Cheukting/mathqa huggingface dataset:
Question (general): a basketball is dropped from a height of 40 feet . if it bounces back up to a height that is exactly half of its previous height , and it stops bouncing after hitting the ground for the fourth time , then how many total feet will the ball have traveled after 3 full bounces .
Options: a ) 50 , b ) 55 , c ) 110 , d ) 75 , e ) 80
Rationale: "initial distance = 40 feet first bounce = 20 feet up + 20 feet down = 40 feet second bouche = 10 feet up + 10 feet down = 20 feet third bounce = 5 feet up and 5 feet down = 10 feet total distance covered = 40 + 40 + 20 + 10 = 110 answer is c"
Answer: c
Question (general): 80 370 860 1550 ? 3530
Options: a ) 900 , b ) 2440 , c ) 750 , d ) 244 , e ) 960
Rationale: "10 ^ 2 - 20 = 80 20 ^ 2 - 30 = 370 30 ^ 2 - 40 = 860 40 ^ 2 - 50 = 1550 50 ^ 2 - 60 = 2440 60 ^ 2 - 70 = 3530 . answer : b"
Answer: b
Question (gain): a sum fetched a total simple interest of 4052.25 at the rate of 9 % . p . a . in 5 years . what is the sum ?
Options: a ) 9005 , b ) 8925 , c ) 2345 , d ) 6474 , e ) 8723
Rationale: principal = ( 100 x 4052.25 ) / ( 9 x 5 ) = 405225 / 45 = 9005 . answer a
Answer: a
Question (physics): a rotameter is a device that measures flow of liquid and gases . when measuring liquid phase flows , 2.5 inches represent 60 liters per minute of liquid . with gas measurements the rotameter moves 50 % of the movement he moves with the liquid phase . how many liters of gas passed through the rotameter if it measured 4 inches ?
Options: a ) 176 , b ) 192 , c ) 202 , d ) 218 , e ) 284
Rationale: in case of liquid - 2.5 inches represents 60 lit / min . in case of gas - 50 % of 2.5 inches represents 60 lit / min 1.25 inches represents 60 lit / min 4 inches will represent 60 * 4 / 1.25 = 192 b is the answer
Answer: b
Question (gain): if a trader sold two cars each at rs . 325475 and gains 10 % on the first and loses 10 % on the second , then his profit or loss percent on the whole is ?
Options: a ) 1.44 % , b ) 1.74 % , c ) 1.84 % , d ) 1.47 % , e ) 1.00 %
Rationale: "sp of each car is rs . 325475 , he gains 10 % on first car and losses 10 % on second car . in this case , there will be loss and percentage of loss is given by = [ ( profit % ) ( loss % ) ] / 100 = ( 10 ) ( 10 ) / 100 % = 1.00 % answer : e"
Answer: e
Question (general): how many positive integers less than 130 are there such that they are multiples of 13 or multiples of 12 but not both ?
Options: a ) 16 , b ) 18 , c ) 20 , d ) 24 , e ) 28
Rationale: "for 13 : 13 . . . 130 = 13 * 10 for 12 : 12 . . . 130 = 12 * 10 but there is one integer 13 * 12 . so n = ( 10 ) + ( 10 ) = 20 c"
Answer: c
Question (other): the marks obtained by polly and sandy are in the ratio 5 : 6 and those obtained by sandy and willy are in the ratio of 3 : 2 . the marks obtained by polly and willy are in the ratio of . . . ?
Options: a ) 3 : 2 , b ) 5 : 4 , c ) 7 : 6 , d ) 9 : 8 , e ) 11 : 10
Rationale: "polly : sandy = 5 : 6 sandy : willy = 3 : 2 = 6 : 4 polly : sandy : willy = 5 : 6 : 4 polly : willy = 5 : 4 the answer is b ."
Answer: b
Question (general): what is the greatest positive integer x such that 3 ^ x is a factor of 9 ^ 7 ?
