euclaise/mathqa_programs 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
Python program:
a = 30 + 1 
b = 30 * a
c = 2 * 30
d = c + 1 
e = b * d
f = 3 + 3 
g = e / f
h = g - 1290
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
Python program:
a = const_60 / 30
b = a * 1
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
Python program:
a = 10 * 1
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
Python program:
a = 22 / 5
b = math.sqrt(a)
c = b * 10
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
Python program:
a = 6 / 3
b = 12 / 4
c = a + b 
d = 12 / 4
e = c + d 
f = e + 4
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
Python program:
a = 10 + 1
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
Python program:
a = 3 * 3
b = 3 * 3
c = b + 1 
d = a * c
e = d + 2 
f = 3 * 3
g = f + 1 
h = 2 * g
i = e / h
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
Python program:
a = 3 + 4 
b = a * 3
c = b + 1 
d = 3 + 4 
e = c / d
f = e * 22
g = f * 2
h = g * 400
i = h / 100
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
Python program:
a = 100 - 5
b = a / 100
c = 100 - 15
d = c / 100
e = b * d
f = 3553 / e
g = math.floor(f)
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
Python program:
a = 1 + 4 
b = a * 1000
c = b * 2
d = c / 3
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
Python program:
a = 27 ** const_0_33
b = 125 ** const_0_33
c = a / b
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
Python program:
a = 160 / 2
b = 16 / 100
c = a / b
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
Python program:
a = 80 * 3
b = 30 * 2
c = a + b 
d = 3 + 2 
e = c / d
f = 100 / 3
g = f - const_0_33
h = e + g
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
Python program:
a = 0 + 8 
b = 0 / 835
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
Python program:
a = 27 * 5
b = 25 * 4
c = a - b
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
Python program:
a = 17 * 3
b = 86 - a
c = b + 3
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
Python program:
a = 4 / 6
b = 4 - a
c = 1/(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
Python program:
a = 1 / 16
b = 1 - a
c = 1 / 16
d = 1 - c
e = d / 15
f = b - e
g = 1 / 16
h = 1 - g
i = 1 / 16
j = 1 - i
k = j / 15
l = h - k
m = 5 / 7
n = l * m
o = f - n
p = 50 / o
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
Python program:
a = 3 / 5
b = 5 / 7
c = 2 / 3
d = b * c
e = a * d
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
Python program:
a = 12 * 5
b = 1 * 100
c = a / b
d = 54 - c
e = d + 104 
f = 350 - e
g = f / 350
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
Python program:
a = 100 + 20 
b = a / 100
c = 36 / b
d = 33 - c
e = 100 + 20 
f = e / 100
g = 36 / f
h = d / g
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
Python program:
a = 12 / 10
b = a - 10
c = 12 / b
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
Python program:
a = 150 / 100
b = a + 1 
c = 550 / b
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
Python program:
a = 20 + 100 
b = a * 60
c = b / 100
d = 100 + 9 
e = c / d
f = 100 * e
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
Python program:
a = 10 - 1
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
Python program:
a = 32 * 25
b = 64 + a 
c = 2 - 64
d = b - c
e = d + 1 
f = e / 50
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
Python program:
a = 24 * 20
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
Python program:
a = 40 / 100
b = a * 600
c = const_60 / 0
d = b / c
e = 40 / 100
f = d * e
g = 40 / 100
h = f / g
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
Python program:
a = 3 + 4 
b = a * 4
c = b * 1000
d = 4 + 1 
e = d * 4
f = e * 1000
g = c - f
h = g * 10
i = h / 100
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
Python program:
a = 8 - 2
b = 60 - a
c = b / 3
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
Python program:
a = 0 * 25
b = a / 2
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
Python program:
a = 8 + 7 
b = 7 / a
c = 2 + 3 
d = 2 / c
e = b - d
f = 8 + 7 
g = 7 / f
h = e / g
i = 8 + 7 
j = h * i
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
Python program:
a = 287 * 287
b = 269 * 269
c = a + b 
d = 2 * 287
e = d * 269
f = c - e
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
Python program:
a = 1 / 12
b = 1 / 15
c = a + b 
d = 1 / c
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
Python program:
a = 12 + 10 
b = a + 5 
c = 1 / 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
Python program:
a = 3 / 4
b = 1 - a
c = 100 / b
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
Python program:
a = 2 / 4
b = a / 100
c = 1 + b 
d = c ** 3
e = 4 * 4
f = e * 100
g = math.sqrt(100)
h = f * g
i = d * h
j = 4 * 4
k = j * 100
l = math.sqrt(100)
m = k * l
n = i - m
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
Python program:
a = 120 / 4
b = 1 + 4 
c = a / b
d = c * 2
e = 1 + 4 
f = 100 / e
g = f / 4
h = d + g 
i = 1 + 4 
j = h + i
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
Python program:
a = 2 / 3
b = 2 / 9
c = 1 - b
d = a + c 
e = 9 / 5
f = d - e