euclaise/mathqa_programs huggingface dataset:
Problem: the h . c . f of two numbers is 11 and their l . c . m is 7700 . if one of the numbers is 275 , then the other is
Options: a ) 344 , b ) 387 , c ) 308 , d ) 299 , e ) 882
Rationale: "explanation : other number = \ inline \ fn _ jvn \ left ( \ frac { 11 \ times 7700 } { 275 } \ right ) = 308 answer : c ) 308"
Answer: c
Python program:
a = 11 * 275
Problem: what is the smallest positive integer x , such that 2160 x is a perfect cube ?
Options: a ) 100 , b ) 6 , c ) 8 , d ) 12 , e ) 18
Rationale: "take out the factors of 2160 that will come 6 ^ 3 * 10 . for perfect cube you need every no . raise to the power 3 . for 2160 to be a perfect cube , you need two 2 and two 5 that means 100 a is the answer ."
Answer: a
Python program:
a = 3 + 4
Problem: a batch of cookies was divided amomg 6 tins : 2 / 3 of all the cookies were placed in either the blue or the green tin , and the rest were placed in the red tin . if 1 / 4 of all the cookies were placed in the blue tin , what fraction of the cookies that were placed in the other tins were placed in the green tin
Options: a ) 15 / 2 , b ) 9 / 4 , c ) 5 / 9 , d ) 5 / 7 , e ) 9 / 7
Rationale: this will help reduce the number of variables you have to deal with : g + b = 2 / 3 r = 1 / 6 b = 1 / 4 we can solve for g which is 5 / 12 what fraction ( let it equal x ) of the cookies that were placed in the other tins were placed in the green tin ? so . . x * ( g + r ) = g x * ( 5 / 12 + 1 / 6 ) = 5 / 12 x = 5 / 7 answer : d
Answer: d
Python program:
a = 2 / 3
b = 1 - a
c = 2 / 3
d = 1 / 4
e = c - d
f = b + e
Problem: in a certain game , a large bag is filled with blue , green , purple and red chips worth 1 , 5 , x and 11 points each , respectively . the purple chips are worth more than the green chips , but less than the red chips . a certain number of chips are then selected from the bag . if the product of the point values of the selected chips is 28160 , how many purple chips were selected ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: "28160 = 1 * 5 * 8 ^ 3 * 11 the factors of 8 must come from the purple point value , so there are 3 purple chips . the answer is c ."
Answer: c
Python program:
a = 5 - 2
Problem: two numbers have a h . c . f of 12 and a product of two numbers is 2460 . find the l . c . m of the two numbers ?
Options: a ) 140 , b ) 150 , c ) 205 , d ) 170 , e ) 180
Rationale: "l . c . m of two numbers is given by ( product of the two numbers ) / ( h . c . f of the two numbers ) = 2460 / 12 = 205 . answer : c"
Answer: c
Python program:
a = 2460 / 12
Problem: if a - b = 9 and a ^ 2 + b ^ 2 = 281 , find the value of ab
Options: a ) 100 , b ) 12 , c ) 15 , d ) 18 , e ) 19
Rationale: "2 ab = ( a ^ 2 + b ^ 2 ) - ( a - b ) ^ 2 = 281 - 81 = 200 = > ab = 100 answer : a"
Answer: a
Python program:
a = math.sqrt(281)
b = a - 9
c = b / 2
d = 9 + c 
e = math.sqrt(281)
f = e - 9
g = f / 2
h = d * g
i = h * 2
Problem: if x and y are sets of integers , x # y denotes the set of integers that belong to set x or set y , but not both . if x consists of 16 integers , y consists of 18 integers , and 6 of the integers are in both x and y , then x # y consists of how many integers ?
Options: a ) 6 , b ) 16 , c ) 22 , d ) 30 , e ) 174
Rationale: "the number of integers that belong to set x only is 16 - 6 = 10 ; the number of integers that belong to set y only is 18 - 6 = 12 ; the number of integers that belong to set x or set y , but not both is 10 + 12 = 22 . answer : c"
Answer: c
Python program:
a = 18 - 6
b = 16 - 6
c = a + b
Problem: a train is moving at 6 / 7 of its usual speed . the train is 15 minutes too late . what is the usual time ( in hours ) for the train to complete the journey ?
