euclaise/mathqa_programs huggingface dataset:
Problem: in an apartment building that has 490 units , 4 out of every 7 units are currently rented , including 1 / 4 of the one - bedroom units . if , of the rented apartments , there is a 6 : 1 ratio of two - bedroom units to one - bedroom units , and the building only consists of two - bedroom and one - bedroom units , how many two - bedroom units are not rented ?
Options: a ) 50 , b ) 90 , c ) 100 , d ) 105 , e ) 140
Rationale: "rented = 4 / 7 * 490 = 280 . rented two - bedroom = 6 / 7 * 280 = 240 ; rented one - bedroom = 1 / 7 * 280 = 40 ( or 280 - 240 = 40 ) . rented one - bedroom units are 1 / 4 of all one - bedroom : 1 / 4 * { all one bedroom } = 40 - - > { all one - bedroom } = 160 { all two - bedroom } = 490 - 160 = 330 two - bedroom not rented = 330 - 240 = 90 b"
Answer: b
Python program:
a = 490 / 7
b = 4 * a
c = 490 / 7
d = 4 * c
e = 4 + 1 
f = d / e
g = 4 * f
h = b - g
i = 1 / 4
j = h / i
k = 490 - j
l = 490 / 7
m = 4 * l
n = 4 + 1 
o = m / n
p = 4 * o
q = k - p
Problem: in how many ways 6 boys and 6 girls can be seated in a row so that boys and girls are alternate ?
Options: a ) 1063800 , b ) 1083600 , c ) 1036800 , d ) 1053800 , e ) 1073600
Rationale: "number of ways = 2 * 6 ! * 6 ! = 1036800 answer is c"
Answer: c
Python program:
a = math.factorial(6)
b = math.factorial(6)
c = a * b
Problem: the cost price of 12 articles is equal to the selling price of 10 articles . what is the profit percent ?
Options: a ) 10 % , b ) 20 % , c ) 30 % , d ) 40 % , e ) 50 %
Rationale: "10 * sp = 12 * cp sp = 1.2 * cp the profit percent is 20 % . the answer is b ."
Answer: b
Python program:
a = 12 * 4
b = 4 + 1 
c = a / b
Problem: a man rows his boat 65 km downstream and 45 km upstream , taking 2 1 / 2 hours each time . find the speed of the stream ?
Options: a ) 5 kmph , b ) 4 kmph , c ) 9 kmph , d ) 8 kmph , e ) 1 kmph
Rationale: "speed downstream = d / t = 65 / ( 2 1 / 2 ) = 26 kmph speed upstream = d / t = 45 / ( 2 1 / 2 ) = 18 kmph the speed of the stream = ( 26 - 18 ) / 2 = 4 kmph answer : b"
Answer: b
Python program:
a = const_60 / 0
b = 45 / 2
c = a - b
d = c / 2
Problem: arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 9 from the product of their ages a year from today . what is their age today ?
Options: a ) 2 . , b ) 4 . , c ) 5 . , d ) 7 , e ) 9 .
Rationale: "ad = ( a + 1 ) ( d + 1 ) - 9 0 = a + d - 8 a + d = 8 a = d ( as they are twin brothers ) a = d = 4 b is the answer"
Answer: b
Python program:
a = 9 - 1
b = a / 2
Problem: 27 * 27 * 27 * 27 = 3 ^ ?
Options: a ) 7 , b ) 12 , c ) 5 , d ) 6 , e ) 8
Rationale: "3 ^ 3 * 3 ^ 3 * 3 ^ 3 * 3 ^ 3 = 3 ^ ( 3 + 3 + 3 + 3 ) = 3 ^ 12 answer : 12 option : b"
Answer: b
Python program:
a = 27 * 27
Problem: there were two candidates in an election . winner candidate received 60 % of votes and won the election by 240 votes . find the number of votes casted to the winning candidate ?
