euclaise/mathqa_programs huggingface dataset:
Problem: if n is a positive integer and n ^ 2 is divisible by 62 , then the largest positive integer that must divide n is
Options: a ) 6 , b ) 12 , c ) 24 , d ) 36 , e ) 48
Rationale: "the question asks aboutthe largest positive integer that must divide n , not could divide n . since the least value of n for which n ^ 2 is a multiple of 72 is 12 then the largest positive integer that must divide n is 12 . complete solution of this question is given above . please ask if anything remains unclear . i spent a few hours on this one alone and i ' m still not clear . i chose 12 at first , but then changed to 48 . i ' m not a native speaker , so here is how i interpreted this question : the largest positive integer that must divide n = the largest positive factor of n . since n is a variable ( i . e . n is moving ) , so is its largest factor . please correct if i ' m wrong here . i know that if n = 12 , n ^ 2 = 144 = 2 * 72 ( satisfy the condition ) . when n = 12 , the largest factor of n is n itself , which is 12 . check : 12 is the largest positive number that must divide 12 - - > true however if n = 48 , n ^ 2 = 48 * 48 = 32 * 72 ( satisfy the condition too ) . when n = 48 , the largest factor of n is n itself , which is 48 . check : 48 is the largest positive number that must divide 48 - - > true so , i also notice that the keyword ismust , notcould . the question is , why is 48 notmust divide 48 , but instead onlycould divide 48 ? i ' m not clear right here . why is 12 must divide 12 ? what ' s the difference between them ? only restriction we have on positive integer n is that n ^ 2 is divisible by 72 . the least value of n for which n ^ 2 is divisible by 72 is 12 , thus nmustbe divisible by 12 ( n is in any case divisible by 12 ) . for all other values of n , for which n ^ 2 is divisible by 72 , n will still be divisible by 12 . this means that n is always divisible by 12 if n ^ 2 is divisible by 72 . now , ask yourself : if n = 12 , is n divisible by 48 ? no . so , n is not always divisible by 48 . b"
Answer: b
Python program:
a = 62 / 2
b = math.sqrt(a)
c = b * 2
Problem: if a and b get profits of rs . 60,000 and rs . 6,000 respectively at the end of year then ratio of their investments are
Options: a ) 4 : 1 , b ) 10 : 1 , c ) 3 : 2 , d ) 2 : 3 , e ) 2 : 5
Rationale: ratio = 60000 / 6000 = 10 : 1 answer : b
Answer: b
Python program:
a = 3 * 2
b = a * 100
c = b * 100
d = 3 * 2
e = d * 100
f = e * 100
g = 2 + 3 
h = g * 2
i = f / h
j = c / i
Problem: what is the 99 th digit after the decimal point in the decimal expansion of 2 / 9 + 3 / 11 ?
Options: a ) 1 , b ) 2 , c ) 4 , d ) 7 , e ) 9
Rationale: 2 / 9 = 0.22222 . . . . = = > 99 th digit is 2 3 / 11 = 0.27272727 . . . . = = > every odd digit is 2 . so , 99 th digit will be 2 . 2 + 2 = 4 answer : c
Answer: c
Python program:
a = 99 / 9
b = a * 2
c = 99 / 11
d = c * 3
e = b + d 
f = e / 99
g = f * 10
h = math.floor(g)
Problem: the ratio of ages of aman , bren , and charlie are in the ratio 5 : 8 : 7 respectively . if 8 years ago , the sum of their ages was 76 , what will be the age of bren 10 years from now ?
Options: a ) 17 , b ) 25 , c ) 27 , d ) 35 , e ) 50
Rationale: let the present ages of aman , bren , and charlie be 5 x , 8 x and 7 x respectively . 5 x - 8 + 8 x - 8 + 7 x - 8 = 76 x = 5 present age of bren = 8 * 5 = 40 bren ' s age 10 years hence = 40 + 10 = 50 answer = e
Answer: e
Python program:
a = 8 * 3
b = a + 76 
c = 5 + 8 
d = c + 7 
e = b / d
f = e * 8
g = f + 10
Problem: a trainer is standing in one corner of a square ground of side 25 m . his voice can be heard upto 140 m . find the area of the ground in which his voice can be heard ?
