euclaise/mathqa_programs huggingface dataset:
Problem: solution a is 20 % salt and solution b is 60 % salt . if you have 30 ounces of solution a and 60 ounces of solution b , in what ratio could you mix solution a with solution b to produce 50 ounces of a 50 % salt solution ?
Options: a ) 6 : 4 , b ) 6 : 14 , c ) 4 : 4 , d ) 4 : 6 , e ) 1 : 3
Rationale: "forget the volumes for the time being . you have to mix 20 % and 80 % solutions to get 50 % . this is very straight forward since 50 is int he middle of 20 and 80 so we need both solutions in equal quantities . if this does n ' t strike , use w 1 / w 2 = ( a 2 - aavg ) / ( aavg - a 1 ) w 1 / w 2 = ( 60 - 50 ) / ( 50 - 20 ) = 1 / 3 so the volume of the two solutions will be equal . answer has to be 1 : 3 e ."
Answer: e
Python program:
a = 60 / 100
b = 50 * a
c = 50 / 100
d = 50 * c
e = b - d
f = 60 / 100
g = 20 / 100
h = f - g
i = e / h
j = 60 / 100
k = 50 * j
l = 50 / 100
m = 50 * l
n = k - m
o = 60 / 100
p = 20 / 100
q = o - p
r = n / q
s = 50 - r
t = i / s
Problem: if 11.25 m of a uniform steel rod weighs 42.75 kg . what will be the weight of 8 m of the same rod ?
Options: a ) 22.8 kg , b ) 25.6 kg , c ) 28 kg , d ) 30.4 kg , e ) none of these
Rationale: explanation : let the required weight be x kg . then , less length , less weight ( direct proportion ) = > 11.25 : 8 : : 42.75 : x = > 11.25 x x = 8 x 42.75 = > x = ( 8 x 42.75 ) / 11.25 = > x = 30.4 answer : d
Answer: d
Python program:
a = 8 * 42
b = a / 11
Problem: the radius of the wheel of a bus is 250 cms and the speed of the bus is 66 km / h , then the r . p . m . ( revolutions per minutes ) of the wheel is
Options: a ) 70 , b ) 125 , c ) 300 , d ) 500 , e ) none of these
Rationale: "radius of the wheel of bus = 250 cm . then , circumference of wheel = 2 ï € r = 500 ï € = 1571.43440 cm distance covered by bus in 1 minute = 66 â  „ 60 ã — 1000 ã — 100 cms distance covered by one revolution of wheel = circumference of wheel = 1571.45 cm â ˆ ´ revolutions per minute = 6600000 / 60 ã — 1571.43 = 70 answer a"
Answer: a
Python program:
a = 1000 * 66
b = 100 * a
c = const_60 * 1
d = b / c
e = 2 * 250
f = 3 * 4
g = 2 + f 
h = 4 * 2
i = 2 + h 
j = i ** 2
k = g / j
l = 3 + k 
m = e * l
n = d / m
Problem: 1000 men have provisions for 21 days . if 800 more men join them , for how many days will the provisions last now ?
Options: a ) 12.9 , b ) 12.5 , c ) 12.6 , d ) 11.6 , e ) 12.1
Rationale: "1000 * 21 = 1800 * x x = 11.6 answer : d"
Answer: d
Python program:
a = 21 * 1000
b = 1000 + 800 
c = a / b
Problem: a rectangular wall is covered entirely with two kinds of decorative tiles : regular and jumbo . 1 / 3 of the tiles are jumbo tiles , which have a length three times that of regular tiles and have the same ratio of length to width as the regular tiles . if regular tiles cover 40 square feet of the wall , and no tiles overlap , what is the area of the entire wall ?
Options: a ) 160 , b ) 220 , c ) 360 , d ) 440 , e ) 560
Rationale: "the number of jumbo tiles = x . the number of regular tiles = 2 x . assume the ratio of the dimensions of a regular tile is a : a - - > area = a ^ 2 . the dimensions of a jumbo tile is 3 a : 3 a - - > area = 9 a ^ 2 . the area of regular tiles = 2 x * a ^ 2 = 40 . the area of jumbo tiles = x * 9 a ^ 2 = 4.5 ( 2 x * a ^ 2 ) = 4.5 * 40 = 180 . total area = 40 + 180 = 220 . answer : b ."
