lcm ( 2 , 7 , 9 ) = 126 answer : c"
Answer: c
Question (general): set # 1 = { a , b } set # 2 = { k , l , m , n , o , p } there are these two sets of letters , and you are going to pick exactly one letter from each set . what is the probability of picking at least one vowel ?
Options: a ) 1 / 6 , b ) 1 / 3 , c ) 7 / 12 , d ) 2 / 3 , e ) 5 / 6
Rationale: so not a vowel in set - 1 : 1 / 2 and not a vowel in ser - 2 : 5 / 6 now , 1 / 2 ∗ 5 / 6 = 5 / 12 this is for not a vowel . then for at least one vowel will be = 1 − 5 / 12 = 7 / 12 answer will be c .
Answer: c
Question (geometry): in a rectangular coordinate system , what is the area of a rectangle whose vertices have the coordinates ( - 7 , 1 ) , ( 1 , 1 ) , ( 1 , - 6 ) and ( - 7 , - 6 ) ?
Options: a ) 112 , b ) 64 , c ) 42 , d ) 24 , e ) 56
Rationale: "length of side 1 = 7 + 1 = 8 length of side 2 = 6 + 1 = 7 area of rectangle = 7 * 8 = 56 e is the answer"
Answer: e
Question (general): in digging a pond 20 m * 12 m * 5 m the volumes of the soil extracted will be ?
Options: a ) 3387 , b ) 1000 , c ) 1200 , d ) 2787 , e ) 1121
Rationale: "20 * 12 * 5 = 1200 answer : c"
Answer: c
Question (general): how many times digit 7 is used while writing numbers from 1 to 300 ?
Options: a ) 28 , b ) 42 , c ) 41 , d ) 35 , e ) 49
Rationale: "in 1 to 100 their are 14 seven ' s in 100 to 200 their are 14 seven ' s in 200 to 300 their are 14 seven ' s so total answer is 42 seven ' s correct option : b"
Answer: b
Question (general): define a = the sum of digits of the number 7 ^ 100 and b = the sum of the digits of a . what is the sum of the digits of b ?
Options: a ) 7 , b ) 16 , c ) 10 , d ) 11 , e ) can not be determined
Rationale: following approach will take you only couple of sec . \ ( 7 ^ { 100 } = ( 7 ^ { 50 } ) ^ 2 \ ) the sum of digits ( digital root ) of a perfect square can only be 1 , 4 , 7 or 9 . for example \ ( 7 ^ 2 = 49 \ ) = > dr = 4 + 9 = 13 = > 1 + 3 = 4 and etc . checking available options . answer a .
Answer: a
Question (geometry): the area of a square is equal to three times the area of a rectangle of dimensions 25 cm * 27 cm . what is the perimeter of the square ?
Options: a ) 180 cm , b ) 190 cm , c ) 170 cm , d ) 150 cm , e ) 160 cm
Rationale: "area of the square = s * s = 3 ( 25 * 27 ) = > s = 3 * 5 * 3 = 45 cm perimeter of the square = 4 * 45 = 180 cm . answer : a"
Answer: a
Question (general): an optometrist charges $ 150 per pair for soft contact lenses and $ 85 per pair for hard contact lenses . last week she sold 5 more pairs of soft lenses than hard lenses . if her total sales for pairs of contact lenses last week were $ 1,925 , what was the total number of pairs of contact lenses that she sold ?
Options: a ) 11 , b ) 13 , c ) 15 , d ) 17 , e ) 19
Rationale: "( x + 5 ) * 150 + x * 85 = 1925 = > x = 5 total lens = 5 + ( 5 + 5 ) = 15 answer c"
Answer: c
Question (other): how many zeros does 1000 ! end with ?
Options: a ) 20 o , b ) 24 o , c ) 249 , d ) 30 o , e ) 325
Rationale: "according to above 1000 ! has 1000 / 5 + 1000 / 25 + 1000 / 125 + 1000 / 625 = 200 + 40 + 8 + 1 = 249 trailing zeros . answer : c"
Answer: c
Question (physics): train e and f , 455 miles apart , are traveling toward each other at constant rates and in the same time zone . if train e left at 4 pm traveling at a speed of 60 miles per hour , and train f left at 5 : 45 pm and traveling at 45 miles per hour , then at what time would they pass each other ?