Options: a ) 1 , b ) 1.5 , c ) 2 , d ) 2.5 , e ) 3
Rationale: "new time = d / ( 6 v / 7 ) = 7 / 6 * usual time 15 minutes represents 1 / 6 of the usual time . the usual time is 1.5 hours . the answer is b ."
Answer: b
Python program:
a = 6 / 7
b = 15 * a
c = 6 / 7
d = 1 - c
e = 1/(d)
f = b * e
g = f / const_60
Problem: exactly 3 / 5 of the people in the room are under the age of 21 , and exactly 5 / 13 of the people in the room are over the age of 65 . if the total number of the people in the room is greater than 50 and less than 100 , how many people in the room are under the age of 21 ?
Options: a ) 21 , b ) 35 , c ) 39 , d ) 60 , e ) 42
Rationale: "the total number of the people in the room must be a multiple of both 5 and 14 ( in order 3 / 5 and 5 / 14 of the number to be an integer ) , thus the total number of the people must be a multiple of lcm of 5 and 14 , which is 70 . since , the total number of the people in the room is greater than 50 and less than 100 , then there are 70 people in the room . therefore there are 3 / 5 * 70 = 42 people in the room under the age of 21 . answer : e ."
Answer: e
Python program:
a = 5 * 13
b = a * 3
c = b / 5
Problem: there are 2 sections a and b in a class , consisting of 60 and 70 students respectively . if the average weight of section a is 60 kg and that of section b is 80 kg , find the average of the whole class ?
Options: a ) 50.78 kg , b ) 49.32 kg , c ) 70.76 kg , d ) 69.15 kg , e ) 70.89 kg
Rationale: "total weight of 60 + 70 students = 60 * 60 + 70 * 80 = 3600 + 5600 average weight of the class is = 9200 / 130 = 70.76 kg answer is c"
Answer: c
Python program:
a = 60 * 60
b = 70 * 80
c = a + b 
d = 60 + 70 
e = c / d
Problem: a furniture store owner decided to drop the price of her recliners by 20 % to spur business . by the end of the week she had sold 80 % more recliners . what is the percentage increase of the gross ?
Options: a ) 10 % , b ) 15 % , c ) 20 % , d ) 25 % , e ) 44 %
Rationale: "say a recliner is actually worth $ 100 if she sells 100 recliners then she earns $ 10000 after the discount of 20 % , she will earn $ 80 per recliner and she sells 80 % more ie . , 180 recliners hence her sales tields 180 * 80 = $ 14400 increase in sales = 14400 - 10000 = $ 4400 so % increase = 4400 * 100 / 10000 = 44 % e is the answer"
Answer: e
Python program:
a = 20 / 100
b = 1 - a
c = 80 / 100
d = 1 + c 
e = b * d
f = e * 100
g = f - 100
Problem: in a family 19 people eat only vegetarian , 9 people eat only non veg . , 12 people eat both veg and non veg . . how many people eat veg in the family ?
Options: a ) 20 , b ) 31 , c ) 9 , d ) 31 , e ) 21
Rationale: "total people eat veg = only veg + both veg and non veg total = 19 + 12 = 31 answer = b"
Answer: b
Python program:
a = 19 + 12
Problem: two friends plan to walk along a 36 - km trail , starting at opposite ends of the trail at the same time . if friend p ' s rate is 25 % faster than friend q ' s , how many kilometers will friend p have walked when they pass each other ?
Options: a ) 19 , b ) 20 , c ) 21 , d ) 22 , e ) 23
Rationale: if q complete x kilometers , then p completes 1.25 x kilometers . x + 1.25 x = 36 2.25 x = 36 x = 16 then p will have have walked 1.25 * 16 = 20 km . the answer is b .
Answer: b
Python program:
a = 25 / 100
b = a + 1 
c = b + 1 
d = 36 / c
e = 25 / 100
f = e + 1 
g = d * f
Problem: for any integer k greater than 1 , the symbol k * denotes the product of all integers between 1 and k , inclusive . if k * is a multiple of 3,675 , what is the least possible value of k ?