Answer: b
Python program:
a = 40 * 3
b = a / 2
c = b * 3
d = 40 + c
Problem: how many positive integers less than 10,000 are such that the product of their digits is 210 ?
Options: a ) 24 , b ) 30 , c ) 48 , d ) 54 , e ) 72
Rationale: "210 = 2 x 5 x 3 x 7 = 5 x 6 x 7 x 1 = 5 x 6 x 7 those are the only sets of digits we can use to for the numbers ( any other combination of factors will have two digit factors ) . numbers using 2,5 , 3,7 = 4 ! numbers using 5,6 , 7,1 = 4 ! numbers using 5 , 6,7 ( 3 - digit numbers ) = 3 ! answer = 24 + 24 + 6 = 54 answer is ( d )"
Answer: d
Python program:
a = 4 + 210 
b = a - 1
c = math.factorial(b)
d = math.factorial(210)
e = 4 - 1
f = math.factorial(e)
g = d * f
h = c / g
Problem: find the last term of a g . p whose first term is 9 and common ratio is ( 1 / 3 ) if the sum of the terms of the g . p is ( 40 / 3 )
Options: a ) 1 / 3 , b ) 2 / 5 , c ) 1 / 4 , d ) 2 / 3 , e ) 4 / 5
Rationale: sum of the g . p . = ( first term - r * last term ) / 1 – r 40 / 3 = 9 – 1 / 3 ( last term ) / 2 / 3 last term = ( - 40 / 3 * 2 / 3 + 9 ) * 3 = - 80 / 3 + 27 = 1 / 3 answer : a
Answer: a
Python program:
a = 1 / 3
Problem: the average of marks obtained by 120 boys was 36 . if the average of marks of passed boys was 39 and that of failed boys was 15 , the number of boys who passed the examination is ?
Options: a ) 105 , b ) 110 , c ) 120 , d ) 130 , e ) 140
Rationale: "let the number of boys who passed = x . then , 39 x x + 15 x ( 120 - x ) = 120 x 36 24 x = 4320 - 1800 = > x = 2520 / 24 x = 105 . hence , the number of boys passed = 105 . answer : a"
Answer: a
Python program:
a = 36 * 120
b = 120 * 15
c = a - b
d = 39 - 15
e = c / d
Problem: shekar scored 76 , 65 , 82 , 62 and 85 marks in mathematics , science , social studies , english and biology respectively . what are his average marks ?
Options: a ) 74 , b ) 69 , c ) 75 , d ) 85 , e ) 90
Rationale: "explanation : average = ( 76 + 65 + 82 + 62 + 85 ) / 5 = 370 / 5 = 74 hence average = 74 answer : a"
Answer: a
Python program:
a = 76 + 65 
b = a + 82 
c = b + 62 
d = c + 85 
e = 1 + 4 
f = d / e
Problem: a rectangular plot measuring 10 meters by 50 meters is to be enclosed by wire fencing . if the poles of the fence are kept 5 meters apart . how many poles will be needed ?
Options: a ) 46 m , b ) 66 m , c ) 26 m , d ) 24 m , e ) 25 m
Rationale: "perimeter of the plot = 2 ( 10 + 50 ) = 120 m no of poles = 120 / 5 = 24 m answer : d"
Answer: d
Python program:
a = 10 + 50 
b = a * 2
c = b / 5
Problem: x can finish a work in 21 days . y can finish the same work in 15 days . y worked for 5 days and left the job . how many days does x alone need to finish the remaining work ?
Options: a ) 12 , b ) 13 , c ) 16 , d ) 14 , e ) 18
Rationale: "work done by x in 1 day = 1 / 21 work done by y in 1 day = 1 / 15 work done by y in 5 days = 5 / 15 = 1 / 3 remaining work = 1 – 1 / 3 = 2 / 3 number of days in which x can finish the remaining work = ( 2 / 3 ) / ( 1 / 21 ) = 14 d"
Answer: d
Python program:
a = 1 / 15
b = 5 * a
c = 1 - b
d = 1 / 21
e = c / d
Problem: country c imposes a two - tiered tax on imported cars : the first tier imposes a tax of 16 % of the car ' s price up to a certain price level . if the car ' s price is higher than the first tier ' s level , the tax on the portion of the price that exceeds this value is 8 % . if ron imported a $ 14,000 imported car and ended up paying $ 1440 in taxes , what is the first tier ' s price level ?