Options: a ) 9.03 pm , b ) 9.05 pm , c ) 9.07 pm , d ) 9.08 pm , e ) 9.10 pm
Rationale: first , since e has a headstart then in that 1 hr 45 min or 1.75 hrs he travels 105 miles then remaining distance to be traveled will be 455 - 105 = 350 miles now , using relative rates ( 105 ) ( t ) = 350 this gives 10 / 3 hours now 5.45 pm + 10 / 3 hours gives us 9.05 pm hence answer is b
Answer: b
Question (general): in the xy - plane , the point ( 1 , 3 ) is the center of a circle . the point ( - 2 , 3 ) lies inside the circle and the point ( 1 , - 2 ) lies outside the circle . if the radius r of the circle is an integer , then r =
Options: a ) 6 , b ) 5 , c ) 4 , d ) 3 , e ) 2
Rationale: "an easy way to solve this question will be just to mark the points on the coordinate plane . you ' ll see that the distance between the center ( 1 , 3 ) and the point inside the circle ( - 2 , 3 ) is 3 units ( both points are on y = 3 line so the distance will simply be 1 - ( - 2 ) = 3 ) so the radius must be more than 3 units . the distance between the center ( 1,3 ) and the point outside the circle ( 1 , - 2 ) is 5 units ( both points are on x = 1 line so the distance will simply be 3 - ( - 2 ) = 5 ) so the radius must be less than 5 units , which implies 3 < r < 5 , thus as r is an integer then r = 4 . answer : c ."
Answer: c
Question (other): the ratio by weight , measured in pounds , of books to clothes to electronics in a suitcase initially stands at 7 : 5 : 3 . someone removes 5 pounds of clothing from the suitcase , thereby doubling the ratio of books to clothes . how many pounds do the electronics in the suitcase weigh ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "the weights of the items in the suitcase are 7 k , 5 k , and 3 k . if removing 5 pounds of clothes doubles the ratio of books to clothes , then 5 pounds represents half the weight of the clothes . 2.5 k = 5 pounds and then k = 2 pounds . the electronics weigh 3 ( 2 ) = 6 pounds . the answer is d ."
Answer: d
Question (general): when 2 / 9 of the votes on a certain resolution have been counted , 3 / 4 of those counted are in favor of the resolution . what fraction w of the remaining votes must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution ?
Options: a ) 11 / 14 , b ) 13 / 18 , c ) 4 / 7 , d ) 3 / 7 , e ) 3 / 14
Rationale: "if we use variable for total votes there will be too many fractions to manipulate with , so pick some smart # : let set total # of votes is 18 . 2 / 9 of the votes on a certain resolution have been counted - - > 4 counted and 18 - 4 = 14 votes left to be counted ; 3 / 4 of those counted are in favor of the resolution - - > 3 in favor and 1 against ; ratio of those who voted against to those who voted for to be 2 to 1 there should be total of 18 * 2 / 3 = 12 people who voted against , so in the remaining 14 votes there should be 12 - 1 = 11 people who voted against . thus w = 11 / 14 of the remaining votes must be against . answer : a ."
Answer: a
Question (general): the total age of a and b is 17 years more than the total age of b and c . c is how many years younger than a ?
Options: a ) 12 , b ) 27 , c ) 21 , d ) 17 , e ) 11
Rationale: "( a + b ) - ( b + c ) = 17 a - c = 17 . c is younger than a by 17 years . answer : d"
Answer: d
Question (physics): a 600 meter long train crosses a signal post in 15 seconds . how long will it take to cross a 3 kilometer long bridge , at the same speed ?
Options: a ) 4 min , b ) 2 min , c ) 8 min , d ) 9 min , e ) 1.5 min
Rationale: "s = 600 / 15 = 40 mps s = 3600 / 40 = 90 sec = 1.5 min answer : e"
Answer: e
Question (gain): a total of $ 20,000 was invested in two certificates of deposit at simple annual interest rates of 3 percent and 6 percent , respectively . if the total interest on the two certificates was $ 1,440 at the end of one year , what fractional part of the $ 20.000 was invested at the higher rate ?