Options: a ) 12300
b ) 14500
c ) 15400
d ) 16700
e ) 18200
Rationale: area covered by goat = pi * r ^ 2 / 4 ( here we divide by 4 because the trainer is standing in a corner of the ground and only in 1 / 4 part , the voice can be heard ) where r = 14 m = length reaching the voice so area = ( 22 / 7 ) * 140 * 140 / 4 = 15400 sq m answer : c
Answer: c
Python program:
a = circle_area / (
Problem: machine m , n , o working simultaneously machine m can produce x units in 3 / 4 of the time it takes machine n to produce the same amount of units . machine n can produce x units in 2 / 7 the time it takes machine o to produce that amount of units . if all 3 machines are working simultaneously , what fraction of the total output is produced by machine n ?
Options: a ) 1 / 2 , b ) 7 / 13 , c ) 4 / 13 , d ) 8 / 29 , e ) 6 / 33
Rationale: now ultimately the speed of every machine is given with respect to mach o . so lets assume the speed of o , say 12 hrs to make x units ( assuming 6 because we can see we will need to divide by 3 and 4 mach o makes x units in 12 hrs so , mach n = 2 / 7 of o = 2 / 7 * 12 = 24 / 7 hrs to make x units and mach m = 3 / 4 of n = 3 / 4 * 24 / 7 = 1 / 6 hrs to make x units no they are running simultaneously . lets see how much each mach makes in 1 hr mach o = x / 12 units mach n = 7 / 24 units mach m = x / 6 units in 1 hr , together they make - x / 12 + 7 / 24 + x / 6 = 13 / 24 so what ratio of this has mach n made ? ( 7 / 24 ) / ( 13 / 24 ) = 7 / 13 ans : b = 7 / 13
Answer: b
Python program:
a = 7 * 2
b = a - 1
c = 7 / b
Problem: a and b can do a work in 2 days , b and c in 6 days and c and a in 8 days . in how many days will the work be completed , if all three of them work together ?
Options: a ) 12 / 28 , b ) 48 / 19 , c ) 16 / 48 , d ) 18 / 48 , e ) 12 / 64
Rationale: "one day work of a and b = 1 / 2 one day work of b and c = 1 / 6 one day work of c and a = 1 / 8 2 ( a + b + c ) = 1 / 2 + 1 / 6 + 1 / 8 2 ( a + b + c ) = 19 / 24 ( a + b + c ) = 19 / 48 number of days required = 48 / 19 days . answer : b"
Answer: b
Python program:
a = 1/(2)
b = 1/(6)
c = a + b 
d = 1/(8)
e = c + d 
f = e / 2
g = 1 / f
Problem: a man is 18 years older than his son . in two years , his age will be twice the age of his son . the present age of this son is
Options: a ) 21 years , b ) 22 years , c ) 23 years , d ) 12 years , e ) 16 years
Rationale: "explanation : let ' s son age is x , then father age is x + 18 . = > 2 ( x + 2 ) = ( x + 18 + 2 ) = > 2 x + 4 = x + 20 = > x = 16 years option e"
Answer: e
Python program:
a = 2 * 2
b = a - 2
c = 18 - b
d = 2 - 1
e = c / d
Problem: ( 0.0088 ) ( 4.5 ) / ( 0.05 ) ( 0.1 ) ( 0.008 ) =
Options: a ) 950 , b ) 940 , c ) 980 , d ) 960 , e ) 990
Rationale: "( 0.0088 ) ( 4.5 ) / ( 0.05 ) ( 0.1 ) ( 0.008 ) = 0.0088 * 450 / 5 * ( 0.1 ) ( 0.008 ) = 0.088 * 90 / 1 * 0.008 = 88 * 90 / 8 = 11 * 90 = 990 answer : e"
Answer: e
Python program:
a = 0 * 88
b = 0 * 5
c = b * 0
d = a / c
Problem: a watch was sold at a loss of 10 % . if it was sold for rs . 210 more , there would have been a gain of 4 % . what is the cost price ?
Options: a ) s . 1000 , b ) s . 1009 , c ) s . 1007 , d ) s . 1006 , e ) s . 1500
Rationale: explanation : 90 % 104 % - - - - - - - - 14 % - - - - 210 100 % - - - - ? = > rs . 1500 answer : e
Answer: e
Python program:
a = 210 * 100
b = 100 + 4 
c = 100 - 10
d = b - c
e = a / d
Problem: ( 4300631 ) - ? = 2535618
Options: a ) 1865113 , b ) 1775123 , c ) 1765013 , d ) 1675123 , e ) none of them
Rationale: "let 4300631 - x = 2535618 then x = 4300631 - 2535618 = 1765013 answer is c"
Answer: c
Python program:
a = 4 + 1 
b = a ** 4
c = 4300631 * b
Problem: two trains 250 m and 500 m long run at the speed of 60 km / hr and 40 km / hr respectively in opposite directions on parallel tracks . the time which they take to cross each other is ?
