euclaise/mathqa_programs huggingface dataset:
Problem: two dogsled teams raced across a 300 mile course in wyoming . team a finished the course in 3 fewer hours than team t . if team a ' s average speed was 5 mph greater than team t ' s , what was team t ' s average mph ?
Options: a ) 12 , b ) 15 , c ) 18 , d ) 20 , e ) 25
Rationale: "this is a very specific format that has appeared in a handful of real gmat questions , and you may wish to learn to recognize it : here we have a * fixed * distance , and we are given the difference between the times and speeds of two things that have traveled that distance . this is one of the very small number of question formats where backsolving is typically easier than solving directly , since the direct approach normally produces a quadratic equation . say team t ' s speed was s . then team t ' s time is 300 / s . team a ' s speed was then s + 5 , and team a ' s time was then 300 / ( s + 5 ) . we need to find an answer choice for s so that the time of team a is 3 less than the time of team t . that is , we need an answer choice so that 300 / ( s + 5 ) = ( 300 / s ) - 3 . you can now immediately use number properties to zero in on promising answer choices : the times in these questions will always work out to be integers , and we need to divide 300 by s , and by s + 5 . so we want an answer choice s which is a factor of 300 , and for which s + 5 is also a factor of 300 . so you can rule out answers a and c immediately , since s + 5 wo n ' t be a divisor of 300 in those cases ( sometimes using number properties you get to the correct answer without doing any other work , but unfortunately that ' s not the case here ) . testing the other answer choices , if you try answer d , you find the time for team t is 15 hours , and for team a is 12 hours , and since these differ by 3 , as desired , d is correct ."
Answer: d
Python program:
a = 300 / 5
b = a / 3
Problem: a boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream . if the speed of the boat in still water is 10 mph , the speed of the stream is :
Options: a ) 4 mph , b ) 2.5 mph , c ) 3 mph , d ) 2 mph , e ) none of these
Rationale: explanation : speed of the boat in still water = 10 mph let speed of the stream be x mph then , speed downstream = ( 10 + x ) mph speed upstream = ( 10 - x ) mph time taken to travel 36 miles upstream - time taken to travel 36 miles downstream = 90 / 60 hours = > 36 / ( 10 − x ) − 36 / ( 10 + x ) = 3 / 2 = > 12 / ( 10 − x ) − 12 / ( 10 + x ) = 1 / 2 = > 24 ( 10 + x ) − 24 ( 10 − x ) = ( 10 + x ) ( 10 − x ) = > 240 + 24 x − 240 + 24 x = ( 100 − x 2 ) = > 48 x = 100 − x 2 = > x 2 + 48 x − 100 = 0 = > ( x + 50 ) ( x − 2 ) = 0 = > x = - 50 or 2 . answer : option d
Answer: d
Python program:
a = 10 ** 2
b = a * 4
c = 90 / const_60
d = 36 / c
e = d * 2
f = e ** 2
g = b + f 
h = math.sqrt(g)
i = 90 / const_60
j = 36 / i
k = j * 2
l = h - k
m = l / 2
Problem: a rectangular field has to be fenced on three sides leaving a side of 25 feet uncovered . if the area of the field is 880 sq . feet , how many feet of fencing will be required ?
Options: a ) 75.4 feet , b ) 98.4 feet , c ) 95.4 feet , d ) 85.4 feet , e ) 92.4 feet
Rationale: "area of the field = 880 sq . feet . length of the adjacent sides are 25 feet and 880 / 25 = 35.2 feet . required length of the fencing = 25 + 35.2 + 35.2 = 95.4 feet answer : c"
Answer: c
Python program:
a = 880 / 25
b = rectangle_perimeter - (
Problem: x + ( 1 / x ) = 2.5 find x ^ 2 + ( 1 / x ^ 2 )
Options: a ) 2.25 , b ) 3.25 , c ) 4.25 , d ) 5.25 , e ) 6.25
Rationale: "squaring on both sides ( x + 1 / x ) ^ 2 = 2.5 ^ 2 x ^ 2 + 1 / x ^ 2 = 6.25 - 2 x ^ 2 + 1 / x ^ 2 = 4.25 answer : c"
Answer: c
Python program:
a = 2 ** 5
b = a - 2
Problem: a = { 0 , 1 , - 3 , 6 , - 8 , - 10 } b = { - 1 , 2 , - 4 , 7 , 6 , - 9 } if a is a number that is randomly selected from set a , and b is a number that is randomly selected from set b , what is the probability that ab < 0 ?
