euclaise/mathqa_programs huggingface dataset:
Problem: a and b can finish a work in 7 days and 8 days respectively . if both do work one day and leave one day . and a start the work then in how much days work will finish ?
Options: a ) 52 / 7 , b ) 8 , c ) 14 / 2 , d ) 6 , e ) 43 / 6
Rationale: ( a + b ) work in 2 days = [ 1 / 7 + 1 / 8 ] = 15 / 56 ( a + b ) work in 6 days = 45 / 56 now , the turn of a , work of a in 1 day = 1 / 7 . till now completed work = [ 45 / 56 + 1 / 7 ] = 53 / 56 , remaining work = [ 1 - 53 / 56 ] = 3 / 56 now , turn of b , 1 / 8 work b do in 1 day so , 3 / 56 part of work b do = [ 8 * 3 / 56 ] = 3 / 7 days total time taken = 6 + 1 + 3 / 7 = 52 / 7 days answer a
Answer: a
Python program:
a = 1 / 7
b = 1 / 8
c = 1 - b
d = a / c
e = 1/(d)
f = 1 + e
Problem: 5 pumps , working 9 hours a day , can empty a tank in 3 days . how many hours a day should 6 pumps work in order to empty the tank in 1 day ?
Options: a ) 20 hours , b ) 22.5 hours , c ) 23 hours , d ) 23.5 hours , e ) 24 hours
Rationale: "let the required hours needed be x more pumps , less hours ( indirect proportion ) more days , less hours ( indirect proportion ) hence we can write as ( pumps ) 5 : 6 } : : x : 9 ( days ) 3 : 1 = > 5 * 3 * 9 = 6 * 1 * x = > x = 45 / 2 = > 22.5 answer : b"
Answer: b
Python program:
a = 5 * 9
b = a * 3
c = b / 6
Problem: if the average ( arithmetic mean ) of ( 2 a + 16 ) and ( 3 a - 8 ) is 79 , what is the value of a ?
Options: a ) 25 , b ) 30 , c ) 28 , d ) 36 , e ) 42
Rationale: "( ( 2 a + 16 ) + ( 3 a - 8 ) ) / 2 = ( 5 a + 8 ) / 2 = 79 a = 30 the answer is b ."
Answer: b
Python program:
a = 79 * 2
b = 16 - 8
c = a - b
d = 2 + 3 
e = c / d
Problem: we run a business that rents out canoes and kayaks . a canoe rental costs $ 12 per day , and a kayak rental costs $ 18 dollars per day . one day , our business rents out 3 canoes for every 2 kayaks and receives a total of $ 504 in revenue . how many more canoes than kayaks were rented out ?
Options: a ) 5 , b ) 6 , c ) 7 , d ) 8 , e ) 9
Rationale: "let x be the number of canoes . then 2 x / 3 is the number of kayaks . 12 x + ( 2 x / 3 ) * 18 = 504 12 x + 12 x = 504 24 x = 504 x = 21 ( canoes ) 2 x / 3 = 14 ( kayaks ) there were 21 - 14 = 7 more canoes rented out . the answer is c ."
Answer: c
Python program:
a = 3 - 2
b = 3 * 12
c = 2 * 18
d = b + c 
e = 504 / d
f = a * e
Problem: the average of runs of a cricket player of 20 innings was 32 . how many runs must he make in his next innings so as to increase his average of runs by 4 ?
Options: a ) 96 , b ) 106 , c ) 122 , d ) 116 , e ) 122
Rationale: "average = total runs / no . of innings = 32 so , total = average x no . of innings = 32 * 20 = 640 now increase in avg = 4 runs . so , new avg = 32 + 4 = 36 runs total runs = new avg x new no . of innings = 36 * 21 = 756 runs made in the 11 th inning = 756 - 640 = 116 answer : d"
Answer: d
Python program:
a = 20 + 1 
b = 4 + 32 
c = a * b
d = 20 * 32
e = c - d
Problem: a cube with its sides numbered 1 through 6 is rolled twice , first landing on a and then landing on b . if any roll of the cube yields an equal chance of landing on any of the numbers 1 through 6 , what is the probability r that a + b is prime ?
