euclaise/mathqa_programs huggingface dataset:
Problem: the mean of 50 observations was 30 . it was found later that an observation 48 was wrongly taken as 23 . the corrected new mean is
Options: a ) 35.2 , b ) 36.1 , c ) 30.5 , d ) 39.1 , e ) none
Rationale: "sol . therefore correct sum = ( 30 × 50 + 48 – 23 ) = 1525 . therefore correct mean = 1525 / 50 = 30.5 . answer c"
Answer: c
Python program:
a = 30 * 50
b = 50 - 2
c = b - 23
d = a + c 
e = d / 50
Problem: a train running at the speed of 60 km / hr crosses a pole in 42 seconds . what is the length of the train ?
Options: a ) 120 metres , b ) 700 metres , c ) 324 metres , d ) 828 metres , e ) 600 metres
Rationale: "speed = 60 x 5 / 18 m / sec = 50 / 3 m / sec . length of the train = ( speed x time ) . length of the train = 50 / 3 x 42 m = 150 m . option b"
Answer: b
Python program:
a = 60 * 1000
b = a / 3600
c = b * 42
Problem: the average weight of 6 person ' s increases by 3.5 kg when a new person comes in place of one of them weighing 47 kg . what might be the weight of the new person ?
Options: a ) 60 kg , b ) 75 kg , c ) 68 kg , d ) 85 kg , e ) 90 kg
Rationale: "total weight increased = ( 6 x 3.5 ) kg = 21 kg . weight of new person = ( 47 + 21 ) kg = 68 kg option c"
Answer: c
Python program:
a = 6 * 3
b = a + 47
Problem: ajay can walk 6 km in 1 hour . in how many hours he can walk 70 km ?
Options: a ) 5 hrs , b ) 11.6 hrs , c ) 15.6 hrs , d ) 20.1 hrs , e ) 30 hrs
Rationale: "1 hour he walk 6 km he walk 70 km in = 70 / 6 * 1 = 11.6 hours answer is b"
Answer: b
Python program:
a = 70 / 6
Problem: the speed of a boat in still water in 18 km / hr and the rate of current is 6 km / hr . the distance travelled downstream in 14 minutes is :
Options: a ) 11.4 km , b ) 10.9 km , c ) 10.4 km , d ) 6.06 km , e ) 12.56 km
Rationale: "explanation : speed downstream = ( 18 + 6 ) = 24 kmph time = 14 minutes = 14 / 60 hour = 7 / 30 hour distance travelled = time × speed = ( 7 / 30 ) × 26 = 6.06 km answer : option d"
Answer: d
Python program:
a = 18 + 6 
b = 14 / const_60
c = a * b
Problem: tea worth rs . 126 per kg are mixed with a third variety in the ratio 1 : 1 : 2 . if the mixture is worth rs . 133 per kg , the price of the third variety per kg will be
Options: a ) rs . 169.50 , b ) rs . 1700 , c ) rs . 175.50 , d ) rs . 155.50 , e ) none
Rationale: "solution since first second varieties are mixed in equal proportions , so their average price = rs . ( 126 + 135 / 2 ) = rs . 130.50 so , the mixture is formed by mixing two varieties , one at rs . 130.50 per kg and the other at say , rs . x per kg in the ratio 2 : 2 , i . e . , 1 : 1 . we have to find x . x - 133 / 22.50 = 1 = â € º x - 133 = 22.50 = â € º x = 155.50 . hence , price of the third variety = rs . 155.50 per kg . answer d"
Answer: d
Python program:
a = 1 + 1 
b = a + 2 
c = 133 * b
d = 126 + 126 
e = c - d
f = e / 2
Problem: find the simple interest on rs . 71,000 at 16 2 / 3 % per year for 9 months .
Options: a ) 7500 , b ) 6500 , c ) 8875 , d ) 9500 , e ) none of them
Rationale: "p = rs . 71000 , r = 50 / 3 % p . a and t = 9 / 12 years = 3 / 4 years . simple interest = ( p * r * t ) / 100 = rs . ( 71,000 * ( 50 / 3 ) * ( 3 / 4 ) * ( 1 / 100 ) ) = rs . 8875 answer is c ."