Options: a ) 1 / 4 , b ) 1 / 3 , c ) 5 / 13 , d ) 5 / 12 , e ) 1 / 2
Rationale: for the product of 2 numbers to be negative either of them must be positive or negative : p ( positive , negative ) = 2 / 6 * 3 / 6 = 6 / 36 ; p ( negative , positive ) = 3 / 6 * 3 / 6 = 9 / 36 p = 6 / 36 + 9 / 36 p = 15 / 36 p = 5 / 12 answer : d
Answer: d
Python program:
a = 3 / 6
b = 3 / 6
c = a * b
d = 2 / 6
e = 3 / 6
f = d * e
g = c + f
Problem: which number need to add to 172835 to get a number exactly divisible by 136 ?
Options: a ) 27 , b ) 25 , c ) 26 , d ) 21 , e ) 24
Rationale: "172835 / 136 = 1270 and reminder = 115 . 136 - 115 = 21 so , the next number divisible by 115 is 21 places in front of 172835 which means 21 + 172835 = 172856 21 should be added to 172835 d"
Answer: d
Python program:
a = 172835 / 136
b = math.floor(a)
c = b + 1 
d = 172835 / 136
e = c - d
f = 136 * e
Problem: simplify : 99 ^ 48 / 49 * 245 .
Options: a ) 23345 , b ) 26695 , c ) 24495 , d ) 25575 , e ) none of them
Rationale: "given expression = ( 100 - 1 / 49 ) * 245 = ( 4899 / 49 ) * 245 = 4899 * 5 = 24495 . answer is c ."
Answer: c
Python program:
a = 99 * 48
b = 49 * 245
c = a + b
Problem: a boy swims downstream 91 km and upstream 21 km taking 7 hours each time , what is the speed of the boy in still water ?
Options: a ) 6 , b ) 7 , c ) 8 , d ) 9 , e ) 10
Rationale: 91 - - - 7 ds = 13 ? - - - - 1 21 - - - - 7 us = 3 ? - - - - 1 m = ? m = ( 13 + 3 ) / 2 = 8 answer : c
Answer: c
Python program:
a = 1 / 2
b = 91 / 7
c = 21 / 7
d = b + c 
e = a * d
Problem: speed of a boat in standing water is 14 kmph and the speed of the stream is 1.2 kmph . a man rows to a place at a distance of 4664 km and comes back to the starting point . the total time taken by him is :
Options: a ) 784 hours , b ) 794 hours , c ) 780 hours , d ) 684 hours , e ) 884 hours
Rationale: "speed downstream = ( 12 + 1.2 ) = 13.2 kmph speed upstream = ( 12 - 1.2 ) = 10.8 kmph total time taken = 4664 / 13.2 + 4664 / 10.8 = 353 + 431 = 784 hours answer : a"
Answer: a
Python program:
a = 14 + 1 
b = 4664 / a
c = 14 - 1
d = 4664 / c
e = b + d
Problem: the hiker walking at a constant rate of 4 miles per hour is passed by a cyclist traveling in the same direction along the same path at 10 miles per hour . the cyclist stops to wait for the hiker 5 minutes after passing her , while the hiker continues to walk at her constant rate , how many minutes must the cyclist wait until the hiker catches up ?
Options: a ) 10 , b ) 15 , c ) 20 , d ) 15 / 2 , e ) 24
Rationale: after passing the hiker the cyclist travels for 5 minutes at a rate of 10 miles / hour . in those 5 mins the cyclist travels a distance of 5 / 6 miles . in those 5 mins the hiker travels a distance of 1 / 2 miles . so the hiker still has to cover 1 / 2 miles to meet the waiting cyclist . the hiker will need 15 / 2 mins to cover the remaining 1 / 2 miles . so the answer is d .
Answer: d
Python program:
a = 10 - 4
b = 5 / const_60
c = a * b
d = c / 4
e = d * const_60
Problem: the average of first five multiples of 2 is :
Options: a ) 3 , b ) 9 , c ) 12 , d ) 15 , e ) e
Rationale: "solution average = 2 ( 1 + 2 + 3 + 4 + 5 ) / 5 = 30 / 5 = 6 answer e"
Answer: e
Python program:
a = 2 + 1
Problem: find the amount on rs . 9000 in 2 years , the rate of interest being 4 % per first year and 5 % for the second year ?
