euclaise/mathqa_programs huggingface dataset:
Problem: the sum of the squares of three numbers is 138 , while the sum of their products taken two at a time is 131 . find the sum ?
Options: a ) 30 , b ) 15 , c ) 40 , d ) 45 , e ) 20
Rationale: "a 2 + b 2 + c 2 = 138 and ( ab + bc + ca ) = 131 a + b + c ^ 2 = 400 root ( 400 ) = 20 answer e"
Answer: e
Python program:
a = 138 - 131
b = a / 2
c = math.sqrt(b)
d = c * 100
e = 138 - 131
f = e / 2
g = 138 - f
h = math.sqrt(g)
i = d + h
Problem: divide rs . 1200 among a , b and c so that a receives 1 / 3 as much as b and c together and b receives 2 / 3 as a and c together . a ' s share is ?
Options: a ) s . 800 , b ) s . 400 , c ) s . 600 , d ) s . 500 , e ) s . 300
Rationale: "a + b + c = 1200 a = 1 / 3 ( b + c ) ; b = 2 / 3 ( a + c ) a / ( b + c ) = 1 / 3 a = 1 / 4 * 1200 = > 300 answer : e"
Answer: e
Python program:
a = 1200 / 3
Problem: 108 . triangle a ’ s base is 9 % greater than the base of triangle b , and a ’ s height is 9 % less than the height of triangle b . the area of triangle a is what percent less or more than the area of triangle b ?
Options: a ) 9 % less , b ) 1 % less , c ) equal to each other , d ) 1.3 % more , e ) 9 % more
Rationale: "wish the question specified that we are talking about corresponding height . base of a = 10 / 9 * base of b height of a = 8 / 9 * height of b area of a = ( 1 / 2 ) * base of a * height of a = 10 / 9 * 8 / 9 * area of b = 80 / 81 * area of b area of a is 1.3 % more than the area of b . answer ( d )"
Answer: d
Python program:
a = 100 * 100
b = 100 + 9 
c = 100 - 9
d = b * c
e = a - d
f = 100 / e
Problem: tourist purchased a total of 30 travelers checks in $ 50 and $ 100 denominations . the total worth of the travelers checks is $ 1800 . how many checks of $ 50 denominations can he spend so that average amount ( arithmetic mean ) of the remaining travelers checks is $ 62.5 ?
Options: a ) 6 , b ) 12 , c ) 15 , d ) 20 , e ) 24
Rationale: "you could set - up a quick table and brute force the answer . a 6 * 50 300 1800 - 300 1500 24 62.50 b 16 * 50 600 1800 - 600 1200 18 66.67 c 15 * 50 750 1800 - 750 1050 15 70.00 d 20 * 50 1000 1800 - 1000 800 10 80.00 e 24 * 50 1200 1800 - 1200 600 6 100.00 answer is a"
Answer: a
Python program:
a = 62 * 5
b = a - 1800
c = 62 - 5
d = b / c
Problem: what will be the compound interest on rs . 40000 after 3 years at the rate of 12 % per annum
Options: a ) rs 16123.20 , b ) rs 16123.30 , c ) rs 16197.12 , d ) rs 16123.50 , e ) none of these
Rationale: "explanation : ( 40000 × ( 1 + 12 / 100 ) 3 ) = > 40000 × 28 / 25 × 28 / 25 × 28 / 25 = > 56197.12 so compound interest will be 56197.12 - 40000 = rs 16197.12 option c"
Answer: c
Python program:
a = 4 * 100
b = a * 100
c = 12 / 100
d = 1 + c 
e = d ** 3
f = b * e
g = 4 * 100
h = g * 100
i = f - h
Problem: how many pieces of 75 cm can be cut from a rope 54 meters long ?
Options: a ) 30 , b ) 40 , c ) 60 , d ) 72 , e ) can not be determined
Rationale: "explanation : total pieces of 75 cm that can be cut from a rope of 54 meters long is = ( 54 meters ) / ( 75 cm ) = ( 54 meters ) / ( 0.75 meters ) = 72 answer : d"
Answer: d
Python program:
a = 54 / 75
Problem: working alone , printers x , y , and z can do a certain printing job , consisting of a large number of pages , in 16 , 10 , and 20 hours , respectively . what is the ratio of the time it takes printer x to do the job , working alone at its rate , to the time it takes printers y and z to do the job , working together at their individual rates ?
