Cheukting/mathqa huggingface dataset:
Question (gain): the owner of a furniture shop charges his customer 24 % more than the cost price . if a customer paid rs . 8587 for a computer table , then what was the cost price of the computer table ?
Options: a ) 6925 , b ) 6887 , c ) 6728 , d ) 6725 , e ) 2871
Rationale: ": cp = sp * ( 100 / ( 100 + profit % ) ) = 8587 ( 100 / 124 ) = rs . 6925 . answer : a"
Answer: a
Question (physics): a train covers a distance of 100 km in 1 hour . if its speed is decreased by 25 km / hr , the time taken by the car to cover the same distance will be ?
Options: a ) 1 hr , b ) 1 hr 20 min , c ) 50 min , d ) 1 hr 30 min , e ) 1 hr 45 min
Rationale: "speed = 100 / 1 = 100 km / hr new speed = 100 - 25 = 75 km / hr time taken = 100 / 75 = 1 hr 20 min answer is b"
Answer: b
Question (probability): how many positive integers less than 600 can be formed using the numbers 1 , 2 , 3 and 5 for the digits ?
Options: a ) 48 , b ) 52 , c ) 66 , d ) 68 , e ) 84
Rationale: "notice that we can find the number of 2 and 3 digit numbers by just assuming the first digit can also be zero : 0 1 1 1 2 2 2 3 3 3 5 5 5 5 5 number of possibilities = 5 * 4 * 4 = 80 . then , just add up the number of 1 digits numbers = 4 , so total is 80 + 4 = 84 . answer : e"
Answer: e
Question (other): in a certain state , the ratio of registered republicans to registered democrats is 3 to 2 , and every registered voter is either a republican or a democrat . if 90 percent of the republicans and 15 percent of the democrats are expected to vote for candidate x , and everyone else is expected to vote for candidate y , by what percent is candidate x expected to win the election ?
Options: a ) 2 % , b ) 5 % , c ) 8 % , d ) 10 % , e ) 20 %
Rationale: "since we were expected to find a percentage figure - it thought that it might be easier to pick a ' smart number ' to represent the total number of voters ( republicans and democrats ) . therefore , i picked 100 ( as the total number of voters ) and thus 30 : 20 represents the number ratio of republicans : democrats . if 90 % of republicans ( which is ( 60 * 0.9 ) = 54 ) and 15 % of democrats ( 40 * 0.15 = 6 ) voted for candidate x , means that out of total of 100 voters ; 60 ( 54 + 6 ) voters voted for candidate x and 40 voted for candidate y . thus we can infer that candidate x is expected to win the election by 20 ( 60 - 40 ) votes . therefore candidate x is expected to win the election by ( 20 / 100 ) votes which is equivalent to 20 % . i think the answer is e ."
Answer: e
Question (physics): a man can row downstream at 22 kmph and upstream at 10 kmph . find the speed of the man in still water and the speed of stream respectively ?
Options: a ) 16 , 2 , b ) 16 , 4 , c ) 16 , 8 , d ) 16 , 6 , e ) 16 , 7
Rationale: "let the speed of the man in still water and speed of stream be x kmph and y kmph respectively . given x + y = 22 - - - ( 1 ) and x - y = 10 - - - ( 2 ) from ( 1 ) & ( 2 ) 2 x = 32 = > x = 16 , y = 6 . answer : d"
Answer: d
Question (general): if | 7 x + 2 | = 16 , then find the product of the values of x ?
Options: a ) - 5.14 , b ) 6.19 , c ) - 7.18 , d ) - 8.62 , e ) 5.69
Rationale: "| 7 x + 2 | = 16 7 x + 2 = 16 or 7 x + 2 = - 16 7 x = 14 or 7 x = - 18 x = 2 or x = - 2.57 product = - 2.57 * 2 = - 5.14 answer is a"
Answer: a
Question (gain): the cost of an article is decreased by 15 % . if the original cost is $ 80 , find the decrease cost .
