regisss/math_qa huggingface dataset:
Problem: the sum of the squares of the first 15 positive integers ( 1 ^ 2 + 2 ^ 2 + 3 ^ 2 + . . . + 15 ^ 2 ) is equal to 1290 . what is the sum of the squares of the second 15 positive integers ( 16 ^ 2 + 17 ^ 2 + 18 ^ 2 + . . . + 30 ^ 2 ) ?
Options: a ) 2480 , b ) 3490 , c ) 6785 , d ) 8265 , e ) 9255
Rationale: you ' ll never need a formula for the sums of squares on the actual gmat . you do n ' t need to use that formula here , though it ' s not all that straightforward to solve without one . two different approaches : 16 ^ 2 + 17 ^ 2 + 18 ^ 2 + . . . + 30 ^ 2 = ( 15 + 1 ) ^ 2 + ( 15 + 2 ) ^ 2 + ( 15 + 3 ) ^ 2 + . . . + ( 15 + 15 ) ^ 2 now we can expand each square ; they are all in the ( x + y ) ^ 2 = x ^ 2 + 2 xy + y ^ 2 pattern . = ( 15 ^ 2 + 2 * 15 + 1 ^ 2 ) + ( 15 ^ 2 + 4 * 15 + 2 ^ 2 ) + ( 15 ^ 2 + 6 * 15 + 3 ^ 2 ) + . . . + ( 15 ^ 2 + 30 * 15 + 15 ^ 2 ) now we have fifteen 15 ^ 2 terms , so adding these gives 15 * 15 ^ 2 = 15 ^ 3 = 3375 . we also have the entire sum 1 ^ 2 + 2 ^ 2 + . . . + 15 ^ 2 , which we know is equal to 1240 . finally adding the middle terms , we have : 2 * 15 + 4 * 15 + 6 * 15 + . . . + 30 * 15 = 15 ( 2 + 4 + 6 + . . . . + 30 ) = 15 * 2 * ( 1 + 2 + 3 + . . . + 15 ) = 15 * 2 * 8 * 15 = 3600 so the sum must be 3375 + 1240 + 3600 = 8215 alternatively , we can use a different factoring pattern . we want to find the value of 30 ^ 2 + 29 ^ 2 + . . . + 17 ^ 2 + 16 ^ 2 . well if we subtract 15 ^ 2 + 14 ^ 2 + . . . . + 2 ^ 2 + 1 ^ 2 from this , the answer will be 1240 less than what we want to find . so if we can find the value of 30 ^ 2 + 29 ^ 2 + . . . + 17 ^ 2 + 16 ^ 2 - ( 15 ^ 2 + 14 ^ 2 + . . . . + 2 ^ 2 + 1 ^ 2 ) then we can add 1240 to get the answer . now grouping the terms above to get differences of squares , we have = ( 30 ^ 2 - 15 ^ 2 ) + ( 29 ^ 2 - 14 ^ 2 ) + . . . + ( 16 ^ 2 - 1 ^ 2 ) and factoring each of these using x ^ 2 - y ^ 2 = ( x + y ) ( x - y ) , we have = 45 * 15 + 43 * 15 + 41 * 15 + . . . + 17 * 15 = 15 ( 45 + 43 + 41 + . . . + 17 ) in brackets we have an equally spaced sum with fifteen terms , which we can evaluate using the familiar formula . so the above equals 15 * 15 * 62 / 2 = 6975 and adding back the 1290 , we get the answer of 8265 . ( ans d )
Answer: d
Problem: if the population of a certain country increases at the rate of one person every 30 seconds , by how many persons does the population increase in 1 hour ?
Options: a ) 100 , b ) 120 , c ) 150 , d ) 180 , e ) 160
Rationale: "answer = 2 * 60 = 120 answer is b"
Answer: b
Problem: the total age of a and b is 10 years more than the total age of b and c . c is how many year younger than
Options: a ) 14 years , b ) 12 years , c ) 56 years , d ) 10 years , e ) 55 years
Rationale: "given that a + b = 10 + b + c = > a â € “ c = 10 + b â € “ b = 10 = > c is younger than a by 10 years answer : d"
Answer: d
Problem: at 1 : 00 pm , there were 10.0 grams of bacteria . the bacteria increased to x grams at 4 : 00 pm , and 22.5 grams at 7 : 00 pm . if the amount of bacteria present increased by the same fraction during each of the 3 - hour periods , how many grams of bacteria were present at 4 : 00 pm ?
Options: a ) 14.4 , b ) 14.7 , c ) 15.0 , d ) 15.3 , e ) 15.6
Rationale: "let x be the factor by which the bacteria increases every three hours . at 4 : 00 pm , the amount of bacteria was 10 x and at 7 : 00 pm it was 10 x ^ 2 . 10 x ^ 2 = 22.5 x ^ 2 = 2.25 x = 1.5 at 4 : 00 pm , the amount of bacteria was 10 ( 1.5 ) = 15 grams . the answer is c ."
Answer: c
Problem: if 6 x ^ 2 + x - 12 = ( ax + w ) ( cx + d ) , then | a | + | w | + | c | + | d | = for a complete solution and more practice problems , see this blog : http : / / magoosh . com / gmat / 2012 / algebra - on . . . to - factor /
Options: a ) 10 , b ) 12 , c ) 15 , d ) 18 , e ) 20
Rationale: 6 x ^ 2 + x - 12 = 6 x ^ 2 + 9 x - 8 x - 12 = > 3 x ( 2 x + 3 ) - 4 ( 2 x + 3 ) = > ( 2 x + 3 ) ( 3 x - 4 ) = ( ax + w ) ( cx + d ) hence a = 2 , w = c = 3 , d = - 4 so , 2 + 3 + 3 + | - 4 | = 2 + 3 + 3 + 4 = 12 answer b .
Answer: b
Problem: the average of first four prime numbers greater than 10 is ?
Options: a ) 10 , b ) 20 , c ) 15 , d ) 40 , e ) 50
Rationale: "11 + 13 + 17 + 19 = 60 / 4 = 15 answer : c"
Answer: c
Problem: a and b started a business investing rs . 60,000 and rs 45,000 respectively . in what ratio the profit earned after 2 years be divided between a and b respectively ?
Options: a ) 3 : 2 , b ) 4 : 3 , c ) 18 : 20 , d ) 1 : 4 , e ) 18 : 4
Rationale: "a : b = 60000 : 45000 = 4 : 3 answer : b"
Answer: b
Problem: the radius of a wheel is 22.4 cm . what is the distance covered by the wheel in making 400 resolutions .
Options: a ) 561.68 m , b ) 562.68 m , c ) 563.68 m , d ) 564.68 m , e ) 565.68 m
Rationale: "in one resolution , the distance covered by the wheel is its own circumference . distance covered in 400 resolutions . = 500 * 2 * 22 / 7 * 22.4 = 56268 cm = 562.68 m answer : b"
Answer: b
Problem: what is the remainder when you divide 2 ^ 200 - 2 by 7 ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: answer is b . this is very similar to the 32 ^ 32 ^ 32 - 2 divided by 7 question .