Options: a ) $ 1600 , b ) $ 6000 , c ) $ 6050 , d ) $ 7050 , e ) $ 4000
Rationale: "let t be the tier price , p be total price = 14000 per the given conditions : 0.16 t + 0.08 ( p - t ) = 1440 - - - - > t = 8000 . e is the correct answer ."
Answer: e
Python program:
a = 2 * 3
b = 3 * a
c = b * 1000
d = 8 / 100
e = c * d
f = 1440 - e
g = 16 / 100
h = 8 / 100
i = g - h
j = f / i
Problem: what is the remainder when the number r = 14 ^ 2 * 15 ^ 8 is divided by 5 ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 4 , e ) 5
Rationale: "14 ^ 2 has units digit 6 15 ^ 8 has units digit 5 thus r = 14 ^ 2 * 15 ^ 8 has units digit 0 and will be divisible by 5 . the remainder will be zero answer : ( a )"
Answer: a
Python program:
a = 5 / 5
Problem: p is able to do a piece of work in 20 days and q can do the same work in 10 days . if they can work together for 2 days , what is the fraction of work left ?
Options: a ) 5 / 10 , b ) 9 / 10 , c ) 7 / 10 , d ) 6 / 10 , e ) 4 / 10
Rationale: "explanation : amount of work p can do in 1 day = 1 / 20 amount of work q can do in 1 day = 1 / 10 amount of work p and q can do in 1 day = 1 / 20 + 1 / 10 = 3 / 20 amount of work p and q can together do in 2 days = 2 × ( 3 / 20 ) = 3 / 10 fraction of work left = 1 – 3 / 10 = 7 / 10 answer : option c"
Answer: c
Python program:
a = 1 / 10
b = 1 / 20
c = a + b 
d = c * 2
e = 1 - d
Problem: carina has 70 ounces of coffee divided into 5 - and 10 - ounce packages . if she has 2 more 5 - ounce packages than 10 - ounce packages , how many 10 - ounce packages does she have ?
Options: a ) 2 , b ) 3 , c ) 4 , d ) 5 , e ) 6
Rationale: "lets say 5 and 10 ounce packages be x and y respectively . given that , 5 x + 10 y = 70 and x = y + 2 . what is the value of y . substituting the x in first equation , 5 y + 10 + 10 y = 70 - > y = 60 / 15 . = 4 c"
Answer: c
Python program:
a = 5 * 2
b = 70 - a
c = 10 + 5 
d = b / c
Problem: the average weight of a , b and c is 45 kg . if the average weight of a and b be 42 kg and that of b and c be 43 kg , then the weight of b is :
Options: a ) 33 kg , b ) 31 kg , c ) 32 kg , d ) 36 kg , e ) 35 kg
Rationale: "let a , b , c represent their respective weights . then , we have : a + b + c = ( 45 x 3 ) = 135 . . . . ( i ) a + b = ( 42 x 2 ) = 84 . . . . ( ii ) b + c = ( 43 x 2 ) = 86 . . . . ( iii ) adding ( ii ) and ( iii ) , we get : a + 2 b + c = 170 . . . . ( iv ) subtracting ( i ) from ( iv ) , we get : b = 35 b ' s weight = 35 kg . e"
Answer: e
Python program:
a = 42 * 2
b = 43 * 2
c = a + b 
d = 45 * 3
e = c - d
Problem: john found that the average of 15 numbers is 40 . if 11 is added to each number then the mean of number is ?
Options: a ) 51 , b ) 45 , c ) 65 , d ) 78 , e ) 64
Rationale: "( x + x 1 + . . . x 14 ) / 15 = 40 51 option a"
Answer: a
Python program:
a = 40 + 11
Problem: a boat goes 100 km downstream in 8 hours , and 75 km upstream in 15 hours . the speed of the stream is ?