Options: a ) 5568 , b ) 9828 , c ) 5460 , d ) 5635 , e ) 6734
Rationale: "9000 * 104 / 100 * 105 / 100 = > 9828 answer : b"
Answer: b
Python program:
a = 100 + 4 
b = 9000 * a
c = b / 100
d = 100 + 5 
e = c * d
f = e / 100
Problem: what decimal fraction is 40 ml of a litre ?
Options: a ) . 4 , b ) . 04 , c ) . 05 , d ) 0.04 , e ) none of these
Rationale: "answer required fraction = 40 / 1000 = 4 / 100 = . 04 correct option : b"
Answer: b
Python program:
a = 40 / 1000
Problem: how many even integers n , such that 20 < = n < = 180 are of the form 3 k + 4 , where k is any natural number ?
Options: a ) 24 , b ) 27 , c ) 30 , d ) 33 , e ) 36
Rationale: the first number is 22 = 16 + 6 ( 1 ) . we can continue adding 6 to make a list : 22 , 28 , 34 , . . . the last number is 278 = 16 + 6 ( 27 ) there are 27 numbers in the list . the answer is b .
Answer: b
Python program:
a = 180 - 2
b = 20 + 2 
c = a - b
d = 2 + 4 
e = c / d
f = e + 1
Problem: two numbers n and 12 have lcm = 48 and gcf = 8 . find n .
Options: a ) 24 , b ) 34 , c ) 44 , d ) 54 , e ) 32
Rationale: "the product of two integers is equal to the product of their lcm and gcf . hence . 12 × n = 48 × 8 n = 48 × 8 / 12 = 32 correct answer e"
Answer: e
Python program:
a = 48 * 8
b = a / 12
Problem: in a hostel , the number of students decreased by 9 % and the price of food increased by 20 % over the previous year . if each student consumes the same amount of food then by how much should the consumption of food be cut short by every student , so that the total cost of the food remains the same as that of the previous year ?
Options: a ) 19 % , b ) 15 % , c ) 25 % , d ) 40 % , e ) 8.4 %
Rationale: "cost of food ( c ) = food consumed per student ( f ) * number of students ( n ) * price of food ( p ) originally , c = fnp when number of students decrease by 8 % , and the price of food increases by 20 % , c = f ( new ) * ( 0.91 n ) * ( 1.2 p ) = > f ( new ) = f / ( 0.91 * 1.2 ) = > f ( new ) = 0.915 f therefore the new cost of food must be 91.5 % of the old cost , or the cost of food must decrease by 8.4 % ( option e )"
Answer: e
Python program:
a = 100 * 100
b = 100 - 9
c = 100 + 20 
d = b * c
e = a / d
f = 1 - e
g = f * 100
Problem: a company produces on an average 4000 items per month for the first 3 months . how many items it must produce on an average per month over the next 9 months , to average of 4375 items per month over the whole ?
Options: a ) 4500 , b ) 4600 , c ) 4680 , d ) 4710 , e ) none
Rationale: sol . required average = ( 4375 x 12 ) - ( 4000 x 3 ) / 9 ⇔ 52500 - 12000 / 9 ⇔ 40500 / 9 = 4500 . answer a
Answer: a
Python program:
a = 9 + 3 
b = 4375 * a
c = 4000 * 3
d = b - c
e = d / 9
Problem: pat , kate and mark charged a total of 144 hours to a certain project . if pat charged twice as much time to the project as kate and 1 / 3 as much times as mark , how many more hours did mark charge to the project than kate .
Options: a ) 18 , b ) 36 , c ) 80 , d ) 90 , e ) 108
Rationale: "let kate charge for x hours , then pat charged for 2 x and mat - for 6 x . so , 2 x + 6 x + x = 144 - total hours charged for , x = 16 . mat charged 6 x - x or 5 x for more hours than kate , or for 80 hours . c is correct"
Answer: c
Python program:
a = 1 + 2 
b = 2 * 3
c = a + b 
d = 144 / c
e = 2 * 3
f = e - 1
g = d * f
Problem: 69 laboratories raise the bacterium , the laboratory have 48 culture dishes on average , which has about 25,075 bacteria each . how many bacteria are there approximately ?