Options: a ) 33 , b ) 11 , c ) 68 , d ) 36 , e ) 91
Rationale: "original cost = $ 80 decrease in it = 15 % of $ 80 = 15 / 100 × 80 = 1200 / 100 = $ 12 therefore , decrease cost = $ 80 - $ 12 = $ 68 answer : c"
Answer: c
Question (general): sandy bought 65 books for $ 1280 from one shop and 55 books for $ 880 from another shop . what is the average price that sandy paid per book ?
Options: a ) $ 12 , b ) $ 14 , c ) $ 16 , d ) $ 18 , e ) $ 20
Rationale: "average price per book = ( 1280 + 880 ) / ( 65 + 55 ) = 2160 / 120 = $ 18 the answer is d ."
Answer: d
Question (gain): a fruit seller had some apples . he sells 35 % and still has 6500 apples . originally , he had ?
Options: a ) 650 apples , b ) 600 apples , c ) 772 apples , d ) 700 apples , e ) none
Rationale: "answer â ˆ µ 65 % of n = 800 â ˆ ´ n = ( 650 x 100 ) / 65 = 1000 correct option : e"
Answer: e
Question (gain): each of the integers from 0 to 8 , inclusive , is written on a separate slip of blank paper and the 10 slips are dropped into hat . if the slips are then drawn one at a time without replacement , how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10 ?
Options: a ) 8 , b ) 9 , c ) 7 , d ) 6 , e ) 5
Rationale: you should consider the worst case scenario : if you pick numbers 0 , 1 , 2 , 3 , 4 and 5 then no two numbers out of these 6 add up to 10 . now , the next , 7 th number whatever it ' ll be ( 6,7 , or 8 ) will guarantee that two number will add up to 10 . so , 7 slips must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10 answer : c
Answer: c
Question (physics): at 15 : 00 there were 20 students in the computer lab . at 15 : 03 and every three minutes after that , 3 students entered the lab . if at 15 : 10 and every ten minutes after that 7 students left the lab , how many students were in the computer lab at 15 : 44 ?
Options: a ) 7 , b ) 14 , c ) 25 , d ) 27 , e ) 30
Rationale: "initial no of students + 3 * ( 1 + no of possible 3 minute intervals between 15 : 03 and 15 : 44 ) - 8 * ( 1 + no of possible 10 minute intervals between 15 : 10 and 15 : 44 ) 20 + 3 * 14 - 8 * 4 = 25 c"
Answer: c
Question (other): an ore contains 25 % of an alloy that has 90 % iron . other than this , in the remaining 75 % of the ore , there is no iron . how many kilograms of the ore are needed to obtain 60 kg of pure iron ?
Options: a ) 250 kg , b ) 275 kg , c ) 300 kg , d ) 266.66 kg , e ) none
Rationale: solution : let there is 100 kg of ore . 25 % ore contains 90 % off iron that means 25 kg contains ; 25 * 90 / 100 = 22.5 kg iron . 22.5 kg iron contains 100 kg of ore . then , 1 kg of iron contains = 25 / 100 kg ore ; hence , 60 kg iron contains = 100 * 60 / 22.5 = 266.66 kg ore . answer : option d
Answer: d
Question (general): how many integers are between 5 and 74 / 5 , inclusive ?
Options: a ) 4 , b ) 5 , c ) 7 , d ) 8 , e ) 10
Rationale: "74 / 5 = 14 . xx we are not concerned about the exact value of 74 / 5 as we just need the integers . the different integers between 5 and 74 / 5 would be 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13,14 . total number of integers = 10 option e"
Answer: e
Question (general): find large number from below question the difference of two numbers is 1365 . on dividing the larger number by the smaller , we get 8 as quotient and the 15 as remainder
Options: a ) 1235 , b ) 1557.9 , c ) 1378 , d ) 1635 , e ) 1489
Rationale: "let the smaller number be x . then larger number = ( x + 1365 ) . x + 1365 = 8 x + 15 7 x = 1350 x = 192.9 large number = 192.9 + 1365 = 1557.9 b"
Answer: b
Question (general): a is the average ( arithmetic mean ) of the first 7 positive multiples of 6 and b is the median of the first 3 positive multiples of positive integer n . if the value of a ^ 2 – b ^ 2 is zero , what is the value of n ?
