euclaise/mathqa_programs huggingface dataset:
Problem: a person spends 1 / 3 rd of the money with him on clothes , 1 / 5 th of the remaining on food and 1 / 4 th of the remaining on travel . now , he is left with rs 100 . how much did he have with him in the beginning ?
Options: a ) rs 200 , b ) rs 250 , c ) rs 300 , d ) rs 450 , e ) rs 550
Rationale: "suppose the amount in the beginning was rs ’ x ’ money spent on clothes = rs 1 x / 3 balance = rs 2 x / 3 money spent on food = 1 / 5 of 2 x / 3 = rs 2 x / 15 balance = 2 x / 3 - 2 x / 15 = rs 8 x / 15 money spent on travel = 1 / 4 of 8 x / 15 = rs 2 x / 15 = 8 x / 15 - 2 x / 15 = 6 x / 15 = rs 2 x / 5 therefore 2 x / 5 = 100 = 250 answer : b"
Answer: b
Python program:
a = 1 / 3
b = 1 - a
c = 1 / 3
d = 1 - c
e = d / 5
f = b - e
g = 1 / 3
h = 1 - g
i = 1 / 3
j = 1 - i
k = j / 5
l = h - k
m = l / 4
n = f - m
o = 100 / n
Problem: marginal cost is the cost of increasing the quantity produced ( or purchased ) by one unit . if the fixed cost for n products is $ 10,000 and the marginal cost is $ 80 , and the total cost is $ 16,000 , what is the value of n ?
Options: a ) 30 , b ) 50 , c ) 60 , d ) 75 , e ) 100
Rationale: "total cost for n products = fixed cost for n products + n * marginal cost - - > $ 16,000 = $ 10,000 + n * $ 80 - - > n = 75 . answer : d ."
Answer: d
Python program:
a = 80 / 10
Problem: 765.765 - ? + 46.82 = 437.785
Options: a ) a ) 374.80 , b ) b ) 752.804 , c ) c ) 714.642 , d ) d ) 629.906 , e ) of these
Rationale: "explanation : 374.80 answer : option a"
Answer: a
Python program:
a = 765 / 765
b = a * 46
c = 1 / 3
d = 765 / 765
e = d * 46
f = c * e
g = b - f
Problem: sam taken a loan rs . 15000 / - from co - operative society with an interest @ 11.5 % per month . at the same time he deposited rs . 10000 / - as fixed deposit with an interest @ 9.5 % per month . after one week sam asked the manager to calculate the interest to be paid . what is the interest amount for 7 days ?
Options: a ) a ) 165 , b ) b ) 220 , c ) c ) 310 , d ) d ) 185 , e ) e ) 181
Rationale: loan amount : rs . 15000 / - @ 11.5 % interest per month = 15000 / - * 11.5 % = rs . 1725 interest for one day = 1725 / 30 = 57.50 interest for 7 days is = 57.50 * 7 = 403 fd amount is = rs . 10000 / - @ 9.5 % interest per month = 10000 * 9.5 % = 950 / - interest for 7 days = 950 / 30 * 7 = 222 interest amount to be paid by sam = 403 - 222 = 181 / - for 7 days answer is e
Answer: e
Python program:
a = 11 / 5
b = 15000 * a
c = 9 / 5
d = 10000 * c
e = b - d
f = 10 * 3
g = e / f
h = 7 * g
Problem: how many integers from 20 to 150 , inclusive , are divisible by 3 but not divisible by 7 ?
Options: a ) 37 , b ) 40 , c ) 42 , d ) 45 , e ) 55
Rationale: "we should find # of integers divisible by 3 but not by 3 * 7 = 21 . # of multiples of 21 in the range from 20 to 150 , inclusive is ( 147 - 21 ) / 21 + 1 = 7 ; 44 - 7 = 37 . answer : a ."