Options: a ) 228 , b ) 720 , c ) 255 , d ) 199 , e ) 231
Rationale: "w = 60 % l = 40 % 60 % - 40 % = 20 % 20 % - - - - - - - - 240 60 % - - - - - - - - ? = > 720 answer : b"
Answer: b
Python program:
a = 100 - 60
b = 60 - a
c = b / 100
d = 240 / c
e = d * 60
f = e / 100
Problem: a factory producing tennis balls stores them in either big boxes , 25 balls per box , or small boxes , 20 balls per box . if 153 freshly manufactured balls are to be stored , what is the least number of balls that can be left unboxed ?
Options: a ) 2 , b ) 1 , c ) 3 , d ) 5 , e ) 4
Rationale: "we have to work with multiples of 20 and 25 . first , we must know the limits of this multiples , so : 152 / 25 = 6 . . . . so the max is 6 152 / 20 = 7 . . . so the max is 7 153 - 150 = 3 ( 6 big box ) answer : c"
Answer: c
Python program:
a = 25 - 20
Problem: what least number should be subtracted from 997 so that the remainder when divided by 5 , 9 , and 11 will leave in each case the same remainder 3 ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: the lcm of 5,9 and 11 is 495 . the next multiple is 2 * 495 = 990 . 990 + { remainder } = 990 + 3 = 993 , which is 4 less than 997 . answer : d .
Answer: d
Python program:
a = 5 * 9
b = a * 11
c = b * 2
d = c + 3 
e = 997 - d
Problem: a garden center sells a certain grass seed in 5 - pound bags at $ 13.82 per bag , 10 - pound bags at $ 20.43 per bag , and 25 - pound bags $ 32.25 per bag . if a customer is to buy at least 65 pounds of the grass seed , but no more than 80 pounds , what is the least possible cost of the grass seed that the customer will buy ?
Options: a ) $ 94.03 , b ) $ 96.75 , c ) $ 98.75 , d ) $ 102.07 , e ) $ 105.3
Rationale: "there can be 2 cases 1 ) 25 + 25 + 10 + 5 = $ 98.75 or 2 ) 25 + 25 + 25 = $ 96.75 c"
Answer: c
Python program:
a = 2 * 32
b = 1 * 20
c = a + b 
d = 1 * 13
e = c + d
Problem: in a group of cows and chickens , the number of legs was 14 more than twice the number of heads . the number of cows was :
Options: a ) 3 , b ) 5 , c ) 9 , d ) 7 , e ) 8
Rationale: d 7 let the number of cows be x and their legs be 4 x . let the number of chicken be y and their legs be 2 x . total number of legs = 4 x + 2 y . total number of heads = x + y . the number of legs was 14 more than twice the number of heads . therefore , 2 × ( x + y ) + 14 = 4 x + 2 y . or , 2 x + 2 y + 14 = 4 x + 2 y . or , 2 x + 14 = 4 x [ subtracting 2 y from both sides ] . or , 14 = 4 x – 2 x [ subtracting 2 x from both sides ] . or , 14 = 2 x . or , x = 7 [ dividing by 2 on both sides ] . therefore , the number of cows = 7 .
Answer: d
Python program:
a = 4 - 2
b = 14 / a
Problem: water consists of hydrogen and oxygen , and the approximate ratio , by mass , of hydrogen to oxygen is 2 : 16 . approximately how many grams of oxygen are there in 153 grams of water ?
Options: a ) 16 , b ) 72 , c ) 112 , d ) 136 , e ) 142
Rationale: "since the ratio , by mass , of hydrogen to oxygen is 2 : 16 , then oxygen is 16 / ( 2 + 16 ) = 8 / 9 of mass of water . therefore there are 153 * 8 / 9 = 136 grams of oxygen in 153 grams of water . answer : d ."
Answer: d
Python program:
a = 2 + 16 
b = 153 / a
c = 2 * b
Problem: how many zeros does 135 ^ 19 x 42 ^ 17 end with ?
