euclaise/mathqa_programs huggingface dataset:
Problem: in a certain company , a third of the workers do not have a retirement plan . 20 % of the workers who do not have a retirement plan are women , and 40 % of the workers who do have a retirement plan are men . if 144 of the workers of that company are men , how many of the workers are women ?
Options: a ) 80 , b ) 95 , c ) 105 , d ) 120 , e ) 126
Rationale: "set up equation : x = total number of workers 144 = 0,4 * 2 / 3 * x + 0,8 * 1 / 3 * x 144 = 16 / 30 x x = 270 270 - 144 = 126 answer e"
Answer: e
Python program:
a = 1 / 3
b = 1 / 3
c = 20 / 100
d = b * c
e = a - d
f = 1 / 3
g = 1 - f
h = 40 / 100
i = g * h
j = e + i 
k = 144 / j
l = 1 / 3
m = 20 / 100
n = l * m
o = 1 / 3
p = 1 - o
q = 1 / 3
r = 1 - q
s = 40 / 100
t = r * s
u = p - t
v = n + u 
w = k * v
Problem: 95 white and black tiles will be used to form a 10 x 10 square pattern . if there must be at least one black tile in every row and at least one white tile in every column , what is the maximum difference between the number of black and white tiles that can be used ?
Options: a ) 75 , b ) 80 , c ) 85 , d ) 90 , e ) 95
Rationale: answer = c please refer diagram below 95 - 10 = 85
Answer: c
Python program:
a = 95 - 10
Problem: in what time will a railway train 20 m long moving at the rate of 36 kmph pass a telegraph post on its way ?
Options: a ) 6 sec , b ) 7 sec , c ) 8 sec , d ) 9 sec , e ) 2 sec
Rationale: "t = 60 / 36 * 18 / 5 = 2 sec answer : e"
Answer: e
Python program:
a = 36 * const_0_2778
b = 20 / a
Problem: what is the rate percent when the simple interest on rs . 800 amount to rs . 400 in 2 years ?
Options: a ) 5 % , b ) 6 % , c ) 2 % , d ) 25 % , e ) 1 %
Rationale: interest for 2 yrs = 400 interest for 1 yr = 200 interest rate = 200 / 800 x 100 = 25 % answer : d
Answer: d
Python program:
a = 100 * 400
b = 800 * 2
c = a / b
Problem: if the area of a square with sides of length 9 centimeters is equal to the area of a rectangle with a width of 3 centimeters , what is the length of the rectangle , in centimeters ?
Options: a ) 4
b ) 27
c ) 12
d ) 16
e ) 18
Rationale: let length of rectangle = l 9 ^ 2 = l * 3 = > l = 81 / 3 = 27 answer b
Answer: b
Python program:
a = 9 ** 2
b = a / 3
Problem: taequan is playing a dice game where he rolls two , fair , 6 - sided dice twice . if he gets the same roll on his second try as his first , he wins . if he rolled an 8 on his first try , what is the probability that he will win with his second roll ?
Options: a ) 1 / 18 , b ) 1 / 12 , c ) 1 / 9 , d ) 1 / 6 , e ) 5 / 36
Rationale: there are 5 ways to roll an 8 : 2 and 6 , 6 and 2 , 3 and 5 , 5 and 3 , 4 and 4 ( 4 and 4 is counted just once ) . there are 6 * 6 = 36 ways to roll two six - sided dice . the dice are fair , so the probability of rolling an 8 is 5 / 36 . e
Answer: e
Python program:
a = 6 - 1
b = 6 * 6
c = a / b
Problem: car a runs at the speed of 100 km / hr & reaches its destination in 5 hr . car b runs at the speed of 80 km / h & reaches its destination in 6 h . what is the respective ratio of distances covered by car a & car b ?