Answer: b
Problem: 5 % people of a village in sri lanka died by bombardment , 15 % of the remainder left the village on account of fear . if now the population is reduced to 3553 , how much was it in the beginning ?
Options: a ) 4400 , b ) 3500 , c ) 8700 , d ) 9000 , e ) 8000
Rationale: "x * ( 95 / 100 ) * ( 85 / 100 ) = 3553 x = 4400 answer a"
Answer: a
Problem: x and y invested in a business . they earned some profit which they divided in the ratio of 2 : 3 . if x invested rs . 40,000 , the amount invested by y is ?
Options: a ) 60229 , b ) 62887 , c ) 60000 , d ) 61298 , e ) 61291
Rationale: "suppose y invested rs . y . then 40000 / y = 2 / 3 or y = 60000 . answer : c"
Answer: c
Problem: two cubes of their volumes in the ratio 27 : 125 . the ratio of their surface area is :
Options: a ) 1.5 : 5 , b ) 2 : 5 , c ) 3 : 5 , d ) 1 : 5 , e ) 4 : 5
Rationale: "the ratio of their surface area is 27 : 125 3 : 5 answer is c ."
Answer: c
Problem: one night 16 percent of the female officers on a police force were on duty . if 160 police officers were on duty that night and half of these were female officers , how many female officers were on the police force ?
Options: a ) 500 , b ) 650 , c ) 800 , d ) 950 , e ) 1100
Rationale: "let x be the number of female police officers on the police force . the number of female police officers on duty was 80 . 0.16 x = 80 x = 500 the answer is a ."
Answer: a
Problem: steve traveled the first 2 hours of his journey at 30 mph and the remaining 3 hours of his journey at 80 mph . what is his average speed for the entire journey ?
Options: a ) 60 mph , b ) 56.67 mph , c ) 53.33 mph , d ) 64 mph , e ) 66.67 mph
Rationale: "distance traveled in 2 hours = 2 * 30 = 60 m distance traveled in 3 hours = 3 * 80 = 240 m total distance covered = 240 + 60 = 300 m total time = 2 + 3 = 5 h hence avg speed = total distance covered / total time taken = 300 / 5 = 60 mph answer : a"
Answer: a
Problem: in what proportion must flour at $ 0.8 per pound be mixed with flour at $ 0.9 per pound so that the mixture costs $ 0.835 per pound ?
Options: a ) 1 : 3 , b ) 1 : 2 , c ) 1 : 1 , d ) 2 : 1 , e ) 3 : 1
Rationale: "using weighted average method : let x be the proportion in which the $ 0.8 per pound flour is mixed with $ 0.9 per pound flour . thus 0.8 * x + 0.9 * ( 1 - x ) = 0.835 0.9 – 0.1 x = 0.835 x = 0.65 thus ratio of both flours is 2 : 1 d"
Answer: d
Problem: the average of 5 numbers id 27 . if one number is excluded , the average becomes 25 . the excluded number is
Options: a ) 25 , b ) 27 , c ) 30 , d ) 35 , e ) none
Rationale: solution excluded number = ( 27 x 5 ) - ( 25 x 4 ) = 135 - 100 = 35 . answer d
Answer: d
Problem: a batsman makes a score of 86 runs in the 17 th inning and thus increases his averages by 3 . find his average after 17 th inning ?
Options: a ) 19 , b ) 29 , c ) 38 , d ) 49 , e ) 59
Rationale: "let the average after 17 th inning = x then average after 16 th inning = ( x - 3 ) therefore 16 ( x - 3 ) + 86 = 17 x therefore x = 38 answer : c"
Answer: c
Problem: a and b can do a piece of work in 6 days . with the help of c they finish the work in 4 days . c alone can do that piece of work in ?
Options: a ) 10 days , b ) 20 days , c ) 25 days , d ) 15 days , e ) 45 days
Rationale: "c = 1 / 5 – 1 / 6 = 1 / 30 = > 30 days answer : b"
Answer: b
Problem: a person deposits 1 / 16 of his income as provident fund and 1 / 15 of the remaining as insurance premium . if he spends 5 / 7 of the balance on domestic needs and deposits an amount of rs . 50 in the bank , his total income would be
Options: a ) 150 , b ) 200 , c ) 250 , d ) 300 , e ) 350
Rationale: ( 1 - ( 1 / 16 ) ) ( 1 - ( 1 / 15 ) ) ( 1 - ( 5 / 7 ) ) of income = 50 hence income = 200 answer : b
Answer: b
Problem: what is 3 / 5 of 5 / 7 of 2 / 3 ?
Options: a ) 2 / 7 , b ) 3 / 8 , c ) 9 / 16 , d ) 5 / 8 , e ) 16 / 9
Rationale: "3 / 5 * 5 / 7 * 2 / 3 = 2 / 7 answer : a"
Answer: a
Problem: in a survey of 350 employees , 104 of them are uninsured , 54 work part time , and 12.5 percent of employees who are uninsured work part time . if a person is to be randomly selected from those surveyed , what is the probability that the person will neither work part time nor be uninsured ?
Options: a ) 7 / 12 , b ) 8 / 41 , c ) 9 / 348 , d ) 1 / 8 , e ) 41 / 70
Rationale: "- - - - - - - - - ui - - - - - - - - - - - - - - - - nui - - - - - - - total pt - - - - ( 12.5 / 100 ) * 104 = 13 - - - - - - - - - - - - - 54 npt - - - 104 - 13 - - - - - - - - - - - - - - x - - - - - - - - 296 total - - 104 - - - - - - - - - - - - - - - - - - - - - - - - - - - - 350 we have to find not part time and not uninsured . in other words not part time and insured = x / 350 = ( 296 - 104 + 13 ) / 350 = 41 / 70 answer is e ."
Answer: e
Problem: a dishonest person wants to make a profit on the selling of milk . he would like to mix water ( costing nothing ) with milk costing 33 $ per litre so as to make a profit of 20 % on cost when he sells the resulting milk and water mixture for 36 $ . in what ratio should he mix the water and milk ?
Options: a ) 1 : 20 , b ) 1 : 10 , c ) 1 : 8 , d ) 1 : 4 , e ) 1 : 2
Rationale: "he needs to make a profit of 20 % , which means he needs to sell the milk + water combination for 39.6 ( 20 % of 33 + 33 ) now , he plans to sell 1 liter of this solution of milk and water for 36 . which means he needs to sell 1.1 liters of this solution to earn 39.6 . ( 36 for 1 liter and so 39.6 for 1.1 liters ) hence the water added would be . 1 liter . therefore the ratio of water to milk is . 1 / 1 = 1 / 10 answer : b"
Answer: b
Problem: a train covers a distance of 12 km in 10 min . if it takes 10 sec to pass a telegraph post , then the length of the train is ?
Options: a ) 200 , b ) 288 , c ) 120 , d ) 776 , e ) 991
Rationale: "speed = ( 12 / 10 * 60 ) km / hr = ( 72 * 5 / 18 ) m / sec = 20 m / sec . length of the train = 20 * 10 = 200 m . answer : a"
Answer: a
Problem: two employees a and b are paid a total of rs . 550 per week by their employer . if a is paid 150 percent of the sum paid to b , how much is b paid per week ?