Answer: c
Python program:
a = 2 * 3
b = a * 100
c = b * 100
d = 3 * 3
e = d * 100
f = 3 + 2 
g = f * 2
h = e * g
i = c + h 
j = 16 * 3
k = j + 2 
l = k / 3
m = i * l
n = 3 * 3
o = 2 * 3
p = 2 * o
q = n / p
r = m * q
s = 1 / 100
t = r * s
Problem: john makes $ 65 a week from his job . he earns a raise andnow makes $ 72 a week . what is the % increase ?
Options: a ) 16 % , b ) 10.76 % , c ) 10.69 % , d ) 10.98 % , e ) 10 %
Rationale: "increase = ( 7 / 65 ) * 100 = ( 7 / 65 ) * 100 = 10.76 % . b"
Answer: b
Python program:
a = 72 - 65
b = a / 65
c = b * 100
Problem: if 20 men take 15 days to to complete a job , in how many days can 20 men finish that work ?
Options: a ) 15 , b ) 16 , c ) 17 , d ) 18 , e ) 19
Rationale: ans . 15 days
Answer: a
Python program:
a = 20 * 15
b = a / 20
Problem: a can do a piece of work 40 days . b can do work in 60 days . in how many days they will complete the work together ?
Options: a ) 15 days , b ) 16 days , c ) 19 days , d ) 17 days , e ) 24 days
Rationale: "lcm = 120 , ratio = 40 : 60 = 2 : 3 no of days = 120 / ( 2 + 3 ) = 90 / 5 = 24 days answer : e"
Answer: e
Python program:
a = 1 / 40
b = 1 / 60
c = a + b 
d = 1 / c
Problem: a profit of rs . 800 is divided between x and y in the ratio of 1 / 2 : 1 / 3 . what is the difference between their profit shares ?
Options: a ) s . 260 , b ) s . 160 , c ) s . 360 , d ) s . 50 , e ) s . 90
Rationale: "a profit of rs . 800 is divided between x and y in the ratio of 1 / 2 : 1 / 3 or 3 : 2 . so profits are 480 and 320 . difference in profit share = 480 - 320 = 160 answer : b"
Answer: b
Python program:
a = 1 / 2
b = 1 / 3
c = a + b 
d = 800 / c
e = d / 2
f = 1 / 2
g = 1 / 3
h = f + g 
i = 800 / h
j = i / 3
k = e - j
Problem: two trains are moving in opposite directions at 60 km / hr and 90 km / hr . their lengths are 1.50 km and 1.0 km respectively . the time taken by the slower train to cross the faster train in seconds is ?
Options: a ) 48 , b ) 9 , c ) 7 , d ) 60 , e ) 15
Rationale: ": relative speed = 60 + 90 = 150 km / hr . = 150 * 5 / 18 = 125 / 3 m / sec . distance covered = 1.50 + 1.0 = 2.5 km = 2500 m . required time = 2500 * 3 / 125 = 60 sec . answer : d"
Answer: d
Python program:
a = 1 * 50
b = 60 * 1000
c = b / 3600
d = a / c
e = 1 * 0
f = 90 * 1000
g = f / 3600
h = e / g
i = d - h
Problem: what is the smallest integer that is multiple of 3 , 5,9
Options: a ) a ) 70 , b ) b ) 45 , c ) c ) 200 , d ) d ) 280 , e ) e ) 140
Rationale: "it is the lcm of 3 , 5 and 9 which is 45 . the answer is b ."
Answer: b
Python program:
a = 3 + 4
Problem: a book is bought for $ 60 and sold for $ 78 . what is the profit in percentage ?
Options: a ) 10 , b ) 20 , c ) 30 , d ) 40 , e ) 50
Rationale: "78 / 60 = 1.3 the answer is c ."
Answer: c
Python program:
a = 78 - 60
b = a / 60
c = b * 100
Problem: product of two natural numbers is 7 . then , the sum of reciprocals of their squares is
Options: a ) 50 / 49 , b ) 1 / 289 , c ) 290 / 90 , d ) 290 / 19 , e ) none of these
Rationale: "explanation : if the numbers are a , b , then ab = 7 , as 17 is a prime number , so a = 1 , b = 7 . 1 / a 2 + 1 / b 2 = 1 / 1 ( 2 ) + 1 / 7 ( 2 ) = 50 / 49 option a"
Answer: a
Python program:
a = 1 / 1
b = a ** 2
c = 1 / 7
d = c ** 2
e = b + d
Problem: in a fuel station the service costs $ 1.15 per car , every liter of fuel costs 0.4 $ . assuming that you own 2 sports cars and 2 executive cars and all fuel tanks are empty . how much will it cost to fuel all cars together if a sports car tank is 32 liters and an executive car tank is 75 % bigger ?
