Cheukting/mathqa huggingface dataset:
Question (physics): two goods trains each 500 m long are running in opposite directions on parallel tracks . their speeds are 45 km / hr and 15 km / hr respectively . find the time taken by the slower train to pass the driver of the faster one ?
Options: a ) 78 , b ) 60 , c ) 26 , d ) 23 , e ) 12
Rationale: "relative speed = 45 + 15 = 60 km / hr . 60 * 5 / 18 = 50 / 3 m / sec . distance covered = 500 + 500 = 1000 m . required time = 1000 * 3 / 50 = 60 sec . answer : b"
Answer: b
Question (probability): if two integers x , y ( x > y ) are selected from - 10 to 11 ( inclusive ) , how many possible cases are there ?
Options: a ) 150 , b ) 180 , c ) 190 , d ) 210 , e ) 231
Rationale: "if two integers x , y ( x > y ) are selected from - 10 to 9 ( inclusive ) , how many possible cases are there ? a . 150 b . 180 c . 190 d . 210 e . 240 - - > 22 c 2 = 22 * 21 / 2 = 231 . therefore , the answer is e ."
Answer: e
Question (gain): an empty fuel tank with a capacity of 200 gallons was filled partially with fuel a and then to capacity with fuel b . fuel a contains 12 % ethanol by volume and fuel b contains 16 % ethanol by volume . if the full fuel tank contains 24 gallons of ethanol , how many gallons of fuel a were added ?
Options: a ) 160 , b ) 150 , c ) 100 , d ) 80 , e ) 200
Rationale: "say there are a gallons of fuel a in the tank , then there would be 200 - a gallons of fuel b . the amount of ethanol in a gallons of fuel a is 0.12 a ; the amount of ethanol in 200 - a gallons of fuel b is 0.16 ( 200 - a ) ; since the total amount of ethanol is 24 gallons then 0.12 a + 0.16 ( 200 - a ) = 24 - - > a = 200 . answer : e ."
Answer: e
Question (physics): a take twice as much time as b or thrice as much time to finish a piece of work . working together , they can finish the work in 2 days . b can do the work alone in ?
Options: a ) 3 hours , b ) 6 hours , c ) 7 hours , d ) 4 hours , e ) 8 hours
Rationale: "b 6 hours suppose a , b and c take x , x / 2 and x / 3 respectively to finish the work . then , ( 1 / x + 2 / x + 3 / x ) = 1 / 2 6 / x = 1 / 2 = > x = 12 so , b takes 6 hours to finish the work ."
Answer: b
Question (geometry): a circle in the coordinate plane passes through points ( - 3 , - 2 ) and ( 1 , 3 ) . what is the smallest possible area of that circle ?
Options: a ) 10.5 π , b ) 26 π , c ) 262 √ π , d ) 52 π , e ) 64 π
Rationale: "the distance between the two points is sqrt ( 41 ) . radius = sqrt ( 41 ) / 2 area = pi * ( sqrt ( 41 ) / 2 ) ^ 2 a . 10.5 π"
Answer: a
Question (gain): a certain candy manufacturer reduced the weight of candy bar m by 10 percent buy left the price unchanged . what was the resulting percent increase in the price per ounce of candy bar m ?
Options: a ) 10 , b ) 11 , c ) 12 , d ) 14 , e ) 9
Rationale: "assume 1 oz candy cost $ 1 before . now price remain same $ 1 but weight of candy reduces to 0.9 oz new price of candy = 1 / 0.9 = 1.11 price increase 11 % b"
Answer: b
Question (physics): a cyclist traveled for two days . on the second day the cyclist traveled 6 hours longer and at an average speed 10 mile per hour slower than she traveled on the first day . if during the two days she traveled a total of 280 miles and spent a total of 12 hours traveling , what was her average speed on the second day ?