Options: a ) 3 , b ) 6.5 , c ) 5.5 , d ) 4 , e ) 3.75
Rationale: "100 - - - 10 ds = 12.5 ? - - - - 1 75 - - - - 15 us = 5 ? - - - - - 1 s = ( 12.5 - 5 ) / 2 = 3.75 kmph . answer : e"
Answer: e
Python program:
a = 100 / 8
b = 75 / 15
c = a - b
d = c / 2
Problem: what is the number of integers from 1 to 1000 ( inclusive ) that are divisible by neither 11 nor by 30 ?
Options: a ) 884 , b ) 890 , c ) 892 , d ) 910 , e ) 945
Rationale: "normally , i would use the method used by bunuel . it ' s the most accurate . but if you are looking for a speedy solution , you can use another method which will sometimes give you an estimate . looking at the options ( most of them are spread out ) , i wont mind trying it . ( mind you , the method is accurate here since the numbers start from 1 . ) in 1000 consecutive numbers , number of multiples of 11 = 1000 / 11 = 90 ( ignore decimals ) in 1000 consecutive numbers , number of multiples of 35 = 1000 / 35 = 28 number of multiples of 11 * 35 i . e . 385 = 1000 / 385 = 2 number of integers from 1 to 1000 that are divisible by neither 11 nor by 35 = 1000 - ( 90 + 28 - 2 ) { using the concept of sets here ) = 884 think : why did i say the method is approximate in some cases ? think what happens if the given range is 11 to 1010 both inclusive ( again 1000 numbers ) what is the number of multiples in this case ? a"
Answer: a
Python program:
a = 1000 / 11
b = 1000 / 30
c = a + b 
d = 11 * 30
e = 1000 / d
f = c - e
g = 1000 - f
Problem: two passenger trains start at the same hour in the day from two different stations and move towards each other at the rate of 26 kmph and 21 kmph respectively . when they meet , it is found that one train has traveled 60 km more than the other one . the distance between the two stations is ?
Options: a ) 288 , b ) 564 , c ) 877 , d ) 278 , e ) 178
Rationale: "1 h - - - - - 5 ? - - - - - - 60 12 h rs = 26 + 21 = 47 t = 12 d = 47 * 12 = 564 answer : b"
Answer: b
Python program:
a = 21 - 26
b = 60 / a
c = b * 26
d = 21 - 26
e = 60 / d
f = e * 21
g = c + f
Problem: what is the smallest no . which must be added to 25268 so as to obtain a sum which is divisible by 11 ?
Options: a ) 5 , b ) 10 , c ) 11 , d ) 20 , e ) 30
Rationale: "for divisibility by 11 , the difference of sums of digits at even and odd places must be either zero or divisible by 11 . for 25268 , difference = ( 2 + 2 + 8 ) - ( 5 + 6 ) = 12 - 11 = 1 . the units digit is at odd place . so we add 10 to the number = > 25268 + 10 = 25278 now , ( 2 + 2 + 8 ) - ( 5 + 7 ) = 12 - 12 = 0 = > 25268 is also divisible by 11 b"
Answer: b
Python program:
a = 25268 * 11
b = a / 25268
Problem: a train 455 m long , running with a speed of 63 km / hr will pass a tree in ?
Options: a ) 22 sec , b ) 16 sec , c ) 17 sec , d ) 39 sec , e ) 12 sec
Rationale: "speed = 63 * 5 / 18 = 35 / 2 m / sec time taken = 455 * 2 / 35 = 39 sec answer : d"
Answer: d
Python program:
a = 63 * 1000
b = 455 / a
c = b * 3600
Problem: the price of an item is discounted 10 percent on day 1 of a sale . on day 2 , the item is discounted another 10 percent , and on day 3 , it is discounted an additional 15 percent . the price of the item on day 3 is what percentage of the sale price on day 1 ?