Options: a ) 37.5 $ , b ) 75 $ , c ) 87.5 $ , d ) 94.5 $ , e ) 98.4 $
Rationale: "total cars = 4 1.15 * 4 = 4.6 - > service cost fuel cost in sports car = 2 * 32 * 0.4 = 25.6 fuel cost in executive car = 25.6 * 7 / 4 = 44.8 total fuel cost = 25.6 + 44.8 = 70.4 cost to fuel car = 70.4 + 4.6 = 75 answer : b"
Answer: b
Python program:
a = 32 * 75
b = a / 100
c = 32 + b 
d = c * 2
e = d * 0
f = 32 * 2
g = f * 0
h = e + g
Problem: a , band c can do a piece of work in 11 days , 20 days and 67 days respectively , working alone . how soon can the work be done if a is assisted by band c on alternate days ?
Options: a ) 7.1 days , b ) 8.1 days , c ) 9.1 days , d ) 10 days , e ) 11 days
Rationale: "( a + b ) ' s 1 day ' s work = 1 / 11 + 1 / 20 = 31 / 220 ( a + c ) ' s 1 day ' s work = 1 / 11 + 1 / 67 = 78 / 737 work done in 2 day ' s = 31 / 220 + 78 / 737 = 19 / 77 19 / 77 th work done in 2 days work done = 77 / 19 * 2 = 8.1 days answer : b"
Answer: b
Python program:
a = 67 / 11
b = 67 / 20
c = a + b 
d = 67 / 11
e = 67 / 67
f = d + e 
g = c + f 
h = g / 2
i = 67 / h
Problem: a telephone company needs to create a set of 3 - digit area codes . the company is entitled to use only digits 6 , 4 and 3 , which can be repeated . if the product of the digits in the area code must be even , how many different codes can be created ?
Options: a ) 20 , b ) 22 , c ) 26 , d ) 24 , e ) 30
Rationale: "total # of codes possible is 3 * 3 * 3 = 27 . oit of those 27 codes only the product of 333 and will be odd , the remaining 26 will have either 2 or 4 in them , which ensures that their product will be even . therefore the number of codes where the product of the digits is even = ( total ) - ( restriction ) = 27 - 1 = 26 . answer : c"
Answer: c
Python program:
a = 3 ** 3
b = a - 1
Problem: in a rectangular axis system , what is the area of a parallelogram with the coordinates : ( 3,5 ) , ( 9,5 ) , ( 2,7 ) , ( 8,7 ) ?
Options: a ) 21 . , b ) 28 . , c ) 35 . , d ) 12 . , e ) 52 .
Rationale: "delta x will give us the dimension of one side of the parallelogram = 9 - 3 = 6 unit delta y will give us the dimension of the other side of parallelogram = 7 - 5 = 2 unit area of parallelogram = 6 * 2 = 12 answer is d"
Answer: d
Python program:
a = 3 + 2
Problem: find the c . i . on a sum of rs . 10000 for 6 months at 25 % per annum , interest being compounded quarterly ?
Options: a ) 10290 , b ) 5290 , c ) 1289 , d ) 1290 , e ) 2290
Rationale: c . i . = 2000 ( 21 / 20 ) ^ 2 - 1800 = 1289 answer : c
Answer: c
Python program:
a = const_0_25 / 4
b = 1 + a 
c = const_0_25 / 4
d = 1 + c 
e = b * d
f = 10000 * e
g = f - 10000
Problem: you collect pens . suppose you start out with 5 . mike gives you another 20 pens . since her father makes pens , cindy decides to double your pens . since you ' re nice , you give sharon 10 pens . how many pens do you have at the end ?