Options: a ) 10.6 , b ) 10.9 , c ) 10.4 , d ) 27 , e ) 10.1
Rationale: "relative speed = 60 + 40 = 100 km / hr . = 100 * 5 / 18 = 250 / 9 m / sec . distance covered in crossing each other = 250 + 500 = 750 m . required time = 750 * 9 / 250 = 27 sec answer : d"
Answer: d
Python program:
a = 250 + 500 
b = 60 + 40 
c = b * const_0_2778
d = a / c
Problem: in a rectangular coordinate system , what is the area of a rectangle whose vertices have the coordinates ( - 3 , 1 ) , ( 1 , 1 ) , ( 1 , - 2 ) and ( - 3 , - 2 ) ?
Options: a ) 12 , b ) 24 , c ) 20 , d ) 30 , e ) 36
Rationale: "length of side 1 = 3 + 1 = 4 length of side 2 = 2 + 1 = 3 area of rectangle = 4 * 3 = 12 a is the answer"
Answer: a
Python program:
a = 3 + 1 
b = 1 + 2 
c = a * b
Problem: in 1970 there were 8,902 women stockbrokers in the united states . by 1978 the number had increased to 22,947 . approximately what was the percent increase ?
Options: a ) 45 % , b ) 125 % , c ) 145 % , d ) 158 % , e ) 225 %
Rationale: "the percent increase is ( 22947 - 8902 ) / 8902 = 14045 / 8902 = 1.58 so the approximate answer is d"
Answer: d
Python program:
a = 22 - 947
b = a / 8
Problem: a can finish a piece of work in 4 days . b can do it in 8 days . they work together for two days and then a goes away . in how many days will b finish the work ?
Options: a ) 4 , b ) 5 , c ) 6 , d ) 2 , e ) 8
Rationale: "2 / 4 + ( 2 + x ) / 8 = 1 = > x = 2 days answer : d"
Answer: d
Python program:
a = 1 / 4
b = a * 2
c = 1 / 8
d = c * 2
e = b + d 
f = 1 - e
g = 1 / 8
h = f / g
Problem: two trains of length 100 m and 200 m are 100 m apart . they start moving towards each other on parallel tracks , at speeds 90 kmph and 72 kmph . in how much time will the trains cross each other ?
Options: a ) 80 / 6 , b ) 80 / 7 , c ) 80 / 9 , d ) 80 / 2 , e ) 80 / 1
Rationale: "relative speed = ( 90 + 72 ) * 5 / 18 = 45 mps . the time required = d / s = ( 100 + 100 + 200 ) / 45 = 400 / 45 = 80 / 9 sec . answer : c"
Answer: c
Python program:
a = 90 + 72 
b = a * const_0_2778
c = 100 / b
Problem: a batsman in his 12 th innings makes a score of 65 and thereby increases his average by 3 runs . what is his average after the 12 th innings if he had never been ‘ not out ’ ?
Options: a ) 32 , b ) 43 , c ) 44 , d ) 45 , e ) 46
Rationale: "let ‘ x ’ be the average score after 12 th innings ⇒ 12 x = 11 × ( x – 3 ) + 65 ∴ x = 32 answer a"
Answer: a
Python program:
a = 12 * 3
b = 65 - a
c = b + 3
Problem: of 30 applicants for a job , 13 had at least 4 years ' experience , 18 had degrees , and 3 had less than 4 years ' experience and did not have a degree . how many of the applicants had at least 4 years ' experience and a degree ?
Options: a ) 14 , b ) 13 , c ) 9 , d ) 6 , e ) 5
Rationale: "d . 6 30 - 3 = 27 27 - 13 - 18 = - 6 then 6 are in the intersection between 4 years experience and degree . answer d"
Answer: d
Python program:
a = 13 + 18 
b = 30 - 3
c = a - b
d = 18 - 13
e = c + d
Problem: a train is 360 meter long is running at a speed of 72 km / hour . in what time will it pass a bridge of 140 meter length ?
