Cheukting/mathqa huggingface dataset:
Question (geometry): a wooden cube whose edge length is 5 inches is composed of smaller cubes with edge lengths of one inch . the outside surface of the large cube is painted red and then it is split up into its smaller cubes . if one cube is randomly selected from the small cubes , what is the probability that the cube will have at least one red face ?
Options: a ) 70.6 % , b ) 74.5 % , c ) 78.4 % , d ) 82.1 % , e ) 86.9 %
Rationale: "there are a total of 5 * 5 * 5 = 125 cubes . all the exterior cubes will have at least one face painted red . the interior is formed by 3 * 3 * 3 = 27 cubes . the number of cubes with at least one side painted red is 125 - 27 = 98 cubes the probability that a cube has at least one side painted red is 98 / 125 which is 78.4 % the answer is c ."
Answer: c
Question (general): in may mrs lee ' s earnings were 60 percent of the lee family ' s total income . in june mrs lee earned 40 percent more than in may . if the rest of the family ' s income was the same both months , then , in june , mrs lee ' s earnings were approximately what percent of the lee family ' s total income ?
Options: a ) 64 % , b ) 68 % , c ) 72 % , d ) 76 % , e ) 80 %
Rationale: "let in may lee family ' s total income = 100 in may mrs lee ' s income = 60 in may rest of the family ' s income = 40 in june mrs lees income = 60 * 140 / 100 = 84 in june total income = 84 + 40 = 124 % of mrs lee ' s income = 84 / 124 = 67.74 ( b )"
Answer: b
Question (general): if bill can buy 3 pairs of jeans and 2 shirts for $ 69 or 2 pairs of jeans and 3 shirts for $ 86 , how much does one shirt cost ?
Options: a ) $ 10 , b ) $ 12 , c ) $ 13.20 , d ) $ 15 , e ) $ 24
Rationale: "3 j + 2 s = 69 2 j + 3 s = 86 - - - - - - - - - - - - - - - - 5 j + 5 s = 155 - - - - ( divide by 5 ) - - - > j + s = 31 3 j + 2 s = j + 2 ( j + s ) = j + 62 = 69 - - - > j = 7 3 * 7 + 2 s = 69 21 + 2 s = 69 2 s = 48 s = 24 answer : e"
Answer: e
Question (gain): if the cost price of 17 articles is same as the selling price of 25 articles . find the gain or loss percentage ?
Options: a ) 30 % gain , b ) 30 % loss , c ) 32 % loss , d ) 40 % loss , e ) 50 % loss
Rationale: "explanation : 17 cp = 25 sp cost price cp = 25 selling price sp = 17 formula = ( sp - cp ) / cp * 100 = ( 17 - 25 ) / 25 * 100 = 32 % loss answer : option c"
Answer: c
Question (general): tough and tricky questions : word problems . if bill ' s salary increased by 16 percent , he would earn $ 812 per month . if instead , his salary were to increase by only 10 percent , how much money r would he earn per month ?
Options: a ) $ 700 , b ) $ 754 , c ) $ 770 , d ) $ 782 , e ) $ 893
Rationale: official solution : ( c ) if bill ’ s salary increased by 16 % he would earn $ 812 . algebraically , this can be written as : $ 812 = 1.16 s , where s is his current salary . then s = $ 812 / 1.16 = $ 700 . now that we know his current salary is $ 700 , we can calculate what his salary would be if it were increased by 10 % . we know that 10 % of $ 700 is $ 70 , so his salary would be : r = $ 700 + $ 70 = $ 770 . the correct answer is choice ( c ) .
Answer: c
Question (physics): a trader sells 85 meters of cloth for rs . 8925 at the profit of rs . 25 per metre of cloth . what is the cost price of one metre of cloth ?