Options: a ) 39 , b ) 40 , c ) 41 , d ) 42 , e ) 43
Rationale: "solution start with 5 pens . mike gives you 20 pens : 5 + 20 = 25 pens . cindy doubles the number of pens you have : 25 ã — 2 = 50 pens . sharon takes 10 pens from you : 50 - 10 = 40 pens . so you have 40 at the end . correct answer : b"
Answer: b
Python program:
a = 20 + 5 
b = a * 2
c = b - 10
Problem: a sum of money at simple interest amounts to rs . 1560 in 4 years and to rs . 1590 in 6 years . the sum is :
Options: a ) rs . 1400 , b ) rs . 1450 , c ) rs . 1500 , d ) rs . 1550 , e ) rs . 1525
Rationale: "s . i . for 2 years = rs . ( 1590 - 1560 ) = rs . 30 . s . i . for 4 years = rs . ( 30 x 2 ) = rs . 60 . principal = rs . ( 1560 - 60 ) = rs . 1500 . answer : option c"
Answer: c
Python program:
a = 1590 - 1560
b = a * 4
c = b / 6
d = 1560 - c
Problem: a lady has fine gloves and hats in her closet - 18 blue , 32 red , and 25 yellow . the lights are out and it is totally dark . in spite of darkness , she can make out the difference between a hat and a glove . she takes out an item out of the closet only if she is sure that it is a glove . how many gloves must she take out to make sure she has a pair of each color ?
Options: a ) 60 , b ) 65 , c ) 70 , d ) 75 , e ) 80
Rationale: "32 r + 24 y + 1 y + 1 b + 2 b = 60 answer : a"
Answer: a
Python program:
a = 25 + 32 
b = a + 2
Problem: of the total amount that jill spent on a shopping trip , excluding taxes , she spent 40 percent on clothing , 30 percent on food , and 30 percent on other items . if jill paid a 4 percent tax on the clothing , no tax on the food , and an 8 percent tax on all other items , then the total tax that she paid was what percent of the total amount that she spent , excluding taxes ?
Options: a ) 2.8 % , b ) 3.6 % , c ) 4 % , d ) 5.2 % , e ) 6.0 %
Rationale: "let amount spent by jill = 100 clothing = 40 , food = 30 , others = 30 tax on clothing = 1.6 tax on others = 2.4 percentage = 4 / 100 = 4 % answer : c"
Answer: c
Python program:
a = 4 / 100
b = 40 * a
c = 8 / 100
d = 30 * c
e = b + d 
f = e / 100
g = f * 100
Problem: a student chose a number , multiplied it by 5 , then subtracted 138 from the result and got 102 . what was the number he chose ?
Options: a ) 48 , b ) 120 , c ) 130 , d ) 140 , e ) 150
Rationale: "solution : let xx be the number he chose , then 5 ⋅ x − 138 = 102 5 x = 240 x = 48 answer a"
Answer: a
Python program:
a = 102 + 138 
b = a / 5
Problem: in a class , 30 students pass in english and 20 students in maths , while some students among these pass in both . how many students do only english as compared to those doing only maths ?
Options: a ) 10 , b ) 15 , c ) 4 , d ) 12 , e ) indeterminate
Rationale: "for doing union or intersection we would need three values . in this question the missing information in this question is total no . of students in the class . answer : e"
Answer: e
Python program:
a = 30 - 20
Problem: if 31 / 198 = 0.1565 , what is the 97 nd digit to the right of the decimal point of the fraction ?
Options: a ) 1 , b ) 2 , c ) 6 , d ) 7 , e ) 9
Rationale: "we are not concerned what 31 / 198 means . . we have to look at the decimal . . 0.1565 means 0.1565656 . . . . so leaving girst and second digit to the right of decimal , all odd numbered are 6 and all even numbered are 5 . . here 97 is odd , so ans is 6 c"
Answer: c
Python program:
a = 2 + 3
Problem: sally has a gold credit card with a certain spending limit , and a platinum card with twice the spending limit of the gold card . currently , she has a balance on her gold card that is 1 / 3 of the spending limit on that card , and she has a balance on her platinum card that is 1 / 8 of the spending limit on that card . if sally transfers the entire balance on her gold card to her platinum card , what portion of her limit on the platinum card will remain unspent ?
Options: a ) 11 / 30 , b ) 29 / 60 , c ) 17 / 24 , d ) 19 / 30 , e ) 11 / 15
Rationale: let s assume the platinum card spending limit = x gold card spending limit will be = x / 2 balance on gold card is = x / 2 * 1 / 3 = x / 6 platinum card unspent limit is = x - 1 / 8 x = 7 / 8 x so if gold card balance is transferred then the rest unspent will be 7 / 8 x - x / 6 = 17 / 24 x so the ans is c
Answer: c
Python program:
a = 1/(3)
b = 1/(2)
c = a * b
d = 1/(8)
e = c + d 
f = 1 - e
Problem: robert spent $ 100 in buying raw materials , $ 125 in buying machinery and 10 % of the total amount he had as cash with him . what was the total amount ?