Options: a ) 12 , b ) 14 , c ) 15 , d ) 21 , e ) 25
Rationale: "we basically need 5,5 , 3,7 and 7 in the number . 14 ! , has 3 , 5 , 7 , 10 , 14 now 10 = 2 x 5 14 = 2 x 7 cross check : k ! = 3675 * m ( m is a positive integer ) k ! = 3 * 5 * 5 * 7 * 7 14 ! has all the numbers above . . therefore b . . ! answer : b"
Answer: b
Python program:
a = 3 / 675
b = a / 3
c = 1 + 4 
d = b / c
Problem: l . c . m of two numbers is 192 and there h . c . f is 16 . if one of them is 48 . find the other
Options: a ) 32 , b ) 64 , c ) 48 , d ) 68 , e ) 78
Rationale: explanation : product of two numbers = product of their l . c . m and h . c . f 48 x a = 192 x 16 a = ( 192 x 16 ) / 48 = 64 answer : option b
Answer: b
Python program:
a = 192 * 16
b = a / 48
Problem: at 3.40 , the hour hand and the minute hand of a clock form an angle of
Options: a ) 378 , b ) 277 , c ) 297 , d ) 267 , e ) 130
Rationale: sol : use formula θ = ∣ ∣ ∣ 30 h − 112 m ∣ ∣ ∣ θ = | 30 h − 112 m | angle = 30 × 3 – 11 / 2 × 40 = 90 – 220 = 130 ° answer : e
Answer: e
Python program:
a = 2 ** 3
b = 3 * a
c = 10 * b
d = 4 + 1 
e = 3 * 4
f = 4 + 1 
g = e - f
h = g - 1
i = h * 10
j = c - i
Problem: x and y started a business with capitals rs . 20000 and rs . 25000 . after few months z joined them with a capital of rs . 30000 . if the share of z in the annual profit of rs . 50000 is rs . 14000 , then after how many months from the beginning did z join ?
Options: a ) 3 , b ) 4 , c ) 6 , d ) 9 , e ) 5
Rationale: investments of x , y and z respectively are rs . 20000 , rs . 25000 and rs . 30000 let investment period of z be x months . ratio of annual investments of x , y and z is ( 20000 * 12 ) : ( 25000 * 12 ) : ( 30000 * x ) = 240 : 300 : 30 x = 8 : 10 : x the share of z in the annual profit of rs . 50000 is rs . 14000 . = > [ x / ( 18 + x ) ] 50000 = 14000 = > [ x / ( 18 + x ) ] 25 = 7 = > 25 x = 7 x + ( 18 * 7 ) = > x = 7 months . z joined the business after ( 12 - 7 ) months . i . e . , 5 months . answer : e
Answer: e
Python program:
a = 14000 / 50000
b = 20000 + 25000 
c = b * 12
d = a * c
e = 14000 / 50000
f = 1 - e
g = d / f
h = g / 30000
i = 12 - h
Problem: if 18 men take 15 days to to complete a job , in how many days can 24 men finish that work ?
Options: a ) 10 , b ) 11 , c ) 12 , d ) 13 , e ) 14
Rationale: ans . 10 days
Answer: a
Python program:
a = 18 * 15
b = a / 24
Problem: a boat goes 100 km downstream in 10 hours , and 75 m upstream in 15 hours . the speed of the stream is ?
Options: a ) 22 1 / 9 kmph , b ) 22 8 / 2 kmph , c ) 22 1 / 8 kmph , d ) 22 1 / 2 kmph , e ) 22 1 / 4 kmph
Rationale: 100 - - - 10 ds = 10 ? - - - - 1 75 - - - - 15 us = 5 ? - - - - - 1 s = ( 10 - 5 ) / 2 = 2 21 / 2 kmph answer : d
Answer: d
Python program:
a = 100 / 10
b = 75 / 15
c = a - b
d = c / 2
Problem: if k is an integer and 121 < k ^ 2 < 225 , then k can have at most how many values ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 8
Rationale: given k is an integer 121 < k ^ 2 < 225 11 < | k | < 15 k = - 12 , - 13 , - 1412 , 1314 answer : d
Answer: d
Python program:
a = math.sqrt(225)
b = math.sqrt(121)
c = a - b
d = c - 1
e = d * 2
Problem: a specialized type of sand consists of 40 % mineral x by volume and 60 % mineral y by volume . if mineral x weighs 2.5 grams per cubic centimeter and mineral y weighs 3 grams per cubic centimeter , how many grams does a cubic meter of specialized sand combination weigh ? ( 1 meter = 100 centimeters )
Options: a ) 5 , 500,000 , b ) 2 , 800,000 , c ) 55,000 , d ) 28,000 , e ) 280