Options: a ) 10 ^ 7 , b ) 10 ^ 8 , c ) 10 ^ 9 , d ) 10 ^ 10 , e ) 10 ^ 11
Rationale: "69 laboratories raise the bacterium , the laboratory have 48 culture dishes on average , which has about 25,075 bacteria each . how many bacteria are there approximately ? a . 10 ^ 7 b . 10 ^ 8 c . 10 ^ 9 d . 10 ^ 10 e . 10 ^ 11 - > due to approximately , 69 = 70 , 48 = 50 , 25,075 = 25,000 are derived , which makes ( 69 ) ( 48 ) ( 25,075 ) = ( 70 ) ( 50 ) ( 25,000 ) = 10 ^ 7 . the answer is a ."
Answer: a
Python program:
a = 69 * 48
b = const_0_25 * 1000
c = b * 100
d = 3 + 4 
e = d * 10
f = c + e 
g = f + 3 
h = a * g
Problem: divide $ 5000 among x , y in the ratio 2 : 8 . how many $ that x get ?
Options: a ) $ 1000 , b ) $ 1050 , c ) $ 1100 , d ) $ 1150 , e ) $ 1200
Rationale: sum of ratio terms = 2 + 8 = 10 x = 5000 * 2 / 10 = $ 1000 answer is a
Answer: a
Python program:
a = 4 * 1000
b = 5000 - a
Problem: a teacher gave the same test to 3 history classes : u , b , and c . the average ( arithmetic mean ) scores for the 3 classes were 65 , 80 , and 77 , respectively . the ratio of the numbers of students in each class who took the test was 4 to 6 to 5 , respectively . what was the average score for the 3 classes combined ?
Options: a ) 74 , b ) 75 , c ) 76 , d ) 77 , e ) 78
Rationale: ans : b ( 75 ) lets say class u indeed has 4 children , b has 6 children and c has 5 children . now , if the average of class u is 65 , hence total marks awarded in the class = 65 * 4 = 260 similarly , class b = 80 * 6 = 480 class c = 77 * 5 = 385 total marks provided = u + b + c = 260 + 480 + 385 = 1125 avg . marks = 1125 / 15 ( total no . of students ) = 75 = b
Answer: b
Python program:
a = 65 * 4
b = 80 * 6
c = a + b 
d = 77 * 5
e = c + d 
f = 4 + 6 
g = f + 5 
h = e / g
Problem: a cube of edge 15 cm is immersed completely in a rectangular vessel containing water . if the dimensions of the base of vessel are 20 cm * 15 cm , find the rise in water level ?
Options: a ) 6 cm , b ) 8.25 cm , c ) 11.25 cm , d ) 15.12 cm , e ) 20.62 cm
Rationale: "increase in volume = volume of the cube = 15 * 15 * 15 cm ^ 3 rise in water level = volume / area = 15 * 15 * 15 / 20 * 15 = 11.25 cm answer is c"
Answer: c
Python program:
a = volume_cube / (
Problem: how many positive integers less than 28 are prime numbers , odd multiples of 5 , or the sum of a positive multiple of 2 and a positive multiple of 4 ?
Options: a ) 27 , b ) 25 , c ) 24 , d ) 22 , e ) 20
Rationale: "9 prime numbers less than 28 : { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 } 3 odd multiples of 5 : { 5 , 15 , 25 } 11 numbers which are the sum of a positive multiple of 2 and a positive multiple of 4 : { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 , 22 , 24 , 26 } notice , that 5 is in two sets , thus total # of integers satisfying the given conditions is 9 + 3 + 11 - 1 = 22 . answer : d ."
Answer: d
Python program:
a = 28 - 2
b = a - 1
c = b - 1
Problem: what are the last two digits of 63 * 35 * 37 * 82 * 71 * 41 * 53 ?