Options: a ) 0 , b ) 1 / 12 , c ) 5 / 12 , d ) 7 / 18 , e ) 4 / 9
Rationale: total # of outcomes is 6 * 6 = 36 ; favorable outcomes : a - b - - > prime 1 - 1 - - > 2 ; 1 - 2 - - > 3 ; 2 - 1 - - > 3 ; 1 - 4 - - > 5 ; 4 - 1 - - > 5 ; 2 - 3 - - > 5 ; 3 - 2 - - > 5 ; 1 - 6 - - > 7 ; 6 - 1 - - > 7 ; 2 - 5 - - > 7 ; 5 - 2 - - > 7 ; 3 - 4 - - > 7 ; 4 - 3 - - > 7 ; 6 - 5 - - > 11 ; 5 - 6 - - > 11 . total of 15 favorable outcomes r = 15 / 36 . answer : c .
Answer: c
Python program:
a = 12 + 3 
b = 1 / 6
c = b ** 2
d = a * c
Problem: a and b started a business in partnership investing rs . 20,000 and rs . 15,000 respectively . after 6 months , c joined them with rs . 20,000 . whatwill be b ' s share in total profit of rs . 22,000 earned at the end of 2 years from the startingof the business ?
Options: a ) s . 5,000 , b ) s . 5,500 , c ) s . 5,700 , d ) s . 6,600 , e ) s . 7,500
Rationale: "a : b : c = ( 20,000 x 24 ) : ( 15,000 x 24 ) : ( 20,000 x 18 ) = 4 : 3 : 3 . b ' s share = rs . 22000 x 3 / 10 = rs . 6,600 . d"
Answer: d
Python program:
a = 4 + 1 
b = a * 4
c = 3 * 4
d = 2 * c
e = b * d
Problem: find the value for x from below equation : x / 3 = - 6 ?
Options: a ) - 16 , b ) 11 , c ) - 12 , d ) - 18 , e ) 14
Rationale: "1 . multiply both sides by 3 : x * 3 / 3 = - 6 / 3 simplify both sides : x = - 18 d"
Answer: d
Python program:
a = 6 - 4
b = a * 3
Problem: x can do a piece of work in 8 hours . y and z together can do it in 6 hours , while x and z together can do it in 4 hours . how long will y alone take to do it ?
Options: a ) 24 hours , b ) 20 hours , c ) 34 hours , d ) 12 hours , e ) 10 hours
Rationale: x ' s 1 hour work = 1 / 8 ( y + z ) ' s 1 hour work = 1 / 6 ( x + z ) ' s 1 hour work = 1 / 4 ( x + y + z ) ' s 1 hour work = ( 1 / 8 + 1 / 6 ) = 7 / 24 y ' s 1 hour ' s work = ( 7 / 24 - 1 / 4 ) = 1 / 24 y alone will take 24 hours to do the work . correct option is a
Answer: a
Python program:
a = 1/(6)
b = 1/(4)
c = 1/(8)
d = b - c
e = a - d
f = 1/(e)
Problem: a , b and c invested rs . 6300 , rs . 4200 and rs . 10500 respectively , in a partnership business . find the share of a in profit of rs . 13000 after a year ?
Options: a ) 3630 , b ) 3900 , c ) 8828 , d ) 2387 , e ) 2813
Rationale: "6300 : 4200 : 10500 3 : 2 : 5 3 / 10 * 13000 = 3900 answer : b"
Answer: b
Python program:
a = 6300 + 4200 
b = a + 10500 
c = 6300 / b
d = c * 13000
Problem: if two students starting from same point , walking in the opposite directions with 6 km / hr and 9 km / hr as average speeds respectively . then the distance between them after 4 hours is ?
Options: a ) 65 km , b ) 55 km , c ) 15 km , d ) 60 km , e ) 75 km
Rationale: explanation : total distance = distance traveled by person a + distance traveled by person b = ( 6 × 4 ) + ( 9 × 4 ) = 24 + 36 = 60 km answer : d
Answer: d
Python program:
a = 6 * 4
b = 9 * 4
c = a + b
Problem: dacid obtained 51 , 65 , 82 , 67 and 85 marks ( out of 100 ) in english , mathematics , physics , chemistry and biology . what are his average marks ?
Options: a ) 29 , b ) 38 , c ) 39 , d ) 70 , e ) 75
Rationale: "average = ( 51 + 65 + 82 + 67 + 85 ) / 5 = 70 answer : d"
Answer: d
Python program:
a = 51 + 65 
b = a + 82 
c = b + 67 
d = c + 85 
e = 10 / 2
f = d / e
Problem: in a mixture of 45 litres the ratio of milk to water is 4 : 1 . additional 23 litres of water is added to the mixture . find the ratio of milk to water in the resulting mixture .