Options: a ) 11 : 5 , b ) 11 : 8 , c ) 13 : 7 , d ) 25 : 24 , e ) 16 : 9
Rationale: "sol . distance travelled by car a = 100 × 5 = 500 km distance travelled by car b = 80 × 6 = 480 km ratio = 500 / 480 = 25 : 24 answer : d"
Answer: d
Python program:
a = 100 * 5
b = 80 * 6
c = a / b
Problem: george baked a total of 100 pizzas for 7 straight days , beginning on saturday . he baked 3 / 5 of the pizzas the first day , and 3 / 5 of the remaining pizzas the second day . if each successive day he baked fewer pizzas than the previous day , what is the maximum number of pizzas he could have baked on wednesday ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "3 / 5 of the 100 pizzas cooked on saturday = 60 pizzas 3 / 5 of the remaining pizzas on sunday = 24 pizzas we ' re left with ( 100 - 60 - 24 ) = 16 pizzas for the remaining 5 days . the prompt tells us that each day has fewer pizzas than the day before it , so we ca n ' t have duplicate numbers . m t w th f 6 4 3 2 1 = 16 w = 3 a"
Answer: a
Python program:
a = 3 / 7
b = 100 * a
c = 100 - b
d = 3 / 7
e = 100 * d
f = 100 - e
g = 3 / 7
h = f * g
i = c - h
j = i - 3
k = 3 / 7
l = 100 * k
m = 100 - l
n = 3 / 7
o = 100 * n
p = 100 - o
q = 3 / 7
r = p * q
s = m - r
t = s - 3
u = j / t
Problem: pipe a usually fills a tank in 2 hours . on account of a leak at the bottom of the tank , it takes pipe a 30 more minutes to fill the tank . how long will the leak take to empty a full tank if pipe a is shut ?
Options: a ) 7 hours , b ) 8 hours , c ) 9 hours , d ) 10 hours , e ) 11 hours
Rationale: pipe a fills the tank normally in 2 hours . therefore , it will fill ½ of the tank in an hour . let the leak take x hours to empty a full tank when pipe a is shut . therefore , the leak will empty of the tank in an hour . the net amount of water that gets filled in the tank in an hour when pipe a is open and when there is a leak = of the tank . — ( 1 ) when there is a leak , the problem states that pipe a takes two and a half hours to fill the tank . i . e . hours . therefore , in an hour , of the tank gets filled . – ( 2 ) equating ( 1 ) and ( 2 ) , we get = > = > x = 10 hours . the problem can also be mentally done as follows . pipe a takes 2 hours to fill the tank . therefore , it fills half the tank in an hour or 50 % of the tank in an hour . when there is a leak it takes 2 hours 30 minutes for the tank to fill . i . e hours to fill the tank or or 40 % of the tank gets filled . on account of the leak , ( 50 - 40 ) % = 10 % of the water gets wasted every hour . therefore , the leak will take 10 hours to drain a full tank . answer : d
Answer: d
Python program:
a = 1 / 2
b = 1 / 2
c = 2 + b 
d = 1 / c
e = a - d
f = e * 100
Problem: walking at 5 / 6 th of its usual speed a cab is 5 mnts late . find its usual time to cover the journey ?
Options: a ) 25 m , b ) 45 m , c ) 32 m , d ) 50 m , e ) 62 m
Rationale: "new speed = 5 / 6 th of usual speed new time = 6 / 5 th of usual time 6 / 5 ut - ut = 5 m ut / 5 = 5 m ut = 25 m answer is a"
Answer: a
Python program:
a = 5 * 5
Problem: what is the 8 th digit to the right of the decimal point in the decimal expansion of 1 / 37 ?
Options: a ) 0 , b ) 2 , c ) 4 , d ) 7 , e ) 9
Rationale: "1 / 37 = 0.027027 . . . so , we have a repeating cycle of 027 . every third digit ( 2 nd , 4 th , 6 th , . . . ) to the right of the decimal point is alternating by 2 , 0,7 , thus 8 th digit to the right of the decimal point is 2 . answer : b ."
Answer: b
Python program:
a = 1 / 37
Problem: find the value of x : 3 x ^ 2 - 6 x + 3 = 0
Options: a ) - 1 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: a = 3 , b = - 6 , c = 3 x 1,2 = ( 6 ± √ ( ( - 6 ) 2 - 4 × 3 × 3 ) ) / ( 2 × 3 ) = ( 6 ± √ ( 36 - 36 ) ) / 6 = ( 6 ± 0 ) / 6 x 1 = x 2 = 1 b
Answer: b
Python program:
a = 6 + 3 
b = 3 ** 2
c = a / b
Problem: in a kilometer race , a beats b by 50 meters or 5 seconds . what time does a take to complete the race ?