Options: a ) 3 / 8 , b ) 21 / 25 , c ) 1 / 2 , d ) 3 / 5 , e ) 3 / 4
Rationale: "x * 3 / 100 * 1 + ( 20000 - x ) * 8 / 100 * 1 = 1440 3 x - 8 x = 144000 - 160000 = > - 5 x = - 16000 = > x = 32000 so 16800 / 20000 = 21 / 25 answer - b"
Answer: b
Question (general): if a = ( 10 ^ 10 ) - 47 , what is the sum of all the digit of a ?
Options: a ) 40 , b ) 45 , c ) 50 , d ) 55 , e ) 80
Rationale: start small and check for any pattern : 10 ^ 2 - 47 = 53 - > number of digits = power of 10 ; number of 9 s = ( power of 10 ) - 2 ; 53 is the last 2 digits 10 ^ 3 - 47 = 953 - > number of digits = power of 10 ; number of 9 s = ( power of 10 ) - 2 ; 53 is the last 2 digits . . . extrapolate 10 ^ 10 - 47 - > number of digits = 10 ; number of 9 s = 8 ; 53 will be the last 2 digits sum , a = 9 * 8 + 5 + 3 = 80 . answer ( e ) .
Answer: e
Question (general): two sets of 7 consecutive positive integers have exactly one integer in common . the sum of the integers in the set with greater numbers is how much greater than the sum of the integers in the other set ?
Options: a ) 24 , b ) 17 , c ) 28 , d ) 42 , e ) it can not be determined from the information given .
Rationale: a = ( 1 , 2,3 , 4,5 , 6,7 ) , sum of this = 28 b = ( 7 , 8,9 , 10,11 , 12,13 ) , sum of this = 70 the differenct between 70 - 28 = 42 hence , 42 is the answer i . e . d
Answer: d
Question (geometry): a lady grows cauliflower in her garden that is in the shape of a square . each cauliflower takes 1 square foot of area in her garden . this year , she has increased her output by 223 cauliflower when compared to last year . the shape of the area used for growing the cauliflower has remained a square in both these years . how many cauliflowers did she produce this year ?
Options: a ) 12541
b ) 12542
c ) 12544
d ) 12545
e ) can not be determined
Rationale: explanatory answer the shape of the area used for growing cauliflower has remained a square in both the years . let the side of the square area used for growing cauliflower this year be x ft . therefore , the area of the ground used for cultivation this year = x 2 sq . ft . let the side of the square area used for growing cauliflower last year be y ft . therefore , the area of the ground used for cultivation last year = y 2 sq . ft . as the number of cauliflower grown has increased by 223 , the area would have increased by 223 sq ft because each cauliflower takes 1 sq ft space . hence , x 2 - y 2 = 223 ( x + y ) ( x - y ) = 223 . 223 is a prime number and hence it will have only two factors . i . e . , 223 and 1 . therefore , 223 can be expressed as product of 2 numbers in only way = 223 * 1 i . e . , ( x + y ) ( x - y ) = 223 * 1 so , ( x + y ) should be 223 and ( x - y ) should be 1 . solving the two equations we get x = 112 and y = 111 . therefore , number of cauliflower produced this year = x 2 = 1122 = 12544 . alternative approach : use answer choices the area in both the years are squares of two numbers . that rules out choice a , b and d . as 12541,12542 and 12545 are not the square of any number . check choice c : if this year ' s produce is 12544 , last year ' s produce would have been 12544 - 223 = 12321 12321 is the square of 111 . so , 12544 is the answer . choice c .
Answer: c
Question (general): 12 . what is the dividend . divisor 17 , the quotient is 9 and the remainder is 9 .
Options: a ) 150 , b ) 162 , c ) 154 , d ) 158 , e ) 144
Rationale: "d = d * q + r d = 17 * 9 + 9 d = 153 + 9 d = 162 answer b"
Answer: b
Question (physics): a boat can travel with a speed of 13 km / hr in still water . if the speed of the stream is 4 km / hr . find the time taken by the boat to go 68 km downstream ?