Options: a ) 5 , b ) 14 , c ) 10 , d ) 20 , e ) 30
Rationale: what is the greatest positive integer x such that 3 ^ x is a factor of 9 ^ 7 ? 9 ^ 7 = ( 3 ^ 2 ) ^ 7 = 3 ^ 14 b . 14
Answer: b
Question (other): if 70 percent of a class answered the first question on a certain test correctly , 55 percent answered the second question on the test correctly , and 20 percent answered neither of the questions correctly , what percent answered both correctly ?
Options: a ) 45 % , b ) 40 % , c ) 35 % , d ) 30 % , e ) 25 %
Rationale: "70 % answered the first question correctly and 20 % answered neither correctly . then 10 % missed the first question but answered the second question correctly . then the percent who answered both correctly is 55 % - 10 % = 45 % . the answer is a ."
Answer: a
Question (general): each machine of type a has 3 steel parts and 2 chrome parts . each machine of type b has 2 steel parts and 4 chrome parts . if a certain group of type a and type b machines has a total of 50 steel parts and 66 chrome parts , how many machines are in the group
Options: a ) 11 , b ) 21 , c ) 31 , d ) 61 , e ) 51
Rationale: "look at the below representation of the problem : steel chrome total a 3 2 50 > > no . of type a machines = 50 / 5 = 10 b 2 4 66 > > no . of type b machines = 66 / 6 = 11 so the answer is 21 i . e b . hope its clear ."
Answer: b
Question (general): a batsman in his 12 th innings makes a score of 75 and thereby increases his average by 1 runs . what is his average after the 12 th innings if he had never been ‘ not out ’ ?
Options: a ) 42 , b ) 43 , c ) 44 , d ) 64 , e ) 46
Rationale: "let ‘ x ’ be the average score after 12 th innings ⇒ 12 x = 11 × ( x – 1 ) + 75 ∴ x = 64 answer d"
Answer: d
Question (general): students at a school were on average 180 cm tall . the average female height was 170 cm , and the average male height was 188 cms . what was the ratio of men to women ?
Options: a ) 5 : 2 , b ) 5 : 1 , c ) 4 : 3 , d ) 5 : 4 , e ) 3 : 1
Rationale: "we ' re given a few facts to work with : 1 ) the average height of the females is 170 cm 2 ) the average height of the males is 188 cm 3 ) the average of the group is 180 cm we ' re asked for the ratio of men to women . w = number of women m = number of men ( 170 w + 188 m ) / ( w + m ) = 180 170 w + 188 m = 180 w + 180 m 8 m = 10 w 4 m = 5 w m / w = 5 / 4 the ratio of men to women is 5 t ổ 4 . d"
Answer: d
Question (general): the average ( arithmetic mean ) of the integers from 100 to 200 , inclusive , is how much lesser than the average of the integers from 150 to 950 , inclusive ?
Options: a ) 350 , b ) 475 , c ) 300 , d ) 425 , e ) 400
Rationale: "for an ap the mean or average of series is average of first and last term . so , average of numbers between 150 to 950 , inclusive = ( 150 + 950 ) / 2 = 550 average of numbers between 100 to 200 , inclusive = ( 100 + 200 ) / 2 = 150 difference = 550 - 150 = 400 answer is e"
Answer: e
Question (general): the present ages of 3 persons are in the proportion of 4 : 7 : 9 . 8 years ago , the sum of their ages was 116 . find their present ages .
Options: a ) 20 , 35,45 , b ) 28 , 49,63 , c ) 16 , 28,36 , d ) 16 , 28,46 , e ) none of these
Rationale: let the present ages of three persons be 4 k , 7 k and 9 k respectively . ( 4 k - 8 ) + ( 7 k - 8 ) + ( 9 k - 8 ) = 116 20 k = 140 k = 7 therefore , then present ages are 28 , 49,63 . answer : b
Answer: b
Question (physics): a car traveling at a certain constant speed takes 5 seconds longer to travel 1 km than it would take to travel 1 km at 90 km / hour . at what speed , in km / hr , is the car traveling ?