Rationale: "let the volume be 1 m ^ 3 = 1 m * 1 m * 1 m = 100 cm * 100 cm * 100 cm = 1 , 000,000 cm ^ 3 by volume 40 % is x = 400,000 cm ^ 3 60 % is y = 600,000 cm ^ 3 by weight , in 1 cm ^ 3 , x is 2.5 gms in 400,000 cm ^ 3 , x = 2.5 * 400,000 = 1 , 000,000 grams in 1 cm ^ 3 , y is 3 gms in 600,000 cm ^ 3 , y = 3 * 600,000 = 1 , 800,000 gms total gms in 1 m ^ 3 = 1 , 000,000 + 1 , 800,000 = 2 , 800,000 answer : b"
Answer: b
Python program:
a = volume_cube / (
b = a * 10
c = b * 2
d = c + 2 
e = volume_cube / (
f = e * 10
g = 2 * 3
h = f * g
i = d - h
Problem: a sum of rs . 2040 has been divided among a , b and c such that a gets of what b gets and b gets of what c gets . b ’ s share is :
Options: a ) rs . 120 , b ) rs . 160 , c ) rs . 240 , d ) rs . 360 , e ) none
Rationale: "explanation let c ’ s share = rs . x then , b ’ s share = rs . x / 4 , a ’ s share = rs . ( 2 / 3 x x / 4 ) = rs . x / 6 = x / 6 + x / 4 + x = 2040 = > 17 x / 12 = 2040 = > 2040 x 12 / 17 = rs . 1440 hence , b ’ s share = rs . ( 1440 / 4 ) = rs . 360 . answer d"
Answer: d
Python program:
a = 2040 / 10
b = a * 2
c = b - 12
d = c - 12
Problem: a boat can travel with a speed of 40 km / hr in still water . if the speed of the stream is 5 km / hr , find the time taken by the boat to go 45 km downstream .
Options: a ) 1 hr , b ) 2 hrs , c ) 3 hrs , d ) 4 hrs , e ) 5 hrs
Rationale: "speed downstream = ( 40 + 5 ) km / hr = 45 km / hr . time taken to travel 45 km downstream = 45 / 45 hrs = 1 hrs . answer : a"
Answer: a
Python program:
a = 40 + 5 
b = 45 / a
Problem: 3 candidates in an election and received 1036 , 4636 and 11628 votes respectively . what % of the total votes did the winning candidate gotin that election ?
Options: a ) 40 % , b ) 55 % , c ) 57 % , d ) 60 % , e ) 67.2 %
Rationale: total number of votes polled = ( 1036 + 4636 + 11628 ) = 17300 so , required percentage = 11628 / 17300 * 100 = 67.2 % e
Answer: e
Python program:
a = 1036 + 4636 
b = a + 11628 
c = 11628 / b
d = c * 100
Problem: find the amount on rs . 5000 in 2 years , the rate of interest being 2 % per first year and 3 % for the second year ?
Options: a ) 3377 , b ) 2678 , c ) 5460 , d ) 1976 , e ) 5253
Rationale: "5000 * 102 / 100 * 103 / 100 = > 5253 answer : e"
Answer: e
Python program:
a = 100 + 2 
b = 5000 * a
c = b / 100
d = 100 + 3 
e = c * d
f = e / 100
Problem: in plutarch enterprises , 60 % of the employees are marketers , 30 % are engineers , and the rest are managers . marketers make an average salary of $ 50,000 a year , and engineers make an average of $ 80,000 . what is the average salary for managers if the average for all employees is also $ 80,000 ?
Options: a ) $ 80,000 , b ) $ 130,000 , c ) $ 260,000 , d ) $ 290,000 , e ) $ 320,000
Rationale: "for sake of ease , let ' s say there are 10 employees : 6 marketers , 3 engineers , and 1 manager . average company salary * number of employees = total company salary > > > $ 80,000 * 10 = $ 800,000 subtract the combined salaries for the marketers ( 6 * $ 50,000 ) and the engineers ( 3 * $ 80,000 ) > > > $ 800,000 - $ 300,000 - $ 240,000 = $ 260,000 . the correct answer is c ."
Answer: c
Python program:
a = 60 + 30 
b = a * 1000
c = b * 100
d = 60 + 30 
e = d * 1000
f = 30 * e
g = c - f
h = 30 / 2
i = h * 30
j = i * 1000
k = j * 60
l = g - k
m = l / 30
Problem: find the missing figures : 0.25 % of ? = 0.04
Options: a ) 16 , b ) 20 , c ) 17 , d ) 12 , e ) 18
Rationale: "let 0.25 % of x = 0.04 . then , 0.25 * x / 100 = 0.04 x = [ ( 0.04 * 100 ) / 0.25 ] = 16 . answer is a ."