Options: a ) 28 % , b ) 40 % , c ) 64.8 % , d ) 76.5 % , e ) 72 %
Rationale: "original price = 100 day 1 discount = 10 % , price = 100 - 10 = 90 day 2 discount = 10 % , price = 90 - 9 = 81 day 3 discount = 15 % , price = 81 - 12.15 = 68.85 which is 68.85 / 90 * 100 of the sale price on day 1 = ~ 76.5 % answer d"
Answer: d
Python program:
a = 15 / 100
b = 1 / 10
c = 1 - b
d = a / c
e = d * 100
f = e + 2
Problem: an aeroplane covers a certain distance at a speed of 120 kmph in 4 hours . to cover the same distance in 1 2 / 3 hours , it must travel at a speed of :
Options: a ) 520 , b ) 288 , c ) 820 , d ) 740 , e ) 720
Rationale: "distance = ( 240 x 5 ) = 480 km . speed = distance / time speed = 480 / ( 5 / 3 ) km / hr . [ we can write 1 2 / 3 hours as 5 / 3 hours ] required speed = ( 480 x 3 / 5 ) km / hr = 288 km / hr answer b ) 288 km / hr"
Answer: b
Python program:
a = 120 * 4
b = 2 / 3
c = 1 + b 
d = a / c
e = d / 2
Problem: the present age of a father is 3 years more than 3 times the age of his son . 5 years hence , father ' s age will be 10 years more than twice the age of the son . find the present age of the father .
Options: a ) 33 , b ) 38 , c ) 27 , d ) 40 , e ) 48
Rationale: if the present age be x years . father ' s will be ( 3 x + 3 ) years . . so , ( 3 x + 3 + 5 ) = 2 ( x + 3 ) + 10 or , x = 8 so the fathers present age = ( 3 x + 3 ) = ( 3 * 8 + 3 ) years = 27 years . . answer : option c
Answer: c
Python program:
a = 2 * 5
b = a + 10 
c = b - 5
d = c - 3
e = 3 * d
f = 3 + e 
g = f - 10
h = g - 1
Problem: the h . c . f . of two numbers is 12 and their l . c . m . is 600 . if one of the number is 20 , find the other ?
Options: a ) 100 , b ) 360 , c ) 120 , d ) 200 , e ) 150
Rationale: "other number = 12 * 600 / 20 = 360 answer is b"
Answer: b
Python program:
a = 12 * 20
Problem: what is the next no . 4 12 84
Options: a ) 632 , b ) 732 , c ) 832 , d ) 850 , e ) 902
Rationale: "3 ^ 0 + 3 = 4 3 ^ 2 + 3 = 12 3 ^ 4 + 3 = 84 3 ^ 6 + 3 = 732 answer : b"
Answer: b
Python program:
a = 4 + reminder
Problem: x and y are positive integers . when x is divided by 9 , the remainder is 2 , and when x is divided by 7 , the remainder is 4 . when y is divided by 11 , the remainder is 3 , and when y is divided by 13 , the remainder is 12 . what is the least possible value of y - x ?
Options: a ) 12 , b ) 13 , c ) 14 , d ) 15 , e ) 16
Rationale: "when x is divided by 9 , the remainder is 2 : so , the possible values of x are : 2 , 11 , 21 , 29 , etc . when x is divided by 7 , the remainder is 4 : so , the possible values of x are : 4 , 11,18 , . . . stop . since both lists include 11 , the smallest possible value of x is 11 . when y is divided by 11 , the remainder is 3 : so , the possible values of y are : 3 , 14 , 25 , etc . when y is divided by 13 , the remainder is 12 : so , the possible values of y are : 12 , 25 , . . . stop . since both lists include 25 , the smallest possible value of y is 25 since the smallest possible values of x and y are 11 and 25 respectively , the smallest possible value of y - x is 14 . so , c is the correct answer to the original question ."
Answer: c
Python program:
a = 11 * 2
b = a + 3 
c = 2 + 9 
d = b - c
Problem: at joes steakhouse the hourly wage for a chef is 22 % greater than that of a dishwasher , and the hourly wage of a dishwasher is half as much as the hourly wage of a manager . if a managers wage is $ 8.50 per hour , how much less than a manager does a chef earn each hour ?