Options: a ) 130 , b ) 140 , c ) 150 , d ) 220 , e ) 280
Rationale: let the amount paid to a per week = x and the amount paid to b per week = y then x + y = 550 but x = 150 % of y = 150 y / 100 = 15 y / 10 ∴ 15 y / 10 + y = 550 ⇒ y [ 15 / 10 + 1 ] = 550 ⇒ 25 y / 10 = 550 ⇒ 25 y = 5500 ⇒ y = 5500 / 25 = rs . 220 d )
Answer: d
Problem: of the families in city x in 1991 , 60 percent owned a personal computer . the number of families in city x owning a computer in 1993 was 20 percent greater than it was in 1991 , and the total number of families in city x was 9 percent greater in 1993 than it was in 1991 . what percent of the families in city x owned a personal computer in 1993 ?
Options: a ) 59 % , b ) 55 % , c ) 63 % , d ) 66 % , e ) 68 %
Rationale: "say a 100 families existed in 1991 then the number of families owning a computer in 1991 - 60 number of families owning computer in 1993 = 60 * 120 / 100 = 72 number of families in 1993 = 109 the percentage = 72 / 109 * 100 = 66 % . answer : d"
Answer: d
Problem: at a local appliance manufacturing facility , the workers received a 10 % hourly pay raise due to extraordinary performance . if one worker decided to reduce the number of hours that he worked so that his overall pay would remain unchanged , by approximately what percent would he reduce the number of hours that he worked ?
Options: a ) 83 % , b ) 80 % , c ) 20 % , d ) 17 % , e ) 9 %
Rationale: let ' s say he works usually 10 hours and earns 100 per hour . 10 * 100 = 1000 10 * 110 = 1100 ( this are the new earnings after the raise ) to figure out how much he needs to work with the new salary in order to earn the original 1000 : 1000 / 110 = 9.09 so he can reduce his work by 0.91 hours . which is > 9 % . answer e
Answer: e
Problem: determine the value of ( 25 / 32 + 50 / 64 ) * 2
Options: a ) 25 / 8 , b ) 25 / 4 , c ) 25 / 6 , d ) 3 , e ) none
Rationale: "solution : both fractions should be reduced before performing arithmetic operations . we get ( 25 / 32 + 2.25 / 2.32 ) * 2 = ( 25 / 32 + 25 / 32 ) * 2 = 2 * ( 25 / 32 ) * 2 = 4 ( 25 / 32 ) = 25 / 8 answer a"
Answer: a
Problem: a cistern has a leak which would empty the cistern in 20 minutes . a tap is turned on which admits 10 liters a minute into the cistern , and it is emptied in 24 minutes . how many liters does the cistern hold ?
Options: a ) 4800 , b ) 1200 , c ) 1680 , d ) 1100 , e ) 1300
Rationale: "1 / x - 1 / 20 = - 1 / 24 x = 120 120 * 10 = 1200 answer : b"
Answer: b
Problem: a school has received 40 % of the amount it needs for a new building by receiving a donation of $ 600 each from people already solicited . people already solicited represent 40 % of the people from whom the school will solicit donations . how much average contribution is requited from the remaining targeted people to complete the fund raising exercise ?
Options: a ) $ 200 , b ) $ 300 , c ) $ 100 , d ) $ 400 , e ) $ 250
Rationale: "let us suppose there are 100 people . 40 % of them donated $ 16000 ( 400 * 40 ) $ 16000 is 40 % of total amount . so total amount = 16000 * 100 / 40 remaining amount is 60 % of total amount . 60 % of total amount = 16000 * ( 100 / 40 ) * ( 60 / 100 ) = 24000 this amount has to be divided by 60 ( remaining people are 60 ) so per head amount is 24000 / 60 = $ 400 answer : d"
Answer: d
Problem: in township k each property is taxed at 10 percent of its assessed value . if the assessed value of a property in township k is increased from $ 20,000 to $ 28,000 , by how much will the property tax increase ?
Options: a ) $ 800 , b ) $ 1000 , c ) $ 600 , d ) $ 400 , e ) $ 500
Rationale: increase in house value = $ 28,000 - $ 20,000 = $ 8000 so , tax increase = 10 % of $ 8000 = $ 800 answer : a
Answer: a
Problem: george is 8 years more than christopher and ford is 2 years younger than christopher . the sum of their ages is 60 . find the ages of george .
Options: a ) 25 , b ) 26 , c ) 27 , d ) 28 , e ) 29
Rationale: "christopher age = x george age , y = x + 8 - - - - - - - - - - > ( 1 ) ford age , z = x - 2 - - - - - - - - - - - - > ( 2 ) sum of their ages , x + y + z = 60 - - - - > ( 3 ) substitute z and y values in equation ( 3 ) therefore , x + ( x + 8 ) + ( x - 2 ) = 60 = > 3 x + 8 - 2 = 60 = > 3 x = 60 - 6 = > 3 x = 54 = > x = 54 / 3 x = 18 ( christopher ' s age ) substitute x value in equation 1 & 2 y = x + 8 y = 18 + 8 y = 26 answer : b"
Answer: b
Problem: if 0.25 : x : : 2 : 6 , then x is equal to
Options: a ) 0.65 , b ) 0.25 , c ) 0.75 , d ) 0.85 , e ) none
Rationale: "sol . ( x × 2 ) = ( 0.25 × 6 ) ⇒ x = 1.5 / 2 = 0.75 . answer c"
Answer: c
Problem: a solution contains 8 parts of water for every 7 parts of lemonade syrup . how many parts of the solution should be removed and replaced with water so that the solution will now contain 10 % lemonade syrup ?
Options: a ) 1.5 , b ) 1.75 , c ) 2.14 , d ) 2.34 , e ) 2.64
Rationale: "let the total solution is 150 l with 80 l water 70 l syrup . to make 10 % syrup solution , the result solution must have 135 l syrup and 15 l syrup . therefore we are taking 55 l of syrup from initial solution and replacing with water . using urinary method : 70 l syrup in 150 l solution 55 l syrup in 117.9 l solution we started by multiplying 10 now to get to the result we need to divide by 55 = > amount of solution to be replaced with water = ( 117.9 / 55 ) = 2.14 . correct option : c"
Answer: c
Problem: 287 x 287 + 269 x 269 - 2 x 287 x 269 = ?
Options: a ) 534 , b ) 446 , c ) 354 , d ) 324 , e ) none of them
Rationale: = a ^ 2 + b ^ 2 - 2 ab where a = 287 and b = 269 = ( a - b ) ^ 2 = ( 287 - 269 ) ^ 2 = ( 18 ) ^ 2 = 324 answer is d
Answer: d
Problem: worker a takes 12 hours to do a job . worker b takes 15 hours to do the same job . how long it take both a & b , working together but independently , to do the same job ?
Options: a ) 40 days , b ) 40 / 9 days , c ) 39 / 9 days , d ) 30 / 9 days , e ) 20 / 3 days
Rationale: "a ' s one hour work = 1 / 12 . b ' s one hour work = 1 / 15 . ( a + b ) ' s one hour work = 1 / 12 + 1 / 15 = 9 / 60 = 3 / 20 . both a & b can finish the work in 20 / 3 days e"
Answer: e
Problem: jamal had a bag of marbles . he had 12 yellow , 10 blue , 5 green and 1 black marbles . he decided to randomly select a ball from the bag . what is the proabilitly that he will draw a black marble ?