Answer: a
Python program:
a = 0 / 25
b = 0 / 4
Problem: how many 6 digit number contain number 4 ?
Options: a ) 317608 , b ) 327608 , c ) 337608 , d ) 427608 , e ) 357608
Rationale: "total 6 digit no . = 9 * 10 * 10 * 10 * 10 * 10 = 900000 not containing 4 = 8 * 9 * 9 * 9 * 9 * 9 = 472392 total 6 digit number contain 4 = 900000 - 472392 = 427608 answer : d"
Answer: d
Python program:
a = 1000 - 10
b = 6 * 6
c = 10 * b
d = 4 * 2
e = c * d
f = a - e
g = f + 10
Problem: cricket match is conducted in us . the run rate of a cricket game was only 3.2 in first 10 over . what should be the run rate in the remaining 40 overs to reach the target of 262 runs ?
Options: a ) 6 , b ) 6.25 , c ) 5.75 , d ) 7.5 , e ) 8
Rationale: "required run rate = 262 - ( 3.2 x 10 ) = 230 = 5.75 40 40 c"
Answer: c
Python program:
a = 3 * 2
b = 262 - a
c = b / 40
Problem: the mean of 40 observations was 36 . it was found later that an observation 34 was wrongly taken as 20 . the corrected new mean is
Options: a ) 76.55 , b ) 36.35 , c ) 46.15 , d ) 16.05 , e ) 20
Rationale: "explanation : correct sum = ( 36 * 40 + 34 - 20 ) = 1454 correct mean = = 1454 / 40 = 36.35 answer : b"
Answer: b
Python program:
a = 36 * 40
b = 40 - 2
c = b - 20
d = a + c 
e = d / 40
Problem: a small pool filled only with water will require an additional 300 gallons of water in order to be filled to 80 % of its capacity . if pumping in these additional 300 gallons of water will increase the amount of water in the pool by 40 % , what is the total capacity of the pool in gallons ?
Options: a ) 1000 , b ) 1312.5 , c ) 1400 , d ) 1600 , e ) 1625
Rationale: "300 gallons of water increases capacity by 40 % that means 40 % is 300 gallons , so 100 % would be = 300 * 100 / 40 = 750 gallons now 750 + 300 gallons is 80 % capacity of tank . so 100 % capacity would be = 1050 * 100 / 80 = 1312.5 b is the answer"
Answer: b
Python program:
a = 300 * 100
b = a / 40
c = b + 300 
d = 80 / 100
e = c / d
Problem: the speed of a car is 10 m / s . what is the its speed in kmph .
Options: a ) 25 kmph , b ) 30 kmph , c ) 50 kmph , d ) 36 kmph , e ) 37 kmph
Rationale: explanation : m / sec to km / hr conversion : x m / sec = [ x x ( 18 / 5 ) ] km / hr 10 x ( 18 / 5 ) 36 kmph answer : option d
Answer: d
Python program:
a = 10 * const_3_6
Problem: total dinning bill of 5 people was $ 139.00 and 10 % tip divided the bill evenly ? what is the bill amount each person shared .
Options: a ) 32.84 , b ) 22.84 , c ) 30.58 , d ) 24.84 , e ) 30.84
Rationale: "dinner bill of 5 person = 139 + 10 % tip so , 10 % of 139 = ( 139 * 10 ) / 100 = 13.9 so , the actual total amount = 139 + 13.9 = $ 152.9 so per head bill = 152.9 / 5 = $ 30.58 answer : c"
Answer: c
Python program:
a = 1 / 10
b = a + 1 
c = 139 * 0
d = c / 5
Problem: at a monthly meeting , 3 / 5 of the attendees were males and 4 / 5 of the male attendees arrived on time . if 5 / 6 of the female attendees arrived on time , what fraction of the attendees at the monthly meeting did not arrive on time ?
Options: a ) 6 / 25 , b ) 11 / 50 , c ) 14 / 75 , d ) 23 / 100 , e ) 31 / 150
Rationale: "males who did not arrive on time are 1 / 5 * 3 / 5 = 3 / 25 of the attendees . females who did not arrive on time are 1 / 6 * 2 / 5 = 1 / 15 of the attendees . the fraction of all attendees who did not arrive on time is 3 / 25 + 1 / 15 = 14 / 75 the answer is c ."
