Cheukting/mathqa huggingface dataset:
Question (physics): a , b , k start from the same place and travel in the same direction at speeds of 30 km / hr , 40 km / hr , 60 km / hr respectively . b starts 6 hours after a . if b and k overtake a at the same instant , how many hours after a did k start ?
Options: a ) 3 , b ) 4.5 , c ) 4 , d ) d ) 12 , e ) e ) 5
Rationale: the table you made does n ' t make sense to me . all three meet at the same point means the distance they cover is the same . we know their rates are 30 , 40 and 60 . say the time taken by b is t hrs . then a takes 6 + t hrs . and we need to find the time taken by k . distance covered by a = distance covered by b 30 * ( 6 + t ) = 40 * t t = 18 hrs distance covered by b = distance covered by k 40 * t = 60 * time taken by k time taken by k = 40 * 18 / 60 = 12 hrs time taken by a = 6 + t = 6 + 18 = 24 hrs time taken by k = 12 hrs so k starts 24 - 12 = 12 hrs after a . ( answer d )
Answer: d
Question (general): in an exam 80 % of the boys and 40 % of the girls passed . the number of girls who passed is 120 , which is 2 / 3 rd of the number of boys who failed . what is the total number of students who appeared for the exam ?
Options: a ) 1200 , b ) 380 , c ) 3800 , d ) 2180 , e ) 3180
Rationale: let the number of boys = x , number of girls = y 40 y / 100 = 120 y = 300 120 = 2 / 3 * 20 x / 100 = 2 x / 15 x = 900 total = x + y = 300 + 900 = 1200 answer : a
Answer: a
Question (general): when 4 is added to half of one - third of one - fifth of a number , the result is one - fifteenth of the number . find the number ?
Options: a ) 32 , b ) 81 , c ) 60 , d ) 120 , e ) 11
Rationale: "explanation : let the number be 4 + 1 / 2 [ 1 / 3 ( a / 5 ) ] = a / 15 = > 4 = a / 30 = > a = 120 answer : d"
Answer: d
Question (general): a certain lab experiments with white and brown mice only . in one experiment , 2 / 3 of the mice are white . if there are 15 brown mice in the experiment , how many mice in total are in the experiment ?
Options: a ) 45 , b ) 33 , c ) 26 , d ) 21 , e ) 10
Rationale: "let total number of mice = m number of white mice = 2 / 3 m number of brown mice = 1 / 3 m = 15 = > m = 45 answer a"
Answer: a
Question (gain): a merchant sells an item at a 20 % discount , but still makes a gross profit of 30 percent of the cost . what percent of the cost would the gross profit on the item have been if it had been sold without the discount ?
Options: a ) 20 % , b ) 40 % , c ) 50 % , d ) 62.5 % , e ) 75 %
Rationale: "original sp = x cost = c current selling price = . 8 x ( 20 % discount ) . 8 x = 1.3 c ( 30 % profit ) x = 1.3 / . 8 * c x = 13 / 8 c original selling price is 1.625 c which is 62.5 % profit answer d"
Answer: d
Question (physics): pipe a fills a tank in 42 minutes . pipe b can fill the same tank 6 times as fast as pipe a . if both the pipes are kept open when the tank is empty , how many minutes will it take to fill the tank ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "a ' s rate is 1 / 42 and b ' s rate is 1 / 7 . the combined rate is 1 / 42 + 1 / 7 = 1 / 6 the pipes will fill the tank in 6 minutes . the answer is d ."
Answer: d
Question (gain): by weight , liquid x makes up 1.5 percent of solution p and 6.5 percent of solution q . if 200 grams of solution p are mixed with 800 grams of solution q , then liquid x accounts for what percent of the weight of the resulting solution ?
Options: a ) 4.5 % , b ) 5.0 % , c ) 5.5 % , d ) 5.8 % , e ) 6.0 %
Rationale: "the number of grams of liquid x is 1.5 ( 200 ) / 100 + 6.5 ( 800 ) / 100 = 3 + 52 = 55 grams . 55 / 1000 = 5.5 % the answer is c ."