Options: a ) 5 mph , b ) 10 mph , c ) 20 mph , d ) 30 mph , e ) 40 mph
Rationale: "solution : d = 280 mi t = 12 hrs đ â y 1 time = t 1 d â y 2 time = t 2 t 2 - t 1 = 4 hrs - - - - - ( i ) t 1 + t 2 = 12 hrs - - - - - ( ii ) adding i and ii , t 2 = 8 hrs and t 1 = 4 hrs d à y 1 rate = r 1 d â y 2 rate = r 2 r 1 - r 2 = 10 mph í . ẹ . r 1 = 10 + r 2 280 = 8 r 2 + 4 r 1 í . ẹ . 280 = 8 r 2 + 4 ( 10 + r 2 ) í . ẹ . r 2 = 20 mph answer : a"
Answer: a
Question (gain): a person buys an article at rs . 500 . at what price should he sell the article so as to make a profit of 60 % ?
Options: a ) 800 , b ) 882 , c ) 772 , d ) 652 , e ) 271
Rationale: "cost price = rs . 500 profit = 60 % of 500 = rs . 300 selling price = cost price + profit = 500 + 300 = 800 answer : a"
Answer: a
Question (general): a man purchased 2 blankets @ rs . 100 each , 5 blankets @ rs . 150 each and two blankets at a certain rate which is now slipped off from his memory . but he remembers that the average price of the blankets was rs . 150 . find the unknown rate of two blankets ?
Options: a ) 278 , b ) 277 , c ) 278 , d ) 450 , e ) 550
Rationale: "10 * 150 = 1500 2 * 100 + 5 * 150 = 950 1500 – 1050 = 550 answer : e"
Answer: e
Question (gain): a trader bought a car at 5 % discount on its original price . he sold it at a 40 % increase on the price he bought it . what percent of profit did he make on the original price ?
Options: a ) 118 , b ) 133 , c ) 112 , d ) 113 , e ) 115
Rationale: "original price = 100 cp = 95 s = 95 * ( 140 / 100 ) = 133 100 - 133 = 5 % answer : b"
Answer: b
Question (gain): what will be the difference between simple and compound interest at 6 % per annum on a sum of rs . 1000 after 4 years ?
Options: a ) 24.19 , b ) 24.12 , c ) 22.1 , d ) 24.1 , e ) 22.5
Rationale: "s . i . = ( 1000 * 6 * 4 ) / 100 = rs . 240 c . i . = [ 1000 * ( 1 + 6 / 100 ) 4 - 1000 ] = rs . 262.5 difference = ( 262.5 - 240 ) = rs . 22.5 answer : e"
Answer: e
Question (general): zinc and copper are melted together in the ratio 9 : 11 . what is the weight of melted mixture , if 28.8 kg of zinc has been consumed in it ?
Options: a ) 58 kg , b ) 60 kg , c ) 64 kg , d ) 70 kg , e ) none
Rationale: sol . for 9 kg zinc , mixture melted = ( 9 + 11 ) kg . for 28.8 kg zinc , mixture , melted = [ 20 / 9 x 28.8 ] kg = 64 kg . answer c
Answer: c
Question (physics): what is the speed of the stream if a canoe rows upstream at 9 km / hr and downstream at 12 km / hr
Options: a ) 1 kmph , b ) 4 kmph , c ) 3 kmph , d ) 2 kmph , e ) 1.5 kmph
Rationale: "sol . speed of stream = 1 / 2 ( 12 - 9 ) kmph = 1.5 kmph . answer e"
Answer: e
Question (physics): . a car covers a distance of 495 km in 5 hours . find its speed ?
Options: a ) 99 , b ) 100 , c ) 102 , d ) 288 , e ) 2761
Rationale: "495 / 5 = 99 kmph answer : a"
Answer: a
Question (general): if a number p is prime , and 2 p + 1 = q , where q is also prime , then the decimal expansion of 1 / q will produce a decimal with q - 1 digits . if this method produces a decimal with 166 digits , what is the units digit of the product of p and q
Options: a ) 1 , b ) 3 , c ) 5 , d ) 7 , e ) 9
Rationale: "3 / 7 = 0.428571 . . . ( a repeating pattern one digit long ) b"
Answer: b
Question (physics): a man is walking at a speed of 10 km per hour . after every kilometre , he takes rest for 3 minutes . how much time will be take to cover a distance of 3 kilometres ?