Options: a ) 70 , b ) 80 , c ) 74 , d ) 75 , e ) 78
Rationale: "time to cover 1 kilometer at 80 kilometers per hour is 1 / 90 hours = 3,600 / 90 seconds = 40 seconds ; time to cover 1 kilometer at regular speed is 40 + 5 = 45 seconds = 45 / 3,600 hours = 1 / 80 hours ; so , we get that to cover 1 kilometer 1 / 80 hours is needed - - > regular speed 80 kilometers per hour ( rate is a reciprocal of time or rate = distance / time ) . answer : b ."
Answer: b
Question (general): if the price of gasoline increases by 25 % and ron intends to spend only 15 % more on gasoline , by how much percent should he reduce the quantity of gasoline that he buys ?
Options: a ) 6.66 % , b ) 8 % , c ) 10 % , d ) 12 % , e ) 12.5 %
Rationale: "let price of original price of gasoline = 10 $ and let original consumption of gasoline = 10 amount spend on gasoline originally = 10 * 10 = 100 $ the price of gasoline increases by 25 % and ron intends to spend only 15 % more on gasoline increased price of gasoline = 12.5 $ now ron intends to spend on gasoline = 1.15 * 100 = 115 $ consumption of gasoline needs to be = 115 / 12.5 = 1150 / 125 = 46 / 5 = 9.2 reduction in consumption of gasoline = 10 - 9.2 = . 8 % reduction in consumption of gasoline = . 8 / 10 * 100 % = 8 % answer b"
Answer: b
Question (physics): a rower can row 7 km / h in still water . when the river is running at 1 km / h , it takes the rower 1 hour to row to big rock and back . how many kilometers is it to big rock ?
Options: a ) 3.19 , b ) 3.28 , c ) 3.35 , d ) 3.43 , e ) 3.56
Rationale: "let x be the distance to big rock . time = x / 6 + x / 8 = 1 x = 48 / 14 = 3.43 km the answer is d ."
Answer: d
Question (physics): a cistern is normally filled in 4 hours but takes two hours longer to fill because of a leak in its bottom . if the cistern is full , the leak will empty it in ?
Options: a ) 33 , b ) 88 , c ) 40 , d ) 99 , e ) 12
Rationale: "1 / 4 - 1 / x = 1 / 6 x = 12 answer : e"
Answer: e
Question (general): 25.25 / 1500 is equal to :
Options: a ) 1.012526333 , b ) 0.012625333 , c ) 0.125263333 , d ) 0.126253333 , e ) 0.016833333
Rationale: "25.25 / 2000 = 2525 / 200000 = 0.016833333 answer : e"
Answer: e
Question (general): if jake loses 20 pounds , he will weigh twice as much as his sister . together they now weigh 224 pounds . what is jake ’ s present weight , in pounds ?
Options: a ) 131 , b ) 135 , c ) 156 , d ) 177 , e ) 188
Rationale: "lets say j is the weight of jack and s is the wt of his sister . if he loses 20 pounds , he s twice as heavy as his sister . j - 20 = 2 * s also , together they weight 224 pounds j + s = 224 solvong the 2 equation , we get j = 156 pounds ! c"
Answer: c
Question (general): in a certain lottery , the probability that a number between 12 and 20 , inclusive , is drawn is 1 / 6 . if the probability that a number 12 or larger is drawn is 1 / 2 , what is the probability that a number less than or equal to 20 is drawn ?
Options: a ) 1 / 18 , b ) 1 / 6 , c ) 2 / 3 , d ) 1 / 2 , e ) 5 / 6
Rationale: "you can simply use sets concept in this question . the formula total = n ( a ) + n ( b ) - n ( a and b ) is applicable here too . set 1 : number 12 or larger set 2 : number 20 or smaller 1 = p ( set 1 ) + p ( set 2 ) - p ( set 1 and set 2 ) ( combined probability is 1 because every number will be either 12 or moreor 20 or lessor both ) 1 / 2 + p ( set 2 ) - 1 / 6 = 2 / 3 p ( set 2 ) = 2 / 3 answer ( c )"
Answer: c
Question (geometry): if the sides of a rectangle are increased by 15 % , what is the percentage increase in the area ?