Options: a ) 10 / 28 , b ) 1 / 28 , c ) 12 / 28 , d ) 28 / 1 , e ) 582 / 695
Rationale: b
Answer: b
Problem: the visitors of a modern art museum who watched a certain picasso painting were asked to fill in a short questionnaire indicating whether they had enjoyed looking at the picture and whether they felt they had understood it . according to the results of the survey , all 100 visitors who did not enjoy the painting also did not feel they had understood the painting , and the number of visitors who enjoyed the painting was equal to the number of visitors who felt they had understood the painting . if 3 / 4 of the visitors who answered the questionnaire both enjoyed the painting and felt they had understood the painting , then how many visitors answered the questionnaire ?
Options: a ) 90 , b ) 120 , c ) 160 , d ) 360 , e ) 400
Rationale: "if we exclude those cases and take the question at face value , then it seems straightforward . group # 1 = ( did n ' t like , did n ' t understand ) = 100 group # 2 = ( likeunderstood ) = 3 / 4 ( 1 / 4 ) n = 100 n = 400 answer = ( e )"
Answer: e
Problem: find the compound interest on rs . 10,000 in 2 years at 4 % per annum , the interest being compounded half - yearly .
Options: a ) rs 824.32 , b ) rs 822.32 , c ) rs 700 , d ) rs 900 , e ) rs 624.00
Rationale: "principal = rs . 10000 ; rate = 2 % per half - year ; time = 2 years = 4 half - years . amount = rs [ 10000 * ( 1 + ( 2 / 100 ) ) 4 ] = rs ( 10000 * ( 51 / 50 ) * ( 51 / 50 ) * ( 51 / 50 ) * ( 51 / 50 ) ) = rs . 10824.32 . : . c . i . = rs . ( 10824.32 - 10000 ) = rs . 824.32 . answer a rs 824.32"
Answer: a
Problem: in an acoustics class , 120 students are male and 100 students are female . 25 % of the male students and 20 % of the female students are engineering students . 20 % of the male engineering students and 25 % of the female engineering students passed the final exam . what percentage of engineering students passed the exam ?
Options: a ) 5 % , b ) 10 % , c ) 16 % , d ) 22 % , e ) none of these
Rationale: explanation : the number of female engineering students in the class is ( 20 / 100 ) × 100 = 20 . now , 25 % of the female engineering students passed the final exam : 25 . hence , the number of female engineering students who passed is 5 . there are 120 male students in the class . and 25 % of them are engineering students . hence , the number of male engineering students is ( 1 / 4 ) × 120 = 30 . now , 20 % of the male engineering students passed the final exam : - 20 . hence , the number of male engineering students who passed is 6 . hence , the total number of engineering students who passed is : ( female engineering students who passed ) + ( male engineering students who passed ) = > 5 + 6 = 11 . the total number of engineering students in the class is : ( number of female engineering students ) + ( number of male engineering students ) = = > 30 + 20 = 50 hence , the percentage of engineering students who passed is : - = > ( total number of engineering students passed / total number of engineering students ) × 100 . = > ( 11 / 50 ) × 100 . = > 22 % . answer : d
Answer: d
Problem: the probability that a computer company will get a computer hardware contract is 2 / 3 and the probability that it will not get a software contract is 5 / 9 . if the probability of getting at least one contract is 4 / 5 , what is the probability that it will get both the contracts ?
Options: a ) 14 / 45 , b ) 4 / 9 , c ) 4 / 5 , d ) 4 / 6 , e ) none of these
Rationale: "explanation : let , a ≡ event of getting hardware contract b ≡ event of getting software contract ab ≡ event of getting both hardware and software contract . p ( a ) = 2 / 3 , p ( ~ b ) = 5 / 9 = > p ( b ) = 1 - ( 5 / 9 ) = 4 / 9 . a and b are not mutually exclusive events but independent events . so , p ( at least one of a and b ) = p ( a ) + p ( b ) - p ( ab ) . = > 4 / 5 = ( 2 / 3 ) + ( 4 / 9 ) - p ( ab ) . = > p ( ab ) = 14 / 45 . hence , the required probability is 14 / 45 . answer : a"
Answer: a
Problem: | 8 - 3 | - | 4 - 6 | = ?
Options: a ) 3 , b ) 2 , c ) 1 , d ) 0 , e ) 4
Rationale: "| 8 - 3 | - | 4 - 6 | = | 5 | - | - 2 | = 5 - 2 = 3 correct answer a"
Answer: a
Problem: if x is an integer and 2.134 × 10 ^ x is less than 240,000 , what is the greatest possible value for x ?
Options: a ) 7 , b ) 6 , c ) 5 , d ) 4 , e ) 3
Rationale: "x is an integer and 2.134 × 10 x is less than 240,000 , what is the greatest possible value for x ? for 2.134 × 10 x is less than 240,000 to remain true , the greatest number is 213,400 , which makes x = 5 c . 5"
Answer: c
Problem: two trains of length 90 m and 100 m are 200 m apart . they start moving towards each other on parallel tracks , at speeds 71 kmph and 89 kmph . after how much time will the trains meet ?
Options: a ) 47 / 10 , b ) 46 / 10 , c ) 45 / 10 , d ) 42 / 10 , e ) 43 / 10
Rationale: "they are moving in opposite directions , relative speed is equal to the sum of their speeds . relative speed = ( 71 + 89 ) * 5 / 18 = 44.4 mps . the time required = d / s = 200 / 44.4 = 45 / 10 sec . answer : c"
Answer: c
Problem: in a class of 40 students , 2 students did not borrow any books from the library , 12 students each borrowed 1 book , 10 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 ) 3 , b ) 5 , c ) 8 , d ) 13 , e ) 15
Rationale: "total # of students = 40 avg # of books per student = 2 total # of books = 40 * 2 = 80 # of student borrowed at least 3 books = 40 - 2 - 12 - 10 = 16 # of books borrowed by above 16 students = 80 - ( 12 * 1 ) - ( 10 * 2 ) = 48 considering that 15 out of above 16 students borrowed only 3 books , # of books borrowed = 15 * 3 = 45 so maximum # of books borrowed by any single student = 48 - 45 = 3 option a"
Answer: a
Problem: of the 20210 employees of the anvil factory , 2 / 7 are journeymen . if half of the journeymen were laid off , what percentage of the total remaining employees would be journeymen ?
