euclaise/mathqa_programs huggingface dataset:
Problem: let p be a prime number greater than 2 , and let n = 32 p . how many odd divisors does n have ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: the answer is a , there is exactly one . since is the largest ( and only ) even prime number , any p larger than 2 is odd . then since 32 = 2 * 2 * 2 * 2 , all of which are even , this means that p must be the only odd prime factor .
Answer: a
Python program:
a = 2 - 1
Problem: the cross - section of a cannel is a trapezium in shape . if the cannel is 12 m wide at the top and 5 m wide at the bottom and the area of cross - section is 344 sq m , the depth of cannel is ?
Options: a ) 39 , b ) 28 , c ) 27 , d ) 40 , e ) 71
Rationale: "1 / 2 * d ( 12 + 5 ) = 344 d = 40 answer : d"
Answer: d
Python program:
a = 12 + 5 
b = a / 2
c = 344 / b
d = c / 5
e = d / 2
Problem: when the number 209 y 913 is exactly divisible by 11 , then the smallest whole number that can replace y ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 5 , e ) 9
Rationale: "the given number = 209 y 913 sum of the odd places = 3 + 9 + 9 + 2 = 23 sum of the even places = 1 + y + 0 ( sum of the odd places ) - ( sum of even places ) = number ( exactly divisible by 11 ) 23 - ( 1 + y ) = divisible by 11 22 � y = divisible by 11 . y must be 0 , to make given number divisible by 11 . a"
Answer: a
Python program:
a = 209 - 209
Problem: pipe p can drain the liquid from a tank in 3 / 4 the time that it takes pipe q to drain it and in 1 / 3 the time that it takes pipe r to do it . if all 3 pipes operating simultaneously but independently are used to drain liquid from the tank , then pipe q drains what portion of the liquid from the tank ?
Options: a ) 9 / 29 , b ) 8 / 23 , c ) 9 / 25 , d ) 17 / 29 , e ) 3 / 4
Rationale: "suppose q can drain in 1 hr . so , rq = 1 / 1 = 1 so , rp = 1 / [ ( 3 / 4 ) rq ] = 4 / 3 also , rp = rr / ( 1 / 3 ) = > 4 / 3 = rr / ( 1 / 3 ) = > rr = 4 / 9 let h is the time it takes to drain by running all 3 pipes simultaneously so combined rate = rc = 1 / h = 1 + 4 / 3 + 4 / 9 = 25 / 9 = 1 / ( 9 / 25 ) thus running simultaneously , pipe q will drain 9 / 25 of the liquid . thus answer = c ."
Answer: c
Python program:
a = 3 * 3
b = 4 * 3
c = b * 2
d = 3 * 3
e = c + d 
f = a / e
Problem: a train of length l is traveling at a constant velocity and passes a pole in t seconds . if the same train travelling at the same velocity passes a platform in 5 t seconds , then what is the length of the platform ?
Options: a ) l , b ) 2 l , c ) 3 l , d ) 4 l , e ) 5 l
Rationale: the train passes a pole in t seconds , so velocity v = l / t ( l + p ) / v = 5 t ( l + p ) / ( l / t ) = 5 t p = 4 l the answer is d .
Answer: d
Python program:
a = 5 - 1
Problem: the h . c . f . of two numbers is 20 and the other two factors of their l . c . m . are 21 and 23 . the larger of the two numbers is
Options: a ) 276 , b ) 299 , c ) 312 , d ) 460 , e ) none
Rationale: "solution clearly , the numbers are ( 20 x 21 ) and ( 20 x 23 ) . larger number = ( 20 x 23 ) = 460 . answer d"
Answer: d
Python program:
a = 20 * 23
Problem: two numbers are in the ratio of 5 : 9 . if 25 be subtracted from each , they are in the ratio of 35 : 59 . find the numbers ?
Options: a ) 150,170 , b ) 150,270 , c ) 50,270 , d ) 180,270 , e ) 150,290
Rationale: "( 5 x - 25 ) : ( 9 x - 25 ) = 35 : 59 x = 30 = > 150,270 answer : b"
Answer: b
Python program:
a = 5 * 25
Problem: there is a rectangular prism made of 1 in cubes that has been covered in tin foil . there are exactly 128 cubes that are not touching any tin foil on any of their sides . if the width of the figure created by these 128 cubes is twice the length and twice the height , what is the measure e in inches of the width of the foil covered prism ?