Options: a ) 8 , b ) 10 , c ) 12 , d ) 16 , e ) 24
Rationale: if a ^ 2 - b ^ 2 = 0 , then let ' s assume that a = b . a must equal the 4 th positive multiple of 4 , thus a = 24 , which also equals b . b is the second positive multiple of n , thus n = 24 / 2 = 12 . the answer is c .
Answer: c
Question (general): cereal a is 11 % sugar by weight , whereas healthier but less delicious cereal b is 2 % sugar by weight . to make a delicious and healthy mixture that is 3 % sugar , what should be the ratio of cereal a to cereal b , by weight ?
Options: a ) 2 : 9 , b ) 3 : 7 , c ) 2 : 5 , d ) 1 : 8 , e ) 1 : 5
Rationale: "2 % is 1 % - points below 3 % and 11 % is 8 % - points above 3 % . the ratio of a : b should be 1 : 8 . the answer is d ."
Answer: d
Question (general): in a class of 20 students , 2 students did not borrow any books from the library , 10 students each borrowed 1 book , 5 students each borrowed 2 books , and the rest borrowed at least 3 books . if the average number of books per student was 2 , what is the maximum number of books any single student could have borrowed ?
Options: a ) 12 , b ) 13 , c ) 14 , d ) 15 , e ) 16
Rationale: "the total number of books the students borrowed is 20 * 2 = 40 . the students who borrowed zero , one , or two books borrowed 10 * 1 + 5 * 2 = 20 books . the 3 students who borrowed at least three books borrowed 40 - 20 = 20 books . if 2 of these students borrowed exactly 3 books , then the maximum that one student could have borrowed is 20 - 6 = 14 books . the answer is c ."
Answer: c
Question (general): a marching band of 240 musicians are to march in a rectangular formation with s rows of exactly t musicians each . there can be no less than 8 musicians per row and no more than 30 musicians per row . how many different rectangular formations g are possible ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) g = 8
Rationale: "the combinations could be { ( 1,240 ) , ( 2,120 ) , ( 3,80 ) , ( 4,60 ) , ( 5,48 ) , ( 6,40 ) , ( 8,30 ) , ( 10,24 ) , ( 12,20 ) , ) 15,16 ) , ( 16,15 ) , ( 20,12 ) , ( 24,10 ) , ( 30,8 ) , ( 40,6 ) , ( 48,5 ) , ( 60,4 ) , ( 80,3 ) , ( 120,2 ) , ( 240,1 ) } of these we are told 8 < = t < = 30 so we can remove these pairs , and we are left only with . { ( 8,30 , ( 10 , 24 ) , ( 12,20 ) , ( 15,16 ) , ( 16,15 ) , ( 20,12 ) , ( 24,10 ) , ( 30,8 ) } hence 8 . . e"
Answer: e
Question (physics): the reciprocal of the hcf and lcm of two are 1 / 15 and 1 / 312 . if one of the number is 24 then other no . is
Options: a ) 126 , b ) 136 , c ) 146 , d ) 156 , e ) 195
Rationale: "reciprocal of the hcf and lcm of two are 1 / 15 and 1 / 312 so , hcf = 15 , lcm = 312 lcm * hcf = product of two numbers = a * b = > b = lcm * hcf / a so , other = 15 * 312 / 24 = 195 answer : e"
Answer: e
Question (gain): x and y start a business with rs . 3000 and rs . 6000 respectively . hoe should they share their profits at the end of one year ?
Options: a ) 3 : 1 , b ) 2 : 1 , c ) 1 : 2 , d ) 1 : 3 , e ) 1 : 4
Rationale: explanation : they should share the profits in the ratio of their investments . the ratio of the investments made by x and y = 3000 : 6000 = > 1 : 2 answer : c
Answer: c
Question (general): in a fuel station the service costs $ 1.75 per car , every liter of fuel costs 0.65 $ . assuming that a company owns 12 cars and that every fuel tank contains 60 liters and they are all empty , how much money total will it cost to fuel all cars ?