Options: a ) 80 , b ) 28 , c ) 90 , d ) 26 , e ) 11
Rationale: "explanation : sp of 1 m of cloth = 8925 / 85 = rs . 105 cp of 1 m of cloth = sp of 1 m of cloth - profit on 1 m of cloth = rs . 105 - rs . 25 = rs . 80 . answer : a"
Answer: a
Question (physics): an aeroplane covers a certain distance of 480 kmph in 4 hours . to cover the same distance in 4 1 / 3 hours , it must travel at a speed of
Options: a ) 440 , b ) 540 , c ) 443 , d ) 740 , e ) 250
Rationale: "speed of aeroplane = 480 kmph distance travelled in 4 hours = 480 * 4 = 1920 km speed of aeroplane to acver 1920 km in 13 / 3 = 1920 * 3 / 13 = 443 km answer c ."
Answer: c
Question (general): tough and tricky questions : decimals . the value of x is derived by summing a , b , and c and then rounding the result to the tenths place . the value of y is derived by first rounding a , b , and c to the tenths place and then summing the resulting values . if a = 5.45 , b = 2.95 , and c = 3.74 , what is y – x ?
Options: a ) - 0.1 , b ) 0 , c ) 0.05 , d ) 0.2 , e ) 0.3
Rationale: to find x , we first sum a , b , and c , then round to the tenths place . 5.45 + 2.95 + 3.74 = 12.14 , which rounds to 12.2 . to find y , we first round a , b , and c to the tenths place and them sum them . so 5.4 + 2.9 + 3.7 = 12.0 . we are looking for y - x , which gives us 12.2 - 12.0 = 0.2 , or answer choice d .
Answer: d
Question (physics): a train covers a distance of 20 km in 30 min . if it takes 18 sec to pass a telegraph post , then the length of the train is ?
Options: a ) 298 m , b ) 125 m , c ) 120 m , d ) 217 m , e ) 200 m
Rationale: "speed = ( 20 / 30 * 60 ) km / hr = ( 40 * 5 / 18 ) m / sec = 100 / 9 m / sec . length of the train = 100 / 9 * 18 = 200 m . answer : e"
Answer: e
Question (physics): the radius of a wheel is 22.4 cm . what is the distance covered by the wheel in making 500 resolutions ?
Options: a ) 794 m , b ) 704 m , c ) 454 m , d ) 186 m , e ) 176 m
Rationale: "in one resolution , the distance covered by the wheel is its own circumference . distance covered in 500 resolutions . = 500 * 2 * 22 / 7 * 22.4 = 70400 cm = 704 m answer : b"
Answer: b
Question (physics): 12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 16 cm height . what is the diameter of each sphere ?
Options: a ) 4 cm , b ) 6 cm , c ) 8 cm , d ) 10 cm , e ) 12 cm
Rationale: "volume of cylinder = pi * r ^ 2 * h volume of a sphere = 4 * pi * r ^ 3 / 3 12 * 4 * pi * r ^ 3 / 3 = pi * r ^ 2 * h r ^ 3 = r ^ 2 * h / 16 = 64 cm ^ 3 r = 4 cm d = 8 cm the answer is c ."
Answer: c
Question (general): the ratio of the ages of mini and minakshi is 4 : 3 . the sum of their ages is 35 years . the ratio of their ages after 8 years will be
Options: a ) 28 : 23 , b ) 6 : 11 , c ) 5 : 4 , d ) 3 : 5 , e ) 7 : 11
Rationale: let mini ’ s age = 4 x and minakshi ’ s age = 3 x then 4 x + 3 x = 35 x = 5 mini ’ s age = 20 years and minakshi ’ s age = 15 years ratio of their ages after 8 years = ( 20 + 8 ) : ( 15 + 8 ) = 28 : 23 answer : a
Answer: a
Question (gain): one fourth of a solution that was 15 % salt by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight . the second solution was what percent salt by weight ?
Options: a ) 24 % , b ) 34 % , c ) 22 % , d ) 19 % , e ) 8.5 %
Rationale: "consider total solution to be 100 liters and in this case you ' ll have : 75 * 0.15 + 25 * x = 100 * 0.16 - - > x = 0.19 . answer : d ."