Options: a ) 10 , b ) 30 , c ) 40 , d ) 70 , e ) 80
Rationale: we know that to find the last two digits , we need to find the remainder we get when we divide the product by 100 . remainder of ( 63 * 35 * 37 * 82 * 71 * 41 ) / 100 note that we can simplify this expression by canceling out the 5 and 2 in the numerator and denominator . but before we do that , here is an important note : note : we can not just cancel off the common terms in the numerator and denominator to get the remainder . but , if we want to cancel off to simplify the question , we can do it , provided we remember to multiply it back again . so say , we want to find the remainder when 14 is divided by 10 i . e . 14 / 10 ( remainder 4 ) . but we cancel off the common 2 to get 7 / 5 . the remainder here will be 2 which is not the same as the remainder obtained by dividing 14 by 10 . but if we multiply 2 back by 2 ( the number we canceled off ) , the remainder will become 2 * 2 = 4 which is correct . take another example to reinforce this – what is the remainder when 85 is divided by 20 ? it is 5 . we might rephrase it as – what is the remainder when 17 is divided by 4 ( cancel off 5 from the numerator and the denominator ) . the remainder in this case is 1 . we multiply the 5 back to 1 to get the remainder as 5 which is correct . so keeping this very important point in mind , let ’ s go ahead and cancel the common 5 and 2 . we need the remainder of ( 63 * 7 * 37 * 41 * 71 * 41 * 5 * 2 ) / 10 * 5 * 2 remainder of ( 63 * 7 * 37 * 41 * 71 * 41 ) / 10 now using concept 2 , let ’ s write the numbers in form of multiples of 10 remainder of ( 60 + 3 ) * 7 * ( 30 + 7 ) * ( 40 + 1 ) * ( 70 + 1 ) * ( 40 + 1 ) / 10 remainder of 3 * 7 * 7 * 1 * 1 * 1 / 10 remainder of 147 / 10 = 7 now remember , we had canceled off 10 so to get the actual remainder so we need to multiply by 10 : 7 * 10 = 70 . when 63 * 35 * 37 * 82 * 71 * 41 is divided by 100 , the remainder is 70 . so the last two digits of 63 * 35 * 37 * 82 * 71 * 41 must be 80 . answer ( e )
Answer: e
Python program:
a = 82 - 2
Problem: in n is a positive integer less than 200 , and 27 n / 60 is an integer , then n has how many different positive prime factors ?
Options: a ) 2 , b ) 3 , c ) 5 , d ) 6 , e ) 8
Rationale: "( a ) . 27 n / 60 must be an integer . = > 9 n / 20 must be an integer . hence n must be a multiple of 2 * 5 . = > n has 2 different prime integers ."
Answer: a
Python program:
a = 1 + 4 
b = 60 / 2
c = b / 2
d = c / 3
e = a / d
f = e + 2
Problem: the ratio of the adjacent angles of a parallelogram is 5 : 10 . also , the ratio of the angles of quadrilateral is 5 : 6 : 7 : 12 . what is the sum of the smaller angle of the parallelogram and the second largest angle of the quadrilateral ?
Options: a ) 168 ° , b ) 228 ° , c ) 144 ° , d ) 224 ° , e ) none of these
Rationale: "the measures of the adjacent angles of a parallelogram add up to be 180 ° given so , 5 x + 10 x = 180 ° or , 15 x = 180 ° or , x = 12 ° hence the angles of the parallelogram are 60 ° and 120 ° further it is given we know sum of all the four angles of a quadrilateral is 360 ° so , 5 y + 6 y + 7 y + 12 y = 360 ° or , 5 y + 6 y + 7 y + 12 y = 360 ° or , 30 y = 360 ° or , y = 12 ° hence the angles of the quadrilateral are 60 ° , 72 , 84 ° and 144 ° will be 60 ° + 84 ° = 144 ° answer : c"
Answer: c
Python program:
a = 5 * 2
b = a * 10
c = b + 100 
d = c * 2
e = 5 + 6 
f = e + 7 
g = f + 12 
h = d / g
i = h * 10
Problem: the population of a city increases by 5 % per year but due to migration it decrease by 1 % per years . what will be the percentage increase in population in 3 years ?
Options: a ) 9 % , b ) 9.27 % , c ) 27 % , d ) 12.48 % , e ) none of these
Rationale: "actual increase in population = 4 % let , earlier population = 100 then the population after 3 years = 100 ( 1 + 4 / 100 ) ^ 3 = 112.4864 ∴ required percentage = 12.48 % answer : d"
Answer: d
Python program:
a = 5 - 1
b = 100 + a 
c = 5 - 1
d = b * c
e = d / 100
f = 5 - 1
g = e + f 
h = 100 + g 
i = 5 - 1
j = h * i
k = j / 100
l = 5 - 1
m = 100 + l 
n = 5 - 1
o = m * n
p = o / 100
q = 5 - 1
r = p + q 
s = k + r
Problem: the sum of the present age of henry and jill is 33 . what is their present ages if 6 years ago henry was twice the age of jill ?