Options: a ) 0 , b ) 2 , c ) 5 , d ) 13 , e ) 17
Rationale: "to know the zeros we need to know how many 10 ' s are there for this need to find number of 2 s and 5 s ( 2 * 5 = 10 ) 135 = ( 5 x 3 ^ 3 ) ^ 19 so number of 5 s are = 19 42 = ( 2 x 7 x 3 ) ^ 17 so number of 2 s = 17 so tot number of 10 s in this number is ( 17 - > 2 s x 17 - > 5 s ) so answer is 17 answer : e"
Answer: e
Python program:
a = 2 + 3 
b = a * 2
c = 135 / 2
d = 135 / c
e = b + d
Problem: due to construction , the speed limit along an 5 - mile section of highway is reduced from 40 miles per hour to 22 miles per hour . approximately how many minutes more will it take to travel along this section of highway at the new speed limit than it would have taken at the old speed limit ?
Options: a ) a ) 3.12 , b ) b ) 8 , c ) c ) 10 , d ) d ) 15 , e ) e ) 6.13
Rationale: "old time in minutes to cross 5 miles stretch = 5 * 60 / 40 = 5 * 3 / 2 = 7.5 new time in minutes to cross 5 miles stretch = 5 * 60 / 22 = 5 * 30 / 11 = 13.63 time difference = 6.13 ans : e"
Answer: e
Python program:
a = 40 + 5 
b = a - 1
c = 5 / 22
d = 5 / 40
e = c - d
f = b * e
g = max(f)
Problem: 2 : 5 = 4 / 3 : x . the value of x is ?
Options: a ) 10 / 3 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: x * 2 = 5 * 4 / 3 x * 2 = 20 / 3 x = 10 / 3 answer : a
Answer: a
Python program:
a = 4 / 3
b = 2 / 5
c = a / b
Problem: how many even integers n , such that 20 < = n < = 200 are of the form 3 k + 4 , where k is any natural number ?
Options: a ) 26 , b ) 30 , c ) 34 , d ) 38 , e ) 42
Rationale: "the first number is 22 = 16 + 6 ( 1 ) . we can continue adding 6 to make a list : 22 , 28 , 34 , . . . the last number is 196 = 16 + 6 ( 30 ) there are 30 numbers in the list . the answer is b ."
Answer: b
Python program:
a = 200 - 2
b = 20 + 2 
c = a - b
d = 2 + 4 
e = c / d
f = e + 1
Problem: find a sum for 1 st 4 prime number ' s ?
Options: a ) 25 , b ) 17 , c ) 30 , d ) 34 , e ) 36
Rationale: "required sum = ( 2 + 3 + 5 + 7 ) = 17 note : 1 is not a prime number option b"
Answer: b
Python program:
a = 2 + 3 
b = 2 + 3 
c = a + b 
d = 2 + 3 
e = d + 2 
f = c + e 
g = 4 + 2 
h = f + g 
i = 4 + 2 
j = i + 2 
k = h + j 
l = 4 + 2 
m = l + 2 
n = m + 4 
o = k + n 
p = 4 + 2 
q = p + 2 
r = q + 4 
s = r + 2 
t = o + s 
u = 4 + 2 
v = u + 2 
w = v + 4 
x = w + 2 
y = x + 4 
z = t + y
Problem: two vessels having volumes in the ratio 3 : 5 are filled with water and milk solutions . the ratio of milk and water in the two vessels are 5 : 1 and 6 : 4 respectively . if the contents of both the vessel are empties into a larger vessel , find the ratio of milk and water in the larger vessel .
Options: a ) 12 : 5 , b ) 11 : 5 , c ) 13 : 5 , d ) 14 : 5 , e ) 9 : 5
Rationale: "vessel a = 300 gallons - - > milk = 250 , water = 50 ; vessel b = 500 gallons - - > milk = 300 , water = 200 ; vessel a + b = 800 gallons - - > milk = 550 , water 250 the ratio = 550 / 250 - - > 11 : 5 answer : b"
Answer: b
Python program:
a = 5 ** 3
Problem: how many seconds will a 750 meter long train moving with a speed of 63 km / hr take to cross a man walking with a speed of 3 km / hr in the direction of the train ?