Options: a ) $ 150 , b ) $ 210 , c ) $ 250 , d ) $ 160 , e ) $ 200
Rationale: let the total amount be x then , ( 100 - 10 ) % of x = 100 + 125 90 % of x = 225 90 x / 100 = 225 x = $ 250 answer is c
Answer: c
Python program:
a = 125 + 125
Problem: a 12 % stock yields 8 % . the market value of the stock is :
Options: a ) rs . 72 , b ) rs . 150 , c ) rs . 112.50 , d ) rs . 116.50 , e ) none of these
Rationale: "solution to obtain rs . 8 , investment = rs . 100 . to obtain rs . 12 , investment = rs . ( 100 / 8 x 12 ) = rs . 150 ∴ market value of rs . 100 stock = rs . 150 answer b"
Answer: b
Python program:
a = 100 / 8
b = a * 12
Problem: how long does a train 130 m long running at the speed of 98 km / hr takes to cross a bridge 160 m length ?
Options: a ) 13.9 sec , b ) 13.1 sec , c ) 17.9 sec , d ) 61.9 sec , e ) 47.98 sec
Rationale: "speed = 98 * 5 / 18 = 27 m / sec total distance covered = 130 + 160 = 290 m . required time = 290 / 22 = 13.1 sec . answer : b"
Answer: b
Python program:
a = 130 + 160 
b = 98 * const_0_2778
c = a / b
Problem: a , band c enter into partnership . a invests 3 times as much as b and b invests two - third of what c invests . at the end of the year , the profit earned is rs . 5500 . what is the share of b ?
Options: a ) 1000 , b ) 800 , c ) 1400 , d ) 1200 , e ) none of them
Rationale: "let c ' s capital = rs . x . then , b ' s capital = rs . ( 2 / 3 ) x a ’ s capital = rs . ( 3 x ( 2 / 3 ) . x ) = rs . 2 x . ratio of their capitals = 2 x : ( 2 / 3 ) x : x = 6 : 2 : 3 . hence , b ' s share = rs . ( 5500 x ( 2 / 11 ) ) = rs . 1000 . answer is a"
Answer: a
Python program:
a = 2 * 3
b = 2 / 3
c = b * 3
d = a + c 
e = d + 3 
f = 2 / e
g = 5500 * f
Problem: if { x } is the product of all even integers from 1 to x inclusive , what is the greatest prime factor of { 14 } + { 12 } ?
Options: a ) 23 , b ) 20 , c ) 15 , d ) 5 , e ) 2
Rationale: "soln : { 14 } + { 12 } = 14 * { 12 } + { 12 } = 15 * { 12 } answer : c"
Answer: c
Python program:
a = 14 + 12 
b = a / 2
c = 1 * 2
d = b + c
Problem: if x + ( 1 / x ) = 5 , what is the value of w = x ^ 2 + ( 1 / x ) ^ 2 ?
Options: a ) w = 21 , b ) w = 22 , c ) w = 23 , d ) w = 24 , e ) 27
Rationale: "squaring on both sides , x ^ 2 + ( 1 / x ) ^ 2 + 2 ( x ) ( 1 / x ) = 5 ^ 2 x ^ 2 + ( 1 / x ) ^ 2 = 23 answer : c"
Answer: c
Python program:
a = 5 ** 2
b = a - 2
Problem: if the price of a certain computer increased 30 percent from d dollars to 351 dollars , then 2 d =
Options: a ) 540 , b ) 570 , c ) 619 , d ) 649 , e ) 700
Rationale: "before price increase price = d after 30 % price increase price = d + ( 30 / 100 ) * d = 1.3 d = 351 ( given ) i . e . d = 351 / 1.3 = $ 270 i . e . 2 d = 2 * 270 = 540 answer : option a"
Answer: a
Python program:
a = 100 + 30 
b = a / 100
c = 351 / b
d = c * 2
Problem: a certain number of men can do a work in 54 days . if there were 6 men more it could be finished in 6 days less . how many men were there in the beginning ?