Options: a ) 48 min . , b ) 50 min . , c ) 45 min . , d ) 55 min . , e ) 24 min .
Rationale: "rest time = number of rest ã — time for each rest = 2 ã — 3 = 6 minutes total time to cover 3 km = ( 3 â  „ 10 ã — 60 ) minutes + 6 minutes = 24 minutes answer e"
Answer: e
Question (general): the average weight of 5 person ' s increases by 5.5 kg when a new person comes in place of one of them weighing 68 kg . what might be the weight of the new person ?
Options: a ) 60 kg , b ) 95.5 kg , c ) 80 kg , d ) 85 kg , e ) 90 kg
Rationale: "total weight increased = ( 5 x 5.5 ) kg = 27.5 kg . weight of new person = ( 68 + 27.5 ) kg = 95.5 kg option b"
Answer: b
Question (gain): in the first 10 overs of a cricket game , the run rate was only 3.2 . what should be the rate in the remaining 40 overs to reach the target of 292 runs ?
Options: a ) 6.5 , b ) 6.22 , c ) 6.29 , d ) 6.39 , e ) 6.13
Rationale: "required run rate = [ 292 - ( 3.2 * 10 ) ] / 40 = 260 / 40 = 6.5 answer : a"
Answer: a
Question (physics): the radius of a semi circle is 5.2 cm then its perimeter is ?
Options: a ) 32.52 , b ) 32.48 , c ) 26.74 , d ) 32.9 , e ) 32.31
Rationale: "36 / 7 r = 5.2 = 26.74 answer : c"
Answer: c
Question (gain): p and q invested in a business . they earned some profit which they divided in the ratio of 2 : 3 . if p invested rs . 50 , 000 , the amount invested by q is :
Options: a ) rs . 65,000 , b ) rs . 70,000 , c ) rs . 75,000 , d ) rs . 90,000 , e ) rs . 60,000
Rationale: q invested = 50000 / 2 * 3 = 75000 answer : c
Answer: c
Question (other): if log 102 = 0.3010 , log 10 3 = 0.4771 , then the number of zeros between the decimal point and the first significant figure in ( 0.0432 ) 10 is ?
Options: a ) 10 , b ) 13 , c ) 14 , d ) 15 , e ) 16
Rationale: x = ( 0.0432 ) 10 = ( 432 / 10000 ) 10 = ( 33.24 / 104 ) 10 therefore , log 10 x = 10 ( log 103 + 4 log 102 - 4 ) = 10 ( 1.4313 + 1.2040 - 4 ) = 10 ( - 1.3647 ) = - 13.647 = - 14.353 therefore , x = antilog ( - 14.053 ) thus , number of zeros between the decimal and the first significant figure = 13 answer : b
Answer: b
Question (physics): eden drove an average speed of 15 miles per hour for the first 20 miles of a tripthen at a average speed of 30 miles / hr for the remaining 20 miles of the trip if she made no stops during the trip what was eden ' s avg speed in miles / hr for the entire trip
Options: a ) 35 , b ) 20 , c ) 45 , d ) 50 , e ) 55
Rationale: avg . speed = total distance / total time total distance = 40 miles total time = 20 / 15 + 15 / 30 = 2 avg . speed = 20 . answer - b
Answer: b
Question (general): the average temperature for monday , tuesday , wednesday and thursday was 48 degrees and for tuesday , wednesday , thursday and friday was 46 degrees . if the temperature on monday was 44 degrees . find the temperature on friday ?
Options: a ) 65 degrees , b ) 73 degrees , c ) 36 degrees , d ) 34 degrees , e ) 74 degrees
Rationale: "m + tu + w + th = 4 * 48 = 192 tu + w + th + f = 4 * 46 = 184 m = 44 tu + w + th = 192 - 44 = 148 f = 184 – 148 = 36 answer : c"
Answer: c
Question (other): if w is the set of all the integers between 49 and 88 , inclusive , that are either multiples of 3 or multiples of 2 or multiples of both , then w contains how many numbers ?