Options: a ) 9 / 8 , b ) 4 / 1 , c ) 2 / 3 , d ) 3 / 4 , e ) 3 / 2
Rationale: "given that milk / water = 4 x / x and 4 x + x = 45 - - > x = 9 . thus milk = 4 x = 36 liters and water = x = 9 liters . new ratio = 36 / ( 9 + 23 ) = 36 / 32 = 9 / 8 . answer : a ."
Answer: a
Python program:
a = 4 + 1 
b = 45 / a
c = 45 - b
d = 4 + 1 
e = 45 / d
f = e + 23 
g = c / f
Problem: 70 percent of the members of a study group are women , and 40 percent of those women are lawyers . if one member of the study group is to be selected at random , what is the probability that the member selected is a woman lawyer ?
Options: a ) 0.28 , b ) 0.27 , c ) 0.25 , d ) 0.32 , e ) 0.35
Rationale: "say there are 100 people in that group , then there would be 0.7 * 0.40 * 100 = 28 women lawyers , which means that the probability that the member selected is a woman lawyer is favorable / total = 28 / 100 . answer : a"
Answer: a
Python program:
a = 5 * 5
b = a * 4
c = 70 / b
d = 5 * 5
e = d * 4
f = 40 / e
g = c * f
Problem: if two integers x , y ( x > y ) are selected from - 5 to 5 ( inclusive ) , how many cases are there ?
Options: a ) 55 , b ) 60 , c ) 65 , d ) 70 , e ) 75
Rationale: "there are 11 integers from - 5 to 5 inclusive . 11 c 2 = 55 . the answer is a ."
Answer: a
Python program:
a = 5 + 5 
b = 5 + 2 
c = a + b 
d = 5 + 1 
e = c + d 
f = e + 5 
g = f + 5 
h = g + 2 
i = h + 1
Problem: if x and y are positive integers and 5 + x + y + xy = 19 , what is the value of x + y ?
Options: a ) 3 , b ) 5 , c ) 6 , d ) 8 , e ) 9
Rationale: "try each answer choices . for a : 5 + 3 + xy = 19 ; xy = 11 ( impossible , 11 prime number . 1 + 11 does n ' t equal 3 ) for b : 5 + 5 + xy = 19 ; xy = 9 ( no combination of xy = 9 and x + y = 5 ) for c : 5 + 6 + xy = 19 ; xy = 8 ( x + y = 6 ; x = 2 , y = 4 or x = 4 , y = 2 ) for d : 5 + 8 + xy = 19 ; xy = 6 ( no combination of xy = 6 and x + y = 8 ) for e : 5 + 9 + xy = 19 ; xy = 5 ( impossible , 5 prime number . 1 + 5 does n ' t equal 9 ) therefore , answer c ."
Answer: c
Python program:
a = 19 + 1
Problem: the cost price of a radio is rs . 1500 and it was sold for rs . 1260 , find the loss % ?
Options: a ) 18 , b ) 16 , c ) 26 , d ) 17 , e ) 11
Rationale: "explanation : 1500 - - - - 240 100 - - - - ? = > 16 % answer : b"
Answer: b
Python program:
a = 1500 - 1260
b = a / 1500
c = b * 100
Problem: find the area of a cuboid of length 12 cm , breadth 6 cm . and height 10 cm .
Options: a ) 400 cm cube , b ) 410 cm cube , c ) 720 cm cube , d ) 730 cm cube , e ) 480 cm cube
Rationale: "area of a cuboid = lxbxh = 12 cm x 6 cm x 10 cm = 720 cm cube answer : c"
Answer: c
Python program:
a = 12 * 6
b = a * 10
Problem: a , b and c started a business with capitals of rs . 8000 , rs . 10000 and rs . 12000 respectively . at the end of the year , the profit share of b is rs . 1900 . the difference between the profit shares of a and c is ?
Options: a ) 288 , b ) 760 , c ) 155 , d ) 600 , e ) 441
Rationale: "ratio of investments of a , b and c is 8000 : 10000 : 12000 = 4 : 5 : 6 and also given that , profit share of b is rs . 1900 = > 5 parts out of 15 parts is rs . 1900 now , required difference is 6 - 4 = 2 parts required difference = 2 / 5 ( 1900 ) = rs . 760 answer : b"
Answer: b
Python program:
a = 12000 / 10000
b = 8000 / 10000
c = a - b
d = c * 1900
Problem: what percent is 520 gm of 1 kg ?