Options: a ) 7 / 4 , b ) 10 / 3 , c ) 12 / 5 , d ) 15 / 7 , e ) 18 / 11
Rationale: "the time it takes printer x is 16 hours . the combined rate of y and z is 1 / 10 + 1 / 20 = 3 / 20 the time it takes y and z is 20 / 3 the ratio of times is 16 / ( 20 / 3 ) = 3 * 16 / 20 = 12 / 5 the answer is c ."
Answer: c
Python program:
a = 1 / 10
b = 1 / 20
c = a + b 
d = 1 / c
e = 16 / d
Problem: the average weight of 10 oarsmen in a boat is increased by 1.8 kg when one of the crew , who weight 53 kg is replaced by a new man . find the weight of the new man .
Options: a ) 59 kg , b ) 62 kg , c ) 74 kg , d ) 85 kg , e ) 71 kg
Rationale: "total weight increased = 1.8 * 10 = 18 kg weight of the new man = 53 + 18 = 71 kg answer is e"
Answer: e
Python program:
a = 1 * 8
b = a + 53
Problem: how many multiples of 13 are less than 6000 , and also multiples of 16 ?
Options: a ) 30 , b ) 29 , c ) 34 , d ) 32 , e ) 33
Rationale: "lcm of 13 & 16 = 208 tried dividing 6000 by 208 got quotient 28.84 ' so b is answer"
Answer: b
Python program:
a = 16 - 6000
b = a / 13
c = b + 1
Problem: two ants , arthur and amy , have discovered a picnic and are bringing crumbs back to the anthill . amy makes twice as many trips and carries one and a half times as many crumbs per trip as arthur . if arthur carries a total of m crumbs to the anthill , how many crumbs will amy bring to the anthill , in terms of m ?
Options: a ) x / 2 , b ) x , c ) 3 m / 2 , d ) mx , e ) mx
Rationale: "lets do it by picking up numbers . let arthur carry 2 crumbs per trip , this means amy carries 3 crumbs per trip . also let arthur make 2 trips and so amy makes 4 trips . thus total crumbs carried by arthur ( m ) = 2 x 2 = 4 , total crumbs carried by amy = 3 x 4 = 12 . 12 is 3 times 4 , so e"
Answer: e
Python program:
a = 1 / 2
b = 1 + a 
c = 2 * b
Problem: a man is walking at the rate of 7 km / hr crosses a bridge in 15 minutes . the length of the bridge is
Options: a ) 1000 meters , b ) 1050 meters , c ) 1200 meters , d ) 1750 meters , e ) none of these
Rationale: "explanation : we need to get the answer in meters . so we will first of change distance from km / hour to meter / sec by multiplying it with 5 / 18 and also change 15 minutes to seconds by multiplying it with 60 . speed = 7 â ˆ — 5 / 18 = 35 / 18 m / sec time = 15 â ˆ — 60 seconds = 900 seconds distance = time â ˆ — speed / distance = 35 / 18 â ˆ — 900 = 1750 meter option d"
Answer: d
Python program:
a = 15 / const_60
b = 7 * a
c = b * 1000
Problem: if two sides of a triangle are 6 and 13 , respectively , which of the following could not be the area of this triangle ?
Options: a ) 40
b ) 6
c ) 17
d ) 29
e ) 38
Rationale: for this question it would be helpful to know the largest area that this triangle could be , given the two sides of 6 and 12 . we know that the area of a triangle will be maximized when two sides are perpendicular to each other ( consult bunuel ' s drawing above ) . thus we have a max area being equal to one half the base times the height , either ( . 5 ) 13 * 6 - or - ( . 5 ) 6 * 13 will result in a maximum area of 39 for the triangle ; therefore , the triangle could never have an area of 40 . a
Answer: a
Python program:
a = 6 * 13
b = a / 2
Problem: average expenditure of a person for the first 3 days of a week is rs . 350 and for the next 4 days is rs . 420 . average expenditure of the man for the whole week is :
Options: a ) 350 , b ) 370 , c ) 390 , d ) 430 , e ) none
Rationale: "explanation : assumed mean = rs . 350 total excess than assumed mean = 4 × ( rs . 420 - rs . 350 ) = rs . 280 therefore , increase in average expenditure = rs . 280 / 7 = rs . 40 therefore , average expenditure for 7 days = rs . 350 + rs . 40 = rs . 390 correct option : c"
Answer: c
Python program:
a = 420 - 350
b = 4 * a
c = 3 + 4 
d = b / c
e = 350 + d
Problem: compound interest of rs . 1000 at 10 % per annum for 1 1 / 2 years will be ( interest compounded half yearly ) .