Options: a ) 14.3 % , b ) 16.67 % , c ) 33 % , d ) 28.6 % , e ) 49.67 %
Rationale: the exam gives us a number that is easily divisible by 7 to pique our curiosity and tempt us into calculating actual numbers ( also because otherwise the ratio would be incorrect ) . since the question is about percentages , the actual numbers will be meaningless , as only the ratio of that number versus others will be meaningful . nonetheless , for those who are curious , each 1 / 7 portion represents ( 14210 / 7 ) 2,030 employees . this in turn means that 4,060 employees are journeymen and the remaining 10,150 are full time workers . if half the journeymen were laid off , that would mean 1 / 7 of the total current workforce would be removed . this statistic is what leads many students to think that since half the journeymen are left , the remaining journeymen would represent half of what they used to be , which means 1 / 7 of the total workforce . if 1 / 7 of the workforce is journeymen , and 1 / 7 is roughly 14.3 % , then answer choice a should be the right answer . in this case , though , it is merely the tempting trap answer choice . what changed between the initial statement and the final tally ? well , you let go of 1 / 7 of the workforce , so the total number of workers went down . the remaining workers are still 1 / 7 of the initial workers , but the group has changed . the new workforce is smaller than the original group , specifically 6 / 7 of it because 1 / 7 was eliminated . the remaining workers now account for 1 / 7 out of 6 / 7 of the force , which if we multiply by 7 gives us 1 out of 6 . this number as a percentage is answer choice b , 49.67 % . using the absolute numbers we calculated before , there were 4,060 journeymen employees out of 14,210 total . if 2,030 of them are laid off , then there are 2,030 journeyman employees left , but now out of a total of ( 14,210 - 2,030 ) 12,180 employees . 2,030 / 12,180 is exactly 1 / 6 , or 16.67 % . the answer will work with either percentages or absolute numbers , but the percentage calculation will be significantly faster and applicable to any similar situation . the underlying principle of percentages ( and , on a related note , ratios ) can be summed up in the brainteaser i like to ask my students : if you ’ re running a race and you overtake the 2 nd place runner just before the end , what position do you end up in ? the correct answer is 2 nd place . percentages , like ratios and other concepts of relative math , depend entirely on the context . whether 100 % more of something is better than 50 % more of something else depends on the context much more than the percentages quoted . when it comes to percentages on the gmat , the goal is to understand them enough to instinctively not fall into the traps laid out for you . e
Answer: e
Problem: find the total number of prime factors in the expression ( 4 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2
Options: a ) 26 , b ) 22 , c ) 25 , d ) 27 , e ) 33
Rationale: ( 4 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 = ( 2 x 2 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 = 2 ^ 13 x 2 ^ 13 x 7 ^ 5 x 11 ^ 2 = 2 ^ 26 x 7 ^ 5 x 11 ^ 2 total number of prime factors = ( 26 + 5 + 2 ) = 33 . answer is e .
Answer: e
Problem: the radius of a wheel is 22.4 cm . what is the distance covered by the wheel in making 650 resolutions ?
Options: a ) 724 m , b ) 704 m , c ) 915.2 m , d ) 278 m , e ) 927 m
Rationale: "in one resolution , the distance covered by the wheel is its own circumference . distance covered in 650 resolutions . = 650 * 2 * 22 / 7 * 22.4 = 91520 cm = 915.2 m answer : c"
Answer: c
Problem: a man is 22 years older than his son . in two years , his age will be twice the age of his son . the present age of the son is ?
Options: a ) 20 , b ) 25 , c ) 27 , d ) 22 , e ) 91
Rationale: "let the son ' s present age be x years . then , man ' s present age = ( x + 22 ) years . ( x + 22 ) + 2 = 2 ( x + 2 ) x + 24 = 2 x + 4 = > x = 20 . answer : a"
Answer: a
Problem: a can run 2 km distance in 2 min . , while b can run this distance in 2 min 40 sec . by how much distance can a beat b ?
Options: a ) 150 m , b ) 100 m , c ) 500 m , d ) 180 m , e ) 190 m
Rationale: a takes time 2 minutes = 120 sec b takes time 2.40 minutes = 160 sec diffrence = 160 - 120 = 40 sec now we are to find distance covered in 40 sec by b 160 sec = 2000 m 1 sec = 25 / 2 m 40 sec = 40 x 25 / 2 = 500 m answer : c
Answer: c
Problem: stacy and heather are 20 miles apart and walk towards each other along the same route . stacy walks at constant rate that is 2 mile per hour faster than heather ' s constant rate of 5 miles / hour . if heather starts her journey 24 minutes after stacy , how far from the original destination has heather walked when the two meet ?
Options: a ) 5 miles , b ) 7 miles , c ) 9 miles , d ) 10 miles , e ) 12 miles
Rationale: "original distance between s and h = 20 miles . speed of s = 5 + 2 = 7 mph , speed of h = 5 mph . time traveled by h = t hours - - - > time traveled by s = t + 24 / 60 = t + 2 / 5 hours . now , the total distances traveled by s and h = 20 miles - - - > 7 * ( t + 2 / 5 ) + 5 * t = 20 - - - > t = 86 / 60 hours . thus h has traveled for 86 / 60 hours giving you a total distance for h = 5 * 86 / 60 = 7 miles . b is thus the correct answer . p . s . : based on the wording of the question , you should calculatehow far from theoriginal destination has heather walkedwhen the two meet . ' original destination ' for h does not make any sense . original destination for h was situated at a distance of 20 miles ."
Answer: b
Problem: two trains of equal length are running on parallel lines in the same directions at 46 km / hr . and 36 km / hr . the faster trains pass the slower train in 36 seconds . the length of each train is :
Options: a ) 82 m , b ) 50 m , c ) 72 m , d ) 80 m , e ) none of these
Rationale: "explanation : the relative speed of train is 46 - 36 = 10 km / hr = ( 10 x 5 ) / 18 = 25 / 9 m / s 10 × 518 = 259 m / s in 36 secs the total distance travelled is 36 x 25 / 9 = 100 m . therefore the length of each train is = 100 / 2 = 50 m . answer b"
Answer: b
Problem: divide rs . 1400 among a , b and c so that a receives 1 / 3 as much as b and c together and b receives 2 / 3 as a and c together . a ' s share is ?
Options: a ) 346 , b ) 368 , c ) 350 , d ) 337 , e ) 639
Rationale: "a + b + c = 1400 a = 1 / 3 ( b + c ) ; b = 2 / 3 ( a + c ) a / ( b + c ) = 1 / 3 a = 1 / 4 * 1400 = > 350 answer : c"
Answer: c
Problem: if the sum of a number and its square is 72 , what is the number ?
Options: a ) 15 , b ) 26 , c ) 8 , d ) 91 , e ) none of these
Rationale: "let the number be x . then , x + x 2 = 72 ( x + 9 ) ( x - 8 ) = 0 x = 8 answer : c"
Answer: c
Problem: sarah is driving to the airport . after driving at 20 miles per hour for one hour , she realizes that if she continues at that same average rate she will be an hour late for her flight . she then travels 50 miles per hour for the rest of the trip , and arrives 30 minutes before her flight departs . how many miles did she drive in all ?
Options: a ) 140 , b ) 175 , c ) 210 , d ) 70 , e ) 280
Rationale: "after driving at 20 miles per hourfor one hour , this distance left to cover is d - 20 . say this distance is x miles . now , we know that the difference in time between covering this distance at 20 miles per hour and 50 miles per hour is 1 + 1 / 2 = 3 / 2 hours . so , we have that x / 20 - x / 50 = 3 / 2 - - > 5 x / 100 - 2 x / 100 = 3 / 2 - - > 3 x / 100 = 3 / 2 - - > x = 50 . total distance = x + 20 = 70 miles . answer : d"
Answer: d
Problem: there are 13 stations between hyderabad and bangalore . how many second class tickets have to be printed , so that a passenger can travel from any station to any other station ?