Options: a ) 50 , b ) 48 , c ) 70 , d ) 40 , e ) 50
Rationale: "explanation : m ( 54 ) = ( m + 6 ) ( 48 ) 9 m - 8 m = 48 m = 48 answer : option b"
Answer: b
Python program:
a = 54 + 6 
b = a * 6
c = b / 6
Problem: in traveling from a dormitory to a certain city , a student went 1 / 2 of the way by foot , 3 / 5 of the way by bus , and the remaining 4 kilometers by car . what is the distance , in kilometers , from the dormitory to the city ?
Options: a ) 20 , b ) 15 , c ) 40 , d ) 10 , e ) 12
Rationale: "whole trip = distance by foot + distance by bus + distance by car x = 1 / 2 x + 3 / 5 x + 4 x - 1 / 2 x - 3 / 5 x = 4 x = 20 km option : a"
Answer: a
Python program:
a = 1 / 2
b = 3 / 5
c = a + b 
d = 1 - c
e = 1/(d)
f = 4 * e
Problem: a small pool filled only with water will require an additional 500 gallons of water in order to be filled to 80 % of its capacity . if pumping in these additional 500 gallons of water will increase the amount of water in the pool by 30 % , what is the total capacity of the pool in gallons ?
Options: a ) 1000 , b ) 1250 , c ) 1300 , d ) 1600 , e ) 1875
Rationale: "since pumping in additional 500 gallons of water will increase the amount of water in the pool by 30 % , then initially the pool is filled with 1,000 gallons of water . so , we have that 1,000 + 500 = 0.8 * { total } - - > { total } = 1,875 answer : e ."
Answer: e
Python program:
a = 500 * 100
b = a / 30
c = b + 500 
d = 80 / 100
e = c / d
Problem: a certain bag contains 60 balls — 22 white , 18 green , 17 yellow , 3 red , and 1 purple . if a ball is to be chosen at random , what is the probability that the ball will be neither red nor purple ?
Options: a ) 0.09 , b ) 0.15 , c ) 0.54 , d ) 0.85 , e ) 0.95
Rationale: "according to the stem the ball can be white , green or yellow , so the probability is ( white + green + yellow ) / ( total ) = ( 22 + 18 + 17 ) / 60 = 57 / 60 = 0.95 . answer : e ."
Answer: e
Python program:
a = 22 + 18 
b = a + 17 
c = b / 60
Problem: if a speaks the truth 40 % of the times , b speaks the truth 20 % of the times . what is the probability that at least one will tell the truth
Options: a ) 1.8 , b ) 2.0 , c ) 1.52 , d ) 0.52 , e ) 1.3
Rationale: "probability of a speaks truth p ( a ) = 4 / 10 ; false = 6 / 10 probability of b speaks truth p ( b ) = 2 / 10 ; false = 8 / 10 . for given qtn ans = 1 - ( neither of them tell truth ) . because a & b are independent events = 1 - [ ( 6 / 10 ) * ( 8 / 10 ) ] = 1 - 48 / 100 = 1 - 0.48 = 0.52 answer : d"
Answer: d
Python program:
a = 4 * 5
b = a * 5
c = 40 / b
d = 4 * 5
e = d * 5
f = 20 / e
g = c * f
Problem: find the number of zeroes in 58 ! ( 58 factorial )
Options: a ) 12 , b ) 13 , c ) 14 , d ) 15 , e ) 16
Rationale: "no of zeroes is 58 / 5 = 11 11 / 5 = 2 11 + 2 = 13 answer : b"
Answer: b
Python program:
a = 2 + 2
Problem: find the cost of fencing around a circular field of diameter 36 m at the rate of rs . 3.50 a meter ?
Options: a ) 438 , b ) 395 , c ) 378 , d ) 279 , e ) 222
Rationale: "2 * 22 / 7 * 18 = 113 113 * 3 1 / 2 = rs . 395 answer : b"
Answer: b
Python program:
a = 36 / 2
b = circumface * (
Problem: a number when divided by 342 gives a remainder 47 . when the same number ift divided by 19 , what would be the remainder ?