Options: a ) 11 , b ) 44 , c ) 48 , d ) 28 , e ) 26
Rationale: "official solution : number of multiples of 3 step 1 . subtract the extreme multiples of 3 within the range ( the greatest is 87 , the smallest is 51 ) : 87 - 51 = 36 step 2 . divide by 3 : 36 / 3 = 12 step 3 . add 1 : 12 + 1 = 13 . so there are 13 multiples of 3 within the range : examples are 51 , 54 , 57 , 60 , etc . number of multiples of 2 step 1 . subtract the extreme multiples of 2 within the range ( the greatest is 88 , the smallest is 50 ) : 88 - 50 = 38 step 2 . divide by 2 : 38 / 2 = 19 step 3 . add 1 : 19 + 1 = 20 . so there are 19 multiples of 2 within the range : examples are 50 , 52 , 54 , 56 , 58 , 60 etc . add the 13 multiples of 3 and the 19 multiples of 2 : 13 + 19 = 32 . however , by adding the multiples of 2 and the multiples of 3 , we are effectively counting several numbers twice : for example , 54 and 60 are parts of both the lists above . so we ca n ' t just take 13 + 19 = 32 . find the number of multiples of 6 ( which are the double counted , as 6 is divisible by both 2 and 3 ) , and subtract it from 32 : step 1 . subtract the extreme multiples of 6 within the range ( the greatest is 72 , the smallest is 54 ) : 84 - 54 = 30 step 2 . divide by 6 : 30 / 6 = 5 step 3 . add 1 : 5 + 1 = 6 . so there are 6 multiples of 6 within the range : we counted 6 numbers twice . subtract the 6 multiples of 6 from the sum of the multiples of 2 and 3 : = 13 + 19 - 6 = 32 - 6 = 26 therefore , the final number of multiples of 2 , 3 or 6 is 26 . hence , this is the correct answer . ( e )"
Answer: e
Question (general): in the xy - coordinate plane , the graph of y = - x ^ 2 + 9 intersects line l at ( p , - 5 ) and ( t , - 7 ) . what is the least possible value of the slope of line l ?
Options: a ) 6 , b ) - 1 , c ) - 2 , d ) - 6 , e ) - 10
Rationale: we need to find out the value of p and l to get to the slope . line l and graph y intersect at point ( p , - 5 ) . hence , x = p and y = - 5 should sactisfy the graph . soliving 5 = - p 2 + 9 p 2 = 4 p = + or - 2 simillarly point ( t , - 7 ) should satisfy the equation . hence x = t and y = - 7 . - 7 = - t 2 + 9 t = + or - 4 considering p = - 2 and t = 4 , the least slope is ( - 7 + 5 ) / ( 4 - 2 ) = - 1 imo option b is correct answer .
Answer: b
Question (physics): two numbers n and 16 have lcm = 52 and gcf = 8 . find n .
Options: a ) 35 , b ) 26 , c ) 76 , d ) 87 , e ) 24
Rationale: "the product of two integers is equal to the product of their lcm and gcf . hence . 16 * n = 52 * 8 n = 52 * 8 / 16 = 26 correct answer b"
Answer: b
Question (general): an ant walks an average of 1000 meters in 30 minutes . a beetle walks 10 % less distance at the same time on the average . assuming the beetle walks at her regular rate , what is its speed in km / h ?
Options: a ) 1.8 , b ) 1.6 , c ) 2.775 . , d ) 1.9 , e ) 3.5 .