Options: a ) 4
b ) 6
c ) 8
d ) 9
e ) 10
Rationale: if the width is w , then length and height would be w / 2 . so , w * w / 2 * w / 2 = 128 = > w ^ 3 = ( 2 ^ 3 ) * 64 = ( 2 ^ 3 ) * ( 4 ^ 3 ) = > w = 2 * 4 = 8 in . along the width of the cuboid , 8 cubes do n ' t touch the tin foil . so the actual width will be non - touching cubes + touching cubes = 8 + 2 = e = 10 ans e .
Answer: e
Python program:
a = 1 * 2
b = 1 * a
c = 128 / b
d = 1/(3)
e = c ** d
f = 1 * 2
g = e * f
h = 1 * 2
i = g + h
Problem: a train 110 m long is running with a speed of 24 km / hr . in what time will it pass a man who is running at 6 km / hr in the direction opposite to that in which the train is going ?
Options: a ) 7 sec , b ) 6 sec , c ) 8 sec , d ) 14 sec , e ) 13.2 sec
Rationale: "speed of train relative to man = 24 + 6 = 30 km / hr . = 30 * 5 / 18 = 8.33 m / sec . time taken to pass the men = 110 / 8.33 = 13.2 sec . answer : e"
Answer: e
Python program:
a = 24 + 6 
b = a * const_0_2778
c = 110 / b
Problem: worker a takes 10 hours to do a job . worker b takes 12 hours to do the same job . how long it take both a & b , working together but independently , to do the same job ?
Options: a ) 40 days , b ) 40 / 9 days , c ) 60 / 11 days , d ) 30 / 9 days , e ) 60 / 9 days
Rationale: "a ' s one hour work = 1 / 10 . b ' s one hour work = 1 / 12 . ( a + b ) ' s one hour work = 1 / 10 + 1 / 12 = 11 / 60 . both a & b can finish the work in 60 / 11 days c"
Answer: c
Python program:
a = 1 / 10
b = 1 / 12
c = a + b 
d = 1 / c
Problem: a women purchased 3 towels @ rs . 100 each , 5 towels @ rs . 150 each and two towels at a certain rate which is now slipped off from his memory . but she remembers that the average price of the towels was rs . 160 . find the unknown rate of two towels ?
Options: a ) a ) 400 , b ) b ) 450 , c ) c ) 500 , d ) d ) 550 , e ) e ) 600
Rationale: "10 * 160 = 1600 3 * 100 + 5 * 150 = 1050 1600 – 1050 = 550 d"
Answer: d
Python program:
a = 3 + 5 
b = a + 2 
c = b * 160
d = 5 * 150
e = c - d
f = 3 * 100
g = e - f
Problem: rs . 1500 is divided into two parts such that if one part is invested at 6 % and the other at 5 % the whole annual interest from both the sum is rs . 80 . how much was lent at 5 % ?
Options: a ) 228 , b ) 299 , c ) 266 , d ) 500 , e ) 1000
Rationale: "( x * 5 * 1 ) / 100 + [ ( 1500 - x ) * 6 * 1 ] / 100 = 80 5 x / 100 + 90 â € “ 6 x / 100 = 80 x / 100 = 10 = > x = 1000 answer : e"
Answer: e
Python program:
a = 5 + 6 
b = a * 100
Problem: if a truck is traveling at a constant rate of 216 kilometers per hour , how many seconds will it take the truck to travel a distance of 600 meters ? ( 1 kilometer = 1000 meters )
Options: a ) 10 , b ) 24 , c ) 30 , d ) 36 , e ) 48
Rationale: "speed = 216 km / hr = > 216,000 m / hr in one minute = > 216000 / 60 = 3600 meters in one sec = > 3600 / 60 = 60 meters time = total distance need to be covered / avg . speed = > 600 / 60 = 10 and hence the answer : a"
Answer: a
Python program:
a = 600 / 1000
b = a / 216
c = b * 3600
Problem: the l . c . m of two numbers is 48 . the numbers are in the ratio 4 : 3 . the sum of numbers is :
Options: a ) 28 , b ) 30 , c ) 40 , d ) 50 , e ) 60
Rationale: "let the numbers be 4 x and 3 x . then , their l . c . m = 12 x . so , 12 x = 48 or x = 4 . the numbers are 16 and 12 . hence , required sum = ( 16 + 12 ) = 28 . answer : a"
Answer: a
Python program:
a = 4 * 48
b = a / 3
Problem: a number a is squared and then multiplied by negative 6 . the result of this operation is equal to 3 times the sum of fourth times a and two . what is one possible value of a ?