Answer: d
Question (general): [ 1 / ( 1 . 2.3 ) ] + [ 1 / ( 2 . 3.4 ) ] + [ 1 / ( 3 . 4.5 ) ] + [ 1 / ( 4 . 5.6 ) ] is equal to
Options: a ) 7 / 30 , b ) 11 / 30 , c ) 13 / 30 , d ) 17 / 30 , e ) none
Rationale: answer given exp . = [ ( 4 x 5 x 6 ) + ( 5 x 6 ) + ( 2 x 6 ) + ( 2 x 3 ) ] / [ ( 2 x 3 x 4 x 5 x 6 ) ] = 168 / ( 24 x 30 ) = 7 / 30 correct option : a
Answer: a
Question (general): a ’ s speed is 21 / 19 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 ) 2 / 21 , b ) 2 / 19 , c ) 1 / 11 , d ) 1 / 7 , e ) 1 / 5
Rationale: let d be the full distance . let x be the fraction of the distance that b runs . let v be the speed at which b runs . the time should be the same for both runners . time = d / ( 21 v / 19 ) = xd / v ( 19 / 21 ) * d / v = x * d / v x = 19 / 21 b should have a head start of 2 / 21 of the full distance . the answer is a .
Answer: a
Question (general): 2.01 ã — 0.001 = ?
Options: a ) 0.201 , b ) 0.000201 , c ) 2.01 , d ) 0.0201 , e ) 0.00201
Rationale: "2.01 ã — 0.001 = ? or , ? = 0.00201 answer e"
Answer: e
Question (general): if there are half as many women as men in a group and an equal number of men and women do not own cars - a group that is 40 % of the total . what fraction of the total is women who own cars ?
Options: a ) 3 ⁄ 20 , b ) 11 ⁄ 60 , c ) 9 ⁄ 40 , d ) 1 ⁄ 5 , e ) 2 ⁄ 15
Rationale: "consider a group of 200 men and 100 women , a total of 300 people . 40 % of them , which is 120 , form a group of people who do n ' t own a car . half of them are men , and the other half are women , more precisely 60 . it means that there are 100 - 60 = 40 women who own a car , and this represents 40 / 300 = 2 / 15 of the total . answer e"
Answer: e
Question (general): the total population of city a and city b is 5000 more than the total population of city b and city c . how much lesser is the population in city c as compared to city a ?
Options: a ) 8500 , b ) 3500 , c ) 4500 , d ) 5000 , e ) 1500
Rationale: ( a + b ) - ( b + c ) = 5000 a - c = 5000 answer : d
Answer: d
Question (geometry): the ratio between the perimeter and the width of a rectangle is 5 : 1 . if the area of the rectangle is 294 sq . cm , what is the length of the rectangle ?
Options: a ) 15 cm , b ) 18 cm , c ) 21 cm , d ) 24 cm , e ) 27 cm
Rationale: "2 l + 2 w = 5 w l = 3 w / 2 w * l = 294 3 w ^ 2 / 2 = 294 w ^ 2 = 196 w = 14 l = 3 ( 14 ) / 2 = 21 the answer is c ."
Answer: c
Question (physics): the length of the bridge , which a train 250 metres long and travelling at 72 km / hr can cross in 30 seconds , is :
Options: a ) 200 m , b ) 225 m , c ) 245 m , d ) 350 m , e ) 240 m
Rationale: "speed = [ 72 x 5 / 18 ] m / sec = 20 m / sec time = 30 sec let the length of bridge be x metres . then , ( 250 + x ) / 30 = 20 = > 250 + x = 600 = > x = 350 m . answer : d"
Answer: d
Question (gain): some of the 50 % solution of acid was removed and this was replaced with an equal amount of 14 % solution of acid . as a result , a 23 % solution of acid was obtained . what fraction of the original solution was replaced ?
Options: a ) 1 / 2 , b ) 1 / 3 , c ) 2 / 3 , d ) 3 / 4 , e ) 4 / 5
Rationale: "let x be the fraction of the original solution that was replaced . 0.5 * ( 1 - x ) + 0.14 ( x ) = 0.23 0.36 x = 0.27 x = 3 / 4 the answer is d ."