Options: a ) rs . 473 , b ) rs . 374 , c ) rs . 495 , d ) rs . 157 , e ) none of the above
Rationale: "10 % interest per annum will be 5 % interest half yearly for 3 terms ( 1 1 / 2 years ) so compound interest = 3000 [ 1 + ( 5 / 100 ) ] ^ 3 - 1000 = 1000 [ ( 21 / 20 ) ^ 3 - 1 ] = 1000 ( 9261 - 8000 ) / 8000 = 1 * 1261 / 8 = 157 answer : d"
Answer: d
Python program:
a = 10 / 2
b = a / 100
c = 1 + b 
d = 1 / 2
e = 1 + d 
f = e * 2
g = c ** f
h = 1000 * g
i = h - 1000
Problem: 4,25 , 49,121 , 169,289 , 361,529 ,
Options: a ) 149 , b ) 169 , c ) 841 , d ) 920 , e ) 219
Rationale: "29 ^ 2 = 841 because follow sequence of square of the prime numbers answer : c"
Answer: c
Python program:
a = negate - (
Problem: what will be the vulgar fraction of 0.70
Options: a ) 3 / 5 , b ) 7 / 10 , c ) 3 / 2 , d ) 3 / 7 , e ) 3 / 8
Rationale: "explanation : 0.70 = 70 / 100 = 7 / 10 option b"
Answer: b
Python program:
a = 0 / 70
b = circle_area / (
Problem: susan drove an average speed of 30 miles per hour for the first 40 miles of a trip then at a average speed of 15 miles / hr for the remaining 40 miles of the trip if she made no stops during the trip what was susan ' s avg speed in miles / hr for the entire trip
Options: a ) 35 , b ) 20 , c ) 45 , d ) 50 , e ) 55
Rationale: avg . speed = total distance / total time total distance = 80 miles total time = 40 / 30 + 40 / 15 = 4 avg . speed = 20 . answer - b
Answer: b
Python program:
a = 40 + 40 
b = 40 / 30
c = 40 / 15
d = b + c 
e = a / d
Problem: if 25 % of x is 15 less than 15 % of 1500 , then x is ?
Options: a ) 720 , b ) 750 , c ) 820 , d ) 840 , e ) 860
Rationale: "25 % of x = x / 4 ; 15 % of 1500 = 15 / 100 * 1500 = 225 given that , x / 4 = 225 - 15 = > x / 4 = 210 = > x = 840 . answer : d"
Answer: d
Python program:
a = 15 / 100
b = 1500 * a
c = b - 15
d = 25 / 100
e = c / d
Problem: two good train each 250 m long , are running in opposite directions on parallel tracks . their speeds are 45 km / hr and 30 km / hr respectively . find the time taken by the slower train to pass the driver of the faster one .
Options: a ) 12 sec , b ) 24 sec , c ) 48 sec , d ) 60 sec , e ) none
Rationale: "sol . relative speed = ( 45 + 30 ) km / hr = ( 75 x 5 / 18 ) m / sec = ( 125 / 6 ) m / sec . distance covered = ( 250 + 250 ) m = 1000 m . required time = ( 500 x 6 / 125 ) sec = 24 sec . answer b"
Answer: b
Python program:
a = 45 + 30 
b = a * const_0_2778
c = 1/(b)
d = 250 * c
e = d * 2
Problem: a is a working partner and b is a sleeping partner in the business . a puts in rs . 3500 and b rs . 1500 , a receives 10 % of the profit for managing the business the rest being divided in proportion of their capitals . out of a total profit of rs . 9600 , money received by a is ?