Options: a ) $ 1.40 , b ) $ 2.40 , c ) $ 3.315 , d ) $ 4.40 , e ) $ 5.40
Rationale: "manager wages per hour = $ 8.50 dishwasher wages per hour = half of manager ' s wages . = 1 / 2 ( $ 8.50 ) = = > $ 4.25 chef wages per hour = 22 % greater than dishwasher wages - - > 22 % of $ 4.25 = ( 22 * ( $ 4.25 ) ) / 100 - - > ( $ 93.5 ) / 100 - - > $ 0.935 therefore , chef wages per hour = $ 4.25 + $ 0.935 = = > $ 5.185 difference of wages between manager and chef = $ 8.50 - $ 5.185 = = > $ 3.315 answer : c"
Answer: c
Python program:
a = 100 + 22 
b = a / 100
c = 1 / 2
d = b * c
e = 1 - d
f = e * 8
Problem: if xerox paper costs 5 cents a sheet and a buyer gets 10 % discount on all xerox paper one buys after the first 2000 papers and 20 % discount after first 10000 papers , how much will it cost to buy 15000 sheets of xerox paper ?
Options: a ) $ 1250 , b ) $ 700 , c ) $ 1350 , d ) $ 900 , e ) $ 1000
Rationale: 30 sec approach - solve it using approximation 15000 sheet at full price , 5 cent = 750 15000 sheet at max discount price , 4 cent = 600 your ans got to be between these two . ans b it is .
Answer: b
Python program:
a = 10000 - 2000
b = 1 / 10
c = 1 - b
d = 1 / 10
e = c * d
f = a * e
Problem: if the sum of two positive integers is 18 and the difference of their squares is 36 , what is the product of the two integers ?
Options: a ) 108 , b ) 80 , c ) 128 , d ) 135 , e ) 143
Rationale: "let the 2 positive numbers x and y x + y = 18 - - 1 x ^ 2 - y ^ 2 = 36 = > ( x + y ) ( x - y ) = 36 - - 2 using equation 1 in 2 , we get = > x - y = 2 - - 3 solving equation 1 and 3 , we get x = 10 y = 8 product = 10 * 8 = 80 answer b"
Answer: b
Python program:
a = 36 / 18
b = 18 - a
c = 36 / 18
d = b / c
e = 36 / 18
f = 18 - e
g = 36 / 18
h = f / g
i = 36 / 18
j = h + i 
k = d * j
Problem: a candidate appearing for an examination has to secure 35 % marks to pass paper i . but he secured only 42 marks and failed by 23 marks . what is the maximum mark for paper i ?
Options: a ) 110 , b ) 120 , c ) 186 , d ) 140 , e ) 150
Rationale: "he secured 42 marks nd fail by 23 marks so total marks for pass the examinatn = 65 let toal marks x x * 35 / 100 = 65 x = 186 answer : c"
Answer: c
Python program:
a = 42 + 23 
b = 35 / 100
c = a / b
Problem: how many odd integers from 1 to 50 ( both inclusive ) have odd number of factors ?
Options: a ) 13 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "integers having odd number of factors will be perfect squares . odd numbers will have odd perfect squares . thus , the possible values for the perfect squares are : 1,9 , 25,49 and the corresponding integers are 1,3 , 5,7 ( more than 3 ) . thus b is the correct answer ."
Answer: b
Python program:
a = 50 / 2
b = a / 2
c = b - 10
d = c - 10
Problem: a rectangular plot measuring 90 metres by 40 metres is to be enclosed by wire fencing . if the poles of the fence are kept 5 metres apart , how many poles will be needed ?
Options: a ) 55 , b ) 56 , c ) 57 , d ) 52 , e ) none of these
Rationale: "solution perimeter of the plot = 2 ( 90 + 40 ) = 260 m . ∴ number of poles = [ 260 / 5 ] = 52 m answer d"
Answer: d
Python program:
a = rectangle_perimeter / (
Problem: find the value for x from below equation : x / 3 = - 2 ?
Options: a ) - 6 , b ) 1 , c ) - 2 , d ) - 3 , e ) 4
Rationale: 1 . multiply both sides by 3 : x * 3 / 3 = - 2 / 3 2 . simplify both sides : x = - 6 a
Answer: a
Python program:
a = 2 - 4
b = a * 3
Problem: dan can do a job alone in 12 hours . annie , working alone , can do the same job in just 9 hours . if dan works alone for 4 hours and then stops , how many hours will it take annie , working alone , to complete the job ?