Options: a ) and 20 , b ) and 24 , c ) and 22 , d ) and 29 , e ) of these
Rationale: "let the age of jill 6 years ago be x , age of henry be 2 x x + 6 + 2 x + 6 = 33 x = 7 present ages will be 13 and 20 answer : a"
Answer: a
Python program:
a = 33 + 6 
b = a / 3
c = 33 - b
Problem: in a division sum , the remainder is 5 and the divisor is 3 times the quotient and is obtained by adding 3 to the thrice of the remainder . the dividend is :
Options: a ) 72 , b ) 76 , c ) 100 , d ) 113 , e ) 112
Rationale: "diver = ( 5 * 3 ) + 3 = 18 3 * quotient = 18 quotient = 6 dividend = ( divisor * quotient ) + remainder dividend = ( 18 * 6 ) + 5 = 113 d"
Answer: d
Python program:
a = 5 * 3
b = a + 3 
c = 5 * 3
d = c + 3 
e = d / 3
f = b * e
g = f + 5
Problem: what is the sum of the multiples of 4 between 34 and 135 inclusive ?
Options: a ) 2000 , b ) 2100 , c ) 2150 , d ) 2200 , e ) 2500
Rationale: the fastest way in an ap is to find the average and multiply with total integers . . between 34 and 135 , the smallest multiple of 4 is 36 and largest = 132 . . average = ( 36 + 132 ) / 2 = 84 . . total numbers = ( 132 - 36 ) / 4 + 1 = 24 + 1 = 25 . . sum = 25 * 84 = 2100 ans b
Answer: b
Python program:
a = 135 - 3
b = 34 + 2 
c = a + b 
d = c / 2
e = 135 - 3
f = 34 + 2 
g = e - f
h = g / 4
i = h + 1 
j = d * i
Problem: a train sets off at 9 : 00 am at the speed of 70 km / h . another train starts at 9 : 30 am in the same direction at the rate of 80 km / h . at what time will the second train catch the first train ?
Options: a ) 12 : 30 , b ) 1 : 00 , c ) 1 : 30 , d ) 2 : 00 , e ) 2 : 30
Rationale: "in thirty minutes the first train travels 35 km . the second train catches the first train at a rate of 80 km / h - 70 km / h = 10 km / h . the second train will catch the first train in 35 / 10 = 3.5 hours , so at 1 : 00 pm . the answer is b ."
Answer: b
Python program:
a = 1 / 2
b = 70 * a
c = 70 + b 
d = 80 - 70
e = c / d
Problem: the area of a rectangular plot is 21 times its breadth . if the difference between the length and the breadth is 10 metres , what is its breadth ?
Options: a ) 10 metres , b ) 5 metres , c ) 11 metres , d ) data inadequate , e ) none of these
Rationale: "l × b = 21 × b ∴ l = 21 m and l – b = 10 ∴ b = 21 – 10 = 11 m answer c"
Answer: c
Python program:
a = 21 - 10
Problem: a man took a loan at rate of 12 % per annum simple interest . after 3 years he had to pay 3600 interest . the principal amount borrowed by him was .
Options: a ) rs 10000 , b ) rs 15000 , c ) rs 16000 , d ) rs 17000 , e ) none of these
Rationale: "explanation : s . i . = p â ˆ — r â ˆ — t / 100 = > p = s . i . â ˆ — 100 / r â ˆ — t = > p = 3600 â ˆ — 100 / 12 â ˆ — 3 = rs 10000 option a"
Answer: a
Python program:
a = 3600 * 100
b = 12 * 3
c = a / b
Problem: a , b and c started a business with a total investment of rs . 72000 . a invests rs . 6000 more than b and b invests rs . 3000 less than c . if the total profit at the end of a year is rs . 8640 , find c ' s share .