Options: a ) 320 $ , b ) 389 $ , c ) 420 $ , d ) 450 $ , e ) 489 $
Rationale: total cost = ( 1.75 * 12 ) + ( 0.65 * 12 * 60 ) = 489 hence answer will be ( e )
Answer: e
Question (general): set x consists of 10 integers and has median of 10 and a range of 10 . what is the value of the greatest possible integer that can be present in the set ?
Options: a ) 32 , b ) 37 , c ) c . 40 , d ) 20 , e ) 50
Rationale: "note that both median and range do not restrict too many numbers in the set . range is only concerned with the smallest and greatest . median only cares about the middle . quick check of each option starting from the largest : ( e ) 50 range of 20 means the smallest integer will be 30 . so 20 can not lie in between and hence can not be the median . ( d ) 43 range of 20 means the smallest integer will be 23 . so 20 can not lie in between and hence can not be the median . ( c ) 40 range of 20 means the smallest integer will be 20 . 20 can lie in between such as : 20 , 20 , 20 , 20 , 20 , 20 , 20 , 20 , 20 , 20 this is possible . hence it is the greatest such number . answer ( d )"
Answer: d
Question (physics): a boy is traveling from his house to school at 3 km / hr and reached school 9 minutes late . next day he traveled 4 km / hr and reached 6 minutes early . then find the distance between house and school ?
Options: a ) 3 km , b ) 4.5 km , c ) 5 km , d ) 2.5 km , e ) 6 km
Rationale: "let distance be x s 1 = 3 km / hr s 2 = 4 km / hr t 1 = x / 3 hr t 2 = x / 4 hr difference in time = 9 + 6 = 15 m = 1 / 4 hr ( x / 3 ) - ( x / 4 ) = 1 / 4 x = 3 km answer is a"
Answer: a
Question (general): a student chose a number , multiplied it by 2 , then subtracted 140 from the result and got 102 . what was the number he chose ?
Options: a ) 90 , b ) 100 , c ) 120 , d ) 160 , e ) 121
Rationale: "solution : let x be the number he chose , then 2 * x * 140 = 102 2 x = 242 x = 121 correct answer e"
Answer: e
Question (physics): a can do a work in 15 days and b in 20 days . if they work on it together for 6 days , then the fraction of the work that is left is
Options: a ) 8 / 17 , b ) 7 / 15 , c ) 3 / 10 , d ) 8 / 15 , e ) 1 / 4
Rationale: "person ( a ) ( b ) ( a + b ) time - ( 15 ) ( 20 ) ( - ) rate - ( 20 ) ( 15 ) ( 35 ) work - ( 300 ) ( 300 ) ( 300 ) therefore a + b requires ( 300 / 35 ) days to complete entire work for 1 st 4 days they work 35 * 6 = 210 remaining work is 300 - 210 = 90 remaining fraction of work is = 90 / 300 = 3 / 10 answer c"
Answer: c
Question (general): a certain quantity is measured on two different scales , the p - scale and the s - scale , that are related linearly . measurements on the p - scale of 6 and 24 correspond to measurements on the s - scale of 30 and 60 , respectively . what measurement on the p - scale corresponds to a measurement of 100 on the s - scale ?
Options: a ) 20 , b ) 36 , c ) 48 , d ) 60 , e ) 84
Rationale: first , we have to understand what linearly means . it ' s not a straight ratio ( since 6 : 30 does not equal 24 : 60 ) . we need to look at the increases in each measurement to see what the scalar actually is . from 6 to 24 we have an increase of 18 . from 30 to 60 we have an increase of 30 . therefore , the increase ratio is 18 : 30 or 3 : 5 . in other words , for every 3 that p increases , s increases by 5 . we know that s is 100 . to get from 60 to 100 , we went up by 40 , or 8 jumps of 5 ; therefore , p will go up by 8 jumps of 3 . 24 + 8 ( 3 ) = 24 + 24 = 48 = c
Answer: c
Question (general): set a contains all the even numbers between 2 and 50 inclusive . set b contains all the even numbers between 102 and 150 inclusive . what is the difference between the sum of elements of set b and the sum of the elements of set a ?