Options: a ) 199 sec , b ) 190 sec , c ) 726 sec , d ) 90 sec , e ) 128 sec
Rationale: "time taken by b run 1000 meters = ( 1000 * 5 ) / 50 = 100 sec . time taken by a = 100 - 10 = 90 sec . answer : d"
Answer: d
Python program:
a = 1 * 1000
b = 50 / 5
c = a / b
d = c - 5
Problem: what is the smallest number which when increased by 3 is divisible by 27 , 35 , 25 , & 21 ?
Options: a ) 4722 , b ) 4725 , c ) 4728 , d ) 4731 , e ) 4733
Rationale: "here smallest number means lcm hence lcm of 27,35 , 25,21 = 3 ^ 3 x 5 ^ 2 x 7 = 4725 therefore required number is 4725 - 3 = 4722 ans - a"
Answer: a
Python program:
a = math.lcm(27, 35)
b = math.lcm(25, 21)
c = math.lcm(a, b)
d = c + 3
Problem: a train 150 m long crosses a platform 130 m long in 20 sec ; find the speed of the train ?
Options: a ) 50 , b ) 60 , c ) 55 , d ) 45 , e ) 53
Rationale: "d = 150 + 130 = 280 t = 20 s = 280 / 20 * 18 / 5 = 50 kmph . answer : a"
Answer: a
Python program:
a = 130 * const_0_2778
b = 20 * a
c = b - 150
Problem: the ages of patrick and michael are in the ratio of 3 : 5 and that of michael and monica are in the ratio of 3 : 5 . if the sum of their ages is 245 , what is the difference between the ages of patrick and monica ?
Options: a ) 27 , b ) 48 , c ) 80 , d ) 72 , e ) 18
Rationale: "ages of p and mi = 3 x : 5 x ages of mi and mo = 3 x : 5 x rationalizing their ages . ratio of their ages will be 9 x : 15 x : 25 x sum = 49 x = 245 x = 5 difference if ages of pa and mo = 25 x - 9 x = 16 x = 16 * 5 = 80 answer c"
Answer: c
Python program:
a = 5 * 5
b = 3 * 3
c = 3 * 5
d = b + c 
e = 5 * 5
f = d + e 
g = 245 / f
h = a * g
i = 3 * 3
j = 3 * 3
k = 3 * 5
l = j + k 
m = 5 * 5
n = l + m 
o = 245 / n
p = i * o
q = h - p
Problem: when 6 + 2 = 85 , 8 + 3 = 115 , 7 + 5 = 125 , then 5 + 8 = ?
Options: a ) 145 , b ) 185 , c ) 245 , d ) 135 , e ) 140
Rationale: "6 + 2 = > 6 + 2 = 8 = > 8 ã — 10 + 5 = 85 8 + 3 = > 8 + 3 = 11 = > 11 ã — 10 + 5 = 115 7 + 5 = > 7 + 5 = 12 = > 12 ã — 10 + 5 = 125 then 5 + 8 = > 5 + 8 = 13 = > 13 ã — 10 + 5 = 135 answer : d"
Answer: d
Python program:
a = 5 * 8
b = a * 10
c = b + 6
Problem: pipe a can fill a tank in 4 hours . due to a leak at the bottom , it takes 8 hours for the pipe a to fill the tank . in what time can the leak alone empty the full tank ?
Options: a ) 13 , b ) 17 , c ) 18 , d ) 8 , e ) 12
Rationale: "let the leak can empty the full tank in x hours 1 / 4 - 1 / x = 1 / 8 = > 1 / x = 1 / 4 - 1 / 8 = ( 2 - 1 ) / 8 = 1 / 8 = > x = 8 . answer : d"
Answer: d
Python program:
a = 8 * 4
b = 8 - 4
c = a / b
Problem: in how many ways 7 boys and 7 girls can be seated in a row so that boys and girls are alternate ?
Options: a ) 50803200 , b ) 50328000 , c ) 50003280 , d ) 50703200 , e ) 50603200
Rationale: "number of ways = 2 * 7 ! * 7 ! = 50803200 answer is a"
Answer: a
Python program:
a = math.factorial(7)
b = math.factorial(7)
c = a * b
Problem: how many two - element subsets of < 1 , 23 , 45 , 67 > are there that do not contain the pair of elements 2 and 4 ?