Options: a ) 32.25 % , b ) 40 % , c ) 37.50 % , d ) 36.75 % , e ) 32 %
Rationale: "say both sides of the rectangle are equal to 100 ( so consider that we have a square ) . in this case the area is 100 * 100 = 10,000 . the area of new rectangle would be 115 * 115 = 13,225 , which is 32.25 % greater than the old area . answer : a ."
Answer: a
Question (gain): a fashion designer sold a pair of jeans to a retail store for 40 percent more than it cost to manufacture the pair of jeans . a customer bought the pair of jeans for 40 percent more than the retailer paid for them . the price the customer paid was what percent greater than the cost of manufacturing the jeans ?
Options: a ) 65 % , b ) 70 % , c ) 75 % , d ) 85 % , e ) 96 %
Rationale: "find the product of the two increases : ( 14 / 10 ) * ( 14 / 10 ) which is 1.96 and a 96 % increase . e"
Answer: e
Question (physics): if 50 apprentices can finish a job in 4 hours , and 30 journeymen can finish the same job in 5,5 hours , how much of the job should be completed by 10 apprentices and 15 journeymen in one hour ?
Options: a ) 1 / 9 , b ) 29 / 180 , c ) 26 / 143 , d ) 1 / 5 , e ) 31 / 220
Rationale: "50 apprentices can finish the job in 4 hours , thus : 10 apprentices can finish the job in 4 * 5 = 20 hours ; in 1 hour 10 apprentices can finish 1 / 20 of the job . 30 journeymen can finish the same job in 4,5 hours , thus : 15 journeymen can finish the job in 5.5 * 2 = 11 hours ; in 1 hour 15 journeymen can finish 1 / 11 of the job . therefore , in 1 hour 10 apprentices and 15 journeymen can finish 1 / 20 + 1 / 11 = 31 / 220 of the job . answer : e"
Answer: e
Question (physics): calculate the l . c . m of 1 / 5 , 6 / 7 , 5 / 6 , 3 / 5 is :
Options: a ) 15 , b ) 30 , c ) 20 , d ) 40 , e ) 10
Rationale: "required l . c . m = l . c . m . of 1 , 6 , 5 , 3 / h . c . f . of 5 , 7 , 6 , 5 = 30 / 1 = 30 answer is b"
Answer: b
Question (general): find the average of all prime numbers between 1 and 10
Options: a ) 2.5 , b ) 1.35 , c ) 4.25 , d ) 6.12 , e ) 7.13
Rationale: "prime numbers between 1 and 10 are 2,3 , 5,7 required average = ( 2 + 3 + 5 + 7 ) / 4 = 17 / 4 = 4.25 answer is c"
Answer: c
Question (general): the average age of a class of 20 students is 19 years . the average increased by 2 when the teacher ' s age also included . what is the age of the teacher ?
Options: a ) 40 years , b ) 41 years , c ) 42 years , d ) 43 years , e ) 44 years
Rationale: "if age of the teacher was 19 , average would not have changed . since average increased by 2 , age of the teacher = 19 + 22 × 1 = 41 answer : b"
Answer: b
Question (general): there were 35 students in a hostel . due to the admission of 7 new students , ; he expenses of the mess were increased by rs . 42 per day while the average expenditure per head diminished by rs 1 . wbat was the original expenditure of the mess ?
Options: a ) 520 , b ) 720 , c ) 425 , d ) 625 , e ) 420
Rationale: "let the original average expenditure be rs . x . then , 42 ( x - 1 ) - 35 x = 42 7 x = 84 x = 12 . original expenditure = rs . ( 35 x 12 ) = rs . 420 . answer is e ."
Answer: e
Question (general): evaluate : | 7 - 8 ( 3 - 12 ) | - | 5 - 11 | = ?