Options: a ) 1978 , b ) 2707 , c ) 7728 , d ) 7008 , e ) 7291
Rationale: "35 : 15 = > 7 : 3 9600 * 10 / 100 = 960 9600 - 960 = 8640 8640 * 7 / 10 = 6048 + 960 = 7008 answer : d"
Answer: d
Python program:
a = 10 / 100
b = 1 - a
c = 9600 * b
d = 3500 + 1500 
e = 1500 / d
f = c * e
g = 9600 - f
Problem: 45 % of major airline companies equip their planes with wireless internet access . 70 % of major airlines offer passengers free on - board snacks . what is the greatest possible percentage of major airline companies that offer both wireless internet and free on - board snacks ?
Options: a ) 25 % , b ) 35 % , c ) 45 % , d ) 55 % , e ) 65 %
Rationale: "to maximize the percentage of companies offering both , let ' s assume that all 45 % of companies which offer wireless internet also offer snacks . the answer is c ."
Answer: c
Python program:
a = 45 * 1
Problem: after 10 % of the inhabitants of a village disappeared , a panic set in during which 25 % of the remaining inhabitants left the village . at that time , the population was reduced to 4725 . what was the number of original inhabitants ?
Options: a ) 5000 , b ) 7000 , c ) 4000 , d ) 8000 , e ) 9000
Rationale: "let the total number of original inhabitants be x . ( 75 / 100 ) * ( 90 / 100 ) * x = 4725 ( 27 / 40 ) * x = 4725 x = 4725 * 40 / 27 = 7000 the answer is b ."
Answer: b
Python program:
a = 10 / 100
b = 1 - a
c = 10 / 100
d = 1 - c
e = 25 / 100
f = d * e
g = b - f
h = 4725 / g
Problem: a and b invests rs . 3000 and rs . 6500 respectively in a business . if a doubles his capital after 6 months . in what ratio should a and b divide that year ' s profit ?
Options: a ) 8 : 8 , b ) 9 : 8 , c ) 9 : 6 , d ) 9 : 5 , e ) 9 : 13
Rationale: "( 3 * 6 + 6 * 6 ) : ( 6.5 * 12 ) 54 : 78 = > 9 : 13 answer : e"
Answer: e
Python program:
a = 3000 * 6
b = 3000 * 2
c = b * 6
d = a + c 
e = 6 + 6 
f = 6500 * e
g = d / f
Problem: if a car went the first third of the distance at 10 kmh , the second third at 24 kmh , and the last third at 48 kmh , what was the average speed of the car for the entire trip ?
Options: a ) 36 kmh , b ) 19 kmh , c ) 42 kmh , d ) 44 kmh , e ) 50 kmh
Rationale: "assume d / 3 = 240 ( this number is convenient because it is divisible by 10 , 24 and 48 ) so : 240 = 10 * t 1 = 24 hrs 240 = 24 * t 2 = 10 hrs 240 = 48 * t 3 = 5 hrs t = t 1 + t 2 + t 3 = 39 hrs d = rt ( 240 * 3 ) = r * 39 r = 19 answer : b"
Answer: b
Python program:
a = 1 / 10
b = 1 / 24
c = a + b 
d = 1 / 48
e = c + d 
f = 3 / e
Problem: if 70 % of an equal to 40 % of b , then ratio between a and b is ?
Options: a ) 4 : 7 , b ) 4 : 9 , c ) 4 : 2 , d ) 4 : 1 , e ) 4 : 5
Rationale: explanation : 70 a = 40 b a : b = 4 : 7 answer : a
Answer: a
Python program:
a = 40 / 70
Problem: a college has classes from 10 : 00 am until 1 : 40 pm . in this duration , there are 5 periods . if 5 minutes are provided between each period to leave one class and enter the next class , how many minutes long is each period ?
Options: a ) 38 min , b ) 41 min , c ) 40 min , d ) 42 min , e ) 52 min
Rationale: the total time is 220 minutes . there are four breaks of five minutes between the classes for a total of 20 minutes . the total class time is 200 minutes . 200 / 5 classes = 40 minutes per class the answer is c .