Answer: c
Question (physics): 900 men have provisions for 15 days . if 200 more men join them , for how many days will the provisions last now ?
Options: a ) 12.88 , b ) 12.6 , c ) 12.55 , d ) 12.21 , e ) 12.27
Rationale: "900 * 15 = 1100 * x x = 12.27 . answer : e"
Answer: e
Question (general): set a { 3 , 3,3 , 4,5 , 5,5 } has a standard deviation of 1 . what will the standard deviation be if every number in the set is multiplied by 2 ?
Options: a ) a ) 1 , b ) b ) 2 , c ) c ) 4 , d ) d ) 8 , e ) e ) 16
Rationale: points to remember - 1 . if oneadd / subtractthe same amont from every term in a set , sd does n ' t change . 2 . if onemultiply / divideevery term by the same number in a set , sd changes by same number . hence the answer to the above question is b
Answer: b
Question (physics): a snail , climbing a 24 feet high wall , climbs up 4 feet on the first day but slides down 2 feet on the second . it climbs 4 feet on the third day and slides down again 2 feet on the fourth day . if this pattern continues , how many days will it take the snail to reach the top of the wall ?
Options: a ) 12 , b ) 16 , c ) 17 , d ) 20 , e ) 21
Rationale: total transaction in two days = 4 - 2 = 2 feet in 20 days it will climb 20 feet on the 21 st day , the snail will climb 4 feet , thus reaching the top therefore , total no of days required = 21 e
Answer: e
Question (general): a ’ s speed is 25 / 18 times that of b . if a and b run a race , what part of the length of the race should a give b as a head start , so that the race ends in a dead heat ?
Options: a ) 1 / 18 , b ) 7 / 18 , c ) 7 / 25 , d ) 3 / 25 , e ) 1 / 25
Rationale: "we have the ratio of a ’ s speed and b ’ s speed . this means , we know how much distance a covers compared with b in the same time . this is what the beginning of the race will look like : ( start ) a _________ b ______________________________ if a covers 25 meters , b covers 18 meters in that time . so if the race is 25 meters long , when a reaches the finish line , b would be 7 meters behind him . if we want the race to end in a dead heat , we want b to be at the finish line too at the same time . this means b should get a head start of 7 meters so that he doesn ’ t need to cover that . in that case , the time required by a ( to cover 25 meters ) would be the same as the time required by b ( to cover 18 meters ) to reach the finish line . so b should get a head start of 7 / 25 th of the race . answer ( c )"
Answer: c
Question (gain): the marked price of a book is 20 % more than the cost price . after the book is sold , the vendor realizes that he had wrongly raised the cost price by a margin of 25 % . if the marked price of the book is rs . 30 , what is the original cost price of the book ?
Options: a ) rs . 30 , b ) rs . 25 , c ) rs . 45 , d ) rs . 20 , e ) rs . 10
Rationale: let the incorrect cost price be c 1 and let the original cost price be c 2 . marked price of book is rs . 30 . it is 20 % more than c 1 . therefore , ( 120 / 100 ) x c 1 = 30 or c 1 = 25 . c 1 is more than c 2 margin of 25 % . or c 1 = ( 125 / 100 ) c 2 therefore , c 2 = ( 100 / 125 ) x 25 = rs 20 answer : d
Answer: d
Question (other): the h . c . f . of two numbers is 18 and the other two factors of their l . c . m . are 11 and 15 . the larger of the two numbers is :
Options: a ) 270 , b ) 300 , c ) 299 , d ) 322 , e ) 345
Rationale: "the numbers are ( 18 x 11 ) and ( 18 x 15 ) . larger number = ( 18 x 15 ) = 270 . answer : a"
Answer: a
Question (other): the ratio of buses to cars on river road is 1 to 3 . if there are 20 fewer buses than cars on river road , how many cars are on river road ?