Options: a ) 2 , b ) 3 , c ) 4 , d ) 1 , e ) 5
Rationale: - 6 * a ^ 2 = 3 ( 4 a + 2 ) a = - 1 or - 1 a = - 1 = b answer : d
Answer: d
Python program:
a = 6 / 3
b = a / 2
Problem: if 20 typists can type 38 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 ) 171 , e ) 400
Rationale: "20 typists can type 38 letters , so 30 typists can type = 38 * 30 / 20 38 * 30 / 20 letters can be typed in 20 mins . in 60 mins typist can type = 38 * 30 * 60 / 20 * 20 = 171 d is the answer"
Answer: d
Python program:
a = 38 * 3
b = a / 20
c = b * 30
Problem: a fair 2 sided coin is flipped 6 times . what is the probability that tails will be the result at least twice , but not more than 4 times ?
Options: a ) 25 / 32 , b ) 13 / 4 , c ) 17 / 8 , d ) 57 / 64 , e ) 15 / 16
Rationale: "at least twice , but not more than 5 timesmeans exactly 2 times , 3 times , 4 times the probability of getting exactly k results out of n flips is nck / 2 ^ n 6 c 2 / 2 ^ 6 + 6 c 3 / 2 ^ 6 + 6 c 4 / 2 ^ 6 = ( 20 + 15 + 15 ) / 2 ^ 6 = 25 / 32 option : a"
Answer: a
Python program:
a = 2 ** 6
b = 1/(a)
c = b * 6
d = 2 ** 6
e = 1/(d)
f = 2 ** 6
g = 1/(f)
h = e + g 
i = c + h 
j = 1 - i
Problem: two cyclist start from the same places in opposite directions . one is going towards north at 10 kmph and the other is going towards south 40 kmph . what time will they take to be 50 km apart ?
Options: a ) 1 hr , b ) 2 hrs , c ) 3 hrs , d ) 5 hrs , e ) 6 hrs
Rationale: to be ( 10 + 40 ) km apart , they take 1 hour to be 50 km apart , they take 1 / 50 * 50 = 1 hrs answer is a
Answer: a
Python program:
a = 10 + 40 
b = 50 / a
Problem: if x is a sum of all even integers on the interval 13 . . . 55 and y is their number , what is the gcd ( x , y ) ?
Options: a ) 1 , b ) 21 , c ) 26 , d ) 52 , e ) 1014
Rationale: "x = 14 + 16 + . . . + 54 = ( largest + smallest ) / 2 * ( # of terms ) = ( 14 + 54 ) / 2 * 21 = 34 * 21 . gcd of 21 and 34 * 21 is 21 answer : b ."
Answer: b
Python program:
a = 13 - 10
b = a + 2 
c = 13 - 10
d = c + 2 
e = b * d
Problem: how many liters of pure alcohol must be added to a 100 - liter solution that is 20 percent alcohol in order to produce a solution that is 30 percent alcohol ?
Options: a ) 100 / 7 , b ) 5 , c ) 20 / 3 , d ) 8 , e ) 39 / 4
Rationale: 20 % alcohol solution means ; in the 100 liter solution , 20 liters of solution is alcohol and 80 liters other solvents . if we addxliters of alcohol to the solution , the solution becomes 100 + xliters and alcohol , which was 20 liters , becomes 20 + x liters . according to the statement ; 20 + x = 30 % of ( 100 + x ) or 20 + x = ( 100 + x ) 3 / 10 200 + 10 x = 300 + 3 x 7 x = 100 x = 100 / 7 ans : a
Answer: a
Python program:
a = 4 + 3 
b = 100 / a
Problem: the banker ' s gain of a certain sum due 2 years hence at 10 % per annum is rs . 24 . the present worth is
Options: a ) rs . 480 , b ) rs . 520 , c ) rs . 600 , d ) rs . 960 , e ) none
Rationale: solution t . d = ( b . g x 100 / rate x time ) = rs . ( 24 x 100 / 10 x 2 ) = rs . 120 . p . w = ( 100 x t . d / rate x time ) = rs . ( 100 x 120 / 10 x 2 ) = rs . 600 . answer c
Answer: c
Python program:
a = 24 * 100
b = 2 * 10
c = a / b
d = 100 * c
e = 2 * 10
f = d / e
Problem: danny is sitting on a rectangular box . the area of the front face of the box is half the area of the top face , and the area of the top face is 1.5 times the area of the side face . if the volume of the box is 3000 , what is the area of the side face of the box ?