Answer: c
Python program:
a = 5 / 6
b = 1 - a
c = 3 / 5
d = 1 - c
e = b * d
f = 4 / 5
g = 1 - f
h = 3 / 5
i = g * h
j = e + i
Problem: the sum of 7 consecutive integers is 980 . how many of them are prime ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: the middle number is 140 and the seven numbers are 137 , 138 , 139 , · · · , 143 . now 138 , 140 , and 142 are even , 141 is divisible by 3 , and 143 - by 11 . the remaining numbers 137 and 139 are prime ( it is easy to check that they are not divisible by 23 , 57 and 11 ) . correct answer c
Answer: c
Python program:
a = 1 + 2 
b = a + 3 
c = 10 - b
d = c - 2
Problem: if n = 8 ^ 8 – 7 , what is the units digit of n ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 7 , e ) 4
Rationale: "8 ^ 8 - 8 = 8 ( 8 ^ 7 - 1 ) = = > 8 ( 2 ^ 21 - 1 ) last digit of 2 ^ 21 is 2 based on what explanation livestronger is saying . 2 ^ 24 - 1 yields 2 - 1 = 1 as the unit digit . now on multiply this with 7 , we get unit digit as 7 answer : d"
Answer: d
Python program:
a = math.log(7)
b = 8 ** 8
c = math.log(b)
d = a / c
Problem: how many prime numbers between 1 and 100 are factors of 210 ?
Options: a ) 5 , b ) 4 , c ) 3 , d ) 2 , e ) 1
Rationale: factor of 210 = 2 * 3 * 5 * 7 - - - 4 prime numbers b
Answer: b
Python program:
a = 4 * 1
Problem: in 1998 the profits of company n were 10 percent of revenues . in 1999 , the revenues of company n fell by 20 percent , but profits were 12 percent of revenues . the profits in 1999 were what percent of the profits in 1998 ?
Options: a ) 80 % , b ) 105 % , c ) 96 % , d ) 124.2 % , e ) 138 %
Rationale: "0,096 r = x / 100 * 0.1 r answer c"
Answer: c
Python program:
a = 20 / 100
b = 1 - a
c = 12 / 100
d = b * c
e = 10 / 100
f = d / e
g = f * 100
Problem: when positive integer x is divided by 7 , the quotient is y and the remainder is 3 . when 2 x is divided by 6 , the quotient is 3 y and the remainder is 2 . what is the value of 11 y – x ?
Options: a ) 4 , b ) 3 , c ) 2 , d ) 1 , e ) 0
Rationale: "( 1 ) x = 7 y + 3 ( 2 ) 2 x = 18 y + 2 ( 2 ) - ( 1 ) : x = 11 y - 1 11 y - x = 1 the answer is d ."
Answer: d
Python program:
a = 2 * 3
b = a - 2
c = 6 * 3
d = 2 * 7
e = c - d
f = b / e
g = 11 * f
h = 2 * 3
i = h - 2
j = 6 * 3
k = 2 * 7
l = j - k
m = i / l
n = 7 * m
o = n + 3 
p = g - o
Problem: n is a positive integer . when n + 1 is divided by 9 , the remainder is 1 . what is the remainder when n is divided by 9 ?
Options: a ) 0 , b ) 5 , c ) 4 , d ) 3 , e ) 2
Rationale: "n + 1 = 9 a + 1 i . e . n + 1 = 10 , 19 , 28 , 37 , . . . etc . i . e . n = 9 , 18 , 27 , 36 , . . . etc . when n is divided by 9 remainder is always 0 answer : a"
Answer: a
Python program:
a = 1 + 9 
b = 9 / a
Problem: how many positive integers less than 23 are prime numbers , odd multiples of 5 , or the sum of a positive multiple of 2 and a positive multiple of 4 ?
Options: a ) 17 , b ) 25 , c ) 24 , d ) 22 , e ) 20
Rationale: "8 prime numbers less than 28 : { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 } 2 odd multiples of 5 : { 5 , 15 } 9 numbers which are the sum of a positive multiple of 2 and a positive multiple of 4 : { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 , 22 } notice , that 5 is in two sets , thus total # of integers satisfying the given conditions is 8 + 1 + 9 - 1 = 17 . answer : a ."