Options: a ) 40 , b ) 50 , c ) 73 , d ) 70 , e ) 80
Rationale: "according to order of operations , inner brackets first . hence | 7 - 8 ( 3 - 12 ) | - | 5 - 11 | = | 7 - 8 * ( - 9 ) | - | 5 - 11 | according to order of operations , multiplication within absolute value signs ( which may be considered as brackets when it comes to order of operations ) next . hence = | 7 + 72 | - | 5 - 11 | = | 79 | - | - 6 | = 79 - 6 = 73 correct answer c ) 73"
Answer: c
Question (physics): if 14 lions can kill 14 deers in 14 minutes how long will it take 100 lions to kill 100 deers ?
Options: a ) 1 minutes , b ) 14 minute , c ) 100 minutes , d ) 10000 minutes , e ) 1000 minutes
Rationale: "we can try the logic of time and work , our work is to kill the deers so 14 ( lions ) * 14 ( min ) / 14 ( deers ) = 100 ( lions ) * x ( min ) / 100 ( deers ) hence answer is x = 14 answer : b"
Answer: b
Question (physics): in a 100 m race , sam beats john by 3 seconds . on the contrary , if sam allowed john to start 24 m ahead of sam , then sam and john reach the finishing point at the same time . how long does sam take to run the 100 m race ?
Options: a ) 4 seconds , b ) 25.33 seconds , c ) 29 seconds , d ) 21 seconds , e ) 6.25 seconds
Rationale: their difference is 3 second but this difference is 0 if john allows sam to start the race from 24 m ahead . that means jhon was 24 m away from finishing line when they started together . so he will cover 24 m in 3 seconds . so his speed = 24 / 3 = 8 metre / second . so time taken = 100 / 3 = 33.33 seconds . so sam took = 25.33 seconds . correct answer = b
Answer: b
Question (general): if | x - 25 | = 50 what is the sum of all the values of x .
Options: a ) 0 , b ) - 100 , c ) - 50 , d ) 50 , e ) 100
Rationale: there will be two cases x - 25 = 50 and x - 25 = - 50 solve for x = > x = 50 + 25 = > x = 75 or x = - 50 + 25 = > x = - 25 the sum of both values will be 75 + - 25 = 50 answer is d
Answer: d
Question (general): product of two co - prime numbers is 117 . their l . c . m should be :
Options: a ) 1 , b ) 117 , c ) 116 , d ) equal to their h . c . f , e ) can not be calculated
Rationale: "h . c . f of co - prime numbers is 1 . so , l . c . m = 117 / 1 = 117 . answer : b"
Answer: b
Question (gain): the price of a tv was increased by 30 percent . the new price was then increased by 40 percent . a single increase of what percent is equivalent to these two successive increases ?
Options: a ) 82 % , b ) 75 % , c ) 65 % , d ) 50 % , e ) 45 %
Rationale: "consider base price - $ 100 30 % increase = 1.30 * 100 = $ 130 another 40 % increase on new price = 1.4 * 130 = $ 182 so final price of radio - $ 182 therefore a 82 % increase correct option - a"
Answer: a
Question (general): terry is having lunch at a salad bar . there are two types of lettuce to choose from , as well as 3 types of tomatoes , and 4 types of olives . he must also choose whether or not to have one of the two types of soup on the side . if terry has decided to have the salad and soup combo and he picks one type of lettuce , one type of tomato , and one type of olive for his salad , how many total options does he have for his lunch combo ?
Options: a ) 9 , b ) 11 , c ) 24 , d ) 48 , e ) 54
Rationale: terry can pick one lunch salad bar . . . . we can choose lettuce 2 c 1 , tomatoes - 3 c 1 and olives - 4 c 1 . . . two types of soups , if he picks one then 2 c 1 ways . terry decided to have the combo : soup + salad . . . hmmm . . . using the above information , we get ( 2 c 1 * 2 c 1 * 3 c 1 * 4 c 1 ) = 48 . . ans option d is correct answer . .