Options: a ) 7 , b ) 8 , c ) 9 , d ) 21 , e ) 14
Rationale: "sol . on dividing the given number by 342 , let k be the quotient and 47 as remainder . then , number – 342 k + 47 = ( 19 x 18 k + 19 x 2 + 9 ) = 19 ( 18 k + 2 ) + 9 . the given number when divided by 19 , gives ( 18 k + 2 ) as quotient and 9 as remainder . option c"
Answer: c
Python program:
a = 19 * 2
b = 47 - a
Problem: which is the least number that must be subtracted from 1100 so that the remainder when divided by 7 , 12 , 16 is 4 ?
Options: a ) 58 , b ) 68 , c ) 78 , d ) 88 , e ) 90
Rationale: "first we need to figure out what numbers are exactly divisible by 7 , 12,16 . this will be the set { lcm , lcmx 2 , lcmx 3 , . . . } lcm ( 7 , 12,16 ) = 48 * 7 = 336 the numbers which will leave remainder 4 will be { 336 + 4 , 336 x 2 + 4 , 336 x 3 + 4 , . . . } the largest such number less than or equal to 1100 is 336 x 3 + 4 or 1012 to obtain this you need to subtract 88 d"
Answer: d
Python program:
a = math.lcm(7, 12)
b = math.lcm(a, 16)
c = math.gcd(1100, b)
d = math.lcm(7, 12)
e = math.lcm(d, 16)
f = c * e
g = 4 + f 
h = 1100 - g
Problem: two integers are in the ratio of 1 to 4 . if 12 is added to the smaller number , the ratio becomes 1 to 1 . find the larger integer .
Options: a ) 8 , b ) 16 , c ) 32 , d ) 48 , e ) 54
Rationale: "one option is to set up the equations and solve : if the ratio of two integers x and y is 1 to 4 , then 4 x = y , where x is the smaller integer . if adding 12 to the smaller integer makes the ratio 1 to 1 , then x + 12 = y . substituting y = 4 x into the second equation yields x + 12 = 4 x . so , x = 4 ( smaller integer ) and , y = 4 x = 16 ( larger integer ) so b is the correct answer . another option is to test the answer choices . a ) the larger number is 8 given : the two integers are in the ratio of 1 to 4 so , the smaller number must be 2 if 12 is added to the smaller number , the ratio becomes 1 to 1 if we add 12 to the smaller number ( 2 ) , we get 14 . so , the new ratio is 14 to 8 which is the same as 7 to 4 . no good . we want a resulting ratio of 1 to 1 eliminate a b ) the larger number is 16 given : the two integers are in the ratio of 1 to 4 so , the smaller number must be 4 if 12 is added to the smaller number , the ratio becomes 1 to 1 if we add 12 to the smaller number ( 4 ) , we get 16 so , the new ratio is 16 to 16 which is 1 to 1 . therefore ( b ) is correct . c ) the larger number is 32 given : the two integers are in the ratio of 1 to 4 so , the smaller number must be 8 if 12 is added to the smaller number , the ratio becomes 1 to 1 if we add 12 to the smaller number ( 8 ) , we get 20 so , the new ratio is 20 to 32 which is 5 to 8 . no good . we want a resulting ratio of 1 to 1 eliminate c d ) the larger number is 48 given : the two integers are in the ratio of 1 to 4 so , the smaller number must be 12 if 12 is added to the smaller number , the ratio becomes 1 to 1 if we add 12 to the smaller number ( 12 ) , we get 24 so , the new ratio is 24 to 48 . this is the same as the ratio of 1 to 2 no good . we want a resulting ratio of 1 to 1 eliminate d e ) the larger number is 54 given : the two integers are in the ratio of 1 to 4 so , the smaller number must be 13.5 ( which is not an integer ) since both numbers are integers , the larger number can not be 54 . eliminate e therefore b is the only correct choice ."
Answer: b
Python program:
a = 1 * 12
b = 4 - 1
c = a / b
d = c * 4
Problem: the salaries of a and b together amount to $ 4000 . a spends 95 % of his salary and b , 85 % of his . if now , their savings are the same , what is a ' s salary ?
Options: a ) $ 1000 , b ) $ 1250 , c ) $ 2500 , d ) $ 3000 , e ) $ 1200
Rationale: let a ' s salary is x b ' s salary = 4000 - x ( 100 - 95 ) % of x = ( 100 - 85 ) % of ( 4000 - x ) x = $ 3000 answer is d
Answer: d
Python program:
a = 100 - 85
b = 100 - 95
c = a / b
d = c + 1 
e = 4000 / d
f = 4000 - e