Answer: d
Question (physics): if 7 spiders make 7 webs in 7 days , then 1 spider will make 1 web in how many days ?
Options: a ) 7 , b ) 6 , c ) 5 , d ) 4 , e ) 3
Rationale: let the required number days be x . less spiders , more days ( indirect proportion ) less webs , less days ( direct proportion ) spiders 1 : 7 webs 7 : 1 1 x 7 x x = 7 x 1 x 7 = > x = 7 answer is 7
Answer: a
Question (general): if p ' s height is 40 % less than that of q , how much percent q ' s height is more than that of p ?
Options: a ) 75.56 % , b ) 66.66 % , c ) 70.05 % , d ) 80 % , e ) 85 %
Rationale: excess of q ' s height over p = ( 40 / ( 100 - 40 ) ) * 100 = = > 66.66 % answer b
Answer: b
Question (physics): a can finish a piece of work in 12 days . b can do it in 20 days . they work together for 9 days and then a goes away . in how many days will b finish the work ?
Options: a ) 6 days , b ) 4 days , c ) 5 days , d ) 3 days , e ) 7 days
Rationale: "9 / 12 + ( 9 + x ) / 20 = 1 = > x = 5 days answer : c"
Answer: c
Question (general): if each participant of a chess tournament plays exactly one game with each of the remaining participants , then 120 games will be played during the tournament . find the number of participants .
Options: a ) 15 , b ) 16 , c ) 17 , d ) 18 , e ) 19
Rationale: "let p be the number of participants . pc 2 = 120 ( p ) ( p - 1 ) = 240 = 16 * 15 p = 16 the answer is b ."
Answer: b
Question (general): a parking garage rents parking spaces for $ 10 per week or $ 35 per month . how much does a person save in a year by renting by the month rather than by the week ?
Options: a ) $ 140 , b ) $ 100 , c ) $ 220 , d ) $ 240 , e ) $ 260
Rationale: 10 $ per week ! an year has 52 weeks . annual charges per year = 52 * 10 = 520 $ 30 $ per month ! an year has 12 months . annual charges per year = 12 * 35 = 420 $ 520 - 420 = 100 ans b
Answer: b
Question (geometry): the cost of the paint is rs . 40 per kg . if 1 kg of paint covers 20 sq . ft , how much will it cost to paint outside of a cube having 30 feet each side
Options: a ) rs . 962 , b ) rs . 672 , c ) rs . 546 , d ) rs . 10800 , e ) none of these
Rationale: "explanation : surface area of a cube = 6 x 30 ^ 2 = 5400 sq . ft quantity of paint required = ( 5400 / 20 ) = 270 kg cost of painting = 40 x 270 = rs . 10800 answer : d"
Answer: d
Question (physics): a rectangular lawn 55 m by 35 m has two roads each 4 m wide running in the middle of it . one parallel to the length and the other parallel to breadth . the cost of graveling the roads at 75 paise per sq meter is
Options: a ) 378
b ) 278
c ) 258
d ) 287
e ) 271
Rationale: explanation : area of cross roads = 55 * 4 + 35 * 4 - 4 * 4 = 344 sq m cost of graveling = 344 * ( 75 / 100 ) = rs . 258 answer : c ) rs . 258
Answer: c
Question (physics): the sum of four consecutive even numbers is 52 , find the third number ?