Options: a ) 2500 , b ) 2550 , c ) 5050 , d ) 6275 , e ) 11325
Rationale: "set a contains 2,4 , 6 . . . 50 set b contains 102 , 104 , 106 . . . 150 number of terms in each set = 25 difference between corresponding terms in set a and b = 100 difference between sum of set b and set a = 100 * 25 = 2500 answer a"
Answer: a
Question (general): a whale goes on a feeding frenzy that lasts for 12 hours . for the first hour he catches and eats 30 kilos of plankton . in every hour after the first , it consumes 2 kilos of plankton more than it consumed in the previous hour . if by the end of the frenzy the whale will have consumed a whopping accumulated total 600 kilos of plankton , how many kilos did he consume on the 10 th hour ?
Options: a ) 36 , b ) 45 , c ) 28 , d ) 48 , e ) 52
Rationale: therefor in 10 th hour he consume x + 2 * 9 = 30 + 18 = 48 correct option is d
Answer: d
Question (general): the average age of 19 students in a group is 17 years . when teacher ’ s age is included to it , the average increases by one . what is the teacher ’ s age in years ?
Options: a ) 36 , b ) 37 , c ) 38 , d ) can not be determined , e ) none of these
Rationale: "age of the teacher = ( 20 × 18 – 19 × 17 ) years = 37 years . answer b"
Answer: b
Question (general): machine x takes 10 hours longer than machine y to produce 1080 widgets . machine y produces 20 percent more widgets in an hour than machine x does in an hour . how many widgets per hour does machine x produce
Options: a ) 100 , b ) 65 , c ) 25 , d ) 11 , e ) 18
Rationale: "machine y produces 20 percent more widgets in an hour than machine x does in an hour . so if machine x produces 100 widgets , then machine y produces 120 widgets . ratio of 120 / 100 = 6 / 5 . this is their speed of work ( y : x ) . i . e . speed of their work ( x : y ) = 5 / 6 now , time is inversely proportional to speed . hence the ratio of the time spent ( x : y ) = 6 / 5 let us assume that they spend 6 x and 5 x hours . given that 6 x - 5 x = 10 so , x = 10 . hence 6 x = 6 * 10 = 60 hours . hence x takes 120 hours to produce 1080 widgets . so , in 1 hour , it can produce ( 1 * 1080 ) / 60 = 18 hence option ( e ) ."
Answer: e
Question (physics): a train is running at a speed of 40 km / hr and it crosses a post in 16.2 seconds . what is the length of the train ?
Options: a ) 190 metres , b ) 160 metres , c ) 200 metres , d ) 120 metres , e ) 250 metres
Rationale: "speed of the train , v = 40 km / hr = 40000 / 3600 m / s = 400 / 36 m / s time taken to cross , t = 16.2 s distance covered , d = vt = ( 400 / 36 ) ã — 16.2 = 160 m distance covered is equal to the length of the train = 160 m correct answer is 160 metres b"
Answer: b
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 30 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 ) 82 % , e ) 95 %
Rationale: find the product of the two increases : ( 14 / 10 ) * ( 13 / 10 ) which is 1.82 and a 82 % increase . d
Answer: d
Question (gain): if 4 % more is gained by selling an article for rs . 350 than by selling it for rs . 340 , the cost of the article is
Options: a ) 250 , b ) 231 , c ) 200 , d ) 288 , e ) 111
Rationale: "explanation : let c . p . be rs . x . then , 4 % of x = 350 - 340 = 10 x / 25 = 10 = > x = 250 answer : a"
Answer: a
Question (gain): on increasing the price of t . v . sets by 50 % , their sale decreases by 20 % . what is the effect on the revenue receipts of the shop ?