Options: a ) 11 , b ) 12 , c ) 14 , d ) 15 , e ) 20
Rationale: 7 c 2 - 1 = 20 answer is e
Answer: e
Python program:
a = 23 - 3
Problem: a certain junior class has 3000 students and a certain senior class has 800 students . among these students , there are 60 siblings pairs each consisting of 1 junior and 1 senior . if 1 student is to be selected at random from each class , what is the probability that the 2 students selected will be a sibling pair ?
Options: a ) 1 / 40000 , b ) 1 / 3600 , c ) 9 / 2000 , d ) 1 / 60 , e ) 1 / 15
Rationale: "let ' s see pick 60 / 3000 first then we can only pick 1 other pair from the 800 so total will be 60 / 800 * 3000 simplify and you get 1 / 40000 answer is a"
Answer: a
Python program:
a = 1 / 3
Problem: there is a sufficient food for 200 men for 20 days after 15 days 100 men leave the place . for how many days will the rest of the food last for the rest of the men ?
Options: a ) 10 days , b ) 5 days , c ) 15 days , d ) 12 days , e ) 10 days
Rationale: explanation : after 13 days , 100 men left , therefore 5 days * 200 men = x days * 100 men x = 10 days answer : option a
Answer: a
Python program:
a = 20 - 15
b = a * 2
Problem: a motorcyclist started riding at highway marker a , drove 120 miles to highway marker b , and then , without pausing , continued to highway marker c , where she stopped . the average speed of the motorcyclist , over the course of the entire trip , was 30 miles per hour . if the ride from marker a to marker b lasted 3 times as many hours as the rest of the ride , and the distance from marker b to marker c was half of the distance from marker a to marker b , what was the average speed , in miles per hour , of the motorcyclist while driving from marker b to marker c ?
Options: a ) 90 , b ) 45 , c ) 50 , d ) 55 , e ) 60
Rationale: "a - b = 120 miles b - c = 60 miles avg speed = 30 miles time taken for a - b 3 t and b - c be t avg speed = ( 120 + 60 ) / total time 30 = 180 / 4 t t = 90 b - c = 90 mph answer a"
Answer: a
Python program:
a = 120 / 2
b = a + 120 
c = b / 30
d = c / 4
e = 120 / 2
f = d * e
Problem: how much 60 % of 50 is greater than 30 % of 30 ?
Options: a ) 21 , b ) 91 , c ) 11 , d ) 17 , e ) 12
Rationale: "( 60 / 100 ) * 50 – ( 30 / 100 ) * 30 30 - 9 = 21 answer : a"
Answer: a
Python program:
a = 60 * 50
b = a / 100
c = 30 * 30
d = c / 100
e = b - d
Problem: the average of 50 numbers id 50 . if two numbers , namely 45 and 55 are discarded , the average of the remaining numbers is :
Options: a ) 22 , b ) 33 , c ) 22 , d ) 41 , e ) 50
Rationale: "explanation : total of 50 numbers = ( 50 × 50 ) = 2500 total of 48 numbers = ( 2500 - ( 45 + 55 ) ] = 2400 required average = 2400 / 48 = 50 answer : e"
Answer: e
Python program:
a = 50 * 50
b = 45 + 55 
c = a - b
d = 50 - 2
e = c / d
Problem: a can do a work in 9 days . b can do the same work in 18 days . if both a & b are working together in how many days they will finish the work ?
Options: a ) 3 , b ) 5 , c ) 4 , d ) 2 , e ) 6
Rationale: "a rate = 1 / 9 b rate = 1 / 18 ( a + b ) rate = ( 1 / 9 ) + ( 1 / 18 ) = 1 / 6 a & b finish the work in 6 days correct option is e"
Answer: e
Python program:
a = 9 * 18
b = 9 + 18 
c = a / b
Problem: how many numbers between 100 and 672 are divisible by 2 , 3 , and 7 together ?