Options: a ) 25 % , b ) 40 % , c ) 52 % , d ) 8 % , e ) 12 %
Rationale: "1 kg = 1000 gm 520 / 1000 ã — 100 = 52000 / 1000 = 52 % answer is c"
Answer: c
Python program:
a = 520 / 1
b = a * 100
Problem: machine a can finish a job in 6 hours , machine в can finish the job in 12 hours , and machine с can finish the job in 8 hours . how many hours will it take for a , b , and с together to finish the job ?
Options: a ) 6 / 5 , b ) 7 / 4 , c ) 8 / 3 , d ) 9 / 2 , e ) 12 / 5
Rationale: the combined rate is 1 / 6 + 1 / 12 + 1 / 8 = 9 / 24 of the job per hour . the time to complete the job is 24 / 9 = 8 / 3 hours . the answer is c .
Answer: c
Python program:
a = 1/(6)
b = 1/(12)
c = a + b 
d = 1/(8)
e = c + d 
f = 1/(e)
Problem: the total marks obtained by a student in physics , chemistry and mathematics is 150 more than the marks obtained by him in physics . what is the average mark obtained by him in chemistry and mathematics ?
Options: a ) 75 , b ) 66 , c ) 55 , d ) 44 , e ) 12
Rationale: let the marks obtained by the student in physics , chemistry and mathematics be p , c and m respectively . p + c + m = 150 + p c + m = 150 average mark obtained by the student in chemistry and mathematics = ( c + m ) / 2 = 150 / 2 = 75 . answer : a
Answer: a
Python program:
a = 150 / 2
Problem: a man buys a cycle for rs . 1400 and sells it at a loss of 18 % . what is the selling price of the cycle ?
Options: a ) rs . 1148 , b ) rs . 1160 , c ) rs . 1190 , d ) rs . 1202 , e ) none
Rationale: "solution s . p = 82 % of rs . 1400 = rs . ( 82 / 100 × 1400 ) rs . 1148 . answer a"
Answer: a
Python program:
a = 100 - 18
b = a * 1400
c = b / 100
Problem: at a contest with 3000 participants , 1 / 2 of the people are aged 8 to 14 . next year , the number of people aged 8 to 14 will increase by 1 / 6 . after this change , what percentage of the total 3000 people will the 8 - to 14 - year - olds represent ?
Options: a ) 53.33 % , b ) 50 % , c ) 58.33 % , d ) 55.33 % , e ) 68 %
Rationale: "i just wanted to mention a couple of things here : * this is a pure ratio question ; the number 3000 is completely irrelevant , and you can ignore it if you like . when we increase something by 1 / 6 , we are multiplying it by 1 + 1 / 6 = 7 / 6 , so the answer here must be ( 1 / 2 ) * ( 7 / 6 ) = 7 / 12 = 58.33 % . answer : c"
Answer: c
Python program:
a = 3000 / 2
b = 1 / 6
c = 1 + b 
d = a * c
e = d / 3000
f = e * 100
Problem: if henry were to add 7 gallons of water to a tank that is already 3 / 4 full of water , the tank would be 7 / 8 full . how many gallons of water would the tank hold if it were full ?
Options: a ) 25 , b ) 40 , c ) 64 , d ) 80 , e ) 56
Rationale: "7 / 8 x - 3 / 4 x = 7 galls 1 / 8 * x = 7 gallons x = 56 gallons answer e"
Answer: e
Python program:
a = 7 / 8
b = 3 / 4
c = a - b
d = 1 / c
e = 7 * d
Problem: the average weight of 16 boys in a class is 50.25 kgs and that of the remaining 8 boys is 45.15 kgs . find the average weight of all the boys in the class .
Options: a ) 47.55 kgs , b ) 48 kgs , c ) 48.55 kgs , d ) 49.25 kgs , e ) none of these
Rationale: "solution required average = ( 50.25 x 16 + 45.15 x 8 / 16 + 8 ) = ( 804 + 361.20 / 24 ) = 1165.20 / 24 = 48.55 answer c"
Answer: c
Python program:
a = 16 * 50
b = 8 * 45
c = a + b 
d = 16 + 8 
e = c / d
Problem: of the 150 employees at company x , 50 are full - time , and 100 have worked at company x for at least a year . there are 20 employees at company x who aren ’ t full - time and haven ’ t worked at company x for at least a year . how many full - time employees of company x have worked at the company for at least a year ?