Options: a ) 100 , b ) 120 , c ) 140 , d ) 30 , e ) 150
Rationale: "b / c = 1 / 3 c - b = 20 . . . . . . . . . > b = c - 20 ( c - 20 ) / c = 1 / 3 testing answers . clearly eliminate abce put c = 30 . . . . . . . . . > ( 30 - 20 ) / 30 = 10 / 30 = 1 / 3 answer : d"
Answer: d
Question (physics): speed of a boat in standing water is 10 kmph and speed of the stream is 2.5 kmph . a man can rows to a place at a distance of 105 km and comes back to the starting point . the total time taken by him is ?
Options: a ) 23.4 hours , b ) 22.4 hours , c ) 21.4 hours , d ) 20.4 hours , e ) 19.4 hours
Rationale: "speed upstream = 7.5 kmph speed downstream = 12.5 kmph total time taken = 105 / 7.5 + 105 / 12.5 = 22.4 hours answer is b"
Answer: b
Question (general): the total of 324 of 20 paise and 25 paise make a sum of rs . 71 . the no of 20 paise coins is
Options: a ) 238 , b ) 277 , c ) 278 , d ) 200 , e ) 288
Rationale: "explanation : let the number of 20 paise coins be x . then the no of 25 paise coins = ( 324 - x ) . 0.20 * ( x ) + 0.25 ( 324 - x ) = 71 = > x = 200 . . answer : d ) 200"
Answer: d
Question (gain): if annual decrease in the population of a town is 5 % and the present number of people is 40000 what will the population be in 2 years ?
Options: a ) 24560 , b ) 26450 , c ) 36100 , d ) 38920 , e ) 45200
Rationale: population in 2 years = 40000 ( 1 - 5 / 100 ) ^ 2 = 40000 * 19 * 19 / 20 * 20 = 36100 answer is c
Answer: c
Question (general): the price of lunch for 15 people was $ 206.00 , including a 15 percent gratuity for service . what was the average price per person , excluding the gratuity ?
Options: a ) $ 11.73 , b ) $ 12.48 , c ) $ 13.80 , d ) $ 14.00 , e ) $ 15.87
Rationale: take the initial price before the gratuity is 100 the gratuity is calculated on the final price , so as we assumed the final bill before adding gratuity is 100 so gratuity is 15 % of 100 is 15 so the total price of meals is 115 so the given amount i . e 206 is for 115 then we have to calculate for 100 for 115 206 for 100 x so by cross multiplication we get 115 x = 100 * 206 = > x = 100 * 206 / 110 by simplifying we get x as 187.27 which is the price of lunch before gratuity so the gratuity is 18.73 so as the question ask the average price person excluding gratuity is 187.27 / 15 = 12.48 so our answer is b )
Answer: b
Question (physics): a man can do a piece of work in 5 days , but with the help of his son , he can finish it in 3 days . in what time can the son do it alone ?
Options: a ) 5 , b ) 5 1 / 2 , c ) 7 1 / 2 , d ) 6 , e ) 9 1 / 2
Rationale: "son ' s 1 day work = 1 / 3 - 1 / 5 = 2 / 15 son alone can do the work in 15 / 2 days = 7 1 / 2 days answer is c"
Answer: c
Question (physics): joe drives 600 miles at 60 miles per hour , and then he drives the next 120 miles at 40 miles per hour . what is his average speed for the entire trip in miles per hour ?
Options: a ) 42 , b ) 55 , c ) 50 , d ) 54 , e ) 56
Rationale: "t 1 = 600 / 60 = 10 hours t 2 = 120 / 40 = 3 hours t = t 1 + t 2 = 13 hours avg speed = total distance / t = 720 / 13 = 55 mph = b"
Answer: b
Question (gain): a pet store regularly sells pet food at a discount of 10 percent to 30 percent from the manufacturer ’ s suggested retail price . if during a sale , the store discounts an additional 20 percent from the discount price , what would be the lowest possible price of a container of pet food that had a manufacturer ’ s suggested retail price o f $ 20.00 ?