Options: a ) 112 , b ) 77 , c ) 267 , d ) 14 , e ) 99
Rationale: "explanation : as the division is by 2 , 3 , 7 together , the numbers are to be divisible by : 2 * 3 * 7 = 42 the limits are 100 and 672 the first number divisible is 42 * 3 = 126 to find out the last number divisible by 42 within 672 : 672 / 42 = 14 hence , 42 * 16 = 672 is the last number divisible by 42 within 672 hence , total numbers divisible by 2 , 3 , 7 together are ( 16 â € “ 2 ) = 14 answer : d"
Answer: d
Python program:
a = 2 * 3
b = a * 7
c = 672 / b
d = 2 * 3
e = d * 7
f = 100 / e
g = c - f
Problem: angelo and isabella are both salespersons . in any given week , angelo makes $ 550 in base salary plus 8 percent of the portion of his sales above $ 3,000 for that week . isabella makes 10 percent of her total sales for any given week . for what amount of weekly sales would angelo and isabella earn the same amount of money ?
Options: a ) 23,500 , b ) 15,500 , c ) 25,500 , d ) 26,500 , e ) 27,500
Rationale: "official solution : the problem asks for the amount of weekly sales it takes for angelo and isabella to earn the same amount of money . you can write an equation that sets angelo ’ s and isabella ’ s weekly earnings equal to each other , with x representing weekly sales . weekly earnings for each salesperson equal base salary plus commission . so angelo ’ s earnings are 550 + ( 0.08 ) ( x – 3,000 ) , and isabella ’ s are 0.10 x . set up the equation and solve : 550 + ( 0.08 ) ( x – 3,000 ) = 0.10 x distribute the 0.08 : 550 + 0.08 x – 240 = 0.10 x combine terms and subtract 0.08 x from both sides : 310 = 0.02 x divide both sides by 0.02 : 15,500 = x your answer is b ."
Answer: b
Python program:
a = 8 / 100
b = 3 * 0
c = 550 - b
d = 10 / 100
e = 8 / 100
f = d - e
g = c / f
h = g / 3
i = math.floor(h)
Problem: a girl scout was selling boxes of cookies . in a month , she sold both boxes of chocolate chip cookies ( $ 1.25 each ) and boxes of plain cookies ( $ 0.75 each ) . altogether , she sold 1,585 boxes for a combined value of $ 1 , 586.75 . how many boxes of plain cookies did she sell ?
Options: a ) 0 , b ) 233 , c ) 500 , d ) 695 , e ) 789
Rationale: "let # plain cookies sold be x then # chocolate cookies = ( total cookies - x ) equating for x ( 0.75 ) * x + ( 1.25 ) * ( 1585 - x ) = 1586.75 = > x = 789"
Answer: e
Python program:
a = 1000 + 586 
b = a / 2
Problem: a man purchased 3 blankets @ rs . 100 each , 5 blankets @ rs . 150 each and two blankets at a certain rate which is now slipped off from his memory . but he remembers that the average price of the blankets was rs . 157 . find the unknown rate of two blankets ?
Options: a ) 420 , b ) 550 , c ) 490 , d ) 450 , e ) 520
Rationale: "10 * 157 = 1570 3 * 100 + 5 * 150 = 1050 1570 – 1050 = 520 answer : e"
Answer: e
Python program:
a = 10 * 150
b = 3 * 100
c = 5 * 150
d = b + c 
e = a - d
Problem: 0.0004 ? = 0.02
Options: a ) 0.02 , b ) 0.2 , c ) 2 , d ) 20 , e ) none of these
Rationale: "explanation : required answer = 0.0004 / 0.02 = 0.04 / 2 = 0.02 . answer : option a"
Answer: a
Python program:
a = 0 / 4
b = a * 100
Problem: a bag contains 10 red jellybeans and 10 blue jellybeans . if 3 jellybeans are removed one at a time , at random and are not replaced , what is the probability t that all 3 jellybeans removed from the bag are blue ?