Options: a ) 20 , b ) 30 , c ) 50 , d ) 80 , e ) 100
Rationale: "150 employees 50 are full - time 100 have worked at company x for at least a year 20 employees at company x who aren ’ t full - time and haven ’ t worked at company x for at least a year . how many full - time employees of company x have worked at the company for at least a year ? 150 - 50 = 100 employees not full time 100 - 20 = 80 employees not full time who worked over a year 100 employees have worked at company x for at least a year - 80 employees not full time who worked over a year = 20 full - time employees of company x have worked at the company for at least a year ans a"
Answer: a
Python program:
a = 150 - 50
b = a - 20
Problem: solution x is 40 % chemical a and 60 % chemical b by volume . solution y is 50 % chemical a and 50 % chemical b by volume . if a mixture of x and y is 47 % chemical a , what percent of the mixture is solution x ?
Options: a ) 15 % , b ) 20 % , c ) 25 % , d ) 30 % , e ) 35 %
Rationale: "the volume of the mixture be x + y . 0.4 x + 0.5 y = 0.47 ( x + y ) x = 3 y / 7 x / ( x + y ) = ( 3 y / 7 ) / ( 10 y / 7 ) = 3 / 10 = 30 % . the answer is d ."
Answer: d
Python program:
a = 50 - 47
b = 47 - 40
c = a / b
d = 50 - 47
e = 47 - 40
f = d / e
g = f + 1 
h = c / g
i = h * 100
Problem: a pet store regularly sells pet food at a discount of 10 percent to 30 percent from the manufacturer ’ s suggested retail price . if during a sale , the store discounts an additional 20 percent from the discount price , what would be the lowest possible price of a container of pet food that had a manufacturer ’ s suggested retail price o f $ 30.00 ?
Options: a ) $ 10.00 , b ) $ 11.20 , c ) $ 14.40 , d ) $ 16.80 , e ) $ 18.00
Rationale: "for retail price = $ 30 first maximum discounted price = 30 - 30 % of 30 = 30 - 9 = 21 price after additional discount of 20 % = 21 - 20 % of 21 = 21 - 4.2 = 16.8 answer : option d"
Answer: d
Python program:
a = 100 - 20
b = a / 100
c = 100 - 30
d = c / 100
e = d * 30
f = b * e
Problem: the simple interest on a sum of money will be rs . 500 after 10 years . if the principal is trebled after 5 years what will be the total interest at the end of the tenth year ?
Options: a ) 2261 , b ) 2888 , c ) 1200 , d ) 2699 , e ) 900
Rationale: "p - - - 10 - - - - 500 p - - - 5 - - - - - 250 3 p - - - 5 - - - - - 750 - - - - - - = > 900 answer : e"
Answer: e
Python program:
a = 500 / 10
b = a * 5
c = b * 3
d = 500 / 10
e = d * 5
f = c + e
Problem: 16 meters of wire is available to fence off a flower bed in the form of a circular sector . what must the radius of the circle in meters be , if we wish to have a flower bed with the greatest possible surface area ?
Options: a ) 2 √ 2 , b ) 4 , c ) 5 , d ) 4 √ 2 , e ) none of these
Rationale: "area of sector , a = x / 360 * pi * r ^ 2 circumference of the sector = 16 = > x / 360 * 2 * pi * r + 2 r = 16 = > 2 a / r + 2 r = 16 = > a = ( r 16 - 2 r ^ 2 ) / 2 = r 8 - r ^ 2 we will now max using derivations max value of a will found at a = 0 i . e 8 - 2 r = 0 r = 4 b"
Answer: b
Python program:
a = 16 / 4
Problem: amar takes as much time in running 21 meters as a car takes in covering 54 meters . what will be the distance covered by amar during the time the car covers 1.8 km ?
Options: a ) 700 m , b ) 500 m , c ) 670 m , d ) 700 m , e ) 640 m
Rationale: "distance covered by amar = 21 / 54 ( 1.8 km ) = 7 / 18 ( 1800 ) = 700 m answer : d"
Answer: d
Python program:
a = 1 * 8
b = 21 * a
c = b / 54
Problem: an alloy weighing 16 ounces is 50 % gold . how many ounces of pure gold must be added to create an alloy that is 80 % gold ?