Options: a ) rs . 3240 , b ) rs . 2520 , c ) rs . 2880 , d ) rs . 3360 , e ) none of these
Rationale: "explanation : let c ' s investment = rs . x b ' s investment = rs . ( x - 3000 ) a ' s investment = rs . ( x - 3000 + 6000 ) = rs . ( x + 3000 ) now , ( a + b + c ) ' s investment = rs . 72000 = > x + ( x - 3000 ) + ( x + 3000 ) = 72000 = > 3 x = 72000 = > x = 24000 hence , a ' s investment = rs . 27000 b ' s investment = rs . 21000 c ' s investment = rs . 24000 ratio of the capitals of a , b and c = 27000 : 21000 : 24000 = 9 : 7 : 8 a ' s share = rs . [ ( 8 / 24 ) ã — 8640 ] = rs . 2880 answer : option c"
Answer: c
Python program:
a = 6000 + 3000 
b = 72000 - a
c = b / 3
d = c + 6000 
e = d / 72000
f = 8640 * e
Problem: ( 3 x + 2 ) ( 2 x - 3 ) = ax ^ 2 + kx + n . what is the value of a - n + k ?
Options: a ) 5 , b ) 7 , c ) 9 , d ) 10 , e ) 11
Rationale: "expanding we have 6 x ^ 2 - 9 x + 4 x - 6 6 x ^ 2 - 5 x - 6 taking coefficients , a = 6 , k = - 5 , n = - 6 therefore a - n + k = 6 - ( - 6 ) - 5 = 12 - 5 = 7 the answer is b ."
Answer: b
Python program:
a = 3 * 2
b = 3 * 2
c = a + b 
d = 2 * 2
e = 3 * 3
f = d - e
g = c + f
Problem: a vendor sells 50 percent of the apples he had and throws away 20 percent of the remainder . the next day , the vendor sells 50 percent of the remaining apples and throws away the rest . in total , what percent of his apples does the vendor throw away ?
Options: a ) 30 , b ) 35 , c ) 40 , d ) 45 , e ) 50
Rationale: "let x be the original number of apples . on day one , the vendor throws away ( 0.2 ) ( 0.5 ) x = 0.1 x . the remaining apples are ( 0.8 ) ( 0.5 ) x = 0.4 x . on day two , the vendor throws away ( 0.5 ) ( 0.4 ) x = 0.2 x . the vendor throws away a total of 0.1 x + 0.2 x = 0.3 x . the vendor throws away 30 percent of the apples . the answer is a ."
Answer: a
Python program:
a = 100 - 50
b = 20 / 100
c = 100 - 50
d = b * c
e = a - d
f = e / 2
g = 20 / 100
h = 100 - 50
i = g * h
j = f + i
Problem: a can do a piece of work in 12 days and b alone can do it in 14 days . how much time will both take to finish the work ?
Options: a ) a ) 3.2222 , b ) b ) 5 , c ) c ) 6.46 , d ) d ) 8.3333 , e ) e ) 9
Rationale: this question can be solved by different methods . we need to conserve time in exams so solving this problem using equations is the good idea . time taken to finish the job = xy / ( x + y ) = 12 x 14 / ( 12 + 14 ) = 150 / 25 = 6.46 days answer : c
Answer: c
Python program:
a = 1 / 12
b = 1 / 14
c = a + b 
d = 1 / c
Problem: a jar full of whisky contains 40 % alcohol . a part of this whisky is replaced by another containg 19 % alcohol and now the percentage of alcohol was found to be 26 % . what quantity of whisky is replaced ?
Options: a ) 1 / 3 , b ) 2 / 3 , c ) 2 / 5 , d ) 3 / 5 , e ) 4 / 5
Rationale: let us assume the total original amount of whiskey = 10 ml - - - > 4 ml alcohol and 6 ml non - alcohol . let x ml be the amount removed - - - > total alcohol left = 4 - 0.4 x new quantity of whiskey added = x ml out of which 0.19 is the alcohol . thus , the final quantity of alcohol = 4 - 0.4 x + 0.19 x - - - - > ( 4 - 0.21 x ) / 10 = 0.26 - - - > x = 20 / 3 ml . per the question , you need to find the x ml removed as a ratio of the initial volume - - - > ( 20 / 3 ) / 10 = 2 / 3 . hence , b is the correct answer .
Answer: b
Python program:
a = 40 - 26
b = 40 - 19
c = a / b