Options: a ) 48 , b ) 36 , c ) 26 , d ) 11 , e ) 45
Rationale: "explanation : here distance d = 750 mts speed s = 63 - 3 = 60 kmph = 60 x 5 / 18 m / s time t = = 45 sec . answer : e"
Answer: e
Python program:
a = 63 - 3
b = a * const_0_2778
c = 750 / b
Problem: a certain country had a total annual expenditure of $ 7.2 x 10 ^ 11 last year . if the population of the country was 240 million last year , what was the per capita expenditure ?
Options: a ) $ 500 , b ) $ 1,000 , c ) $ 2,000 , d ) $ 3,000 , e ) $ 5,000
Rationale: "total expenditure / population = per capita expenditure hence , ( 7,2 x 10 ^ 11 ) / 240 000 000 = ( 7,2 x 10 ^ 11 ) / ( 2,4 x 10 ^ 8 ) = 3 x 10 ^ ( 11 - 8 ) = 3 x 10 ^ 3 = 3000 . answer is d ."
Answer: d
Python program:
a = 10 ** 11
b = 7 * 2
c = 4 + 2 
d = 10 ** c
e = 240 * d
f = b / e
Problem: a certain manufacturer produces items for which the production costs consist of annual fixed costs totaling $ 130,000 and variables costs averaging $ 5 per item . if the manufacturer ’ s selling price per item is $ 15 , how many items the manufacturer produce and sell to earn an annual profit of $ 150,000 ?
Options: a ) 2,858 , b ) 18,667 , c ) 28,000 , d ) 35,000 , e ) 40,000
Rationale: "let the items manufactured or sold bex 130000 + 5 x = 15 x - 150000 10 x = 280000 x = 28000 ans : c"
Answer: c
Python program:
a = 3 + 10 
b = a + 15 
c = 100 ** 2
d = b * c
e = 15 - 5
f = d / e
g = 4 * 2
h = f / g
i = h * 3
j = i / 1000
Problem: what should be added to 4529 so that it may become a perfect square ?
Options: a ) 94 , b ) 95 , c ) 96 , d ) 97 , e ) 98
Rationale: "68 x 68 = 4624 4624 - 4529 = 95 if added to 95 get perfect square answer = b"
Answer: b
Python program:
a = 10 + 10 
b = 10 + 10 
c = a + b 
d = 10 + 10 
e = c + d 
f = 4 + 3 
g = e + f 
h = 10 + 10 
i = 10 + 10 
j = h + i 
k = 10 + 10 
l = j + k 
m = 4 + 3 
n = l + m 
o = g * n
p = o - 4529
Problem: barbata invests $ 2400 in the national bank at 5 % . how much additional money must she invest at 8 % so that the total annual income will be equal to 8 % of her entire investment ?
Options: a ) 1200 , b ) 3000 , c ) 1000 , d ) 3600 , e ) 2400
Rationale: "let the additional invested amount for 8 % interest be x ; equation will be ; 2400 + 0.05 * 2400 + x + 0.08 x = 2400 + x + 0.08 ( 2400 + x ) 0.05 * 2400 + 0.08 x = 0.06 x + 0.08 * 2400 0.02 x = 2400 ( 0.08 - 0.05 ) x = 2400 * 0.03 / 0.02 = 3600 ans : d"
Answer: d
Python program:
a = 8 / 100
b = a * 2400
c = 5 / 100
d = 2400 * c
e = b - d
f = 8 / 100
g = 8 / 100
h = f - g
i = e / h
Problem: if x : y = 3 : 4 , find ( 4 x + 5 y ) : ( 5 x - 2 y ) .