Answer: d
Question (general): what will be the reminder when ( 31 ^ 31 + 31 ) is divided by 32 ?
Options: a ) 72 , b ) 43 , c ) 91 , d ) 40 , e ) 30
Rationale: "( x ^ n + 1 ) will be divisible by ( x + 1 ) only when n is odd ; ( 31 ^ 31 + 1 ) will be divisible by ( 31 + 1 ) ; ( 31 ^ 31 + 1 ) + 30 when divided by will give 30 as remainder . correct option : e"
Answer: e
Question (general): if m equals the sum of the odd integers from 1 to 111 , inclusive , and t equals the sum of the even integers from 2 to 50 , inclusive , what is the value of m + t ?
Options: a ) 400 , b ) 5550 , c ) 6922 , d ) 3786 , e ) 5553
Rationale: use following formulae for such problems : sum of evenly spaced integers = ( # of integers ) * ( mean of integers ) # of integers = [ ( last - first ) / 2 ] + 1 mean of integers = ( last + first ) / 2 in above problem : # of integers = [ ( 111 - 1 ) / 2 ] + 1 = 56 and [ ( 50 - 2 ) / 2 ] + 1 = 25 mean of integers = ( 111 + 1 ) / 2 = 56 and ( 50 + 2 ) / 2 = 26 sum of integers = ( 56 * 56 ) = 3136 and ( 25 * 26 ) = 650 thus their sum ( m + t ) = 3786 answer : d
Answer: d
Question (probability): a meeting has to be conducted with 6 managers . find the number of ways in which the managers be selected from among 10 managers , if 2 managers will not attend the meeting together ?
Options: a ) 120 , b ) 110 , c ) 105 , d ) 140 , e ) 135
Rationale: "we can either choose all 6 people from 8 manager who have no problems or choose 5 from the 8 and 1 from the 2 managers who have a problem sitting together so 8 c 6 + ( 8 c 5 * 2 c 1 ) this is 28 + 112 = 140 answer : d"
Answer: d
Question (general): the smallest value of n , for which 2 n + 3 is not a prime number , is
Options: a ) 4,5 , b ) 3,4 , c ) 2,6 , d ) 5,6 , e ) 2,3
Rationale: "( 2 ã — 1 + 3 ) = 5 . ( 2 ã — 2 + 3 ) = 7 . ( 2 ã — 3 + 3 ) = 9 . ( 2 ã — 4 + 3 ) = 15 . which is not prime , n = 3,4 . answer : b"
Answer: b
Question (general): the average age of a class of 12 students is 19 years . the average increased by 2 when the teacher ' s age also included . what is the age of the teacher ?
Options: a ) 40 years , b ) 34 years , c ) 42 years , d ) 43 years , e ) 44 years
Rationale: "if age of the teacher was 12 , average would not have changed . since average increased by 2 , age of the teacher = 12 + 22 × 1 = 34 answer : b"
Answer: b
Question (general): a certain basketball team that has played 2 / 3 of its games has a record of 14 wins and 6 losses . what is the greatest number of the remaining games that the team can lose and still win at least 3 / 4 of all of its games ?
Options: a ) 1 , b ) 6 , c ) 5 , d ) 4 , e ) 3
Rationale: "14 wins , 6 losses - total 20 games played . the team has played 2 / 3 rd of all games so total number of games = 30 3 / 4 th of 30 is 22.5 so the team must win 23 games and can afford to lose at most 7 total games . it has already lost 6 games so it can lose another 1 at most . answer ( a )"
Answer: a
Question (physics): a train , 125 meters long travels at a speed of 45 km / hr crosses a bridge in 30 seconds . the length of the bridge is
Options: a ) 250 m , b ) 245 m , c ) 235 m , d ) 220 m , e ) 240 m
Rationale: "explanation : assume the length of the bridge = x meter total distance covered = 125 + x meter total time taken = 30 s speed = total distance covered / total time taken = ( 125 + x ) / 30 m / s = > 45 ã — ( 10 / 36 ) = ( 125 + x ) / 30 = > 45 ã — 10 ã — 30 / 36 = 125 + x = > 45 ã — 10 ã — 10 / 12 = 125 + x = > 15 ã — 10 ã — 10 / 4 = 125 + x = > 15 ã — 25 = 125 + x = 375 = > x = 375 - 125 = 250 answer : option a"
Answer: a
Question (general): in a division sum , the quotient is 36 , the divisor 85 and the remainder 26 , find the dividend ?