Options: a ) 14 , b ) 10 , c ) 12 , d ) 16 , e ) 20
Rationale: "let the first number be n let the second number be n + 2 let the third number be n + 4 let the third number be n + 6 n + n + 2 + n + 4 + n + 6 = 52 combine like terms : 4 n + 12 = 52 4 n + 12 = 52 4 n = 52 - 12 4 n = 40 n = 10 n + 2 = 12 n + 4 = 14 n + 6 = 16 the third number is 14 answer : a"
Answer: a
Question (physics): earl can stuff advertising circulars into envelopes at the rate of 36 envelopes per minutes and ellen requires a minutes and half to stuff the same number of envelops . working together , how long will it take earl and ellen to stuff 60 envelopes
Options: a ) 1 minute , b ) 5 minutes , c ) 7 minutes , d ) 3 minutes , e ) 4 minutes
Rationale: "earl takes 1 min . for 36 envelopes . ellen takes 3 / 2 mins for the same . so ellen can stuff ( ( 36 ) / ( 3 / 2 ) ) in 1 min . i . e . , 24 envlpes a min . so both of them when work together can stuff 36 + 24 = 60 envelopes in 1 min . for 60 envelopes they will take 60 / 60 mins . i . e . , 1 mins . answer : a"
Answer: a
Question (general): if 5395 / 4.15 = 1300 , then 539.5 / 41.5 is equal to ?
Options: a ) 17 , b ) 13 , c ) 15 , d ) 14 , e ) 16
Rationale: "answer given expression 539.5 / 41.5 = 5395 / 415 = 5395 / ( 415 x 100 ) = 1300 / 100 = 13 correct option : b"
Answer: b
Question (gain): in a certain math department , students are required to enroll in either calculus or trigonometry , each of which is offered in beginner and advanced courses . the number of students enrolled in trigonometry is 50 % greater than the number of students enrolled in calculus , and 80 % of calculus students are enrolled in the beginner course . if 4 / 5 of students are in the beginner courses , and one student is selected at random , what is the probability that an advanced trigonometry student is selected ?
Options: a ) 8 % , b ) 12 % , c ) 16 % , d ) 20 % , e ) 24 %
Rationale: "let x be the number of students in calculus . then the number of students in trigonometry is 1.5 x the number of students on beginner calculus is 0.8 x the number of students in beginner trigonometry is 4 / 5 * ( 2.5 x ) - 0.8 x = 1.2 x the number of students in advanced trigonometry is 0.3 x the percentage in advanced trigonometry is 0.3 x / 2.5 x = 12 % the answer is b ."
Answer: b
Question (physics): pipe a can fill a tank in 15 minutes and pipe b cam empty it in 24 minutes . if both the pipes are opened together after how many minutes should pipe b be closed , so that the tank is filled in 30 minutes ?
Options: a ) 18 , b ) 27 , c ) 98 , d ) 27 , e ) 24
Rationale: "let the pipe b be closed after x minutes . 30 / 15 - x / 24 = 1 = > x / 24 = 30 / 15 - 1 = 1 = > x = 1 * 24 = 24 . answer : e"
Answer: e
Question (general): a train starts full of passengers at the first station it drops 1 / 3 of the passengers and takes 280 more at the second station it drops one half the new total and takes 12 more . on arriving at the third station it is found to have 242 passengers . find the no of passengers in the beginning ?
Options: a ) 292 , b ) 270 , c ) 192 , d ) 144 , e ) 168
Rationale: let no of passengers in the beginning be x after first station no passengers = ( x - x / 3 ) + 280 = 2 x / 3 + 280 after second station no passengers = 1 / 2 ( 2 x / 3 + 280 ) + 12 1 / 2 ( 2 x / 3 + 280 ) + 12 = 242 2 x / 3 + 280 = 2 * 230 = 460 - 280 = 180 2 x / 3 = 180 x = 270 . so the answer is option b ) 270 .
Answer: b
Question (physics): a farming field can be ploughed by 6 tractors in 4 days . when 6 tractors work together , each of them ploughs 120 hectares a day . if two of the tractors were moved to another field , then the remaining 4 tractors could plough the same field in 5 days . how many hectares a day would one tractor plough then ?