Answer: a
Python program:
a = 150 / 10
Problem: on a two - dimensional coordinate plane , the line d = x ^ 2 - x ^ 3 touches the x - axis in how many places ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: "apparently it ' s d = x ^ 2 - x ^ 3 instead of d = x ^ 2 - d ^ 3 . in this case : the x - intercept is the value ( s ) of x for d = 0 . 0 = x ^ 2 - x ^ 3 ; 0 = x ^ 2 ( 1 - x ) ; x = 0 or x = 1 . answer : c ."
Answer: c
Python program:
a = max(3)
b = a - 1
Problem: how many even integers from 1 to 100 ( both inclusive ) have even number of factors ?
Options: a ) 13 , b ) 14 , c ) 5 , d ) 6 , e ) 7
Rationale: integers having even number of factors will be perfect squares . even numbers will have even perfect squares . thus , the possible values for the perfect squares are : 4 , 16,36 , 64,100 and the corresponding integers are 2 , 4,6 , 8,10 ( more than 4 ) . thus c is the correct answer .
Answer: c
Python program:
a = math.sqrt(100)
b = a / 2
Problem: the ratio of the number of females to males at a party was 1 : 2 but when 4 females and 4 males left , the ratio became 1 : 3 . how many people were at the party originally ?
Options: a ) 20 , b ) 22 , c ) 24 , d ) 26 , e ) 28
Rationale: "the total number of people are x females + 2 x males . 3 * ( x - 4 ) = 2 x - 4 x = 8 there were 3 x = 24 people at the party originally . the answer is c ."
Answer: c
Python program:
a = 4 * 2
b = a * 3
c = 4 * 2
d = b - c
e = d / 2
f = 4 * 2
g = f * 3
h = 4 * 2
i = g - h
j = e + i
Problem: if 7875 / 5.25 = 1500 , then 787.5 / 52.5 is equal to ?
Options: a ) 15 , b ) 19 , c ) 12 , d ) 14 , e ) 16
Rationale: "answer given expression 787.5 / 52.5 = 7875 / 525 = 7875 / ( 525 x 100 ) = 1500 / 100 15 correct option : a"
Answer: a
Python program:
a = 1500 * 5
b = a / 7875
Problem: two trains , one from howrah to patna and the other from patna to howrah , start simultaneously . after they meet , the trains reach their destinations after 18 hours and 8 hours respectively . the ratio of their speeds is
Options: a ) 4 : 9 , b ) 4 : 3 , c ) 4 : 12 , d ) 4 : 8 , e ) 4 : 5
Rationale: let us name the trains as a and b . then , ( a ' s speed ) : ( b ' s speed ) = b : a = 8 : 18 = 4 : 9 . answer : a
Answer: a
Python program:
a = math.sqrt(8)
b = math.sqrt(18)
c = a / b
Problem: when the price of an article is reduced by 30 % , the sales increases by 50 % . the percentage change in the total amount of receipts is ?
Options: a ) 5 % decrease , b ) 5 % increase , c ) 10 % decrease , d ) 15 % decrease , e ) none of these
Rationale: we have , receipts = price x sales therefore , net % change in receipts = ( x + y + xy / 100 ) % = [ - 30 + 50 + ( - 30 x 50 ) / 100 ] % = 5 % increase . answer : b
Answer: b
Python program:
a = 100 - 30
b = 100 + 50 
c = a * b
d = c / 100
e = d - 100
Problem: find the ratio in which rice at rs . 7.20 a kg be mixed with rice at rs . 5.60 a kg to produce a mixture worth rs . 6.30 a kg
Options: a ) 2 : 0 , b ) 2 : 3 , c ) 2 : 1 , d ) 2 : 2 , e ) 7 : 9
Rationale: "by the rule of alligation : cost of 1 kg rice of 1 st kind cost of 1 kg rice of 2 nd kind required ratio = 70 : 90 = 7 : 9 answer : e"
Answer: e
Python program:
a = 6 - 30
b = 7 - 20
c = a / b
Problem: two trains of equal lengths take 10 sec and 15 sec respectively to cross a telegraph post . if the length of each train be 140 m , in what time will they cross other travelling in opposite direction ?