Answer: c
Python program:
a = 3 * const_60
b = 40 + a 
c = 5 * 4
d = b - c
e = d / 5
Problem: a card game called “ high - low ” divides a deck of 52 playing cards into 2 types , “ high ” cards and “ low ” cards . there are an equal number of “ high ” cards and “ low ” cards in the deck and “ high ” cards are worth 2 points , while “ low ” cards are worth 1 point . if you draw cards one at a time , how many ways can you draw “ high ” and “ low ” cards to earn 5 points if you must draw exactly 3 “ low ” cards ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: to get a 5 , you need one high and three lows ( you could have had 2 highs and one low , but the constraint is that you must have three low cards ) hlll = 4 ! 3 ! = 4 4 ! is the number of ways you can arrange these four spaces . divide by 3 ! because you you repeat three low cards ans : d
Answer: d
Python program:
a = 4 * 3
b = a * 2
c = b * 1
d = 3 * 2
e = d * 1
f = c / e
Problem: john distributes his pencil among his 4 friends rose , mary , ranjan , and rohit in the ratio 1 / 2 : 1 / 3 : 1 / 3 : 1 / 5 . what is the minimum no . of pencils that the person should have ?
Options: a ) 41 , b ) 65 , c ) 70 , d ) 77 , e ) 98
Rationale: rakesh : rahul : ranjan : rohit = 1 / 2 : 1 / 3 : 1 / 3 : 1 / 5 step 1 : at first we need to do is lcm of 2 , 3,3 and 5 is 30 . step 2 : then pencil are distributed in ratio among friends , rakesh = ( 1 / 2 x 30 ) = 15 . rahul = ( 1 / 3 x 30 ) = 10 . ranjan = ( 1 / 3 x 30 ) = 10 . rohit = ( 1 / 5 x 30 ) = 6 . step 3 : total number of pencils are ( 15 x + 10 x + 10 x + 6 x ) = 41 x . for minimum number of pencils x = 1 . the person should have at least 41 pencils . a )
Answer: a
Python program:
a = math.lcm(2, 3)
b = math.lcm(a, 3)
c = math.lcm(b, 5)
d = c / 2
e = math.lcm(2, 3)
f = math.lcm(e, 3)
g = math.lcm(f, 5)
h = g / 3
i = d + h 
j = math.lcm(2, 3)
k = math.lcm(j, 3)
l = math.lcm(k, 5)
m = l / 3
n = i + m 
o = math.lcm(2, 3)
p = math.lcm(o, 3)
q = math.lcm(p, 5)
r = q / 5
s = n + r
Problem: the jogging track in a sports complex is 726 meters in circumference . deepak and his wife start from the same point and walk in opposite directions at 4.5 km / hr and 3.75 km / hr respectively . they will meet for the first time in ?
Options: a ) 450 min , b ) 528 min , c ) 610 min , d ) 714 min , e ) 359 min
Rationale: "clearly , the two will meet when they are 726 m apart to be 4.5 + 3.75 = 8.25 km apart , they take 1 hour to be 726 m apart , they take 100 / 825 * 726 / 1000 = 242 / 2750 * 60 = 528 min . answer is b"
Answer: b
Python program:
a = 4 + 5
Problem: the marks obtained by vijay and amith are in the ratio 4 : 7 and those obtained by amith and abhishek in the ratio of 3 : 2 . the marks obtained by vijay and abhishek are in the ratio of ?
Options: a ) 6 : 8 , b ) 6 : 1 , c ) 6 : 5 , d ) 6 : 7 , e ) 6 : 3
Rationale: "4 : 7 3 : 2 - - - - - - - 12 : 21 : 14 12 : 14 6 : 7 answer : d"
Answer: d
Python program:
a = 4 * 3
b = 7 * 2
c = a / b
Problem: there are , in a certain league , 10 teams , and each team face another team for a total of 10 times . how many games are played in the season ?
Options: a ) 1000 , b ) 450 , c ) 720 , d ) 180 , e ) 400
Rationale: "by using the formula , t [ n ( n - 1 ) / 2 ] , where t = no . of games between two teams and n = total no . of teams , we get : 450 option a ."