Options: a ) $ 10.00 , b ) $ 11.20 , c ) $ 14.40 , d ) $ 16.00 , e ) $ 18.00
Rationale: "for retail price = $ 20 first maximum discounted price = 20 - 30 % of 20 = 20 - 6 = 14 price after additional discount of 20 % = 14 - 20 % of 14 = 14 - 2.8 = 11.2 answer : option b"
Answer: b
Question (general): if 20 liters of chemical x are added to 80 liters of a mixture that is 15 % chemical x and 85 % chemical y , then what percentage of the resulting mixture is chemical x ?
Options: a ) 30 % , b ) 32 % , c ) 35 % , d ) 38 % , e ) 40 %
Rationale: "the amount of chemical x in the solution is 20 + 0.15 ( 80 ) = 32 liters . 32 liters / 100 liters = 32 % the answer is b ."
Answer: b
Question (general): a set consists of 12 numbers , all are even or multiple of 5 . if 4 numbers are even and 10 numbers are multiple of 5 , how many numbers is multiple of 10 ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 5
Rationale: "{ total } = { even } + { multiple of 5 } - { both } + { nether } . since { neither } = 0 ( allare even or multiple of 5 ) then : 12 = 4 + 10 - { both } + 0 ; { both } = 2 ( so 1 number is both even and multiple of 5 , so it must be a multiple of 10 ) . answer : c ."
Answer: c
Question (general): two employees x and y are paid a total of rs . 506 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 . 250 , b ) s . 280 , c ) s . 290 , d ) s . 299 , e ) s . 230
Rationale: "let the amount paid to x per week = x and the amount paid to y per week = y then x + y = 506 but x = 120 % of y = 120 y / 100 = 12 y / 10 ∴ 12 y / 10 + y = 506 ⇒ y [ 12 / 10 + 1 ] = 506 ⇒ 22 y / 10 = 506 ⇒ 22 y = 5060 ⇒ y = 5060 / 22 = 460 / 2 = rs . 230 e"
Answer: e
Question (general): for any integer k greater than 1 , the symbol k * denotes the product of all the fractions of the form 1 / t , where t is an integer between 1 and k , inclusive . what is the value of 3 * / 4 * ?
Options: a ) 5 , b ) 5 / 4 , c ) 4 / 5 , d ) 1 / 4 , e ) 1 / 5
Rationale: "when dealing with ' symbolism ' questions , it often helps to ' play with ' the symbol for a few moments before you attempt to answer the question that ' s asked . by understanding how the symbol ' works ' , you should be able to do the latter calculations faster . here , we ' re told that k * is the product of all the fractions of the form 1 / t , where t is an integer between 1 and k , inclusive . based on this definition . . . . if . . . . k = 2 k * = ( 1 / 1 ) ( 1 / 2 ) = 1 / 2 if . . . . k = 3 k * = ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) = 1 / 6 we ' re asked to find the value of 5 * / 4 * now that we know how the symbol ' works ' , solving this problem should n ' t be too difficult . you can actually choose to do the math in a couple of different ways . . . . 5 * = ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) ( 1 / 5 ) do n ' t calculate this just yet though . . . . since we ' re dividing by 4 * , many of those fractions will ' cancel out . ' 4 * = ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) we ' re looking for the value of : ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) ( 1 / 5 ) / ( 1 / 1 ) ( 1 / 2 ) ( 1 / 3 ) ( 1 / 4 ) since the first four fraction in the numerator and denominator cancel out , we ' re left with just one fraction : 5 / 4 b"
Answer: b
Question (gain): find the missing figures : ? % of 25 = 20125
Options: a ) 4.5 , b ) 6.5 , c ) 2.5 , d ) 7.5 , e ) 8.5
Rationale: "let x % of 25 = 2.125 . then , ( x / 100 ) * 25 = 2.125 x = ( 2.125 * 4 ) = 8.5 . answer is e ."