Options: a ) 17 sec , b ) 12 sec , c ) 16 sec , d ) 14 sec , e ) 18 sec
Rationale: "speed of the first train = 120 / 10 = 12 m / sec . speed of the second train = 120 / 5 = 8 m / sec . relative speed = 12 + 8 = 20 m / sec . required time = ( 140 + 140 ) / 20 = 14 sec . answer : d"
Answer: d
Python program:
a = 140 * 2
b = speed + ( 
c = a / b
Problem: at a special sale , 5 tickets can be purchased for the price of 3 tickets . if 5 tickets are purchased at the sale , the amount saved will be what percent of the original price of the 5 tickets ?
Options: a ) 20 % , b ) 33.3 % , c ) 40 % , d ) 60 % , e ) 66.6 %
Rationale: "let the price of a ticket be rs . 100 , so 3 tickets cost 300 & 5 tickets cost 500 5 tickets purchased at price of 3 tickets ie . , for 300 , so amount saved s rs . 200 , % of 5 tickets = ( 200 / 500 ) * 100 = 40 % answer : c"
Answer: c
Python program:
a = 5 * 5
b = 3 * 5
c = a - b
d = c * 100
e = 5 * 5
f = d / e
Problem: the average salary of all the workers in a workshop is rs . 8000 . the average salary of 7 technicians is rs . 20000 and the average salary of the rest is rs . 6000 . the total number of workers in the workshop is
Options: a ) 20 , b ) 49 , c ) 22 , d ) 23 , e ) none
Rationale: "sol . let the total number of workers be x . then , 8000 x = ( 20000 × 7 ) + 6000 ( x – 7 ) ‹ = › 2000 x = 98000 ‹ = › x = 49 . answer b"
Answer: b
Python program:
a = 20000 - 8000
b = 7 * a
c = 8000 - 6000
d = b / c
e = 7 + d
Problem: find the average of all the numbers between 6 and 34 which are divisible by 7 .
Options: a ) 18 , b ) 20 , c ) 17.5 , d ) 30 , e ) 32
Rationale: "solution average = ( 7 + 14 + 21 + 28 ) / 4 ) = 70 / 4 = 17.5 . answer c"
Answer: c
Python program:
a = 6 + 4 
b = 34 - 4
c = a + b 
d = c / 2
Problem: at what price must an article costing rs . 47.50 be marked in order that after deducting 6 % from the list price . it may be sold at a profit of 25 % on the cost price ?
Options: a ) 63.16 , b ) 62.6 , c ) 62.1 , d ) 62.7 , e ) 62.2
Rationale: "cp = 47.50 sp = 47.50 * ( 125 / 100 ) = 59.375 mp * ( 94 / 100 ) = 59.375 mp = 63.16 answer : a"
Answer: a
Python program:
a = 47 * 50
b = a / 100
c = 47 + 50 
d = c * 100
e = 100 - 6
f = d / e
Problem: if a mixture is 3 ⁄ 5 alcohol by volume and 4 ⁄ 5 water by volume , what is the ratio of the volume of alcohol to the volume of water in this mixture ?
Options: a ) 1 / 3 , b ) 3 / 2 , c ) 3 / 4 , d ) 3 , e ) 3 / 5
Rationale: "should be a sub - 600 level q . . volume = { 3 / 5 } / { 4 / 5 } = 3 / 4 c"
Answer: c
Python program:
a = 3 / 5
b = 4 / 5
c = a / b
Problem: what is difference between biggest and smallest fraction among 2 / 3 , 3 / 4 , 4 / 5 and 5 / 6
Options: a ) 2 / 5 , b ) 3 / 5 , c ) 1 / 6 , d ) 1 / 7 , e ) none of these
Rationale: "explanation : 2 / 3 = . 66 , 3 / 4 = . 75 , 4 / 5 = . 8 and 5 / 6 = . 833 so biggest is 5 / 6 and smallest is 2 / 3 their difference is 5 / 6 - 2 / 3 = 1 / 6 option c"
Answer: c
Python program:
a = 4 / 5
b = 2 / 3
c = a - b
Problem: a reduction of 25 % in the price of salt enables a lady to obtain 10 kgs more for rs . 800 , find the original price per kg ?