Options: a ) 65 seconds , b ) 46 seconds , c ) 25 seconds , d ) 97 seconds , e ) 26 seconds
Rationale: "speed = 72 km / hr = 72 * ( 5 / 18 ) m / sec = 20 m / sec total distance = 360 + 140 = 500 meter time = distance / speed = 500 * ( 1 / 20 ) = 25 seconds answer : c"
Answer: c
Python program:
a = 360 + 140 
b = 72 * 1000
c = b / 3600
d = a / c
Problem: a certain university will select 2 of 3 candidates eligible to fill a position in the mathematics department and 4 of 5 candidates eligible to fill 2 identical positions in the computer science department . if none of the candidates is eligible for a position in both departments , how many different sets of 3 candidates are there to fill the 3 positions ?
Options: a ) 42 , b ) 10 , c ) 40 , d ) 65 , e ) 15
Rationale: "3 c 2 * 5 c 4 = 3 * 5 = 15 the answer is ( e )"
Answer: e
Python program:
a = 5 * 3
b = a * 3
Problem: if 9 spiders make 9 webs in 9 days , then how many days are needed for 1 spider to make 1 web ?
Options: a ) 5 days , b ) 7 days , c ) 8 days , d ) 9 days , e ) 6 days
Rationale: "let , 1 spider make 1 web in a days . more spiders , less days ( indirect proportion ) more webs , more days ( direct proportion ) hence we can write as ( spiders ) 9 : 1 } : : a : 9 ( webs ) 1 : 9 ⇒ 7 × 1 × 7 = 1 × 7 × a ⇒ a = 7 answer : b"
Answer: b
Python program:
a = 1 * 9
Problem: a bullet train 220 m long is running with a speed of 59 kmph . in what time will it pass a man who is running at 7 kmph in the direction opposite to that in which the bullet train is going ?
Options: a ) 23 sec , b ) 15 sec , c ) 12 sec , d ) 11 sec , e ) 16 sec
Rationale: "c 12 sec speed of the bullet train relative to man = ( 59 + 7 ) kmph = 66 * 5 / 18 m / sec = 55 / 3 m / sec . time taken by the bullet train to cross the man = time taken by it to cover 220 m at ( 55 / 3 ) m / sec = ( 220 * 3 / 55 ) sec = 12 sec"
Answer: c
Python program:
a = 59 + 7 
b = a * 1000
c = b / 3600
d = 220 / c
Problem: at the wholesale store you can buy an 8 - pack of hot dogs for $ 1.55 , a 20 - pack for $ 3.05 , and a 250 - pack for $ 22.95 . what is the greatest number of hot dogs you can buy at this store with $ 208 ?
Options: a ) 1,108 , b ) 2,100 , c ) 2,108 , d ) 2,160 , e ) 2,256
Rationale: "i think it should be d . i can buy 8 250 - pack for rs 22.95 * 8 = $ 183.60 now , i can buy 8 20 - pack for 3.05 * 8 = $ 24.40 now , i am left with only $ 1.15 . i can not but anything with this . hence total hotdogs = 250 * 8 + 20 * 5 = 2160"
Answer: d
Python program:
a = 208 / 22
b = a * 250
Problem: two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 26 seconds . the ratio of their speeds is ?
Options: a ) 3 / 1 , b ) 9 / 1 , c ) 3 / 3 , d ) 3 / 5 , e ) 5 / 2
Rationale: "let the speeds of the two trains be x m / sec and y m / sec respectively . then , length of the first train = 27 x meters , and length of the second train = 17 y meters . ( 27 x + 17 y ) / ( x + y ) = 26 = = > 27 x + 17 y = 26 x + 26 y = = > 1 x = 9 y = = > x / y = 9 / 1 . answer : b"
Answer: b
Python program:
a = 27 - 26
b = 26 - 17
c = a / b
Problem: the spherical ball of lead 3 cm in diameter is melted and recast into 3 spherical balls . the diameters of two of these are 1 ½ cm and 2 cm respectively . the diameter of third ball is ?