Answer: a
Python program:
a = 10 - 1
b = a * 10
c = 10 / 2
d = b * c
Problem: in 10 years , a will be twice as old 5 as b was 10 years ago . if a is now 8 years older than b , the present age of b is
Options: a ) 35 , b ) 37 , c ) 38 , d ) 41 , e ) 42
Rationale: "explanation : let b ' s age = x years . then , as age = ( x + 8 ) years . ( x + 8 + 10 ) = 2 ( x — 10 ) hence x = 38 . present age of b = 38 years answer : option c"
Answer: c
Python program:
a = 2 * 10
b = 8 + 10 
c = a + b
Problem: an assembly line produces 36 cogs per hour until an initial order of 60 cogs is completed . the speed of the assembly line is then immediately increased so that it can produce 60 cogs per hour until another 60 cogs are produced . what is the overall average output , in cogs per hour , for the assembly line during this whole time ?
Options: a ) 38 , b ) 40 , c ) 42 , d ) 45 , e ) 50
Rationale: the time to produce the first 60 cogs is 60 / 36 = 5 / 3 hours . the time to produce the next 60 cogs is 60 / 60 = 1 hour . the average output is 120 cogs / ( 8 / 3 ) hours = 45 cogs per hour . the answer is d .
Answer: d
Python program:
a = 60 * 2
b = 60 / 36
c = b + 1 
d = a / c
Problem: if the number is decreased by 5 and divided by 7 the result is 7 . what would be the result if 2 is subtracted and divided by 13 ?
Options: a ) 4 , b ) 7 , c ) 8 , d ) 5 , e ) 3
Rationale: "explanation : let the number be x . then , ( x - 5 ) / 7 = 7 = > x - 5 = 49 x = 54 . : ( x - 2 ) / 13 = ( 54 - 2 ) / 13 = 4 answer : option a"
Answer: a
Python program:
a = 7 * 7
b = a + 5 
c = b - 2
d = c / 13
Problem: a ladder 14 feet long is leaning against a wall that is perpendicular to level ground . the bottom of the ladder is 5 feet from the base of the wall . if the top of the ladder slips down 4 feet , how many feet will the bottom of the ladder slip ?
Options: a ) 4 , b ) 5 , c ) 8 , d ) 10.72 , e ) 15
Rationale: "14 ^ 2 - 5 ^ 2 = 171 it means that the height is equal to 13.07 ~ = 13 . since the top of the ladder slips down 4 feet , then the height of the wall = 13 - 4 = 9 the bottom = sqrt ( 14 ^ 2 - 9 ^ 2 ) = sqrt ( 196 - 81 ) = 10.72 ans is d"
Answer: d
Python program:
a = 14 ** 2
b = 14 ** 2
c = 5 ** 2
d = b - c
e = math.sqrt(d)
f = e - 4
g = f ** 2
h = a - g
i = math.sqrt(h)
Problem: find the simple interest on rs . 78000 at 15 ( 2 / 5 ) % per annum for 9 months .
Options: a ) rs . 9009 , b ) rs . 9008 , c ) rs . 9002 , d ) rs . 9202 , e ) rs . 9001
Rationale: "explanation : p = rs . 78000 , r = 77 / 5 % p . a and t = 9 / 12 years = ¾ years therefore , s . i = ( p * r * t ) / 100 = ( 78000 * 77 / 5 * ¾ * 1 / 100 ) = rs . 9009 answer : a"
Answer: a
Python program:
a = 2 * 3
b = a * 100
c = b * 100
d = 3 * 3
e = d * 100
f = 5 + 2 
g = f * 2
h = e * g
i = c + h 
j = 15 * 5
k = j + 2 
l = k / 5
m = i * l
n = 5 * 5
o = 2 * 5
p = 2 * o
q = n / p
r = m * q
s = 1 / 100
t = r * s
Problem: the instructions state that cheryl needs 4 / 9 square yards of one type of material and 2 / 3 square yards of another type of material for a project . she buys exactly that amount . after finishing the project , however , she has 6 / 12 square yards left that she did not use . what is the total amount of square yards of material cheryl used ?
Options: a ) 1 / 12 , b ) 1 / 9 , c ) 2 / 3 , d ) 1 1 / 9 , e ) 4 / 9
Rationale: "total bought = 4 / 9 + 2 / 3 left part 6 / 12 - - - > 2 / 3 so used part 4 / 9 + 2 / 3 - 2 / 3 = 4 / 9 ans e"
Answer: e
Python program:
a = 4 / 9
b = 2 / 3
c = a + b 
d = 6 / 12
e = c - d
Problem: how many different pairs of positive integers ( a , b ) satisfy the equation 1 / a + 1 / b = 32 / 51 ?