Options: a ) 25.67 , b ) 26.67 , c ) 27.2 , d ) 26.98 , e ) 27.76
Rationale: "answer : 800 * ( 25 / 100 ) = 200 - - - 10 ? - - - 1 = > rs . 20 800 - - - 600 ? - - - 20 = > rs . 26.67 . answer : b"
Answer: b
Python program:
a = 25 / 100
b = a * 800
c = b / 10
d = 25 / 100
e = d * 800
f = c / e
g = f * 100
Problem: the compound and the simple interests on a certain sum at the same rate of interest for two years are rs . 11730 and rs . 10200 respectively . find the sum .
Options: a ) 33498 , b ) 17000 , c ) 2665 , d ) 28876 , e ) 313
Rationale: "the simple interest for the first year is 10200 / 2 is rs . 5100 and compound interest for first year also is rs . 5100 . the compound interest for second year on rs . 5100 for one year so rate of the interest = ( 100 * 1530 ) / ( 5100 * 1 ) = 30 % p . a . so p = ( 100 * 10200 ) / ( 30 * 2 ) = rs . 17000 answer : b"
Answer: b
Python program:
a = 100 * 10200
b = 11730 - 10200
c = b * 100
d = 10200 / 2
e = c / d
f = e * 2
g = a / f
Problem: the length of a room is 5.5 m and width is 3.75 m . what is the cost of paying the floor by slabs at the rate of $ 300 per sq . metre .
Options: a ) $ 2587.50 , b ) $ 3587.50 , c ) $ 4187.50 , d ) $ 6187.50 , e ) $ 8587.50
Rationale: "area = 5.5 × 3.75 sq . metre . cost for 1 sq . metre . = $ 300 hence , total cost = 5.5 × 3.75 × 300 = $ 6187.50 d"
Answer: d
Python program:
a = 5 * 5
b = 300 * a
Problem: of the 150 people at a party , 70 were women , and 30 women tried the appetizer . if 60 people did not try the appetizer , what is the total number of men who tried the appetizer ?
Options: a ) 40 , b ) 50 , c ) 60 , d ) 70 , e ) 80
Rationale: "total people at party = 150 women = 70 so men 150 - 70 = 80 no . of pple who tried appetizer = 150 - 60 ( given info ) = 90 no of women who tried appetizer = 30 so remaining ppl ( men ) who tried the appetizer = 90 - 30 = 60 correct option c"
Answer: c
Python program:
a = 150 - 60
b = a - 30
Problem: a man swims downstream 28 km and upstream 12 km taking 2 hours each time , what is the speed of the man in still water ?
Options: a ) 10 , b ) 8 , c ) 5 , d ) 2 , e ) 4
Rationale: "28 - - - 2 ds = 14 ? - - - - 1 12 - - - - 2 us = 6 ? - - - - 1 m = ? m = ( 14 + 6 ) / 2 = 10 answer : a"
Answer: a
Python program:
a = 12 / 2
b = 28 / 2
c = a + b 
d = c / 2
Problem: in a restaurant , the profit is 130 % of the cost . if the cost increases by 12 % but the selling price remains constant , approximately what percentage of the selling price is the profit ?