Options: a ) 300 , b ) 150 , c ) 180 , d ) 109 , e ) 200
Rationale: "lets suppose length = l , breadth = b , depth = d front face area = l * w = 1 / 2 w * d ( l = 1 / 2 d or d = 2 l ) top face area = w * d side face area = w * d = 1.5 d * l ( w = 1.5 l ) volume = l * w * d = 3000 l * 1.5 l * 2 l = 3000 l = 10 side face area = l * d = l * 2 l = 10 * 2 * 10 = 200 e is the answer"
Answer: e
Python program:
a = 3000 ** 2
b = 3 * a
c = 1 / 3
d = b ** c
e = d / 1
Problem: the salary of a person was reduced by 20 % . by what percent should his reduced salary be raised so as to bring it at par with his original salary ?
Options: a ) 50 % , b ) 32 % , c ) 25 % , d ) 43 % , e ) 29 %
Rationale: "let the original salary be $ 100 new salary = $ 80 increase on 80 = 20 increase on 100 = 20 / 80 * 100 = 25 % answer is c"
Answer: c
Python program:
a = 20 / 100
b = 100 * a
c = 20 / 100
d = 100 * c
e = 100 - d
f = b / e
g = f * 100
Problem: if 10 % of 30 % of 50 % of a number is 90 , then what is the number ?
Options: a ) 4000 , b ) 6000 , c ) 4400 , d ) 4500 , e ) none of these
Rationale: "let the number be a given , 10 / 100 * 30 / 100 * 50 / 100 * a = 90 = > 1 / 10 * 3 / 10 * 1 / 2 * a = 90 = > a = 10 * 20 * 10 * 2 = 6000 . answer : b"
Answer: b
Python program:
a = 50 / 100
b = 30 / 100
c = a * b
d = 10 / 100
e = c * d
f = 90 / e
Problem: the average of non - zero number and its square is 5 times the number . the number is ?
Options: a ) 9 , b ) 8 , c ) 6 , d ) 4 , e ) 3
Rationale: "let the number be x . then , ( x + x 2 ) / 2 = 5 x = > x 2 - 9 x = 0 = > x ( x - 9 ) = 0 = > x = 0 or x = 9 so , the number is 9 . answer : a"
Answer: a
Python program:
a = 5 * 2
b = a - 1
Problem: preethi has 6 flavors of ice cream in his parlor . how many options are there for dorathy to pick a one - flavor , two - flavor , 3 - flavor , 4 - flavor , 5 flavor or 6 flavor order ?
Options: a ) 26 , b ) 36 , c ) 64 , d ) 67 , e ) 63
Rationale: 6 c 1 + 6 c 2 + 6 c 3 + 6 c 4 + 6 c 5 + 6 c 6 = 63 . answer : e
Answer: e
Python program:
a = math.factorial(6)
b = math.factorial(5)
c = a / b
d = math.factorial(6)
e = math.factorial(4)
f = 2 * e
g = d / f
h = c + g 
i = math.factorial(6)
j = math.factorial(3)
k = math.factorial(3)
l = j * k
m = i / l
n = h + m 
o = math.factorial(6)
p = math.factorial(4)
q = 2 * p
r = o / q
s = n + r 
t = math.factorial(6)
u = math.factorial(5)
v = t / u
w = s + v
Problem: the average weight of 2 person ' s increases by 4.5 kg when a new person comes in place of one of them weighing 65 kg . what might be the weight of the new person ?
Options: a ) 74 kg , b ) 76.5 kg , c ) 85 kg , d ) 78 , e ) none of these
Rationale: "explanation : total weight increased = ( 2 x 4.5 ) kg = 9 kg . weight of new person = ( 65 + 9 ) kg = 74 kg . answer : a"
Answer: a
Python program:
a = 2 * 4
b = a + 65
Problem: john started a business , investing rs . 18000 . after 3 months and 4 months respectively , rose and tom joined him with capitals of 12000 and 9000 . at the end of the year the total profit was rs . 4070 . what is the difference between rose ’ s and tom ’ s share in the profit ?