Options: a ) 2 hours , b ) 3 hours , c ) 4 hours , d ) 5 hours , e ) none
Rationale: "solution speed downstream = ( 13 + 4 ) km / hr = 17 km / hr . time taken to travel 68 km downstream = ( 68 / 17 ) hrs = 4 hrs . answer c"
Answer: c
Question (physics): find the cost of fencing around a circular field of diameter 42 m at the rate of rs . 3 a meter ?
Options: a ) 138 , b ) 390 , c ) 393 , d ) 279 , e ) 222
Rationale: "2 * 22 / 7 * 21 = 131 131 * 3 = rs . 393 answer : c"
Answer: c
Question (geometry): the volumes of two cones are in the ratio 1 : 15 and the radii of the cones are in the ratio of 1 : 2 . what is the length of the wire ?
Options: a ) 3 : 5 , b ) 3 : 9 , c ) 3 : 2 , d ) 2 : 2 , e ) 3 : 8
Rationale: "the volume of the cone = ( 1 / 3 ) π r 2 h only radius ( r ) and height ( h ) are varying . hence , ( 1 / 3 ) π may be ignored . v 1 / v 2 = r 1 ^ 2 . h 1 / r 2 ^ 2 . h 2 = > 1 / 15 = ( 1 ) ^ 2 h 1 / ( 2 ) ^ 2 h 2 = > h 1 / h 2 = 3 / 5 i . e . h 1 : h 2 = 3 : 5 answer : a"
Answer: a
Question (general): the average salary of all the workers in a workshop is rs . 6750 . the average salary of 7 technicians is rs . 12000 and the average salary of the rest is rs . 6000 . the total number of workers in the workshop is ?
Options: a ) a ) 23 , b ) b ) 21 , c ) c ) 52 , d ) d ) 56 , e ) e ) 12
Rationale: "let the total number of workers be x . then , 6750 x = ( 12000 * 7 ) + 6000 ( x - 7 ) = > 750 x = 42000 = x = 56 . answer : d"
Answer: d
Question (gain): each light bulb at a hotel is either incandescent or fluorescent . at a certain moment , forty percent of the incandescent bulbs are switched on , and eighty percent of the fluorescent bulbs are switched on . if 70 percent of all the bulbs are switched on at this moment , what percent of the bulbs that are switched on are incandescent ?
Options: a ) 10 % , b ) 14 % , c ) 18 % , d ) 22 % , e ) 26 %
Rationale: "let i be the number of incandescent bulbs . let f be the number of fluorescent bulbs . 0.4 i + 0.8 f = 0.7 ( i + f ) 0.1 f = 0.3 i f = 3 i this means that for every 1 incandescent bulb , there are 3 fluorescent bulbs . the percent of bulbs that are switched on which are incandescent is : 0.4 i / ( 0.4 i + 0.8 f ) = 0.4 i / ( 0.4 i + 0.8 * 3 i ) = 0.4 i / 2.8 i = 1 / 7 which is about 14 % . the answer is b ."
Answer: b
Question (general): on dividinng 109 by a number , the quotient is 9 and the remainder is 1 . find the divisor .
Options: a ) 12 , b ) 14 , c ) 15 , d ) 16 , e ) 17
Rationale: d = ( d - r ) / q = ( 109 - 1 ) / 9 = 108 / 9 = 18 a
Answer: a
Question (gain): what is the compound interest paid on a sum of rs . 700 for the period of 2 years at 10 % per annum .
Options: a ) 130 , b ) 620 , c ) 147 , d ) 120 , e ) none of these
Rationale: "solution = interest % for 1 st year = 10 interest % for 2 nd year = 10 + 10 % of 10 = 10 + 10 * 10 / 100 = 11 total % of interest = 10 + 11 = 21 total interest = 21 % 700 = 700 * ( 21 / 100 ) = 147 answer c"
Answer: c
Question (physics): a train covers a distance of 15 km in 10 min . if it takes 10 sec to pass a telegraph post , then the length of the train is ?