Options: a ) 3066 , b ) 3586 , c ) 3086 , d ) 3968 , e ) 3286
Rationale: "explanation : 36 * 85 + 26 = 3086 answer : c"
Answer: c
Question (other): a company conducted a survey about its two brands , a and b . x percent of respondents liked product a , ( x – 20 ) percent liked product b , 23 percent liked both products , and 23 percent liked neither product . what is the minimum number q of people surveyed by the company ?
Options: a ) 46 , b ) q . 80 , c ) q . 90 , d ) q . 100 , e ) 200
Rationale: "100 = x + x - 20 + 23 - 23 x = 60 , so , product a = 60 % , product b = 40 % , both = 23 % , neither = 23 % 23 % of the total no . of people should be an integer . so , a , bc are out . 60 % of d and 40 % of d are both integers . so , d satisfies all conditions . so , answer is d ."
Answer: d
Question (physics): two trains 141 meters and 165 meters in length respectively are running in opposite directions , one at the rate of 80 km and the other at the rate of 65 kmph . in what time will they be completely clear of each other from the moment they meet ?
Options: a ) 7.16 , b ) 7.59 , c ) 7.12 , d ) 7.15 , e ) 7.11
Rationale: "t = ( 141 + 165 ) / ( 80 + 65 ) * 18 / 5 t = 7.59 answer : b"
Answer: b
Question (general): a pharmaceutical company received $ 6 million in royalties on the first $ 20 million in sales of the generic equivalent of one of its products and then $ 9 million in royalties on the next $ 108 million in sales . by approximately what percent did the ratio of royalties to sales decrease from the first $ 20 million in sales to the next $ 108 million in sales ?
Options: a ) 8 % , b ) 15 % , c ) 45 % , d ) 52 % , e ) 72 %
Rationale: "solution : this is a percent decrease problem . we will use the formula : percent change = ( new – old ) / old x 100 to calculate the final answer . we first set up the ratios of royalties to sales . the first ratio will be for the first 20 million in sales , and the second ratio will be for the next 108 million in sales . because all of the sales are in millions , we do not have to express all the trailing zeros in our ratios . first 20 million royalties / sales = 6 / 20 = 3 / 10 next 108 million royalties / sales = 9 / 108 = 1 / 12 because each ratio is not an easy number to use , we can simplify each one by multiplying each by the lcm of the two denominators , which is 60 . keep in mind that we are able to do this only because our answer choices are expressed in percents . first 20 million royalties / sales = ( 6 / 20 ) x 60 = 18 next 108 million royalties / sales = 9 / 108 = ( 1 / 12 ) x 60 = 5 we can plug 18 and 5 into our percent change formula : ( new – old ) / old x 100 [ ( 5 – 18 ) / 18 ] x 100 - 650 / 9 x 100 at this point we can stop and consider the answer choices . since we know that 650 / 9 is just a bit less than ½ , we know that - 650 / 9 x 100 is about a 72 % decrease . answer e ."
Answer: e
Question (physics): one pipe can fill a tank three times as fast as another pipe . if together the two pipes can fill the tank in 86 minutes , then the slower pipe alone will be able to fill the tank in
Options: a ) 81 min , b ) 108 min , c ) 144 min , d ) 192 min , e ) none
Rationale: "solution let the slower pipe alone fill the tank in x minutes . then , faster pipes will fill it in x / 3 minutes . therefore , 1 / x + 3 / x = 1 / 36 ‹ = › 4 / x = 1 / 36 ‹ = › x = 144 min . answer c"
Answer: c