Options: a ) 24 , b ) 18 , c ) 9 , d ) 27 , e ) 18
Rationale: "an alloy of 16 oz which is 50 % gold means there is 8 oz of gold . to get to an alloy that is 80 % gold , let ' s use this expression : ( 8 + x ) / ( 16 + x ) = 0.8 with x representing the amount of pure gold that must be added to get to 80 % . the expression we are using represents the new total weight of pure gold over the new total new weight of the alloy and this fraction should represent 80 % or 0.8 . you will see that 24 is the correct answer , as 32 / 40 = 0.8 choose a"
Answer: a
Python program:
a = 80 - 50
b = 16 * a
c = b / 12
d = c - 16
Problem: what percent of 12.4 kg is 20 gms ?
Options: a ) 25 , b ) 16 , c ) 18 , d ) 19 , e ) 17
Rationale: "explanation : required percentage = ( 20 / 12400 * 100 ) % = 4 / 25 % = 0.16 % answer : b ) . 16 %"
Answer: b
Python program:
a = 12 / 4
b = a * 100
Problem: from 4 officers and 8 jawans in how many can 6 be chosen to include exactly one officer ?
Options: a ) 150 , b ) 164 , c ) 259 , d ) 311 , e ) 224
Rationale: required number of ways = 4 c 1 * 8 c 5 = 224 answer is e
Answer: e
Python program:
a = math.factorial(8)
b = 6 - 1
c = math.factorial(b)
d = a / c
e = math.factorial(3)
f = d / e
g = f * 4
Problem: if each side of a rectangle is increased by 100 % , by what % the area increases ?
Options: a ) 100 %
b ) 200 %
c ) 250 %
d ) 300 %
e ) 400 %
Rationale: if length is l and width is b then area = l * b now new l and b is 2 l and 2 b respectively . so new area is = 4 * l * b % increase = 300 % answer : d
Answer: d
Python program:
a = 100 / 100
b = a * 2
c = b ** 2
d = c - 1
e = d * 100
Problem: a train 560 m long is running at a speed of 45 km / hr . in what time will it pass a bridge 140 m long ?
Options: a ) 40 , b ) 56 , c ) 41 , d ) 42 , e ) 34
Rationale: "speed = 45 * 5 / 18 = 25 / 2 m / sec total distance covered = 560 + 140 = 700 m required time = 700 * 2 / 25 = 40 sec answer : option b"
Answer: b
Python program:
a = 45 - 140
b = a * const_0_2778
c = 560 / b
Problem: a boat sails 15 km of a river towards upstream in 5 hours . how long will it take to cover the same distance downstream , if the speed of current is one - fourth the speed of the boat in still water ?
Options: a ) 7 h , b ) 3 h , c ) 4 h , d ) 2 h , e ) 9 h
Rationale: explanation : upstream speed = b - s downstream speed = b + s b - s = 15 / 5 = 3 km / h again b = 4 s therefore b - s = 3 = 3 s = > s = 1 and b = 4 km / h therefore b + s = 5 km / h therefore , time during downstream = 15 / 5 = 3 h answer : b
Answer: b
Python program:
a = 15 / 5
Problem: the tax on a commodity is diminished by 22 % and its consumption increased by 11 % . the effect on revenue is ?
Options: a ) 8 , b ) 6 , c ) 5 , d ) 4 , e ) 2
Rationale: "100 * 100 = 10000 88 * 111 = 9768 - - - - - - - - - - - 10000 - - - - - - - - - - - 232 100 - - - - - - - - - - - ? = > 2 % decrease answer : e"
Answer: e
Python program:
a = 11 / 100
b = 1 + a 
c = 22 / 100
d = 1 - c
e = b * d
f = e * 100
g = 100 - f
Problem: if an integer n is to be chosen at random from the integers 1 to 96 , inclusive , what is the probability that n ( n + 1 ) ( n + 2 ) will be divisible by 16 ?