Answer: e
Question (general): the average age of 15 students of a class is 15 years . out of these , the average age of 4 students is 14 years and that of the other 10 students is 16 years . the age of the 15 th student is
Options: a ) 9 years , b ) 11 years , c ) 14 years , d ) 21 years , e ) 9 years
Rationale: "solution age of the 15 th student = [ 15 x 15 - ( 14 x 4 + 16 x 10 ) ] = ( 225 - 216 ) = 9 years . answer e"
Answer: e
Question (general): a certain sum of money is divided among a , b and c so that for each rs . a has , b has 65 paisa and c 40 paisa . if c ' s share is rs . 40 , find the sum of money ?
Options: a ) 288 , b ) 262 , c ) 72 , d ) 205 , e ) 267
Rationale: "a : b : c = 100 : 65 : 40 = 20 : 13 : 8 8 - - - - 40 41 - - - - ? = > rs . 205 answer : d"
Answer: d
Question (general): two trains leave the train station at the same time . one train , on the blue line , heads east - while the other , on the red line , heads west . if the train on the blue line averages 40 km / hr and the other train averages 40 km / hr - how long will it take for the trains to be 100 km apart ?
Options: a ) 2 hours , b ) 2.25 hours , c ) 1 hour , d ) 1.25 hours , e ) not enough information
Rationale: each train is averaging 40 km / hour in an opposite direction . after 1 hour , they will be 80 km apart , and after 1.25 hours , they will be 100 km apart . ( 80 * 1.25 = 100 ) answer is d
Answer: d
Question (gain): a certain college ' s enrollment at the beginning of 1992 was 30 percent greater than it was at the beginning of 1991 , and its enrollment at the beginning of 1993 was 10 percent greater than it was at the beginning of 1992 . the college ' s enrollment at the beginning of 1993 was what percent greater than its enrollment at the beginning of 1991 ?
Options: a ) 43 % , b ) 45 % , c ) 50 % , d ) 35 % , e ) 38 %
Rationale: "suppose enrollment in 1991 was 100 then enrollment in 1992 will be 130 and enrollment in 1993 will be 130 * 1.1 = 143 increase in 1993 from 1991 = 143 - 100 = 43 answer : a"
Answer: a
Question (physics): what is the greatest possible length which can be used to measure exactly the lengths 10 m 50 cm , 14 m 55 cm and 50 cm ?
Options: a ) 20 cm , b ) 24 cm , c ) 30 cm , d ) 10 cm , e ) 5 cm
Rationale: "required length = hcf of 1050 cm , 1455 cm , 50 cm = 5 cm answer is e"
Answer: e
Question (general): for any integer p , * p is equal to the product of all the integers between 1 and p , inclusive . how many prime numbers are there between * 9 + 3 and * 9 + 9 , inclusive ?
Options: a ) none , b ) one , c ) two , d ) three , e ) four
Rationale: "generally * p or p ! will be divisible by all numbers from 1 to p . therefore , * 9 would be divisible by all numbers from 1 to 9 . = > * 9 + 3 would give me a number which is a multiple of 3 and therefore divisible ( since * 9 is divisible by 3 ) in fact adding anyprimenumber between 1 to 9 to * 9 will definitely be divisible . so the answer is none ( a ) ! supposing if the question had asked for prime numbers between * 9 + 3 and * 9 + 11 then the answer would be 1 . for * 9 + 3 and * 9 + 13 , it is 2 and so on . . . a"
Answer: a
Question (physics): a can give b 120 meters start and c 200 meters start in a kilometer race . how much start can b give c in a kilometer race ?