Rationale: the ant walks an average of 1000 meters in 30 minutes 1000 meters in 1 / 2 hours the beetle walks 10 % less distance = 1000 - 10 = 900 meters in 30 minutes 0.900 km in 30 / 60 = 1 / 2 hours speed = 0.900 * 2 = 1.8 km / h i guess option a should be 1.8
Answer: a
Question (physics): two trains 151 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 ) 7.19 , b ) 7.18 , c ) 7.16 , d ) 7.84 , e ) 7.12
Rationale: "t = ( 151 + 165 ) / ( 80 + 65 ) * 18 / 5 t = 7.84 answer : d"
Answer: d
Question (probability): joe has candys on his pocket , 4 of those candies have lemon flavor , and 3 have caramel flavor . if joe take one , what is the probability that he picks one caramel flavor ?
Options: a ) 1 , b ) 2 , c ) none , d ) 3 / 7 , e ) 3 / 4
Rationale: the total number of candies is 7 and the number of candies with lemon flavor is 3 then , in one chance the probability to pick one candie with lemon flavor is 3 / 7 answer d
Answer: d
Question (general): john bought a total of 24 mangoes and oranges . each mango costs 80 cents and each orange costs 60 cents . if the average price of the 24 mangoes and oranges that john originally purchased was 65 cents , then how many oranges needs to return to raise the average price of his purchase to 72 cents ?
Options: a ) 4 , b ) 5 , c ) 6 , d ) 7 , e ) 14
Rationale: "let number of mangoes be x , number of oranges be 24 - x 0.80 x + ( 24 - x ) 0.60 / 24 = 0.65 solving for x , we get x = 6 - - > mangoes 6 , oranges 18 now , number of oranges to be returned be y 0.80 * 6 + ( 18 - y ) * 0.60 / 24 - y = 0.72 solving for y , y = 14 ans : e"
Answer: e
Question (general): the difference between the ages of two persons is 10 years . fifteen years ago , the elder one was twice as old as the younger one . the present age of the elder person is
Options: a ) 25 years , b ) 30 years , c ) 35 years , d ) 40 years , e ) none
Rationale: "sol . let their ages of x years and ( x + 10 ) years respectively . then , ( x + 10 ) - 15 = 2 ( x - 15 ) ⇔ x - 5 = 2 x - 30 ⇔ x = 25 . ∴ present age of the elder person = ( x + 10 ) = 35 years . answer c"
Answer: c
Question (general): how many 3 - digit numerals begin with a digit that represents a prime and end with a digit that represents a odd prime number ?
Options: a ) 16 , b ) 120 , c ) 160 , d ) 180 , e ) 240
Rationale: "prime digits 2 , 3,5 and 7 . three digit numbers _ _ _ 1 st place can be filled in 4 ways 2 nd place can be filled in 10 ways 3 rd place can be filled in 3 ways total = 4 * 10 * 3 = 120 ans : b"
Answer: b
Question (general): an automobile parts supplier charges $ 20 per package of gaskets . when a customer orders more than 10 packages of gaskets , the supplier charges 4 / 5 the price for each package in excess of 10 . during a certain week , the supplier sold 60 packages of gaskets . if 20 percent of the gaskets went to company x , 15 percent to company y , and the rest to company z , what was the total amount , in dollars , that the parts supplier received in payment for the gaskets ?
Options: a ) 1005 , b ) 1045 , c ) 1075 , d ) 1015 , e ) 1096
Rationale: "$ 25 per packet of gasket in case a customer orders less than 10 in case a customer orders > 10 price per gasket = 20 * 4 / 5 = 16 a certain week the supplier sold 60 gasket 1 . he sold 20 % of the gaskets to x = 12 gaskets = 20 * 10 + 16 * 2 = 200 + 32 = 232 2 . he sold 15 % of the gaskets to y = 9 gaskets = 20 * 9 = 180 3 . he sold remaining 65 % to z = 39 gaskets = 20 * 10 = 200 + 16 * 29 = 684 thus , total money earned 232 + 180 + 684 = 1096 answer is e"
Answer: e
Question (geometry): the length of a rectangle is increased by 25 % and its breadth is decreased by 15 % . what is the effect on its area ?