Options: a ) 4 , b ) 5 , c ) 56 , d ) 20 , e ) 7
Rationale: "explanation : let the price be = rs . 100 , and number of units sold = 100 then , sale value = rs . ( 100 × 100 ) = rs . 10000 new sale value = rs . ( 150 × 80 ) = rs . 12000 increase % = 2000 / 10000 × 100 = 20 % answer : d"
Answer: d
Question (general): p software has coding line 5 % more than n , n software has coding line 4 / 13 more than m . m software has 78 lines of coding . find p lines .
Options: a ) 106 , b ) 107 , c ) 108 , d ) 109 , e ) 110
Rationale: "m s / w has 78 line of code n s / w has = 78 + 78 * 4 / 13 = 102 line of code p s / w 5 % more n ' code 102 + 5.1 = 107.1 or 107 line of code answer : b"
Answer: b
Question (gain): a , b and c started a business with capitals of rs . 8000 , rs . 10000 and rs . 12000 respectively . at the end of the year , the profit share of b is rs . 1600 . the difference between the profit shares of a and c is ?
Options: a ) 288 , b ) 266 , c ) 155 , d ) 600 , e ) 640
Rationale: ratio of investments of a , b and c is 8000 : 10000 : 12000 = 4 : 5 : 6 and also given that , profit share of b is rs . 1600 = > 5 parts out of 15 parts is rs . 1600 now , required difference is 6 - 4 = 2 parts required difference = 2 / 5 ( 1600 ) = rs . 640 answer : e
Answer: e
Question (physics): machine a can finish a job in 4 hours , machine в can finish the job in 12 hours , and machine с can finish the job in 6 hours . how many hours will it take for a , b , and с together to finish the job ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: "the combined rate is 1 / 4 + 1 / 12 + 1 / 6 = 1 / 2 of the job per hour . the time to complete the job is 2 / 1 = 2 hours . the answer is b ."
Answer: b
Question (geometry): a rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered . if the area of the field is 720 sq . feet , how many feet of fencing will be required ?
Options: a ) 34 , b ) 92 , c ) 68 , d ) 88 , e ) none
Rationale: "explanation we have : l = 20 ft and lb = 720 sq . ft . so , b = 36 ft . length of fencing = ( l + 2 b ) = ( 20 + 72 ) ft = 92 ft . answer b"
Answer: b
Question (general): positive integer y is 50 percent of 50 percent of positive integer x , and y percent of x equals 4 . what is the value of x ?
Options: a ) 40 , b ) 100 , c ) 200 , d ) 400 , e ) 600
Rationale: "y = 0.5 * 0.5 * x = x / 4 y % * x = 4 ( y / 100 ) * x = 4 ( x / 400 ) * x = 4 x ^ 2 = 4 * 400 x = 40 the answer is a ."
Answer: a
Question (physics): walking with 3 / 5 of my usual speed , i miss the bus by 5 minutes . what is my usual time ?
Options: a ) 16 min , b ) 26 min , c ) 34 min , d ) 20 min , e ) 15 min
Rationale: "speed ratio = 1 : 3 / 5 = 5 : 3 time ratio = 3 : 5 1 - - - - - - - - 5 3 - - - - - - - - - ? è 15 answer : e"
Answer: e
Question (general): [ ( 3.241 x 14 ) / 100 ] = ?
Options: a ) 0.045374 , b ) 4.5388 , c ) 453.88 , d ) 473.88 , e ) none of these
Rationale: "answer multiplying 3.241 x 14 = 4.5374 now divide 4.5374 by 100 so , 4.5374 ÷ 100 = 0.045374 ∴ shift the decimal two places to the left as 100 correct option : a"
Answer: a
Question (gain): a person borrows rs . 4000 for 2 years at 4 % p . a . simple interest . he immediately lends it to another person at 6 p . a for 2 years . find his gain in the transaction per year .
Options: a ) 100 rs , b ) 80 rs , c ) 160 rs , d ) 180 rs , e ) 200 rs
Rationale: "gain in 2 years = [ ( 4000 * 6 * 2 ) / 100 ] - [ ( 4000 * 4 * 2 ) / 100 ] 480 - 320 = 160 gain in 1 year = ( 160 / 2 ) = 80 rs answer : b"
Answer: b
Question (general): 365 : 90 : : 623 : ?