Options: a ) 32 / 7 , b ) 13 / 7 , c ) 2 / 7 , d ) 32 / 17 , e ) 2 / 17
Rationale: ". x / y = 3 / 4  ( 4 x + 5 y ) / ( 5 x + 2 y ) = ( 4 ( x / y ) + 5 ) / ( 5 ( x / y ) - 2 ) = ( 4 ( 3 / 4 ) + 5 ) / ( 5 ( 3 / 4 ) - 2 ) = ( 3 + 5 ) / ( 7 / 4 ) = 32 / 7 answer a 32 / 7"
Answer: a
Python program:
a = 4 ** 5
b = a + 5 
c = 3 + 4 
d = c + 5 
e = b / d
Problem: find two integers , neither of which ends in a zero , and whose product is exactly 1 , 000,000
Options: a ) 64 and 15,625 , b ) 60 and 15,625 , c ) 64 and 15,620 , d ) 64 and 15,635 , e ) 64 and 16,625
Rationale: "1 , 000,000 = 10 ^ 6 = 10 x 10 x 10 x 10 x 10 x 10 = ( 2 x 5 ) x ( 2 x 5 ) x ( 2 x 5 ) x ( 2 x 5 ) x ( 2 x 5 ) x ( 2 x 5 ) = ( 2 ^ 6 ) x ( 5 ^ 6 ) = 64 x 15625 so the numbers are 64 and 15,625 answer : a"
Answer: a
Python program:
a = 100 * 100
b = a * 100
c = 100 * 100
d = c * 100
e = 3 * 12
f = e - 4
g = d / f
h = b / g
Problem: the average of 17 numbers is 59 . average of the first 9 of them is 56 and that of the last 9 is 63 . find the 9 th number ?
Options: a ) 64 , b ) 68 , c ) 65 , d ) 66 , e ) 67
Rationale: sum of all the 17 numbers = 17 * 59 = 1,003 sum of the first 9 of them = 9 * 56 = 504 sum of the last 9 of them = 9 * 63 = 567 so , the 8 th number = 504 + 567 - 1,003 = 68 . answer : b
Answer: b
Python program:
a = 9 * 63
b = 17 * 59
c = 56 * 9
d = b - c
e = a - d
Problem: a boat can travel with a speed of 24 km / hr in still water . if the speed of the stream is 4 km / hr , find the time taken by the boat to go 56 km downstream
Options: a ) 2 hours , b ) 4 hours , c ) 5 hours , d ) 6 hours , e ) 7 hours
Rationale: "explanation : speed of the boat in still water = 24 km / hr speed of the stream = 4 km / hr speed downstream = ( 22 + 5 ) = 28 km / hr distance travelled downstream = 56 km time taken = distance / speed = 56 / 28 = 2 hours answer : option a"
Answer: a
Python program:
a = 24 + 4 
b = 56 / a
Problem: carl is facing very difficult financial times and can only pay the interest on a $ 10,000 loan he has taken . the bank charges him a quarterly compound rate of 7 % . what is the approximate interest he pays annually ?
Options: a ) 1000 , b ) 1500 , c ) 3000 , d ) 2500 , e ) 2400
Rationale: "an easy way too attempt this is by approximation : for the first quarter , he pays 7 % of 10,000 which is $ 700 . so for the four quarters in the year , he will approximately play 700 * 4 = 2800 over the entire year . because of compound interest it will be more than 2800 . approx . option c is correct . answer : c"
Answer: c
Python program:
a = 100 * 100
b = 7 / 100
c = 1 + b 
d = c ** 4
e = a * d
f = 100 * 100
g = e - f
Problem: a alone can do a piece of work in 6 days and b alone in 8 days . a and b undertook to do it for rs . 3200 . with the help of c , they completed the work in 3 days . how much is to be paid to c ?