Options: a ) 9 / 100 , b ) 2 / 19 , c ) 1 / 8 , d ) 3 / 20 , e ) 3 / 10
Rationale: "method - 1 10 red jellybeans and 10 blue jellybeans total outcomes = no . of ways to choose 3 jelly bean at random out of a total 20 jellybeans = 20 c 3 = 1140 favourable outcomes = no . of ways to choose 3 jelly bean such that they are all blue out of 10 blue = 10 c 3 = 120 probability = favourable outcomes / total outcomes = 10 c 3 / 20 c 3 probability t = 120 / 1140 = 2 / 19 answer : option b method - 2 probability of first jelly bean to be blue = 10 / 20 [ total 10 blue out of total 20 jellybeans ] probability of second jelly bean to be blue = 9 / 19 [ total 9 blue remaining out of total 19 jellybeans remaining ] probability of third jelly bean to be blue = 8 / 18 [ total 8 blue remaining out of total 18 jellybeans remaining ] required probability = ( 10 / 20 ) * ( 9 / 19 ) * ( 8 / 18 ) = 2 / 19 answer : option b"
Answer: b
Python program:
a = math.comb(10, 3)
b = 10 + 10 
c = math.comb(b, 3)
d = a / c
Problem: the length of a rectangle is reduced by 25 % . by what % would the width have to be increased to maintainthe original area ?
Options: a ) 15 % , b ) 20 % , c ) 25 % , d ) 33 % , e ) 40 %
Rationale: "sol . required change = ( 25 * 100 ) / ( 100 - 25 ) = 25 % d"
Answer: d
Python program:
a = 100 - 25
b = a / 100
c = 1 - b
d = 100 - 25
e = d / 100
f = c / e
g = f * 100
Problem: a survey was sent to 95 customers , 7 of whom responded . then the survey was redesigned and sent to another 63 customers , 9 of whom responded . by approximately what percent did the response rate increase from the original survey to the redesigned survey ?
Options: a ) 2 % , b ) 5 % , c ) 7 % , d ) 28 % , e ) 63 %
Rationale: "case 1 : ( 7 / 95 ) = x / 100 x = 7 % case 2 : ( 9 / 63 ) = y / 100 y = 14 % so percent increase is = ( y - x ) = ( 14 - 7 ) % = 7 % answer is c"
Answer: c
Python program:
a = 9 / 63
b = 7 / 95
c = a - b
d = c * 100
e = math.floor(d)
Problem: tough and tricky questions : word problems . ak car company wants to manufacture a new car known as model x , and it is trying to determine how many cars it needs to sell in order to make an annual profit of $ 30 , 500000 . the annual fixed costs for model x total $ 50 , 200000 . in addition , each model x car has an average cost of $ 5000 per vehicle to manufacture . if the company forecasts it will sell 20000 model x cars this year , at what price must the company sell the car to achieve the desired annual profit ?
Options: a ) $ 4035 , b ) $ 4036 , c ) $ 9035 , d ) $ 16140 , e ) $ 36140
Rationale: the ak company plans to make a profit of $ 30 , 500000 and it needs to earn the fixed costs of $ 50 , 200000 , so it needs to earn $ 80 , 700000 on top of the costs for the production of the 20000 cars . therefore , price of each car needs to be the unit costs plus $ 80 , 700000 / 20000 . after removing the 0 ' s , we end with the unit costs plus $ 8070 / 2 , which is $ 5000 plus $ 4035 . therefore , answer c is correct .
Answer: c
Python program:
a = 1000 * 1000
b = 50 * a
c = b + 200000 
d = 20000 * 5000
e = c + d 
f = 1000 * 1000
g = 30 * f
h = g + 500000 
i = e + h 
j = i / 20000
Problem: according to the directions on a packet of smoothie mix , 1 3 - ounce packet of smoothie mix is to be combined with 15 ounces of water to make a smoothie . how many 3 - ounce packets of smoothie mix are required to prepare 150 12 - ounce smoothies ?
Options: a ) 120 , b ) 150 , c ) 180 , d ) 240 , e ) 600
Rationale: "this question was n ' t particularly grueling , but i think it ' s the first where i had the opportunity to solve it via theory andinspectionthat many on this board suggest as strategy on the gmat . it actually came to me by accident . basically , if we thought that the 3 packets of powder were included in the 12 ounces of water , that would mean we would need 150 packets of smoothie mix ( along with 12 ( 150 ) ounces of water for a total of 150 packets . however , we know , after a more careful reading of the stimulus , that the 3 ounces are not included in the 12 ounces . as such , the answer has to be less than 150 packets , since 150 would be too much powder considering you already have 150 ( 12 ) ounces of water and need less packets than water to make a smoothie . as such , the only answer less than 150 is 120 , a . does this make sense ? or am i way off base ? c"
Answer: c
Python program:
a = 3 * 10
b = 150 + a
Problem: how many 3 - digit numbers are divisible by 6 in all ?