Options: a ) 30 % , b ) 51 % , c ) 90 % , d ) 100 % , e ) none of these
Rationale: "explanation : let c . p . = rs . 100 . then , profit = rs . 130 , s . p . = rs . 230 . new c . p . = 112 % of rs . 100 = rs . 112 new s . p . = rs . 230 . profit = rs . ( 230 - 112 ) = rs . 118 . required percentage = ( 118 / 230 * 100 ) % = 51 % appox answer : b"
Answer: b
Python program:
a = 130 / 100
b = 100 * a
c = b + 100 
d = 12 / 100
e = 1 + d 
f = 100 * e
g = c - f
h = 130 / 100
i = 100 * h
j = i + 100 
k = g / j
l = 100 * k
Problem: a dog takes 5 leaps for every 6 leaps of a hare . if one leap of the dog is equal to 3 leaps of the hare , the ratio of the speed of the dog to that of the hare is :
Options: a ) 1 : 6 , b ) 5 : 6 , c ) 6 : 1 , d ) 5 : 1 , e ) 9 : 1
Rationale: "explanation : dog : hare = ( 5 * 3 ) leaps of hare : 6 leaps of hare = 15 : 3 - - > 5 : 1 answer : d"
Answer: d
Python program:
a = 5 * 3
b = a / 6
Problem: a mathematics teacher tabulated the marks secured by 35 students of 8 th class . the average of their marks was 72 . if the marks secured by reema was written as 36 instead of 66 then find the correct average marks up to two decimal places .
Options: a ) 72.85 , b ) 74.31 , c ) 72.43 , d ) 73.43 , e ) can not be determined
Rationale: total marks = 35 x 72 = 2520 corrected total marks = 2520 - 36 + 66 = 2550 correct average = 2550 / 35 = 72.85 answer : a
Answer: a
Python program:
a = 35 * 72
b = 66 - 36
c = a - b
d = c / 35
Problem: a bag contains 6 green balls and 7 white balls . if two balls are drawn simultaneously , what is the probability that both balls are the same colour ?
Options: a ) 2 / 7 , b ) 3 / 10 , c ) 5 / 11 , d ) 6 / 13 , e ) 11 / 21
Rationale: "the total number of ways to draw two balls is 13 c 2 = 78 the number of ways to draw two green balls is 6 c 2 = 15 the number of ways to draw two white balls is 7 c 2 = 21 p ( two balls of the same colour ) = 36 / 78 = 6 / 13 the answer is d ."
Answer: d
Python program:
a = 6 + 7 
b = 7 / a
c = 7 - 1
d = 6 + 7 
e = d - 1
f = c / e
g = b * f
h = 6 - 1
i = 6 + 7 
j = i - 1
k = h / j
l = 6 + 7 
m = 6 / l
n = k * m
o = g + n
Problem: in the biology lab of ` ` jefferson ' ' high school there are 0.036 * 10 ^ 5 germs , equally divided among 75000 * 10 ^ ( - 3 ) petri dishes . how many germs live happily in a single dish ?
Options: a ) 10 , b ) 20 , c ) 30 , d ) 40 , e ) 48
Rationale: "0.036 * 10 ^ 5 can be written as 3600 75000 * 10 ^ ( - 3 ) can be written as 75 required = 3600 / 75 = 48 answer : e"
Answer: e
Python program:
a = 1000 * 100
b = a * 0
c = 75000 / 1000
d = b / c
Problem: set x consists of the integers from 1 to 12 , inclusive , while set y consists of the integers from 0 to 20 , inclusive . how many distinct integers do belong to the both sets at the same time ?
Options: a ) 12 , b ) 15 , c ) 10 , d ) 20 , e ) 8
Rationale: x = { 1,2 , 3,4 , 5,6 , 7 , 8 , 9 , 10 , 11 , 12 } y = { 0 , 1,2 , 3,4 , 5,6 , 7 , 8 , 9 , 10 , 11,12 , 13,14 , 15,16 , 17,18 , 19,20 } common elements = { 1,2 , 3,4 , 5,6 , 7,8 , 9,10 , 11,12 } = 12 elements answer : option a .
Answer: a
Python program:
a = 12 - 1
b = a + 1
Problem: a train 160 m long running at 72 kmph crosses a platform in 25 sec . what is the length of the platform ?