Options: a ) 144 , b ) 255 , c ) 148 , d ) 265 , e ) 320
Rationale: if each of 66 tractors ploughed 120120 hectares a day and they finished the work in 44 days , then the whole field is : 120 ⋅ 6 ⋅ 4 = 720 ⋅ 4 = 2880120 ⋅ 6 ⋅ 4 = 720 ⋅ 4 = 2880 hectares . let ' s suppose that each of the four tractors ploughed xx hectares a day . therefore in 5 days they ploughed 5 ⋅ 4 ⋅ x = 20 ⋅ x 5 ⋅ 4 ⋅ x = 20 ⋅ x hectares , which equals the area of the whole field , 2880 hectares . so , we get 20 x = 288020 x = 2880 x = 288020 = 144 x = 288020 = 144 . hence , each of the four tractors would plough 144 hectares a day . correct answer is a ) 144
Answer: a
Question (general): o ( x ) represents the least odd integer greater than x , whereas o ( x ) represents the greatest odd integer less than x . likewise , e ( x ) represents the least even integer greater than x , whereas e ( x ) represents the greatest even integer less than x . according to these definitions , the value of o ( 3.2 ) + e ( – 1.7 ) + o ( – 3.3 ) + e ( 4.7 ) is :
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "o ( 3.2 ) + e ( – 1.7 ) + o ( – 3.3 ) + e ( 4.7 ) = 5 + 0 + ( - 5 ) + 4 = 4 the answer is b ."
Answer: b
Question (general): a mixture of 125 gallons of wine and water contains 20 % of water . how much water must be added to the mixture in order to increase the percentage of water to 25 % of the new mixture ?
Options: a ) 10 gallons , b ) 8.5 gallons , c ) 8 gallons , d ) 8.33 gallons , e ) 9.33 gallons
Rationale: in 125 gallons of the solution there are 0.2 ∗ 125 = 25 gallons of water . we want to add w gallons of water to 125 gallons of solution so that 25 + w gallons of water to be 25 % of new solution : 25 + w = 0.25 ( 125 + w ) - - > w = 253 ≈ 8.33 . answer : d
Answer: d
Question (general): barbata invests $ 3000 in the national bank at 5 % . how much additional money must she invest at 8 % so that the total annual income will be equal to 6 % of her entire investment ?
Options: a ) 1200 , b ) 3000 , c ) 1000 , d ) 3600 , e ) 1500
Rationale: "let the additional invested amount for 8 % interest be x ; equation will be ; 3000 + 0.05 * 3000 + x + 0.08 x = 3000 + x + 0.06 ( 3000 + x ) 0.05 * 3000 + 0.08 x = 0.06 x + 0.06 * 3000 0.02 x = 3000 ( 0.06 - 0.05 ) x = 3000 * 0.01 / 0.02 = 1500 ans : ` ` e ' '"
Answer: e
Question (gain): the price of a jacket is reduced by 15 % . during a special sale the price of the jacket is reduced another 30 % . by approximately what percent must the price of the jacket now be increased in order to restore it to its original amount ?
Options: a ) 60 , b ) 65 , c ) 69 , d ) 68.25 , e ) 67.5
Rationale: 1 ) let the price of jacket initially be $ 100 . 2 ) then it is decreased by 15 % , therefore bringing down the price to $ 85 . 3 ) again it is further discounted by 30 % , therefore bringing down the price to $ 59.5 4 ) now 59.5 has to be added byx % in order to equal the original price . 59.5 + ( x % ) 59.5 = 100 . solving this eq for x , we get x = 68.25 ans is d
Answer: d
Question (general): village p ’ s population is 800 greater than village q ' s population . if village q ’ s population were reduced by 700 people , then village p ’ s population would be 7 times as large as village q ' s population . what is village q ' s current population ?
Options: a ) 910 , b ) 920 , c ) 930 , d ) 940 , e ) 950
Rationale: "p = q + 800 . p = 7 ( q - 700 ) . 7 ( q - 700 ) = q + 800 . 6 q = 5700 . q = 950 . the answer is e ."
Answer: e
Question (other): half of the workers in palabras bookstore have read the latest book by j . saramago , and 5 / 8 of the workers have read the latest book by h . kureishi . the number of workers that have read neither book is one less than the number of the workers that have read the latest saramago book and have not read the latest kureishi book . if there are 56 workers in the palabras bookstore , how many of them have read both books ?