Options: a ) 2.66 cm , b ) 2.5 cm , c ) 3 cm , d ) 3.5 cm , e ) 4 cm
Rationale: 4 / 3 π * 3 * 3 * 3 = 4 / 3 π [ ( 3 / 2 ) 3 + 23 + r 3 ] r = 1.25 d = 2.5 answer : b
Answer: b
Python program:
a = 3 / 2
b = a ** 3
c = 1 / 2
d = 1 + c 
e = d / 2
f = e ** 3
g = b - f
h = g - 1
i = 1/(3)
j = h ** i
k = j * 2
Problem: find the number which when multiplied by 15 is increased by 196
Options: a ) 10 , b ) 12 , c ) 14 , d ) 16 , e ) 17
Rationale: "explanation : let the number be x . then , 15 x = x + 196 = › 14 x = 196 = › x = 14 . answer : option c"
Answer: c
Python program:
a = 15 - 1
b = 196 / a
Problem: a certain university will select 1 of 5 candidates eligible to fill a position in the mathematics department and 4 of 7 candidates eligible to fill 2 identical positions in the computer science department . if none of the candidates is eligible for a position in both departments , how many different sets of 3 candidates are there to fill the 3 positions ?
Options: a ) 42 , b ) 70 , c ) 175 , d ) 165 , e ) 315
Rationale: "5 c 1 * 7 c 4 = 5 * 35 = 175 the answer is ( c )"
Answer: c
Python program:
a = 7 * 3
b = a * 5
Problem: if the speed of a man is 51 km per hour , then what is the distance traveled by him in 30 seconds ?
Options: a ) 275 m , b ) 360 m , c ) 375 m , d ) 425 m , e ) 440 m
Rationale: "the distance traveled in 30 sec = 51 * ( 5 / 18 ) * 30 = 425 m answer : d"
Answer: d
Python program:
a = 51 * const_0_2778
b = a * 30
Problem: the ratio between the sale price and the cost price of an article is 6 : 4 . what is the ratio between the profit and the cost price of that article ?
Options: a ) 23 , b ) 1 : 2 , c ) 2 : 5 , d ) 3 : 5 , e ) 25
Rationale: "c . p . = rs . 4 x and s . p . = rs . 6 x . then , gain = rs . 2 x required ratio = 2 x : 4 x = 1 : 2 b"
Answer: b
Python program:
a = 6 - 4
b = a / 4
Problem: a person want to give his money of $ 5400 to his 3 children a , b , c in the ratio 2 : 3 : 4 . what is the b ' s share ?
Options: a ) $ 2000 , b ) $ 1800 , c ) $ 2500 , d ) $ 1500 , e ) $ 1600
Rationale: "b ' s share = 5400 * 3 / 9 = $ 1800 answer is b"
Answer: b
Python program:
a = 2 + 3 
b = a + 4 
c = 3 / b
d = c * 5400
Problem: what is the product of first three whole no .
Options: a ) 0 , b ) 1 , c ) 6 , d ) 2 , e ) 3
Rationale: "first three whole no . are 0 , 1,2 . remember whole no . start from 0 and natural no . from 1 so product will 0 only . answer a"
Answer: a
Python program:
a = 3 + 3 
b = math.comb(a, 3)
c = 2 / b
Problem: what is 2 2 / 3 - 1 1 / 4 divided by 1 / 4 - 1 / 6 ?
Options: a ) 17 / 36 , b ) 36 / 17 , c ) 17 / 6 , d ) 17 / 1 , e ) 51 / 4
Rationale: "2 2 / 3 - 1 1 / 4 = 8 / 3 - 5 / 4 = ( 32 - 15 ) / 12 = 17 / 12 1 / 4 - 1 / 6 = ( 6 - 4 ) / 24 = 2 / 24 = 1 / 12 so 17 / 12 / 1 / 12 = 17 / 12 * 12 = 17 / 1 answer - d"
Answer: d
Python program:
a = 10 * 2
b = a + 2 
c = b / 3
d = 10 + 1 
e = d / 4
f = c - e
Problem: income and expenditure of a person are in the ratio 10 : 7 . if the income of the person is rs . 10000 , then find his savings ?
Options: a ) 3005 , b ) 3098 , c ) 3300 , d ) 3010 , e ) 3000
Rationale: "let the income and the expenditure of the person be rs . 10 x and rs . 7 x respectively . income , 10 x = 10000 = > x = 1000 savings = income - expenditure = 10 x - 7 x = 3 x = 3 ( 1000 ) so , savings = rs . 3000 . answer : e"
Answer: e
Python program:
a = 7 / 10
b = a * 10000
c = 10000 - b
Problem: if the price of an article went up by 15 % , then by what percent should it be brought down to bring it back to its original price ?