Options: a ) 443 m , b ) 354 m , c ) 340 m , d ) 350 m , e ) 250 m
Rationale: "c 340 c = ( 72 * 5 / 18 ) * 25 - 160 = 340"
Answer: c
Python program:
a = 72 * const_0_2778
b = 25 * a
c = b - 160
Problem: a cube of edge 5 cm is cut into cubes each of edge 1 cm . the ratio of the total surface area of one of the small cubes to that of the large cube is equal to :
Options: a ) 1 : 25
b ) 1.225
c ) 1 : 52
d ) 1 : 522
e ) none
Rationale: sol . required ratio = 6 * 1 * 1 / 6 * 5 * 5 = 1 / 25 = 1 : 25 . answer a
Answer: a
Python program:
a = 4 / 100
Problem: the difference b / w the compound interest payble half yearly and the simple interest on a certain sum lent out at 10 % p . a for 1 year is rs 25 . what is the sum ?
Options: a ) rs . 15000 , b ) rs . 12000 , c ) rs . 10000 , d ) rs . 20000 , e ) none of these
Rationale: in simple words s . i = ptr / 100 = ( p * 1 * 10 ) / 100 p / 10 here amount a = p + s . i = p + p / 10 = 11 p / 10 . . . . now for compound interest when calculated half yearly amount = p [ 1 + r / 200 ] ^ 2 t = p [ 1 + 10 / 200 ] ^ 2 * 1 = 441 p / 400 here its given in the question dat diff btwn simple interest and compound interest half yearly . . . is 25 = > 441 p / 400 - 11 p / 10 = 25 = > p / 400 = 25 = > p = 400 * 25 = 10000 so sum = 10000 answer : c
Answer: c
Python program:
a = 10 / 2
b = a / 100
c = b + 1 
d = c ** 2
e = d - 1
f = 10 / 100
g = e - f
h = 25 / g
Problem: how many bricks , each measuring 25 cm x 11 cm x 6 cm , will be needed to build a wall of 2 m x 3 m x 2 cm ?
Options: a ) 5600 , b ) 6000 , c ) 73 , d ) 7200 , e ) 8600
Rationale: "number of bricks = volume of the wall / volume of 1 brick = ( 200 x 300 x 2 ) / ( 25 x 11 x 6 ) = 72.7 = 73 . answer : option c"
Answer: c
Python program:
a = 2 * 100
b = 3 * 100
c = a * b
d = c * 2
e = 25 * 11
f = e * 6
g = d / f
Problem: 60 percent of movie theatres in town x have 2 screens or less . 20 % of those theatres sell an average of more than $ 300 worth of popcorn per showing . 56 percent of all the movie theatres in town x sell $ 300 or less of popcorn per showing . what percent of all the stores on the street have 4 or more screens and sell an average of more than $ 300 worth of popcorn per day ?
Options: a ) 12 , b ) 18 , c ) 32 , d ) 40 , e ) 44
Rationale: "lets take numbers here . assume that the total number of movie theaters in the town = 100 then number of movie theaters with 3 screens or less = 60 = > number of movie theaters with 4 screens or more = 40 movie theaters with 3 screens or less selling popcorn at more than $ 300 = 20 % of 60 = 12 number of movie theaters selling popcorn at $ 300 or less = 56 = > number of movie theaters selling popcorn at more than $ 300 = 100 - 56 = 44 of these 44 theaters , 12 are those with 3 screens or less therefore 40 ( 44 - 12 ) must be those with four screens or more d is the answer"
Answer: d
Python program:
a = 20 / 300
b = 20 / 300
c = b * 300
d = a * c
Problem: if w is the set of all the integers between 69 and 99 , inclusive , that are either multiples of 3 or multiples of 2 or multiples of both , then w contains how many numbers ?