Options: a ) rs . 450 , b ) rs . 600 , c ) rs . 800 , d ) rs . 375 , e ) rs . 400
Rationale: "c ' s 1 day ' s work = 1 / 3 - ( 1 / 6 + 1 / 8 ) = 1 / 24 . a ' s wages : b ' s wages : c ' s wages = 1 / 6 : 1 / 8 : 1 / 24 = 4 : 3 : 1 . c ' s share ( for 3 days ) = rs . ( 3 x 1 / 24 x 3200 ) = rs . 400 . correct option ` ` e ' ' ."
Answer: e
Python program:
a = 1/(3)
b = 1/(8)
c = 1/(6)
d = b + c 
e = a - d
f = e * 3200
g = f * 3
Problem: 1 / 2 - [ ( 2 / 5 * 5 / 32 ) + 1 ] + 9 / 16 =
Options: a ) 29 / 16 , b ) 19 / 16 , c ) 15 / 16 , d ) 9 / 13 , e ) 0
Rationale: "1 / 2 - [ ( 2 / 5 * 5 / 32 ) + 1 ] + 9 / 16 = 1 / 2 - [ ( 1 / 16 ) + 1 ] + 9 / 16 = 1 / 2 - [ 17 / 16 ] + 9 / 16 = 8 / 16 - 17 / 16 + 9 / 16 = 0 the answer is e ."
Answer: e
Python program:
a = 9 / 16
Problem: mariah has decided to hire three workers . to determine whom she will hire , she has selected a group of 20 candidates . she plans to have one working interview with 3 of the 20 candidates every day to see how well they work together . how many days will it take her to have working interviews with all the different combinations of job candidates ?
Options: a ) 520 , b ) 120 , c ) 300 , d ) 30 , e ) 333
Rationale: 520 . answer a
Answer: a
Python program:
a = math.factorial(20)
b = 20 - 3
c = math.factorial(b)
d = a / c
e = math.factorial(3)
f = d / e
g = f / 2
h = g - 20
i = h - 20
j = i - 10
Problem: when positive integer x is divided by 10 , the quotient is y and the remainder is 3 . when 2 x is divided by 7 , the quotient is 3 y and the remainder is 1 . what is the value of 11 y – x ?
Options: a ) 2 , b ) - 2 , c ) 3 , d ) - 3 , e ) 4
Rationale: "( 1 ) x = 10 y + 3 ( 2 ) 2 x = 21 y + 1 ( 2 ) - ( 1 ) : x = 11 y - 2 11 y - x = 2 the answer is a ."
Answer: a
Python program:
a = 2 * 3
b = a - 1
c = 7 * 3
d = 2 * 10
e = c - d
f = b / e
g = 11 * f
h = 2 * 3
i = h - 1
j = 7 * 3
k = 2 * 10
l = j - k
m = i / l
n = 10 * m
o = n + 3 
p = g - o
Problem: at an upscale fast - food restaurant , shin can buy 3 burgers , 7 shakes , and one cola for $ 120 . at the same place it would cost $ 160.50 for 4 burgers , 10 shakes , and one cola . how much would it cost for a meal of one burger , one shake , and one cola ?
Options: a ) $ 21 , b ) $ 27 , c ) $ 31 , d ) $ 39 , e ) it can not be determined
Rationale: "let ' s suppose that the price of a burger is bb , of a shake - ss and that of a cola is cc . we can then construct these equations : 3 b + 7 s + c = 120 4 b + 10 s + c = 160.5 subtracting the first equation from the second gives us b + 3 s = 40.5 now if we subtract the new equation two times from first or 3 times from second we will get b + s + c = 39 . in any case , there is no necessity to know each item ' s price , just the sum . answer : d"
Answer: d
Python program:
a = 160 - 50
b = 120 + a 
c = 160 - 50
d = c * 3
e = b - d
Problem: if 300 ! / 10 ^ n is an integer , what is the largest possible value of n ?