Options: a ) s . 340 , b ) s . 362 , c ) s . 370 , d ) s . 382 , e ) s . 390
Rationale: john : rose : tom ratio of their investments = 18000 × 12 : 12000 × 9 : 9000 × 8 = 6 : 3 : 2 the difference between rose ’ s and tom ’ s share = 1 share : . i . e . = rs . 4070 × 1 / 11 = rs . 370 . c )
Answer: c
Python program:
a = 3 * 4
b = a - 3
c = 12000 * b
d = 10 * 100
e = c / d
f = 4070 * e
g = 3 * 4
h = g * 18000
i = 10 * 100
j = h / i
k = 3 * 4
l = k - 3
m = 12000 * l
n = 10 * 100
o = m / n
p = j + o 
q = 3 * 4
r = q - 4
s = 9000 * r
t = 10 * 100
u = s / t
v = p + u 
w = f / v
x = 3 * 4
y = x - 4
z = 9000 * y
A = 10 * 100
B = z / A
C = 4070 * B
D = 3 * 4
E = D * 18000
F = 10 * 100
G = E / F
H = 3 * 4
I = H - 3
J = 12000 * I
K = 10 * 100
L = J / K
M = G + L 
N = 3 * 4
O = N - 4
P = 9000 * O
Q = 10 * 100
R = P / Q
S = M + R 
T = C / S
U = w - T
Problem: if each year the population of the country grows by 20 % , how many years will elapse before the population of the country doubles ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "till year 2000 , population is 100 . year 2001 : population becomes 120 . . . . . . . . . . . . . 1 year elapsed year 2002 : population becomes 144 . . . . . . . . . . . . . 2 year elapsed year 2003 : population becomes 172 . . . . . . . . . . . . . 3 year elapsed year 2004 : population > 200 . . . . . . . . . . . . . . . . . . 4 year elapsed answer : b"
Answer: b
Python program:
a = 4 + 4
Problem: peter invests a sum of money and gets back an amount of $ 815 in 3 years . david invests an equal amount of money and gets an amount of $ 854 in 4 years . if both amounts were invested at the same rate ( simple interest ) what was the sum of money invested ?
Options: a ) 670 , b ) 664 , c ) 698 , d ) 744 , e ) 700
Rationale: "since both peter and david invested the same amount of money at the same rate , they would earn same interest per year . david invested for one year more than peter and hence he got interest amount for one more year . interest earned per year = amount received by david - amount received by peter = 854 - 815 = 39 interest earned for 3 years = 39 * 3 = 117 amount invested = 815 - 117 = 698 answer : c"
Answer: c
Python program:
a = 854 - 815
b = 4 / 100
c = 3 / 100
d = b - c
e = a / d
f = 3 / 100
g = e * f
h = 815 - g
Problem: andrew purchased 10 kg of grapes at the rate of 82 per kg and 6 kg of mangoes at the rate of 62 per kg . how much amount did he pay to the shopkeeper ?
Options: a ) 1000 , b ) 1055 , c ) 1378 , d ) 1075 , e ) 1080
Rationale: "cost of 10 kg grapes = 82 × 10 = 820 . cost of 6 kg of mangoes = 62 × 6 = 558 . total cost he has to pay = 820 + 558 = 1378 c"
Answer: c
Python program:
a = 10 * 82
b = 6 * 62
c = a + b
Problem: maths , physics and chemistry books are stored on a library shelf that can accommodate 25 books . currently , 20 % of the shelf spots remain empty . there are twice as many maths books as physics books and the number of physics books is 4 greater than that of chemistry books . among all the books , 12 books are soft cover and the remaining are hard - cover . if there are a total of 7 hard - cover books among the maths and physics books . what is the probability r , that a book selected at random is either a hard cover book or a chemistry book ?