Options: a ) 1288 , b ) 1299 , c ) 10625 , d ) 10000 , e ) 2887
Rationale: "100 * 100 = 10000 125 * 85 = 10625 answer : c"
Answer: c
Question (general): suraj has a certain average of runs for 8 innings . in the 9 th innings he scores 90 runs thereby increasing his average by 6 runs . what is his average after the 9 th innings ?
Options: a ) 42 , b ) 64 , c ) 36 , d ) 72 , e ) 27
Rationale: "to improve his average by 6 runs per innings he has to contribute 8 x 6 = 48 runs for the previous 8 innings . thus , the average after the 9 th innings = 90 - 48 = 42 . answer : a"
Answer: a
Question (physics): 20 men or 25 women can finish a work in 25 days . how many days 38 men and 15 women will take to finish this job ?
Options: a ) 12 days , b ) 10 days , c ) 5 days , d ) 6 days , e ) 15 days
Rationale: 20 men = 25 women therefore , 1 women = 20 / 25 men = 4 / 5 men 15 women = 4 / 5 * 15 men = 12 men i . e . 38 men + 15 women = 38 men + 12 men = 50 men 1 work done by men = 20 * 25 20 * 25 = 50 * ? number of days = 20 * 25 / 50 = 10 days . b
Answer: b
Question (general): two heavily loaded sixteen - wheeler transport trucks are 940 kilometers apart , sitting at two rest stops on opposite sides of the same highway . driver a begins heading down the highway driving at an average speed of 90 kilometers per hour . exactly one hour later , driver b starts down the highway toward driver a , maintaining an average speed of 80 kilometers per hour . how many kilometers farther than driver b , will driver a have driven when they meet and pass each other on the highway ?
Options: a ) 90 , b ) 130 , c ) 150 , d ) 140 , e ) 450
Rationale: "i ' ve been reading the website for a while and i ' m always keen to see different approaches so i would like to share one that works for me : short version : truck a travels for an hour . distance remaining = 940 - 90 = 850 k ratio of speeds 9 : 8 - > 850 / 17 = 50 truck a = 90 + 50 * 9 = 540 truck b = 50 * 8 = 400 delta = 140 km answer d"
Answer: d
Question (general): in an examination , 32 % of total students failed in hindi , 56 % failed in english and 12 % in both . the percentage of these who passed in both the subjects is :
Options: a ) 24 , b ) 37 , c ) 28 , d ) 40 , e ) 81
Rationale: "explanation : formula n ( a â ˆ ª b ) = n ( a ) + n ( b ) â ˆ ’ n ( a â ˆ © b ) fail in hindi or english = 32 + 56 â € “ 12 = 76 therefore students who passed = 100 â € “ 76 = 24 . answer : a"
Answer: a
Question (general): of the 250 employees at company x , 80 are full - time , and 175 have worked at company x for at least a year . there are 20 employees at company x who aren ’ t full - time and haven ’ t worked at company x for at least a year . how many full - time employees of company x have worked at the company for at least a year ?
Options: a ) 25 , b ) 30 , c ) 50 , d ) 80 , e ) 100
Rationale: "250 employees 80 are full - time 175 have worked at company x for at least a year 20 employees at company x who aren ’ t full - time and haven ’ t worked at company x for at least a year . how many full - time employees of company x have worked at the company for at least a year ? 250 - 80 = 170 employees not full time 170 - 20 = 150 employees not full time who worked over a year 175 employees have worked at company x for at least a year - 150 employees not full time who worked over a year = 25 full - time employees of company x have worked at the company for at least a year ans a"
Answer: a
Question (general): a farmer has an apple orchard consisting of fuji and gala apple trees . due to high winds this year 10 % of his trees cross pollinated . the number of his trees that are pure fuji plus the cross - pollinated ones totals 221 , while 3 / 4 of all his trees are pure fuji . how many of his trees are pure gala ?