Options: a ) 36 , b ) 45 , c ) 123 , d ) 63 , e ) none of these
Rationale: "365 - - - 3 * 6 * 5 . . . . 90 623 - - - 6 * 2 * 3 . . . . 36 answer a"
Answer: a
Question (geometry): the volumes of two cones are in the ratio 1 : 20 and the radii of the cones are in the ratio of 1 : 2 . what is the length of the wire ?
Options: a ) 4 : 5 , b ) 4 : 9 , c ) 4 : 2 , d ) 5 : 2 , e ) 2 : 8
Rationale: "the volume of the cone = ( 1 / 3 ) π r 2 h only radius ( r ) and height ( h ) are varying . hence , ( 1 / 3 ) π may be ignored . v 1 / v 2 = r 1 ^ 2 . h 1 / r 2 ^ 2 . h 2 = > 1 / 20 = ( 1 ) ^ 2 h 1 / ( 2 ) ^ 2 h 2 = > h 1 / h 2 = 4 / 5 i . e . h 1 : h 2 = 4 : 5 answer : a"
Answer: a
Question (general): let p and q be the root of the quadratic equation x 2 - ( ∝ - 2 ) x - ∝ - 1 = 0 . what is the minimum possible value of p 2 + q 2 ?
Options: a ) 0 , b ) 3 , c ) 4 , d ) 5 , e ) 6
Rationale: explanation : p + q = α – 2 and pq = – α – 1 ( p + q ) 2 = p 2 + q 2 + 2 pq , thus ( α – 2 ) 2 = p 2 + q 2 + 2 ( – α – 1 ) p 2 + q 2 = α 2 – 4 α + 4 + 2 α + 2 p 2 + q 2 = α 2 – 2 α + 6 p 2 + q 2 = α 2 – 2 α + 1 + 5 p 2 + q 2 = ( α – 1 ) 2 + 5 thus , minimum value of p 2 + q 2 is 5 . answer : d
Answer: d
Question (geometry): in a rectangular coordinate system , what is the area of a rhombus whose vertices have the coordinates ( 0 , 3.5 ) , ( 9 , 0 ) , ( 0 , - 3.5 ) , ( - 9 , 0 ) ?
Options: a ) 63 , b ) 88 , c ) 112 , d ) 116 , e ) 120
Rationale: "ares of rhombus = 1 / 2 * d 1 * d 2 length of 1 st diagonal = 9 + 9 = 18 length of 2 nd diagonal = 3.5 + 3.5 = 7 area = 1 / 2 * 18 * 7 = 63 a is the answer"
Answer: a
Question (geometry): each of the 35 points is placed either inside or on the surface of a perfect sphere . if 88 % or fewer of the points touch the surface , what is the maximum number of segments which , if connected from those points to form chords , could be the diameter of the sphere ?
Options: a ) 7 , b ) 11 , c ) 13 , d ) 14 , e ) 15
Rationale: "maximum number of points on the surface is 88 % * 35 = 30.8 . . . or 30 since it has to be an integer now note that if two points form a diameter , they can not be part of any other diameter . so in the best case we can pair up the points we have 30 points , so at best we can form 15 pairs ( 30 ) . so , answer is ( e )"
Answer: e
Question (general): 6 years ago , p was half of q in age . if the ratio of their present ages is 3 : 4 , what will be the total of their present ages
Options: a ) a ) 21 , b ) b ) 34 , c ) c ) 45 , d ) d ) 25 , e ) e ) 26
Rationale: explanation : let the present age of p and q be 3 x and 4 x respectively . six years ago , p was half of q in age = > 2 ( 3 x – 6 ) = ( 4 x – 6 ) = > 6 x – 12 = 4 x – 6 = > 2 x = 6 = > x = 3 7 x = 7 * 3 = 21 answer : option a
Answer: a