Options: a ) 17.39 % , b ) 2 / 8 % , c ) 2 / 1 % , d ) 1 / 3 % , e ) 2 / 7 %
Rationale: "let the price of the article be rs . 100 . 15 % of 100 = 15 new price = 100 + 15 = rs . 115 required percentage = ( 115 - 100 ) / 115 * 100 = 20 / 115 * 100 = 17.39 % . answer : a"
Answer: a
Python program:
a = 100 + 15 
b = 15 / a
c = b * 100
Problem: solution x is 10 percent alcohol by volume , and solution y is 30 percent alcohol by volume . how many milliliters of solution y must be added to 100 milliliters of solution x to create a solution that is 25 percent alcohol by volume ?
Options: a ) 250 / 3 , b ) 500 / 3 , c ) 300 , d ) 480 , e ) 600
Rationale: "we know that x is 10 % , y is 30 % and w . avg = 25 % . what does this mean with respect to w . avg technique ? w . avg is 1 portion away from y and 3 portion away from x so for every 1 portion of x we will have to add 3 portions of y . if x = 100 then y = 300 answer : c"
Answer: c
Python program:
a = 25 - 10
b = 30 - 25
c = a / b
d = c * 100
Problem: one millisecond is 0.001 of a second . the costs for a single run of a computer program are $ 1.07 for operating - system overhead , $ 0.023 per millisecond of computer time , and $ 5.35 for the mounting of a data tape . what is the total of these 3 costs for 1 run of a program that requires 1.5 seconds of computer time ?
Options: a ) $ 7.15 , b ) $ 8.87 , c ) $ 28.96 , d ) $ 35.57 , e ) $ 40.92
Rationale: operating system overhead for 1 run = 1.07 $ mounting of data tape = 5.35 $ cost per 1 millisecond of computer time = . 023 $ total cost of 1 run of a program that requires 1.5 seconds of computer time = 1.07 + ( . 023 * 1500 ) + 5.35 = 1.07 + 34.5 + 5.35 = 40.92 $ answer e
Answer: e
Python program:
a = 1 + 7 
b = 1 / 5
c = 0 * 23
d = a + c
Problem: if 20 typists can type 46 letters in 20 minutes , then how many letters will 30 typists working at the same rate complete in 1 hour ?
Options: a ) 63 , b ) 72 , c ) 144 , d ) 216 , e ) 207
Rationale: "20 typists can type 46 letters , so 30 typists can type = 46 * 30 / 20 46 * 30 / 20 letters can be typed in 20 mins . in 60 mins typist can type = 46 * 30 * 60 / 20 * 20 = 207 e is the answer"
Answer: e
Python program:
a = 46 * 3
b = a / 20
c = b * 30
Problem: the figure above shows the dimensions of a semicircular cross section of a one - way tunnel . the single traffic lane is 12 feet wide and is equidistant from the sides of the tunnel . if vehicles must clear the top of the tunnel by at least ½ foot when they are inside the traffic lane , what should be the limit h on the height of vehicles that are allowed to use the tunnel ?
Options: a ) 5 ½ ft , b ) 7 ½ ft , c ) 8 ½ ft , d ) 9 ½ ft , e ) 10 ft
Rationale: "let ' s label the midpoint of the circle o . since the base of the semi - circle is 20 , we know that the diameter is 20 and , accordingly , the radius is 10 . we also know that the traffic lane is 12 feet long and there ' s an equal amount of space on either side , so the traffic lane extends 6 feet on either side of o . let ' s call the leftmost point on the base of the traffic lane a . so , the distance oa is 6 . now draw a line straight up from a to the top of the tunnel . let ' s label the point at which the line intersects the circle b . the answer to the question will , therefore , be the height ab - . 5 feet ( we need to leave . 5 feet of clearance ) . here ' s the key to solving the question : if we draw a line from o to b , that line is a radius of the circle and , therefore , has length 10 . we now have right triangle oab ( the right angle is at point a ) , with leg oa = 6 and hypotenuse ob = 10 . we can now solve for leg ab = 8 ( either by applying the pythagorean theorum or by applying the 3 / 4 / 5 special right triangle ratio ) . finally : ab = 8 , so the correct answer h is 8 - . 5 = 7.5 . . . choose ( b ) ! from a strategic guessing point of view , as soon as we realize that the height of the tunnel is 10 in the middle , we should quickly eliminate ( d ) and ( e ) as too big ; worse case you have a 1 / 3 shot at picking up the points . b"
Answer: b
Python program:
a = 12 / 2
b = 1 / 2
c = a - b
Problem: if 4 x + y = 34 , 2 x - y = 20 , for integers of x and y , y ^ 2 = ?