Options: a ) 90.09 meters , b ) 111.12 meters , c ) 112.12 meters , d ) 113.12 meters , e ) none of these
Rationale: "explanation : a runs 1000 meters while b runs 880 meters and c runs 800 meters . therefore , b runs 880 meters while c runs 800 meters . so , the number of meters that c runs when b runs 1000 meters = ( 1000 x 800 ) / 880 = 909.09 meters thus , b can give c ( 1000 - 909.09 ) = 90.09 meters start answer : a"
Answer: a
Question (physics): convert 40 miles into yards ?
Options: a ) 25630 yards , b ) 35200 yards , c ) 39520 yards , d ) 42560 yards , e ) 70400 yards
Rationale: "1 mile = 1760 yards 40 miles = 40 * 1760 = 70400 yards answer is e"
Answer: e
Question (gain): a merchant marks his goods up by 30 % and then offers a discount of 20 % on the marked price . what % profit does the merchant make after the discount ?
Options: a ) 8 % , b ) 10 % , c ) 21 % , d ) 15 % , e ) 4 %
Rationale: "let the price be 100 . the price becomes 130 after a 30 % markup . now a discount of 20 % on 130 . profit = 104 - 100 4 % answer e"
Answer: e
Question (gain): the owner of a furniture shop charges his customer 42 % more than the cost price . if a customer paid rs . 8300 for a computer table , then what was the cost price of the computer table ?
Options: a ) rs . 5725 , b ) rs . 5845 , c ) rs . 6275 , d ) rs . 6725 , e ) none of these
Rationale: cp = sp * ( 100 / ( 100 + profit % ) ) = 8300 ( 100 / 142 ) = rs . 5845 . answer : b
Answer: b
Question (gain): an article with cost price of 320 is sold at 18 % profit . what is the selling price ?
Options: a ) 198 , b ) 200 , c ) 204 , d ) 378 , e ) 347
Rationale: "sp = 1.18 * 320 = 378 answer : d"
Answer: d
Question (general): if it takes a tub 5 minutes to drain 5 / 7 of its content , how much more time will it take for the tub to be empty ?
Options: a ) 48 seconds , b ) 1 minute , 12 seconds , c ) 1 minute , 50 seconds , d ) 2 minutes , 00 seconds , e ) 4 minutes , 12 seconds
Rationale: "if 5 / 7 of tub ' s content is drained 2 / 7 th of tub still needs to be drained . if it takes 5 minutes to drain 5 / 7 th of tub it takes 5 * ( 7 / 5 ) minutes to drain the entire tub and 5 * ( 7 / 5 ) * ( 2 / 7 ) min to drain 2 / 7 th of the tub which is 2 minutes so answer is d"
Answer: d
Question (general): a collection of books went on sale , and 2 / 3 of them were sold for $ 3.25 each . if none of the 20 remaining books were sold , what was the total amount received for the books that were sold ?
Options: a ) $ 130 , b ) $ 185 , c ) $ 175 , d ) $ 165 , e ) $ 155
Rationale: "if 20 books constitute 1 / 3 rd of the total , then 2 / 3 rd of the total = 40 books amount received for sold books = 40 * 3.25 = $ 130 answer : a"
Answer: a
Question (physics): there are two cars . one is 160 miles north of the other . simultaneously , the car to the north is driven westward at 10 miles per hour and the other car is driven eastward at 30 miles per hour . how many miles apart are the cars after 3 hours ?