Options: a ) 240 m , b ) 150 m , c ) 120 m , d ) 200 m , e ) 165 m
Rationale: "speed = ( 15 / 10 * 60 ) km / hr = ( 90 * 5 / 18 ) m / sec = 25 m / sec . length of the train = 20 * 10 = 200 m . answer : d"
Answer: d
Question (gain): fresh grapes contain 65 % by weight while dried grapes contain 10 % water by weight . what is the weight of dry grapes available from 400 kg of fresh grapes ?
Options: a ) 105.00 kg , b ) 115.60 kg , c ) 125.00 kg , d ) 130.50 kg , e ) 155.55 kg
Rationale: "from the question we know : 400 kg * 65 % = 260 kg of water in the fresh grapes 400 kg - 260 kg of water = 140 kg of non - water mass we are looking for the weight of the dry grapes ( x ) . since the question tells us that 10 % of the weight of the dry graps is water and we know that 140 kg is non - water mass we can set up the following equation : x = 1 / 10 ( x ) + 140 kg 9 / 10 ( x ) = 140 kg x = 155.55 kg answer - e"
Answer: e
Question (general): in a fuel station the service costs $ 2.20 per vehicle and every liter of fuel costs $ 0.70 . assuming that you fill up 4 mini - vans and 2 trucks , what will be the total cost , if a mini - van ' s tank is 65 liters and a truck ' s tank is 120 % bigger and they are all empty ?
Options: a ) $ 382.10 , b ) $ 395.40 , c ) $ 404.30 , d ) $ 416.80 , e ) $ 426.50
Rationale: "the service cost of 4 vans and 2 trucks is 6 * 2.20 $ 13.20 the fuel in 4 vans is 4 * 65 = 260 liters the fuel in 2 trucks is 2 * 65 * 2.2 = 286 liters the total fuel ( vans + trucks ) = 546 liters the total fuel cost is 546 * 0.7 = $ 382.20 the total cost is $ 382.20 + $ 13.20 = $ 395.40 the answer is b ."
Answer: b
Question (general): a group of n students can be divided into equal groups of 4 with 1 student left over or equal groups of 7 with 4 students left over . what is the sum of the two smallest possible values of n ?
Options: a ) 66 , b ) 70 , c ) 74 , d ) 78 , e ) 82
Rationale: "n = 4 k + 1 = 7 j + 4 let ' s start at 1 = 4 ( 0 ) + 1 and keep adding 4 until we find a number in the form 7 j + 4 . 1 , 5 , 9 , 13 , 17 , 21 , 25 = 7 ( 3 ) + 4 the next such number is 25 + 4 * 7 = 53 . 25 + 53 = 78 the answer is d ."
Answer: d
Question (general): if an integer n is to be chosen at random from the integers 1 to 84 , inclusive , what is the probability that n ( n + 1 ) ( n + 2 ) will be divisible by 8 ?
Options: a ) 9 / 14 , b ) 3 / 8 , c ) 1 / 2 , d ) 5 / 8 , e ) 3 / 4
Rationale: "n ( n + 1 ) ( n + 2 ) will be divisible by 8 when n is a multiple of 2 or when ( n + 1 ) is a multiple of 8 . thus when n is even , this whole expression will be divisible by 8 . from 1 to 84 , there are 42 even integers . now when ( n + 1 ) is multiple by 8 , we have 12 such values for ( n + 1 ) probability that n ( n + 1 ) ( n + 2 ) will be divisible by 8 = ( 42 + 12 ) / 84 = 54 / 84 = 9 / 14 ans is a"
Answer: a
Question (general): an auto assembly plant performs 6 functions with each frame that arrives : add axles , add wheels to the axles , install the windshield to the frame , install the instrument panel , install the steering wheel , and install the interior seating . once those 6 tasks are performed , each car goes to a separate building for finishing touches . if these tasks can be arranged along a linear assembly line in any order , except that the axles must be installed before the wheels can be added , how many y ways can the assembly line be arranged ?
Options: a ) 120 , b ) 240 , c ) 360 , d ) 480 , e ) 720
Rationale: c ) 360 short way : there are 6 c ! ways to do the six tasks . half will have wheels before axles and half will have axles before wheels . so we want y = 6 c ! / 2 - > 720 / 2 = 360
Answer: c