Options: a ) 13 , b ) 12 , c ) 9 , d ) 8 , e ) 17
Rationale: "there are total 56 workers . half of the workers in palabras bookstore have read the latest book by j . saramago , so 28 have read saramago . 5 / 8 of the workers have read the latest book by h . kureishi . so ( 5 / 8 ) * 56 = 35 have read kureishi the number of workers that have read neither book is one less than the number of the workers that have read the latest saramago book and have not read the latest kureishi book if b workers have read both books , 20 - b have read saramago but not kureishi . so , ( 28 - b - 1 ) have read neither . total = n ( a ) + n ( b ) - both + neither 56 = 28 + 35 - b + ( 28 - b - 1 ) b = 17 answer ( e )"
Answer: e
Question (general): for integers x and y , if ( x + 2 ) ( y + 3 ) is even then 4 xy must be divisible by :
Options: a ) 4 , b ) 8 , c ) 9 , d ) 10 , e ) 12
Rationale: just looking at the question , ( x + 2 ) ( y + 3 ) could be even * even , odd * even , or even * odd in order for the whole expression to be even . however , just looking at 4 xy , since x * y will always be an integer , it will always be divisible by 4 . 4 divided by 4 is 1 , hence preserving x * y . answer : a . 4
Answer: a
Question (other): two integers are in the ratio of 1 to 3 . if 10 is added to the smaller number , the ratio becomes 1 to 1 . find the larger integer .
Options: a ) 6 , b ) 9 , c ) 10 , d ) 12 , e ) 15
Rationale: "one option is to set up the equations and solve : if the ratio of two integers x and y is 1 to 3 , then 3 x = y , where x is the smaller integer . if adding 10 to the smaller integer makes the ratio 1 to 1 , then x + 10 = y . substituting y = 3 x into the second equation yields x + 10 = 3 x . so , x = 5 ( smaller integer ) and , y = 3 x = 15 ( larger integer ) so e is the correct answer . another option is to test the answer choices . a ) the larger number is 6 given : the two integers are in the ratio of 1 to 3 so , the smaller number must be 2 if 10 is added to the smaller number , the ratio becomes 1 to 1 if we add 10 to the smaller number ( 2 ) , we get 12 . so , the new ratio is 12 to 6 which is the same as 2 to 1 . no good . we want a resulting ratio of 1 to 1 eliminate a b ) the larger number is 9 given : the two integers are in the ratio of 1 to 3 so , the smaller number must be 3 if 10 is added to the smaller number , the ratio becomes 1 to 1 if we add 10 to the smaller number ( 3 ) , we get 13 so , the new ratio is 13 to 9 . no good . we want a resulting ratio of 1 to 1 eliminate b c ) the larger number is 10 given : the two integers are in the ratio of 1 to 3 so , the smaller number must be 3.33 ( which is not an integer ) since both numbers are integers , the larger number can not be 10 . eliminate c d ) the larger number is 12 given : the two integers are in the ratio of 1 to 3 so , the smaller number must be 4 if 10 is added to the smaller number , the ratio becomes 1 to 1 if we add 10 to the smaller number ( 4 ) , we get 14 so , the new ratio is 14 to 12 which is 7 to 6 . no good . we want a resulting ratio of 1 to 1 eliminate d e ) the larger number is 15 given : the two integers are in the ratio of 1 to 3 so , the smaller number must be 5 if 10 is added to the smaller number , the ratio becomes 1 to 1 if we add 10 to the smaller number ( 5 ) , we get 15 so , the new ratio is 15 to 15 . this is the same as the ratio of 1 to 1 therefore e is the only correct choice ."
Answer: e
Question (general): how many integers are divisible by 9 between 10 ! and 10 ! + 100 inclusive ?
Options: a ) 6 , b ) 7 , c ) 8 , d ) 9 , e ) 10
Rationale: "e - 7 10 ! is divisible by 9 there are 11 numbers between 10 ! and 10 ! + 100 that are divisible by 9 . hence 10"
Answer: e