Options: a ) 2 , b ) 4 , c ) 0 , d ) 10 , e ) 3
Rationale: 4 x + y = 34 2 x - y = 20 6 x = 54 x = 9 y = - 2 y ^ 2 = 4 answer is b
Answer: b
Python program:
a = 34 + 20 
b = 4 + 2 
c = a / b
d = 2 * c
e = d - 20
f = e ** 2
Problem: what is the sum of mean and median of 17 , 45 , 59 , 113 , 76 , 95 , 59 , 35 , 1236 ?
Options: a ) 50.2 , b ) 45.8 , c ) 54.7 , d ) 61.8 , e ) 10.5
Rationale: mean = ( 17 + 45 + 59 + 113 + 76 + 95 + 59 + 35 + 12 + 36 ) / 10 = 547 / 10 = 54.7 answer is c
Answer: c
Python program:
a = 113 - 59
Problem: if a / b = 5 / 2 , b / c = 1 / 2 , c / d = 1 , d / e = 3 / 2 and e / f = 4 / 3 , then what is the value of abc / def ?
Options: a ) 15 / 8 , b ) 14 / 3 , c ) 41 / 7 , d ) 3 / 18 , e ) 18 / 4
Rationale: say a = 2 . then : a / b = 5 / 2 - - > b = 2 / 5 ; b / c = 1 / 2 - - > c = 4 / 5 ; c / d = 1 - - > d = 4 / 5 ; d / e = 3 / 2 - - > e = 8 / 15 ; e / f = 4 / 3 - - > f = 2 / 5 . abc / def = ( 1 * 2 / 5 * 4 / 5 ) / ( 4 / 5 * 8 / 15 * 2 / 5 ) = 15 / 8 . answer : a .
Answer: a
Python program:
a = 5 * 1
b = a * 1
c = 2 * 2
d = c * 1
e = b / d
f = 1 * 1
g = f * 3
h = 2 * 1
i = h * 2
j = g / i
k = e * j
l = 1 * 3
m = l * 4
n = 1 * 2
o = n * 3
p = m / o
q = k * p
Problem: a computer manufacturer produces a certain electronic component at a cost of $ 75 per component . shipping costs for delivering the components are $ 2 per unit . further , the manufacturer has costs of $ 16,500 a month related to the electronic component regardless of how many it produces . if the manufacturer produces and sells 150 components a month , what is the lowest price it can sell them for such that the costs do n ' t exceed the revenues ?
Options: a ) $ 28 , b ) $ 187 , c ) $ 110 , d ) $ 138 , e ) $ 192
Rationale: "this question can be solved easily using a equation model . 150 * p = 16500 ( fixed cost ) + 150 * 75 + 150 * 2 . = 16500 + 11250 + 300 p = $ 187 b"
Answer: b
Python program:
a = 4 * 4
b = a * 1000
c = 2 * 100
d = b + c 
e = 75 + 2 
f = e * 150
g = d + f 
h = g / 150
Problem: if paint costs $ 3.20 per quart , and a quart covers 10 square feet , how much will it cost to paint the outside of a cube 10 feet on each edge ?
Options: a ) $ 1.60 , b ) $ 16.00 , c ) $ 96.00 , d ) $ 108.00 , e ) $ 192.00
Rationale: "total surface area = 6 a ^ 2 = 6 * 10 * 10 = 600 each quart covers 20 sqr ft thus total number of quarts = 600 / 10 = 60 cost will be 60 * 3.2 = $ 192 ans : e"
Answer: e
Python program:
a = 3 / 20
b = a * surface_cube
Problem: walking 7 / 6 of his usual rate , a boy reaches his school 3 min early . find his usual time to reach the school ?
Options: a ) 21 , b ) 28 , c ) 99 , d ) 77 , e ) 66
Rationale: "speed ratio = 1 : 7 / 6 = 6 : 7 time ratio = 7 : 6 1 - - - - - - - - 7 3 - - - - - - - - - ? = 21 m answer : a"
Answer: a
Python program:
a = 3 * 7