Options: a ) 288 , b ) 267 , c ) 261 , d ) 211 , e ) 210
Rationale: "the total number of stations = 15 from 15 stations we have to choose any two stations and the direction of travel ( i . e . , hyderabad to bangalore is different from bangalore to hyderabad ) in 15 p ₂ ways . 15 p ₂ = 15 * 14 = 210 . answer : e"
Answer: e
Problem: a student committee on academic integrity has 42 ways to select a president and vice president from a group of candidates . the same person can not be both president and vice president . how many candidates are there ?
Options: a ) 7 , b ) 8 , c ) 9 , d ) 10 , e ) 11
Rationale: "xc 1 * ( x - 1 ) c 1 = 42 x ^ 2 - x - 42 = 0 ( x - 7 ( x + 6 ) = 0 x = 7 , - 6 - 6 ca n ' t possible . hence 7 should be the answer a"
Answer: a
Problem: the difference between the local value and the face value of 4 in the numeral 98625147 is
Options: a ) 36 , b ) 63 , c ) 34 , d ) 30 , e ) 33
Rationale: "explanation : ( local value of 4 ) - ( face value of 4 ) = ( 40 - 4 ) = 36 a )"
Answer: a
Problem: a boat moves upstream at the rate of 1 km in 30 minutes and down stream 1 km in 12 minutes . then the speed of the current is :
Options: a ) 1.5 kmph , b ) 2 kmph , c ) 3 kmph , d ) 2.5 kmph , e ) 3.5 kmph
Rationale: "rate upstream = ( 1 / 30 * 60 ) = 2 kmph rate down stream = 1 / 12 * 60 = 5 kmph rate of the current = ½ ( 5 - 2 ) = 1.5 kmph answer : a"
Answer: a
Problem: if five machines working at the same rate can do 3 / 4 of a job in 30 minutes , how many minutes would it take two machines working at the same rate to do 3 / 5 of the job ?
Options: a ) 45 , b ) 60 , c ) 75 , d ) 80 , e ) 100
Rationale: using the std formula m 1 d 1 h 1 / w 1 = m 2 d 2 h 2 / w 2 substituting the values we have 5 * 1 / 2 * 4 / 3 = 2 * 5 / 3 * x ( converted 30 min into hours = 1 / 2 ) 10 / 3 = 10 / 3 * x x = 1 hour so 60 minutes answer : b
Answer: b
Problem: a circular racetrack is 3 miles in length and has signs posted to indicate each 1 / 10 mile increment . if a race car starts at a random location on the track and travels exactly one half mile , what is the probability that the car ends within a half mile of the sign indicating 2 1 / 2 miles ?
Options: a ) 1 / 6 , b ) 3 / 10 , c ) 1 / 3 , d ) 1 / 2 , e ) 2 / 3
Rationale: the car ends within a half mile of the sign indicating 2 1 / 2 miles means that the car will end in one mile interval , between the signs indicating 2 and 3 miles . now , it does n ' t matter where the car starts or what distance it travels , the probability will be p = ( favorable outcome ) / ( total # of outcomes ) = 1 / 3 ( as the car starts at random point end travels some distance afterwards we can consider its end point as the point where he randomly appeared , so the probability that the car appeared within 1 mile interval out of total 3 miles will be 1 / 3 ) . answer : c .
Answer: c
Problem: the cost price of a radio is rs . 2400 and it was sold for rs . 2100 , find the loss % ?
Options: a ) 12.5 % , b ) 11 % , c ) 13 % , d ) 15 % , e ) 12.8 %
Rationale: "2400 - - - - 300 100 - - - - ? = > 12.5 % answer : a"
Answer: a
Problem: the ages of two persons differ by 20 years . if 5 years ago , the older one be 5 times as old as the younger one , then their present ages , in years , are
Options: a ) 25 , 5 , b ) 30 , 10 , c ) 35 , 15 , d ) 50 , 30 , e ) none of these
Rationale: "let the ages be ’ x ’ and ’ y ’ years now . then x - y = 20 . . . ( i ) ( x - 5 ) = 5 ( y - 5 ) . . . ( ii ) from ( i ) and ( ii ) , 20 + y - 5 = 5 y - 25 y = 10 and x = 30 . answer : b"
Answer: b
Problem: the sum of 4 th and 20 th term of a . p . is equal to the sum of 8 th , 15 th and 12 th term . find the term which is 0
Options: a ) 10 , b ) 7 , c ) 11 , d ) 5 , e ) 3
Rationale: t 4 + t 20 = t 8 + t 15 + t 12 = > a + 3 d + a + 19 d = a + 7 d + a + 14 d + a + 11 d = > a + 10 d = 0 = > t 11 = 0 i . e . 11 th term is zero . answer : c
Answer: c
Problem: a particular library has 75 books in a special collection , all of which were in the library at the beginning of the month . these book are occasionally loaned out through an inter - library program . if , by the end of the month , 80 percent of books that were loaned out are returned and there are 64 books in the special collection at that time , how many books of the special collection were loaned out during that month ?
Options: a ) 20 , b ) 30 , c ) 35 , d ) 40 , e ) 55
Rationale: "there are 11 books less ( 75 - 64 ) which represents 20 % of the loaned books ( 100 - 80 ) so total loaned out books = 55 answer e"
Answer: e
Problem: a 6 - liter solution is 20 % alcohol . how many liters of pure alcohol must be added to produce a solution that is 50 % alcohol ?
Options: a ) 2.4 , b ) 2.7 , c ) 3.0 , d ) 3.3 , e ) 3.6
Rationale: "let x be the amount of pure alcohol required . 0.2 ( 6 ) + x = 0.5 ( x + 6 ) 0.5 x = 3 - 1.2 x = 3.6 liters the answer is e ."
Answer: e
Problem: how many multiples of 2 are there between 1 and 52 , exclusive ?
Options: a ) 21 , b ) 22 , c ) 25 , d ) 26 , e ) 28
Rationale: "25 multiples of 2 between 1 and 52 exclusive . from 2 * 1 upto 2 * 25 , ( 1,2 , 3,4 , . . . , 25 ) . hence , 25 multiples ! correct option is c"
Answer: c
Problem: the length of a room is 6 m and width is 4.75 m . what is the cost of paying the floor by slabs at the rate of rs . 900 per sq . metre .
Options: a ) 25650 , b ) 25750 , c ) 26550 , d ) 26750 , e ) 25725
Rationale: "area = 6 × 4.75 sq . metre . cost for 1 sq . metre . = rs . 900 hence total cost = 6 × 4.75 × 900 = 6 × 4275 = rs . 25650 answer is a ."
Answer: a
Problem: nails and screws are manufactured at a uniform weight per nail and a uniform weight per screw . if the total weight of one screw and one nail is half that of 6 screws and one nail , then the total weight of 3 screws , and 3 nails is how many times that of 2 screws and 4 nails ?