Options: a ) 20 , b ) 32 , c ) 33 , d ) 34 , e ) 35
Rationale: "multiples of 2 from 69 to 99 = multiples of 2 from 1 to 99 - multiples of 2 from 1 to 68 = [ 99 / 2 ] - [ 68 / 2 ] = 49 - 34 = 15 multiples of 3 from 69 to 99 = multiples of 3 from 1 to 99 - multiples of 3 from 1 to 68 = [ 99 / 3 ] - [ 68 / 3 ] = 33 - 23 = 10 multiples of 2 and 3 bothi . e . 6 from 69 to 99 = multiples of 6 from 1 to 99 - multiples of 6 from 1 to 68 = [ 99 / 6 ] - [ 68 / 6 ] = 16 - 11 = 5 these 8 numbers have been counted twice in both the above calculation while calculating multiples of 2 and 3 i . e . total numbers in w = 15 + 10 - 5 = 20 answer option a"
Answer: a
Python program:
a = 99 - 69
b = a / 3
c = math.floor(b)
d = 99 - 69
e = d / 2
f = c + e 
g = 99 - 69
h = 2 * 3
i = g / h
j = math.floor(i)
k = f - j
Problem: 75 boys can complete a work in 26 days . how many men need to complete twice the work in 20 days
Options: a ) 160 , b ) 195 , c ) 180 , d ) 190 , e ) 200
Rationale: "one man can complete the work in 26 * 75 = 1950 days = one time work to complete the work twice it will be completed in let m be the no . of worker assign for this therefore the eqn becomes m * 20 = 2 * 1950 m = 195 workers answer : b"
Answer: b
Python program:
a = 26 * 2
b = 75 * a
c = b / 20
Problem: a trader bought a car at 20 % discount on its original price . he sold it at a 55 % increase on the price he bought it . what percent of profit did he make on the original price ?
Options: a ) 17 % , b ) 24 % , c ) 12 % , d ) 82 % , e ) 16 %
Rationale: "original price = 100 cp = 80 s = 80 * ( 155 / 100 ) = 124 100 - 124 = 24 % answer : b"
Answer: b
Python program:
a = 100 - 20
b = 100 + 55 
c = a * b
d = c / 100
e = d / 100
f = e - 1
g = f * 100
Problem: how many shots of 1 cm radius can be prepared from a sphere of 2 cm radius ?
Options: a ) 3 , b ) 8 , c ) 27 , d ) 88 , e ) 99
Rationale: "4 / 3 ï € * 2 * 2 * 2 = 4 / 3 ï € * 1 * 1 * 1 * x x = 8 answer : b"
Answer: b
Python program:
a = 2 ** 2
b = a * 2
Problem: machine a and machine b are each used to manufacture 660 sprockets . it takes machine a 10 hours longer to produce 660 sprockets than machine b . machine b produces 10 percent more sprockets per hour than machine a . how many sprockets per hour does machine a produces ?
Options: a ) 6 , b ) 6.6 , c ) 60 , d ) 100 , e ) 110
Rationale: b takes x hours a takes x + 10 hours rate of a = 660 / x + 10 rate of b = 660 / x thus , 660 / x = ( 660 / x + 10 ) * 1.10 x = 100 so b = 100 a = 110 sprockets per hour 660 / 110 = 6 answer : a
Answer: a
Python program:
a = 10 * 660
b = 100 + 10 
c = b / 100
d = a * c
e = 100 + 10 
f = e / 100
g = 660 * f
h = g - 660
i = d / h
j = 660 / i
Problem: what is 50 % of 40 % of 1200 grams ?
Options: a ) 450 gms , b ) 100 gms , c ) 300 gms , d ) 240 gms , e ) none of these
Rationale: "50 / 100 × 40 / 100 × 1200 = 240 answer : d"
Answer: d
Python program:
a = 3 + 2 
b = a * 2
c = 3 * 4
d = c * 100
e = b * d
f = 3 + 4 
g = 3 + 2 
h = f * g
i = 3 + 2 
j = i * 2
k = h * j
l = e + k 
m = 3 + 3 
n = l + m 
o = 50 * n
p = o / 100
Problem: the sum of the squares of the first 15 positive integers ( 1 ^ 2 + 2 ^ 2 + 3 ^ 2 + . . . + 15 ^ 2 ) is equal to 1250 . what is the sum of the squares of the second 15 positive integers ( 16 ^ 2 + 17 ^ 2 + 18 ^ 2 + . . . + 30 ^ 2 ) ?