Options: a ) 75 , b ) 74 , c ) 79 , d ) 77 , e ) 72
Rationale: "the question actually asks the highest power of 10 which divides 300 ! ( for a number to be an integer - without any remainder all the trailing zeroe ' s must be divided by the denominator ) 10 = 2 x 5 300 factorial will have 74 as - 300 / 5 = 60 60 / 5 = 12 12 / 5 = 2 so answer will be ( d ) 74"
Answer: d
Python program:
a = 4 + 1 
b = 300 / a
c = 4 + 1 
d = b / c
e = 4 + 1 
f = 300 / e
g = d + f
Problem: 15.06 * 0.0001 = ?
Options: a ) 15060000 , b ) 0.001506 , c ) 0.01506 , d ) 0.1506 , e ) none of these
Rationale: "explanation : clearly after decimal 6 digits should be there . option b"
Answer: b
Python program:
a = 15 / 6
b = a * 100
Problem: addison high school ’ s senior class has 200 boys and 160 girls . if 60 % of the boys and 75 % of the girls attended college , what percentage of the total class that did not attend college ?
Options: a ) 33.33 , b ) 60 , c ) 80 , d ) 66.67 , e ) 64
Rationale: "number of boys attending the college : 0.60 * 200 = 120 number of girls attending the college : 0.75 * 160 = 120 total attending = 120 + 120 = 240 total people = 200 + 160 = 360 % of people attending = 240 / 360 = 66.67 % . d is the correct answer"
Answer: d
Python program:
a = 60 / 100
b = a * 200
c = 75 / 100
d = 1 - c
e = d * 160
f = b + e 
g = 200 + 160 
h = f / g
i = h * 100
Problem: find the fraction which has the same ratio to 1 / 13 that 5 / 34 has to 7 / 48 .
Options: a ) 110 / 1547 , b ) 120 / 1547 , c ) 140 / 1547 , d ) 160 / 1547 , e ) 220 / 1547
Rationale: p : 1 / 13 = 5 / 34 : 7 / 48 as the product of the means is equal to the product of the extremes . p * 7 / 48 = 1 / 13 * 5 / 34 p * 7 / 48 = 5 / 442 p = 240 / 3094 = > p = 120 / 1547 answer : b
Answer: b
Python program:
a = 5 / 34
b = 7 / 48
c = a / b
d = 1 / 13
e = c * d
Problem: two numbers are such that the square of one is 224 less than 8 times the square of the other . if the numbers be in the ratio of 3 : 4 , the numbers are
Options: a ) 6 , 8 , b ) 9 , 12 , c ) 12 , 16 , d ) 16 , 20 , e ) none
Rationale: solution let the numbers be 3 x and 4 x . then , ( 4 x ) 2 = 8 x ( 3 x ) 2 - 224 ‹ = › 16 x 2 = 72 x 2 - 224 ‹ = › 56 x 2 = 224 . ‹ = › x 2 = 4 . ‹ = › x = 2 . so , the numbers are 6 and 8 . answer a
Answer: a
Python program:
a = 3 ** 2
b = a * 224
c = 3 ** 2
d = c * 224
e = 3 ** 2
f = 8 * e
g = 4 ** 2
h = f - g
i = math.gcd(d, h)
j = b / i
k = math.sqrt(j)
Problem: a and b invests rs . 3500 and rs . 4000 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 ) 21 : 5 , b ) 21 : 16 , c ) 21 : 2 , d ) 9 : 9 , e ) 21 : 11
Rationale: "( 3.5 * 6 + 7 * 6 ) : ( 4 * 12 ) 63 : 48 = > 21 : 16 answer : b"
Answer: b
Python program:
a = 3500 * 6
b = 3500 * 2
c = b * 6
d = a + c 
e = 6 + 6 
f = 4000 * e
g = d / f
Problem: a searchlight on top of the watchtower makes 2 revolutions per minute . what is the probability that a man appearing near the tower will stay in the dark for at least 10 seconds ?