Options: a ) 1 / 10 , b ) 3 / 20 , c ) 1 / 5 , d ) 1 / 4 , e ) 9 / 20
Rationale: "first phase of this problem requires you to determine how many mathematics and chemistry books are even on the shelf . to do so , you have the equations : m + p + c = 20 ( since 4 / 5 of the 25 spots are full of books ) m = 2 p p = 4 + c from that , you can use substitution to get everything down to one variable . c = p - 4 m = 2 p p = p then ( p - 4 ) + 2 p + p = 20 , so 4 p = 24 and p = 6 . that means that there are 12 math , 6 physics , and 2 chemistry books on the shelf . with those numbers , you also know that there are 8 total hardcovers , 1 of which is chemistry . so if your goal is to get either a hardcover or a chemistry , there are 9 ways towin - either one of the 7 hardcovers that are n ' t chemistry or the two chemistry books . so out of the 20 total , r = 9 provide the desired outcome , making the answer e ."
Answer: e
Python program:
a = 2 + 4 
b = 2 + a 
c = b + 12 
d = 2 / c
e = 2 + 4 
f = 2 + e 
g = 2 + 4 
h = 2 + g 
i = h + 12 
j = f / i
k = d + j 
l = 2 + 4 
m = 2 + l 
n = m + 12 
o = 2 / n
p = 2 + 4 
q = 2 + p 
r = 2 + 4 
s = 2 + r 
t = s + 12 
u = q / t
v = o * u
w = k - v
Problem: tom and linda stand at point a . linda begins to walk in a straight line away from tom at a constant rate of 2 miles per hour . one hour later , tom begins to jog in a straight line in the exact opposite direction at a constant rate of 7 miles per hour . if both tom and linda travel indefinitely , what is the positive difference , in minutes , between the amount of time it takes tom to cover half of the distance that linda has covered and the amount of time it takes tom to cover twice the distance that linda has covered ?
Options: a ) 60 , b ) 56 , c ) 84 , d ) 90 , e ) 108
Rationale: "b is the answer . . . . d = ts where d = distance , t = time and s = speed to travel half distance , ( 2 + 2 t ) = 7 t = = > t = 2 / 5 = = > 24 minutes to travel double distance , 2 ( 2 + 2 t ) = 7 t = = > 2 = = > 80 minutes difference , 56 minutes b"
Answer: b
Python program:
a = 2 * 2
b = 2 * 2
c = 7 - b
d = a / c
e = 7 - 2
f = 2 / e
g = d - f
h = g * const_60
Problem: 64 is divided into two parts in such a way that seventh part of first and ninth part of second are equal . find the smallest part ?
Options: a ) 66 , b ) 26 , c ) 42 , d ) 28 , e ) 11
Rationale: "x / 7 = y / 9 = > x : y = 7 : 9 7 / 16 * 64 = 28 answer : d"
Answer: d
Python program:
a = 4 + 3 
b = 4 + 3 
c = b + 2 
d = a / c
e = d * 64
f = e / 2
Problem: a man walking at the rate of 6 km / hr crosses a bridge in 15 minutes . the length of the bridge is ______ .
Options: a ) 1000 m , b ) 1250 m , c ) 1500 m , d ) 1800 m , e ) 2300 m
Rationale: "hint : to find the answer in meter , we will first convert distance from km / hour to meter / sec by multiplying it with 5 / 18 . also , change 15 minutes to seconds by multiplying it with 60 . distance = speed x time 1 . convert speed into m / sec : 6 x 5 / 18 m / s = 1.66 m / s 2 . convert time from minutes into seconds = 15 x 60 s = 900 sec 3 . calculate : distance = 1.66 x 900 = 1500 m answer is c"
Answer: c
Python program:
a = 6 * 1000
b = a / const_60
c = b * 15
Problem: if w = x ^ 4 + y ^ 4 = 100 , then the greatest possible value of x is between
Options: a ) 0 and 3 , b ) 3 and 6 , c ) 6 and 9 , d ) 9 and 12 , e ) 12 and 15
Rationale: my attempt : if w = x ^ 4 + y ^ 4 = 100 , then the greatest possible value of x would be when y is minimum . let y ^ 4 be 0 . now x ^ 4 = 100 . x should be definitely greater than 3 but less than 4 . the only option that fits this range is b hence answer is - - b ) 3 and 6 .
Answer: b
Python program:
a = 1 / 4
b = 100 ** a
c = math.floor(b)
d = c * 10
e = 1 / 4
f = 100 ** e
g = math.floor(f)
h = g * 2
i = d + h
Problem: if m and n are positive integers and m ^ 2 + n ^ 2 = 89 , what is the value of m ^ 3 + n ^ 3 ?