Options: a ) a . 4 , b ) b . 5 / 2 , c ) c . 2 , d ) d . 3 / 2 , e ) e . 5 / 4
Rationale: "let the weight of nail be n and that of screw be s . . so s + w = 1 / 2 * ( 6 s + 1 n ) . . . or 1 n = 4 s . . lets see the weight of 3 s and 3 n = 3 s + 3 * 4 s = 15 s . . and weight of 2 s and 4 n = 2 s + 4 * 2 s = 10 s . . ratio = 15 s / 10 s = 15 / 10 = 3 / 2 d"
Answer: d
Problem: a box contains 10 tablets of medicine a and 14 tablets of medicine b . what is the least number of tablets that should be taken from the box to ensure that at least two tablets of each kind are among the extracted .
Options: a ) 12 , b ) 16 , c ) 17 , d ) 19 , e ) 21
Rationale: the worst case scenario will be if we remove all 14 tablets of medicine b first . the next 2 tablets we remove have to be of medicine a , so to guarantee that at least two tablets of each kind will be taken we should remove minimum of 14 + 2 = 16 tablets . answer : b .
Answer: b
Problem: a and b together can do a work in 6 days . if a alone can do it in 15 days . in how many days can b alone do it ?
Options: a ) 10 , b ) 99 , c ) 77 , d ) 55 , e ) 21
Rationale: "1 / 6 – 1 / 15 = 1 / 10 = > 10 answer : a"
Answer: a
Problem: the sum of present age of abe and the age before 7 years is 27 . find the present age of abe . what will be his age after 7 years ?
Options: a ) 25 , b ) 24 , c ) 27 , d ) 28 , e ) 29
Rationale: "present age = x before 7 yrs , y = x - 7 after 7 yrs , z = x + 7 by the qn , x + ( x - 7 ) = 27 2 x - 7 = 27 2 x = 27 + 7 x = 34 / 2 x = 17 z = x + 7 = 17 + 7 = 24 answer : b"
Answer: b
Problem: a , b and c invested rs . 6600 , rs . 4400 and rs . 11000 respectively , in a partnership business . find the share of a in profit of rs . 12100 after a year ?
Options: a ) rs . 3300 , b ) rs . 3100 , c ) rs . 3500 , d ) rs . 4200 , e ) rs . 3400
Rationale: "6600 : 4400 : 11000 3 : 2 : 5 3 / 10 * 11000 = rs . 3300 answera"
Answer: a
Problem: a man drive his car to the office at 20 miles / hr . after reaching office , he realize that its a new year holiday so he went back home at a speed of 30 miles / hr . discounting the time spent in the stoppage what was his average speed of his journey ?
Options: a ) 23 miles , b ) 25 miles , c ) 22 miles , d ) 24 miles , e ) 26 miles
Rationale: d 24 miles / hr ( not 25 miles / hr which might be guessed by many ) d : distance traveled t 1 : time of going to office t 2 : time of returning back y : average speed d = 20 * t 1 t 1 = d / 20 d = 30 * t 2 t 2 = d / 30 2 d = y ( t 2 + t 1 ) 2 d = y ( d / 20 + d / 30 ) 2 d = y ( 3 d / 60 + 2 d / 60 ) 2 d = y ( 5 d / 60 ) y = 120 d / 5 d = > y = 24
Answer: d
Problem: walking 7 / 6 of his usual rate , a boy reaches his school 4 min early . find his usual time to reach the school ?
Options: a ) 11 , b ) 15 , c ) 16 , d ) 28 , e ) 19
Rationale: "speed ratio = 1 : 7 / 6 = 6 : 7 time ratio = 7 : 6 1 - - - - - - - - 7 4 - - - - - - - - - ? è 28 m answer : d"
Answer: d
Problem: find the principle on a certain sum of money at 5 % per annum for 2 2 / 5 years if the amount being rs . 896 ?
Options: a ) 1000 , b ) 800 , c ) 2889 , d ) 2777 , e ) 2991
Rationale: "896 = p [ 1 + ( 5 * 12 / 5 ) / 100 ] p = 800 answer : b"
Answer: b
Problem: a number is said to be prime saturated if the product of all the different positive prime factors of g is less than the square root of g . what is the greatest two digit prime saturated integer ?
Options: a ) 99 , b ) 98 , c ) 97 , d ) 96 , e ) 95
Rationale: "g = 96 = 3 * 32 = 3 * 2 ^ 5 answer is d ."
Answer: d
Problem: a person saved $ 10 in buying an item on sale . if he spent $ 600 for the item , approximately how much percent he saved in the transaction ?
Options: a ) 8 % , b ) 1.6 % , c ) 2 % , d ) 5 % , e ) 6 %
Rationale: "actual price = 600 + 10 = $ 610 saving = 10 / 610 * 100 = 100 / 61 = 1.6 % approximately answer is b"
Answer: b
Problem: an article is bought for rs . 705 and sold for rs . 900 , find the gain percent ?
Options: a ) 27.65 % , b ) 33 1 / 2 % , c ) 33 1 / 3 % , d ) 33 5 / 3 % , e ) 35 1 / 3 %
Rationale: "705 - - - - 195 100 - - - - ? = > 27.65 % answer : a"
Answer: a
Problem: what will be the percentage increase in the area of the cube ' s surface if each of the cube ' s edges grows by 10 % ?
Options: a ) 25 % , b ) 28 % , c ) 21 % , d ) 30 % , e ) 32 %
Rationale: "the question is very easy . my logic is the following : a surface = 6 * a ^ 2 after 10 % increase a surface = 6 * ( ( 1.1 a ) ^ 2 ) = 6 * 1.21 * a ^ 2 the increase in the surface area = ( 6 * 1.21 * a ^ 2 - 6 * a ^ 2 ) / 6 * a ^ 2 = ( 6 * a ^ 2 ( 1.21 - 1 ) ) / ( 6 * a ^ 2 ) = 1.21 - 1 = 0.21 = 21 % answer : c"
Answer: c
Problem: how many positive integers less than 500 can be formed using the numbers 1 , 2,3 , and 5 for digits , each digit being used only once .
Options: a ) 34 , b ) 677 , c ) 197 , d ) 177 , e ) 191
Rationale: "single digit numbers = 4 double digit numbers = 4 x 3 = 12 three digit numbers = 3 x 3 x 2 x 1 = 18 total = 34 answer : a"
Answer: a
Problem: a certain clock marks every hour by striking a number of times equal to the hour , and the time require for a stroke is exactly equal to the time interval between strokes . at 6 : 00 the time lapse between the beginning of the first stoke and the end of the last stroke is 22 seconds . at 11 : 00 , how many seconds elapse between the beginning of the first stroke and the end of the last stroke ?
Options: a ) 34 , b ) 38 , c ) 36 , d ) 40 , e ) 42
Rationale: "at 6 ' o clock , there would be 6 strikes . first strike , then a short interval , the second strike , then a short interval and so on till the 6 th strike . so there would be in all 5 intervals between 6 strikes . similarly , between 11 strikes , there would be 10 intervals . according to the question , the time spent in the strike and the interval is same . at 6 ' o clock , the 6 strikes and the 5 intervals together take 22 sec so each strike and each interval takes 2 secs . at 11 ' o clock , the 11 strikes and 10 intervals will take 2 * ( 11 + 10 ) = 42 secs e"
Answer: e
Problem: the length of a rectangle is 5 more than the width . what are the dimensions of the rectangle if the perimeter is 34 ?