Options: a ) 5 / 6 , b ) 3 / 4 , c ) 2 / 3 , d ) 1 / 2 , e ) 3 / 5
Rationale: "the searchlight completes one revolution every 30 seconds . the probability that the man ' s area will be lit up is 10 / 30 = 1 / 3 . the probability that he will stay in the dark is 1 - 1 / 3 = 2 / 3 the answer is c ."
Answer: c
Python program:
a = 10 * 3
b = 10 / a
c = 1 - b
Problem: tom , working alone , can paint a room in 6 hours . peter and john , working independently , can paint the same room in 3 hours and 4 hours , respectively . tom starts painting the room and works on his own for one hour . he is then joined by peter and they work together for an hour . finally , john joins them and the three of them work together to finish the room , each one working at his respective rate . what fraction of the whole job was done by peter ?
Options: a ) 7 / 15 , b ) 9 / 20 , c ) 11 / 25 , d ) 13 / 27 , e ) 15 / 31
Rationale: "tom paints 1 / 6 of the room in the first hour . tom and peter paint 1 / 6 + 1 / 3 = 1 / 2 of the room in the next hour for a total of 4 / 6 . the three people then paint the remaining 2 / 6 in a time of ( 2 / 6 ) / ( 3 / 4 ) = 4 / 9 hours peter worked for 13 / 9 hours so he painted 13 / 9 * 1 / 3 = 13 / 27 of the room . the answer is d ."
Answer: d
Python program:
a = 4 * 4
b = a + 1 
c = 4 / b
Problem: the distance from city a to city b is 30 miles . while driving from city a to city b , bob drives at a constant speed of 40 miles per hour . alice leaves city a 30 minutes after bob . what is the minimum constant speed in miles per hour that alice must exceed in order to arrive in city b before bob ?
Options: a ) 45 , b ) 88 , c ) 120 , d ) 152 , e ) 154
Rationale: "the time it takes bob to drive to city b is 30 / 40 = 0.75 hours . alice needs to take less than 0.25 hours for the trip . alice needs to exceed a constant speed of 30 / 0.25 = 120 miles per hour . the answer is c ."
Answer: c
Python program:
a = 30 / 40
b = 30 / const_60
c = a - b
d = 30 / c
Problem: if x dollars is invested at 10 percent for one year and y dollars is invested at 8 percent for one year , the annual income from the 10 percent investment will exceed the annual income from the 8 percent investment by $ 83 . if $ 2,000 is the total amount invested , how much is invested at 8 percent ?
Options: a ) $ 550 , b ) $ 650 , c ) $ 750 , d ) $ 850 , e ) $ 950
Rationale: "0.1 x = 0.08 ( 2000 - x ) + 83 0.18 x = 243 x = 1350 then the amount invested at 8 % is $ 2000 - $ 1350 = $ 650 the answer is b ."
Answer: b
Python program:
a = 100 * 10
b = a * 2
c = 10 * 8
d = c * 2
e = d + 83 
f = 10 / 100
g = 8 / 100
h = f + g 
i = e / h
j = b - i
Problem: line q has the equation 5 y – 3 x = 90 . if line s is perpendicular to q , has an integer for its y - intercept , and intersects q in the second quadrant , then how many possible line s ’ s exist ? ( note : intersections on one of the axes do not count . )
Options: a ) 58 , b ) 61 , c ) 64 , d ) 67 , e ) 70
Rationale: 5 y - 3 x = 90 and so y = 3 x / 5 + 18 when x = 0 , then y = 18 . when y = 0 , then x = - 30 the slope is 3 / 5 , so the slope of line s is - 5 / 3 . through the point ( - 30 , 0 ) , 0 = - 5 ( - 30 ) / 3 + c the y - intercept is c = - 50 . thus the perpendicular line s can have y - intercepts from - 49 up to 17 . the number of possible lines is 49 + 17 + 1 = 67 the answer is d .
Answer: d
Python program:
a = 90 / 5
b = a - 2
c = 90 / 3
d = 90 / 5
e = c + d 
f = b + e