Options: a ) 2 , b ) 4 , c ) 6 , d ) 8 , e ) 10
Rationale: "dan can complete 1 / 12 of the job per hour . in 4 hours , dan completes 4 ( 1 / 12 ) = 1 / 3 of the job . annie can complete 1 / 9 of the job per hour . to complete the job , annie will take 2 / 3 / 1 / 9 = 6 hours . the answer is c ."
Answer: c
Python program:
a = 4 / 12
b = 1 - a
c = b * 9
Problem: the measurement of a rectangular box with lid is 25 cmx 18 cmx 18 cm . find the volume of the largest sphere that can be inscribed in the box ( in terms of π cm 3 ) . ( hint : the lowest measure of rectangular box represents the diameter of the largest sphere )
Options: a ) 288 , b ) 48 , c ) 72 , d ) 972 , e ) 964
Rationale: "d = 18 , r = 9 ; volume of the largest sphere = 4 / 3 π r 3 = 4 / 3 * π * 9 * 9 * 9 = 972 π cm 3 answer : d"
Answer: d
Python program:
a = 4 / 3
b = 3 ** 3
c = a * b
Problem: the probability that a computer company will get a computer hardware contract is 3 / 4 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 ) 11 / 30 , b ) 31 / 60 , c ) 41 / 80 , d ) 51 / 120 , e ) 71 / 180
Rationale: "let , a ≡ event of getting hardware contract b ≡ event of getting software contract ab ≡ event of getting both hardware and software contract . p ( a ) = 3 / 4 , 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 = ( 3 / 4 ) + ( 4 / 9 ) - p ( ab ) . = > p ( ab ) = 71 / 180 . hence , the required probability is 71 / 180 . the answer is e ."
Answer: e
Python program:
a = 5 / 4
b = 3 / 0
c = 1 - b
d = a + c 
e = 9 / 5
f = d - e
Problem: working alone at its constant rate , machine a produces x boxes in 10 minutes and working alone at its constant rate , machine b produces 3 x boxes in 5 minutes . how many minutes does it take machines a and b , working simultaneously at their respective constant rates , to produce 3 x boxes ?
Options: a ) 3 minutes , b ) 4 minutes , c ) 4.2 minutes , d ) 6 minutes , e ) 12 minutes
Rationale: "rate = work / time given rate of machine a = x / 10 min machine b produces 3 x boxes in 5 min hence , machine b produces 4 x boxes in 10 min . rate of machine b = 6 x / 10 we need tofind the combined time that machines a and b , working simultaneouslytakeat their respective constant rates let ' s first find the combined rate of machine a and b rate of machine a = x / 10 min + rate of machine b = 6 x / 10 = 7 x / 10 now combine time = combine work needs to be done / combine rate = 3 x / 7 x * 10 = 4.2 min ans : c"
Answer: c
Python program:
a = 3 * 10
b = speed + ( 
c = 3 * 10
d = a / b
Problem: a student chose a number , multiplied it by 4 , then subtracted 138 from the result and got 102 . what was the number he chose ?
Options: a ) 60 , b ) 120 , c ) 130 , d ) 140 , e ) 150
Rationale: solution : let xx be the number he chose , then 4 ⋅ x − 138 = 102 4 x = 240 x = 60 answer a
Answer: a
Python program:
a = 102 + 138 
b = a / 4
Problem: what is the speed of the stream if a canoe rows upstream at 4 km / hr and downstream at 12 km / hr
Options: a ) 1 kmph , b ) 4 kmph , c ) 3 kmph , d ) 2 kmph , e ) 1.9 kmph
Rationale: "sol . speed of stream = 1 / 2 ( 12 - 4 ) kmph = 4 kmph . answer b"
Answer: b
Python program:
a = 12 - 4
b = a / 2
Problem: when asked what the time is , a person answered that the amount of time left is 3 / 5 of the time already completed . what is the time .
Options: a ) 2 pm , b ) 9 pm , c ) 3 pm , d ) 8 pm , e ) 6 pm
Rationale: "a day has 24 hrs . assume x hours have passed . remaining time is ( 24 - x ) 24 − x = 3 / 5 x ⇒ x = 15 time is 3 pm answer : c"
Answer: c
Python program:
a = 3 * 12
b = 3 * 12
c = 3 / 5
d = c + 1 
e = b / d
f = a - e