Options: a ) 72 , b ) 224 , c ) 320 , d ) 512 , e ) 637
Rationale: "you need to integers which squared are equal 89 . which could it be ? let ' s start with the first integer : 1 ^ 2 = 1 2 ^ 2 = 4 3 ^ 2 = 9 4 ^ 2 = 16 5 ^ 2 = 25 6 ^ 2 = 36 7 ^ 2 = 49 8 ^ 2 = 64 stop . the integers ca n ' t be greater than 8 or we will score above 89 . the second integer need to be picked up the same way . 1 ^ 2 = 1 2 ^ 2 = 4 3 ^ 2 = 9 4 ^ 2 = 16 5 ^ 2 = 25 6 ^ 2 = 36 7 ^ 2 = 49 8 ^ 2 = 64 the only pair that matches is 8 ^ 2 + 5 ^ 2 = 89 . so 8 ^ 3 + 5 ^ 3 = 637 . answer e . )"
Answer: e
Python program:
a = math.gcd(89, 4)
Problem: the ratio of the number of red cars in a certain parking lot to the number of black cars is 3 to 8 . if there are 70 black cars in the lot , how many red cars are there in the lot ?
Options: a ) 11 , b ) 26 , c ) 24 , d ) 27 , e ) 32
Rationale: "b is correct r / b = 3 / 8 and b = 70 r = 70 * 3 / 8 = 26"
Answer: b
Python program:
a = 3 / 8
b = a * 70
c = math.floor(b)
Problem: if the sum of a number and its square is 272 , what is the number ?
Options: a ) 14 , b ) 16 , c ) 19 , d ) 21 , e ) none of these
Rationale: "explanation : let the integer be x . then , x + x 2 = 272 x 2 + x - 272 = 0 ( x + 17 ) ( x – 16 ) = 0 x = 16 answer : b"
Answer: b
Python program:
a = math.sqrt(272)
b = math.floor(a)
Problem: a , b , c and d enter into partnership . a subscribes 1 / 3 of the capital b 1 / 4 , c 1 / 5 and d the rest . how much share did a get in a profit of rs . 2460 ?
Options: a ) 28888 , b ) 27789 , c ) 2777 , d ) 14000 , e ) 2881
Rationale: "25 * 12 : 30 * 12 : 35 * 8 15 : 18 : 14 14 / 47 * 47000 = 14000 answer : d"
Answer: d
Python program:
a = 1 / 3
b = 2460 * a
Problem: volume of a hemisphere is 19404 cu . cm . its radius is :
Options: a ) 19 cm
b ) 20 cm
c ) 21 cm
d ) 30 cm
e ) none
Rationale: sol . let the radius be r cm . then , 2 / 3 * 22 / 7 * r ³ = 19404 ⇔ r ³ = [ 19404 * 21 / 44 ] = ( 21 ) ³ ⇔ r = 21 cm . answer c
Answer: c
Python program:
a = 2 * math.pi
b = a / 3
c = 19404 / b
d = 1 / 3
e = c ** d
Problem: a hiker walked for 3 days . she walked 18 miles on the first day , walking 3 miles per hour . on the second day she walked for one less hour but she walked one mile per hour , faster than on the first day . on the third day she walked at 4 miles per hour for 5 hours . how many miles in total did she walk ?
Options: a ) 24 , b ) 44 , c ) 58 , d ) 60 , e ) 62
Rationale: "first day - 18 miles with 3 miles per hours then total - 6 hours for that day second day - 4 miles per hour and 5 hours - 20 miles third day - 4 miles per hour and 5 hours - 20 miles total 18 + 20 + 20 = 58 answer : option c"
Answer: c
Python program:
a = 4 * 4
b = 18 + a 
c = 4 * 5
d = b + c
Problem: 2 corner most boxes of a chess board ( diagonally opposite ) haven been cut out there ' s a rectangular block = 2 sqaures of chess board , how many such blocks can be placed on the chess board ? ”
Options: a ) 27 , b ) 28 , c ) 29 , d ) 30 , e ) 31
Rationale: chess board has 64 squares . so we can place 32 rectangular blocks , but 2 are cut off from the corner . hence 32 - 2 = 30 blocks answer : d
Answer: d
Python program:
a = 2 * 2
b = 2 * 2
c = a * b
d = c * 2
e = d - 2