Options: a ) 149 , b ) 150 , c ) 151 , d ) 166 , e ) none
Rationale: solution required numbers are 102 , 018114 , . . . . 996 . this is an a . p with a = 102 , d = 6 . let the number of its terms be n . then a + ( n - 1 ) d ‹ = › 102 + ( n - 1 ) × 6 = 996 ‹ = › n = 150 . answer b
Answer: b
Python program:
a = 1000 / 6
b = 100 / 6
c = a - b
Problem: two pipes can fill a tank in 10 minutes and 25 minutes . an outlet pipe can empty the tank in 45 minutes . if all the pipes are opened when the tank is empty , then how many minutes will it take to fill the tank ?
Options: a ) 30 minutes , b ) 17 minutes , c ) 7 minutes , d ) 10 minutes , e ) 12 minutes
Rationale: "part of the filled by all the three pipes in one minute = 1 / 10 + 1 / 25 - 1 / 45 = ( 45 + 18 - 10 ) / 450 = 53 / 450 so , the tank becomes full in 7 minutes . answer : c"
Answer: c
Python program:
a = 1 / 10
b = 1 / 25
c = a + b 
d = 1 / 45
e = c - d
Problem: if 7 ^ k = 2 , then 7 ^ ( 4 k + 2 ) =
Options: a ) 729 , b ) 754 , c ) 784 , d ) 783 , e ) 108
Rationale: "7 ^ k = 2 7 ^ 4 k = 2 ^ 4 7 ^ 4 k = 16 7 ^ ( 4 k + 2 ) = 7 ^ 4 k * 7 ^ 2 = 16 * 49 = 784 answer : c"
Answer: c
Python program:
a = 2 ** 4
b = 7 ** 2
c = a * b
Problem: one day a car rental agency rented 3 / 4 of its cars , including 3 / 5 of its cars with cd players . if 3 / 5 of its cars have cd players , what percent of the cars that were not rented had cd players ?
Options: a ) 29 / 30 , b ) 24 / 25 , c ) 19 / 20 , d ) 14 / 15 , e ) 9 / 10
Rationale: "the cars with cd players which were not rented is ( 2 / 5 ) ( 3 / 5 ) = 6 / 25 of all the cars . the cars which were not rented is 1 / 4 of all the cars . the percent of non - rented cars which had cd players is ( 6 / 25 ) / ( 1 / 4 ) = 24 / 25 the answer is b ."
Answer: b
Python program:
a = 4 * 3
b = a * 5
c = 4 * b
d = c / 5
e = 4 * 3
f = e * 5
g = 4 * f
h = g / 5
i = h * 4
j = i / 3
k = d - j
l = 4 * 3
m = l * 5
n = 4 * 3
o = n * 5
p = 3 * o
q = p / 4
r = m - q
s = k / r
t = s * 100
Problem: little john had $ 5.10 . he spent $ 1.05 on sweets and gave to his two friends $ 1.00 each . how much money was left ?
Options: a ) $ 2.15 , b ) $ 2.05 , c ) $ 2.45 , d ) $ 2.25 , e ) $ 2.85
Rationale: "john spent and gave to his two friends a total of 1.05 + 1.00 + 1.00 = $ 3.05 money left 5.10 - 3.05 = $ 2.05 answer : b"
Answer: b
Python program:
a = 1 + 0 
b = 1 + 5 
c = 5 - 10
Problem: a fair coin is tossed 3 times . what is the probability of getting at least 2 heads ?
Options: a ) 3 / 4 , b ) 3 / 2 , c ) 1 / 4 , d ) 1 / 2 , e ) 1 / 5
Rationale: "let ' s find the probability of 2 heads , 3 heads p ( hhh ) = ( ( 1 / 2 ) ^ 3 = 1 / 8 . p ( hht ) = ( 3 ! / 2 ! ) * ( 1 / 2 ) ^ 3 = 3 / 8 total probablity = 1 / 8 + 3 / 8 = 1 / 2 answer ( d )"
Answer: d
Python program:
a = math.comb(3, 2)
b = math.comb(3, 3)
c = a + b 
d = math.comb(3, 4)
e = c + d 
f = math.comb(3, 3)
g = e + f 
h = 2 ** 3
i = g / h