Options: a ) 1 / 4 , b ) 3 / 8 , c ) 1 / 2 , d ) 9 / 16 , e ) 3 / 4
Rationale: "i get 5 / 8 as well 1 to 96 inclusive means we have 48 odd and 48 even integers e o e / 16 = integer , therefore we have 48 / 96 numbers divisible by 8 o e o / 16 = not integer we can not forget multiples of 16 from 1 to 96 we have 6 numbers that are multiple of 16 therefore , 48 / 96 + 6 / 96 = 54 / 96 = 9 / 16 answer : d"
Answer: d
Python program:
a = 96 / 2
b = 96 / 16
c = a + b 
d = c / 96
Problem: there was two trains from calcutta to kanyakumari one train is fast service travels with a speed 75 km per hour another travels with a speed of 44 km per hour the time taken to reach from calcutta to kanyakumari is 4 hours less than the first train . . . find the distance b / w calcatta to kanyakumari
Options: a ) 415.8 km , b ) 425.8 km , c ) 435.8 km , d ) 445.8 km , e ) 455.8 km
Rationale: let distance b / w calcutta to kanyakumari is = x km then eqn is x / 44 - x / 75 = 4 = > x * ( 75 - 44 ) = 4 * 75 * 44 = > x = 425.8 km answer : b
Answer: b
Python program:
a = 75 * 44
b = 4 * a
c = 75 - 44
d = b / c
Problem: in a group of 10 doctors , 3 doctors are only pediatricians ; the others are surgeons or general practitioners - but not both . a team of 3 doctors is to be chosen which must have at least 1 pediatrician , how many different teams can be chosen ?
Options: a ) 85 , b ) 140 , c ) 60 , d ) 80 , e ) 75
Rationale: the problem asks for a combination , since order does n ' t matter . now , selecting r items from a set of n gives the combination formula : ncr = n ! / r ! ( n - r ) ! n = 10 r = 3 so , total teams is 10 c 3 = 10 ! / ( 3 ! ( 10 - 3 ) ! ) = 120 , and n = 10 - 3 = 7 r = 3 for teams without a pediatrician is 7 c 3 = 7 ! / ( 3 ! ( 7 - 3 ) ! ) = 35 , so , teams with at least 1 pediatrician = 120 - 35 = 85 answer : a
Answer: a
Python program:
a = math.factorial(10)
b = 10 - 3
c = math.factorial(b)
d = math.factorial(3)
e = c * d
f = a / e
g = 10 - 3
h = 10 * g
i = h / 2
j = f - i
Problem: if the length of a certain rectangle is decreased by 4 cm and the width is increased by 3 cm , a square with the same area as the original rectangle would result . find the perimeter of the original rectangle ?
Options: a ) 76 , b ) 27 , c ) 19 , d ) 50 , e ) 11
Rationale: "let x and y be the length and breadth of the rectangle respectively . then , x - 4 = y + 3 or x - y = 7 - - - - ( i ) area of the rectangle = xy ; area of the square = ( x - 4 ) ( y + 3 ) ( x - 4 ) ( y + 3 ) = xy < = > 3 x - 4 y = 12 - - - - ( ii ) solving ( i ) and ( ii ) , we get x = 16 and y = 9 . perimeter of the rectangle = 2 ( x + y ) = [ 2 ( 16 + 9 ) ] cm = 50 cm . answer : d"
Answer: d
Python program:
a = 3 + 4 
b = a * 3
c = 4 * 3
d = b - c
e = 3 + 4 
f = e * 3
g = 4 * 3
h = f - g
i = 3 + 4 
j = h + i 
k = d + j 
l = k * 2
Problem: a football player scores 4 goals in his fifth match thus increasing his average goals score by 0.2 . the total number of goals in his 5 matches would be
Options: a ) 4 , b ) 6 , c ) 8 , d ) 10 , e ) 16
Rationale: "while this question can be solved with a rather straight - forward algebra approach ( as the other posters have noted ) , it can also be solved by testing the answers . one of those numbers must be the total number of goals . . . from a tactical standpoint , it ' s best to test either answer b or answer d , so if the answer is not correct , then you would have a gauge for whether you should gohigherorlowerwith your next test . here , i ' ll start with answer e = 16 goals if . . . . total goals = 16 goals 5 th game = 4 goals 1 st 4 games = 12 goals avg . for 1 st 4 games = 12 / 4 = 3 goal / game avg . for all 5 games = 6 / 5 = 3.2 goals / game this is an exact match for what we ' re told in the prompt , so answer e must be the answer ."
Answer: e
Python program:
a = 4 * 4
b = 4 * 5
c = b * 0
d = a - c
e = d + 4