Options: a ) 6 , b ) 3 , c ) 2 , d ) 1 , e ) 0
Rationale: "there is no certain way to solve 2 unknown with 1 equation . the best way is to look at the question and retrospect the most efficient way . in this question , a and b are only positive integers . so that is a big relief . now , we can start with putting a = 1,2 , . . and so on till the time we are confident about one of the options . so , we start with a = 1 , we get b as - ve . out a = 2 , we get b as 6 . yes ( now ( a , b ) = ( 2,6 ) . we can directly see that ( a , b ) = ( 6,2 ) will also satisfy . so we have 2 possible solutions ) a = 3 , we get b as 3 . yes ( now we have 3 possible solutions ) a = 4 , we get b as fraction . out a = 5 , we get b again as some fraction . out a = 6 already taken . we have a , b options left . c , d , e are out . a is 6 . to have 6 as the answer , we will need one more pair like 2,6 and one more solution where a = b . when a = b , we have only 1 solution = 2 . so , one more solution , where a = b is not possible . so , answer will be c ."
Answer: c
Python program:
a = 32 / 51
Problem: a , b and c rents a pasture for rs . 870 . a put in 12 horses for 8 months , b 16 horses for 9 months and 18 horses for 6 months . how much should b pay ?
Options: a ) 360 , b ) 227 , c ) 268 , d ) 198 , e ) 176
Rationale: "12 * 8 : 16 * 9 = 18 * 6 8 : 12 : 9 12 / 29 * 870 = 360 answer : a"
Answer: a
Python program:
a = 12 * 8
b = 16 * 9
c = a + b 
d = 18 * 6
e = c + d 
f = 870 / e
g = 16 * 9
h = f * g
Problem: the surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 6 cm each . the radius of the sphere is
Options: a ) 3 cm , b ) 4 cm , c ) 6 cm , d ) 8 cm , e ) none
Rationale: "solution 4 î r 2 = 2 î 3 x 6 â ‡ ’ r 2 = ( 3 x 6 / 2 ) â ‡ ’ 9 â ‡ ’ r = 3 cm . answer a"
Answer: a
Python program:
a = 6 / 2
b = 6 * a
c = math.pi * b
d = c * 2
e = math.pi * 4
f = d / e
g = math.sqrt(f)
Problem: the speed of a railway engine is 96 km per hour when no compartment is attached , and the reduction in speed is directly proportional to the square root of the number of compartments attached . if the speed of the train carried by this engine is 24 km per hour when 9 compartments are attached , the maximum number of compartments that can be carried by the engine is :
Options: a ) 19 , b ) 18 , c ) 16 , d ) 17 , e ) 14
Rationale: "the reduction in speed is directly proportional to the square root of the number of compartments attached doesreductionmean amount subtracted ? or percentage decrease ? there are at least two interpretations , and the wording does not provide a clear interpretation between them . evidently what the question intends is the subtraction interpretation . what is subtracted from the speed is directly proportional to the square root of the number of compartments attached . in other words , if s = speed , and n = number of compartments , then s = 42 - k * sqrt ( n ) wherekis a constant of the proportionality . in general , if a is directly proportional to b , we can write a = k * b and solve for k . if n = 9 , then s = 24 24 = 96 - k * sqrt ( 9 ) = 96 - 3 k k = 24 now , we need to know : what value of n makes s go to zero ? 0 = 96 - 24 * sqrt ( n ) 24 * sqrt ( n ) = 96 sqrt ( n ) = 4 n = 4 ^ 2 = 16 thus , 16 is the maximum number of cars the engine can pull and still move . c"
Answer: c
Python program:
a = 96 - 24
b = math.sqrt(9)
c = a / b
d = 96 / c
e = d ** 2
Problem: average of money that group of 4 friends pay for rent each month is $ 800 . after one persons rent is increased by 20 % the new mean is $ 870 . what was original rent of friend whose rent is increased ?
Options: a ) 800 , b ) 1400 , c ) 1000 , d ) 1100 , e ) 1200
Rationale: "0.2 x = 4 ( 870 - 800 ) 0.2 x = 280 x = 1400 answer b"
Answer: b
Python program:
a = 870 - 800
b = a * 4
c = 20 / 100
d = b / c