Options: a ) 2480 , b ) 3490 , c ) 6785 , d ) 8225 , e ) 9255
Rationale: "you ' ll never need a formula for the sums of squares on the actual gmat . you do n ' t need to use that formula here , though it ' s not all that straightforward to solve without one . two different approaches : 16 ^ 2 + 17 ^ 2 + 18 ^ 2 + . . . + 30 ^ 2 = ( 15 + 1 ) ^ 2 + ( 15 + 2 ) ^ 2 + ( 15 + 3 ) ^ 2 + . . . + ( 15 + 15 ) ^ 2 now we can expand each square ; they are all in the ( x + y ) ^ 2 = x ^ 2 + 2 xy + y ^ 2 pattern . = ( 15 ^ 2 + 2 * 15 + 1 ^ 2 ) + ( 15 ^ 2 + 4 * 15 + 2 ^ 2 ) + ( 15 ^ 2 + 6 * 15 + 3 ^ 2 ) + . . . + ( 15 ^ 2 + 30 * 15 + 15 ^ 2 ) now we have fifteen 15 ^ 2 terms , so adding these gives 15 * 15 ^ 2 = 15 ^ 3 = 3375 . we also have the entire sum 1 ^ 2 + 2 ^ 2 + . . . + 15 ^ 2 , which we know is equal to 1240 . finally adding the middle terms , we have : 2 * 15 + 4 * 15 + 6 * 15 + . . . + 30 * 15 = 15 ( 2 + 4 + 6 + . . . . + 30 ) = 15 * 2 * ( 1 + 2 + 3 + . . . + 15 ) = 15 * 2 * 8 * 15 = 3600 so the sum must be 3375 + 1240 + 3600 = 8215 alternatively , we can use a different factoring pattern . we want to find the value of 30 ^ 2 + 29 ^ 2 + . . . + 17 ^ 2 + 16 ^ 2 . well if we subtract 15 ^ 2 + 14 ^ 2 + . . . . + 2 ^ 2 + 1 ^ 2 from this , the answer will be 1240 less than what we want to find . so if we can find the value of 30 ^ 2 + 29 ^ 2 + . . . + 17 ^ 2 + 16 ^ 2 - ( 15 ^ 2 + 14 ^ 2 + . . . . + 2 ^ 2 + 1 ^ 2 ) then we can add 1240 to get the answer . now grouping the terms above to get differences of squares , we have = ( 30 ^ 2 - 15 ^ 2 ) + ( 29 ^ 2 - 14 ^ 2 ) + . . . + ( 16 ^ 2 - 1 ^ 2 ) and factoring each of these using x ^ 2 - y ^ 2 = ( x + y ) ( x - y ) , we have = 45 * 15 + 43 * 15 + 41 * 15 + . . . + 17 * 15 = 15 ( 45 + 43 + 41 + . . . + 17 ) in brackets we have an equally spaced sum with fifteen terms , which we can evaluate using the familiar formula . so the above equals 15 * 15 * 62 / 2 = 6975 and adding back the 1250 , we get the answer of 8225 . ( ans d )"
Answer: d
Python program:
a = 30 + 1 
b = 30 * a
c = 2 * 30
d = c + 1 
e = b * d
f = 3 + 3 
g = e / f
h = g - 1250
Problem: there were two candidates in an election . winner candidate received 52 % of votes and won the election by 288 votes . find the number of votes casted to the winning candidate ?
Options: a ) 776 , b ) 3744 , c ) 299 , d ) 257 , e ) 125
Rationale: "w = 52 % l = 48 % 52 % - 48 % = 4 % 4 % - - - - - - - - 288 52 % - - - - - - - - ? = > 3744 answer : b"
Answer: b
Python program:
a = 100 - 52
b = 52 - a
c = b / 100
d = 288 / c
e = d * 52
f = e / 100