Options: a ) 180 , b ) 200 , c ) 220 , d ) 240 , e ) 250
Rationale: "here , drawing a quick sketch of the ' actions ' described will end in a diagonal line that you canbuilda right triangle around : the right triangle will have a base of 120 and a height of 160 . the hidden pattern here is a 3 / 4 / 5 right triangle ( the 120 lines up with the ' 3 ' and the 160 lines up with the ' 4 ' ) . in this way , you can deduce that each side is ' 40 times ' bigger than it ' s corresponding side : 3 / 4 / 5 becomes 120 / 160 / 200 thus the distance between the two cars is the length of the hypotenuse of this larger right triangle . . . final answer : b"
Answer: b
Question (general): a batsman scored 120 runs which included 5 boundaries and 8 sixes . what % of his total score did he make by running between the wickets
Options: a ) 43.33 % , b ) 50 % , c ) 65 % , d ) 70 % , e ) 75 %
Rationale: "number of runs made by running = 120 - ( 5 x 4 + 8 x 6 ) = 120 - ( 68 ) = 52 now , we need to calculate 60 is what percent of 120 . = > 52 / 120 * 100 = 43.33 % a"
Answer: a
Question (general): given that 100.48 = x , 100.70 = y and xz = y 2 , then the value of z is close to :
Options: a ) 2.2 , b ) 8.2 , c ) 2.9 , d ) 2.1 , e ) 2.6
Rationale: "xz = y 2 10 ( 0.48 z ) = 10 ( 2 x 0.70 ) = 101.40 0.48 z = 1.40 z = 140 = 35 = 2.9 ( approx . ) 48 12 answer : c"
Answer: c
Question (general): how many digits are required to number a book containing 250 pages ?
Options: a ) 756 , b ) 642 , c ) 492 , d ) 372 , e ) 250
Rationale: "9 pages from 1 to 9 will require 9 digits . 90 pages from 10 to 99 will require 90 * 2 = 180 digits . 250 - ( 90 + 9 ) = 151 pages will require 151 * 3 = 453 digits . the total number of digits is 9 + 180 + 453 = 642 . the answer is b ."
Answer: b
Question (general): at scratch and dents rent - a - car , it costs $ 34.95 a day plus $ 0.23 per mile to rent a car . at rent - a - lemon , the charge is $ 25.00 a day plus $ 1.31 per mile . if you need to rent a car for 3 days , how many miles ( to nearest tenth ) must you drive for a car from both agencies to cost the same amount ?
Options: a ) 25.7 miles , b ) 26.2 miles , c ) 27.6 miles , d ) 27.9 miles , e ) 29.9 miles
Rationale: for sad : saddaily = $ 34.95 / day sadmile = $ 0.23 / mile for ral : raldaily = $ 25.00 / day ralmile = $ 1.31 / mile we want the raltotal = sadtotal , so we get ( raldaily * days ) + ( ralmile * miles ) = ( saddaily * days ) + ( sadmile * miles ) = > miles = ( ( saddaily * days ) - ( raldaily * days ) ) / ( ralmiles - sadmiles ) = ( ( saddaily - raldaily ) * days ) / ( ralmiles - sadmiles ) miles = ( ( $ 34.95 * 3 ) - ( $ 25.00 * 3 ) ) / ( $ 1.31 - $ 0.23 ) = 27.6 miles c . 27.6 miles
Answer: c
Question (physics): pipe a can fill a tank in 9 hours . due to a leak at the bottom , it takes 12 hours for the pipe a to fill the tank . in what time can the leak alone empty the full tank ?
Options: a ) 36 , b ) 88 , c ) 18 , d ) 26 , e ) 12
Rationale: "let the leak can empty the full tank in x hours 1 / 9 - 1 / x = 1 / 12 = > 1 / x = 1 / 9 - 1 / 12 = 1 / 12 = > x = 36 . answer : a"
Answer: a
Question (general): if p is a prime number greater than 3 , find the remainder when p ^ 2 + 16 is divided by 12 .
Options: a ) 5 , b ) 1 , c ) 0 , d ) 8 , e ) 7
Rationale: every prime number greater than 3 can be written 6 n + 1 or 6 n - 1 . if p = 6 n + 1 , then p ^ 2 + 16 = 36 n ^ 2 + 12 n + 1 + 16 = 36 n ^ 2 + 12 n + 12 + 5 if p = 6 n - 1 , then p ^ 2 + 16 = 36 n ^ 2 - 12 n + 1 + 16 = 36 n ^ 2 - 12 n + 12 + 5 when divided by 12 , it must leave a remainder of 5 . the answer is a .
Answer: a