Options: a ) 11
b ) 12
c ) 13
d ) 14
e ) none
Rationale: pretend width = 1 , then length = 6 ( 1 + 5 ) 2 × 1 + 2 × 7 = 2 + 14 = 16 . notice that 16 is far from a perimeter of 34 try much bigger number . how about if we . . . pretend width = 4 , then length = 9 ( 4 + 5 ) 2 × 4 + 2 × 9 = 6 + 18 = 24 . we are getting closer to a perimeter of 34 pretend width = 5 , then length = 10 ( 5 + 5 ) 2 × 5 + 2 × 10 = 10 + 30 = 30 . pretend width = 7 , then length = 12 ( 7 + 5 ) 2 × 7 + 2 × 12 = 14 + 24 = 38 . this is higher than a perimeter of 34 . so width should be higher than 5 and smaller than 7 . may be a width of 6 will work . pretend width = 6 , then length = 11 ( 6 + 5 ) 2 × 6 + 2 × 11 = 12 + 22 = 34 . use of algebra : let width = x let length = x + 5 p = 2 × l + 2 × w 34 = 2 × ( x + 5 ) + 2 × x 34 = 2 x + 10 + 2 x 34 = 4 x + 10 34 - 10 = 4 x + 10 - 10 24 = 4 x 24 / 4 = 4 x / 4 6 = x therefore , width = 6 and length = x + 5 = 6 + 5 = 11 answer a
Answer: a
Problem: if there are only 2 wheelers and 4 wheelers parked in a school located at the heart of the city , find the number of 4 wheelers parked there if the total number of wheels is 54 ?
Options: a ) 11 , b ) 12 , c ) 13 , d ) 14 , e ) 15
Rationale: four wheeler = 13 * 4 = 52 ( max ) 2 wheel = 1 so no of 4 wheeler = 13 answer : c
Answer: c
Problem: a train passes a train standing in platform in 30 sec and a man standing on the platform in 12 sec . if the speed of the train is 36 km / hr . what is the length of the platform ?
Options: a ) 250 , b ) 200 , c ) 300 , d ) 240 , e ) 180
Rationale: speed = 36 * 5 / 18 = 10 m / sec . length of the train = 10 * 12 = 120 m . let the length of the platform be x m . then , ( x + 120 ) / 30 = 10 = > x = 180 m . answer : e
Answer: e
Problem: what is the tenth digit of ( 4 ! * 3 ! + 4 ! * 2 ! ) / 3 ?
Options: a ) 5 , b ) 7 , c ) 9 , d ) 1 , e ) 0
Rationale: "( 4 ! * 3 ! + 4 ! * 2 ! ) / 2 = 4 ! ( 3 ! + 2 ! ) / 2 = 24 ( 6 + 2 ) / 2 = 96 units digit of the above product will be equal to 9 answer c"
Answer: c
Problem: bullock likes to keep a spare tyre in his car every time . on a certain day , he travels 1 , 60,000 km and just to make the most of all the tyres , he changes the tyres between his journey such that each tyre runs the same distance . what is the distance traveled by each tyre ?
Options: a ) 70,000 , b ) 60,000 , c ) 128,000 , d ) 90,000 , e ) 10,000
Rationale: the distance traveled by each tyre : 4 / 5 * 1 , 60 , 000 km = 128,000 km . c
Answer: c
Problem: what is the remainder when 10 ^ 49 + 2 is divided by 11 ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 5 , e ) 7
Rationale: "with exponent - based problems and huge numbers , it ' s often helpful to try to establish a pattern using small numbers . on this problem , while 1049 + 2 is a massive number that you ' d never want to try to perform calculations with , you can start by using smaller numbers to get a feel for what it would look like : 10 ^ 2 + 2 = 102 , which when divided by 11 produces a remainder of 3 ( as 9 * 11 = 99 , leaving 3 left over ) 10 ^ 3 + 2 = 1002 , which when divided by 11 produces a remainder of 1 ( as 9 * 110 = 990 , and when you add one more 11 to that you get to 1001 , leaving one left ) 10 ^ 4 + 2 = 10002 , which when divided by 11 produces a remainder of 3 ( as you can get to 9999 as a multiple of 11 , which would leave 3 left over ) 10 ^ 5 + 2 = 100002 , which when divided by 11 produces a remainder of 1 ( as you can get to 99990 and then add 11 more , bringing you to 100001 leaving one left over ) . by this point , you should see that the pattern will repeat , meaning that when 10 has an even exponent the remainder is 3 and when it has an odd exponent the remainder is 1 . therefore , since 10 has an odd exponent in the problem , the remainder will be 1 . answer is a"
Answer: a
Problem: 39 ! is divided by 41 then find the remainder ?
Options: a ) 1 , b ) 3 , c ) 5 , d ) 6 , e ) 7
Rationale: "substituting p = 41 in the wilson ' s theorem , we get 40 ! + 141 = 0 40 × 39 ! + 141 = 0 − 1 × 39 ! 41 = − 1 cancelling - 1 on both sides , 39 ! 41 = 1 a"
Answer: a
Problem: evaluate : 60 - 9 ÷ 3 × 6 =
Options: a ) 52 , b ) 18 , c ) 42 , d ) 102 , e ) 36
Rationale: "according to order of operations , 9 ÷ 3 × 6 ( division and multiplication ) is done first from left to right 9 ÷ 3 × 6 = 3 × 6 = 18 hence 60 - 9 ÷ 3 × 6 = 60 - 18 = 42 correct answer c ) 42"
Answer: c
Problem: a trader mixes 80 kg of tea at 15 per kg with 20 kg of tea at cost price of 20 per kg . in order to earn a profit of 20 % , what should be the sale price of the mixed tea ?
Options: a ) 23.75 , b ) 22 , c ) 20 , d ) 19.2 , e ) none of these
Rationale: "c . p . of mixture = 80 × 15 + 20 × 20 / 80 + 20 = 16 ∴ s . p . = ( 100 + 20 ) / 100 × 16 = 19.2 answer d"
Answer: d
Problem: the difference between the value of a number increased by 12.5 % and the value of the original number decreased by 25 % is 30 . what is the original number w ?
Options: a ) 60 , b ) 80 , c ) 40 , d ) 120 , e ) 160
Rationale: "( 1 + 1 / 8 ) x - ( 1 - 1 / 4 ) x = 30 ( 9 / 8 ) x - ( 3 / 4 ) x = 30 x = 80 = w answer : b"
Answer: b
Problem: uncle bruce is baking chocolate chip cookies . he has 36 ounces of dough ( with no chocolate ) and 12 ounces of chocolate . how many ounces of chocolate are left over if he uses all the dough but only wants the cookies to consist of 20 % chocolate ?
Options: a ) 3 , b ) 6 , c ) 7.2 , d ) 7.8 , e ) 9
Rationale: "answer is a . x / x + 36 = 1 / 5 x = 9 12 - 9 = 3"
Answer: a