Options: a ) 22 , b ) 33 , c ) 55 , d ) 77 , e ) 39
Rationale: "let f = pure fuji , g = pure gala and c - cross pollinated . c = 10 % of x where x is total trees . c = . 1 x also 3 x / 4 = f and c + f = 221 = > . 1 x + 3 / 4 x = 221 = > x = 260 260 - 221 = pure gala = 39 . e"
Answer: e
Question (geometry): the ratio between the perimeter and the breadth of a rectangle is 5 : 1 . if the area of the rectangle is 216 sq . cm , what is the length of the rectangle ?
Options: a ) 18 cm
b ) 20 cm
c ) 22 cm
d ) 24 cm
e ) 25 cm
Rationale: 2 ( l + b ) = 5 b 1 2 l + 2 b = 5 b 3 b = 2 l b = 2 l 3 then , area = 216 cm 2 l x b = 216 l x 2 l = 216 3 l 2 = 324 l = 18 cm . a
Answer: a
Question (general): 60 percent of movie theatres in town x have 3 screens or less . 20 % of those theatres sell an average of more than $ 300 worth of popcorn per showing . 50 percent of all the movie theatres in town x sell $ 300 or less of popcorn per showing . what percent of all the stores on the street have 4 or more screens and sell an average of more than $ 300 worth of popcorn per day ?
Options: a ) 12 , b ) 18 , c ) 32 , d ) 40 , e ) 44
Rationale: "lets take numbers here . assume that the total number of movie theaters in the town = 100 then number of movie theaters with 3 screens or less = 60 = > number of movie theaters with 4 screens or more = 40 movie theaters with 3 screens or less selling popcorn at more than $ 300 = 20 % of 60 = 12 number of movie theaters selling popcorn at $ 300 or less = 56 = > number of movie theaters selling popcorn at more than $ 300 = 100 - 56 = 44 of these 44 theaters , 12 are those with 3 screens or less therefore 44 ( 44 - 12 ) must be those with four screens or more e is the answer"
Answer: e
Question (general): on a two - dimensional coordinate plane , the line q = x ^ 2 - x ^ 3 touches the x - axis in how many places ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: "apparently it ' s q = x ^ 2 - x ^ 3 instead of q = x ^ 2 - q ^ 3 . in this case : the x - intercept is the value ( s ) of x for q = 0 . 0 = x ^ 2 - x ^ 3 ; 0 = x ^ 2 ( 1 - x ) ; x = 0 or x = 1 . answer : c ."
Answer: c
Question (general): if x and y are integers such that | y + 3 | ≤ 3 and 2 y – 3 x + 6 = 0 , what is the least possible value w of the product xy ?
Options: a ) - 12 , b ) - 3 , c ) 0 , d ) 2 , e ) none of the above
Rationale: "how to deal with inequalities involving absolute values ? first example shows us the so callednumber case in this case we have | y + 3 | ≤ 3 which is generalized | something | ≤ some number . first we solve as if there were no absolute value brackets : y + 3 ≤ 3 y ≤ 0 so y is 0 or negative second scenario - remove the absolute value brackets . put a negative sign around the other side of the inequality , andflip the sign : y + 3 > = - 3 y > = - 6 therefore we have a possible range for y : - 6 = < y < = 0 ok , so far so good , we ' re half way through . what about x ? here ' s the formula : 2 y – 3 x + 6 = 0 , rewrite it as 2 y + 6 = 3 x . you can say that 2 y + 6 is a multiple of 3 ( = 3 x ) . so all values which must be integer must also satisfy this constraint . i ' m just saying that , so it ' s easier to evaluate all the possible numbers w ( - 6 , - 3 , 0 ) . if you plug in y = 0 , x will be 2 and xy = 0 as the lowest possible value . hence , answer choice c is the one to go ."
Answer: c
Question (gain): there is 60 lit of milk and water in which milk forms 84 % . howmuch water must be added to this solution to make it solution in which milk forms 60 %
Options: a ) 18.75 , b ) 19.75 , c ) 20.75 , d ) 21.75 , e ) 24
Rationale: "60 * 84 / 100 = 50.40 lit milk that is 9.60 lit water let x lit water will be added then ( 60 + x ) * 60 / 100 = 50.40 so x = 18.75 answer : e"
Answer: e