Answer: a
Python program:
a = 23 - 2
b = a - 1
c = b - 1
Problem: we need to carve out 125 identical cubes from a cube . what is the minimum number of cuts needed ?
Options: a ) 8
b ) 9
c ) 10
d ) 11
e ) 12
Rationale: total no . of cubes = n ^ 3 here n ^ 3 = 125 which makes n = 5 , also minimum no . of cuts required = 3 ( n - 1 ) hence , 3 ( 5 - 1 ) = 12 cuts . answer : e
Answer: e
Python program:
a = 125 ** const_0_33
b = a - 1
c = 3 * b
Problem: if a : b : : 3 : 7 , then what is ( 5 a + 6 b ) : ( a - 2 b ) ?
Options: a ) 57 : 11 , b ) - 57 : 11 , c ) 11 : 10 , d ) - 11 : 10 , e ) - 1 : 10
Rationale: a / b = 3 / 7 dividing numerator & denominator of ' ( 5 a + 6 b ) / ( a - 2 b ) ' by b , [ 5 ( a / b ) + 6 ] / [ ( a / b ) - 2 ] = [ 5 * ( 3 / 7 ) + 6 ] / [ ( 3 / 7 ) - 2 ] = - 57 / 11 answer : b
Answer: b
Python program:
a = 5 * 3
b = 7 * 6
c = a + b 
d = 2 * 7
e = 3 - d
f = c / e
Problem: a block of wood has dimensions 10 cm x 10 cm x 50 cm . the block is painted red and then cut evenly at the 25 cm mark , parallel to the sides , to form two rectangular solids of equal volume . what percentage of the surface area of each of the new solids is not painted red ?
Options: a ) 5.5 % , b ) 8.3 % , c ) 11.6 % , d ) 14.2 % , e ) 17.5 %
Rationale: "the area of each half is 100 + 4 ( 250 ) + 100 = 1200 the area that is not painted is 100 . the fraction that is not painted is 100 / 1200 = 1 / 12 = 8.3 % the answer is b ."
Answer: b
Python program:
a = 4 * 100
b = a * 4
c = b + 100 
d = c + 100 
e = 100 / d
f = e * 100
Problem: a mobile battery in 1 hour charges to 20 percent . how much time ( in minute ) will it require more to charge to 30 percent .
Options: a ) 145 , b ) 90 , c ) 175 , d ) 160 , e ) 130
Rationale: 1 hr = 20 percent . thus 15 min = 5 percent . now to charge 30 percent 90 min . answer : b
Answer: b
Python program:
a = 30 / 20
b = a * const_60
Problem: how many 4 digit numbers are there , if it is known that the first digit is even , the second is odd , the third is prime , the fourth ( units digit ) is divisible by 3 , and the digit 5 can be used only once ?
Options: a ) 20 , b ) 150 , c ) 225 , d ) 300 , e ) 304
Rationale: "4 options for the first digit : 2 , 4 , 6 , 8 ; 5 options for the second digit : 1 , 3 , 5 , 7 , 9 ; 4 options for the third digit : 2 , 3 , 5 , 7 ; 4 options for the fourth digit : 0 , 3 , 6 , 9 . four digit # possible without the restriction ( about the digit 2 ) : 4 * 5 * 4 * 4 = 320 numbers with five 5 - s , 5 x 5 x 4 * 1 * 1 * 4 = 16 . thus there are 320 - 16 = 304 such numbers . answer : e ."
Answer: e
Python program:
a = 4 + 1 
b = 4 + 1 
c = a * b
d = 4 * 4
e = c * d
f = 4 + 1 
g = 4 + 1 
h = f * g
i = h * 4
j = e - i
Problem: two brothers ram and ravi appeared for an exam . the probability of selection of ram is 6 / 7 and that of ravi is 1 / 5 . find the probability that both of them are selected .
Options: a ) 2 / 35 , b ) 2 / 3 , c ) 6 / 35 , d ) 5 / 7 , e ) 7 / 5
Rationale: "let a be the event that ram is selected and b is the event that ravi is selected . p ( a ) = 6 / 7 p ( b ) = 1 / 5 let c be the event that both are selected . p ( c ) = p ( a ) x p ( b ) as a and b are independent events : = 6 / 7 x 1 / 5 = 6 / 35 answer : c"
Answer: c
Python program:
a = 6 / 7
b = 1 / 5
c = a * b