euclaise/mathqa_programs huggingface dataset:
Problem: for any integer k > 1 , the term “ length of an integer ” refers to the number of positive prime factors , not necessarily distinct , whose product is equal to k . for example , if k = 24 , the length of k is equal to 4 , since 24 = 2 × 2 × 2 × 3 . if x and y are positive integers such that x > 1 , y > 1 , and x + 3 y < 980 , what is the maximum possible sum of the length of x and the length of y ?
Options: a ) 16 , b ) 14 , c ) 18 , d ) 10 , e ) 20
Rationale: "we know that : x > 1 , y > 1 , and x + 3 y < 980 , and it is given that length means no of factors . for any value of x and y , the max no of factors can be obtained only if factor is smallest no all factors are equal . hence , lets start with smallest no 2 . 2 ^ 1 = 2 2 ^ 2 = 4 2 ^ 3 = 8 2 ^ 4 = 16 2 ^ 5 = 32 2 ^ 6 = 64 2 ^ 7 = 128 2 ^ 8 = 256 2 ^ 9 = 512 2 ^ 10 = 1024 ( it exceeds 1000 , so , x ca n ' t be 2 ^ 10 ) so , max value that x can take is 2 ^ 9 , for which has length of integer is 9 . now , since x = 512 , x + 3 y < 980 so , 3 y < 468 = = > y < 156 so , y can take any value which is less than 162 . and to get the maximum no of factors of smallest integer , we can say y = 2 ^ 7 for 2 ^ 7 has length of integer is 7 . so , combined together : 9 + 7 = 16 . a"
Answer: a
Python program:
a = 4 + 3 
b = 4 + 4 
c = b + 1 
d = a + c
Problem: how long will a boy take to run round a square field of side 55 meters , if he runs at the rate of 9 km / hr ?
Options: a ) 88 sec , b ) 45 sec , c ) 1 min , d ) 32 sec , e ) 25 sec
Rationale: "speed = 9 km / hr = 9 * 5 / 18 = 5 / 2 m / sec distance = 55 * 4 = 220 m time taken = 220 * 2 / 5 = 88 sec answer is a"
Answer: a
Python program:
a = 55 * 4
b = 1000 / 3600
c = 9 * b
d = a / c
Problem: a contractor undertook to do a piece of work in 15 days . he employed certain number of laboures but 5 of them were absent from the very first day and the rest could finish the work in only 20 days . find the number of men originally employed ?
Options: a ) 25 , b ) 20 , c ) 19 , d ) 23 , e ) 18
Rationale: "let the number of men originally employed be x . 15 x = 20 ( x â € “ 5 ) or x = 20 answer b"
Answer: b
Python program:
a = 20 * 5
b = 20 - 15
c = a / b
Problem: the temperature of a certain cup of coffee 2 minutes after it was poured was 120 degrees fahrenheit . if the temperature f of the coffee t minutes after it was poured can be determined by the formula f = 120 * 2 ^ ( - at ) + 60 , where f is in degrees fahrenheit and a is a constant . then the temperature of the coffee 30 minutes after it was poured was how many degrees fahrenheit ?
Options: a ) 65 , b ) 60.003662 , c ) 80.2 , d ) 85 , e ) 90
Rationale: "first , we have to find a . we know that after t = 2 minutes the temperature f = 120 degrees . hence : 120 = 120 * ( 2 ^ - 2 a ) + 60 60 = 120 * ( 2 ^ - 2 a ) 60 / 120 = 2 ^ - 2 a 1 / 2 = 2 ^ - 2 a 2 ^ - 1 = 2 ^ - 2 a - 1 = - 2 a 1 / 2 = a now we need to find f after t = 30 minutes : f = 120 * ( 2 ^ - 1 / 2 * 30 ) + 60 f = 120 * ( 2 ^ - 15 ) + 60 f = 120 * ( 1 / 2 ^ 15 ) + 60 f = 120 * 1 / 32768 + 60 f = 0.003662 + 60 = 60.003662 answer b !"
Answer: b
Python program:
a = 60 / 2
b = 1 - 2
c = a * b
d = 2 ** c
e = d * 120
f = e + 60
Problem: 10 % of ( 50 % of $ 500 ) is ?
Options: a ) $ 15 , b ) $ 20 , c ) $ 25 , d ) $ 30 , e ) $ 22
Rationale: "10 % of ( 50 % of $ 500 ) = 10 / 100 ( 50 / 100 * 500 ) = $ 25 answer is c"
Answer: c
Python program:
a = 500 * 50
b = a / 100
c = b * 10
d = c / 100
Problem: a chemical supply company has 60 liters of a 20 % hno 3 solution . how many liters of pure undiluted hno 3 must the chemists add so that the resultant solution is a 50 % solution ?
Options: a ) 12 , b ) 15 , c ) 20 , d ) 24 , e ) 36
Rationale: "60 liters of a 20 % hno 3 solution means hno 3 = 12 liters in 60 liters of the solution . now , let x be the pure hno 3 added . as per question , 12 + x = 50 % of ( 60 + x ) or x = 36 . hence , e"
Answer: e
Python program:
a = 60 / 2
b = 60 * 20
c = b / 100
d = a - c
e = d * 2
Problem: a can finish a piece of work in 5 days . b can do it in 25 days . they work together for four days and then a goes away . in how many days will b finish the work ?
Options: a ) 3 days , b ) 1 day , c ) 2 days , d ) 4 days , e ) 5 days
Rationale: "4 / 5 + ( 4 + x ) / 25 = 1 = > x = 1 day answer : b"
Answer: b
Python program:
a = 1 / 5
b = a * 2
c = 1 / 25
d = c * 2
e = b + d 
f = 1 - e
g = 1 / 25
h = f / g
Problem: 4 women can complete a work in 7 days and 10 children take 14 days to complete the work . how many days will 5 women and 10 children take to complete the work ?
Options: a ) 4 days , b ) 6 days , c ) 7 days , d ) 9 days , e ) 1 days
Rationale: "1 women ' s 1 day work = 1 / 28 1 child ' s 1 day work = 1 / 140 ( 5 women + 10 children ) ' s 1 day work = ( 5 / 28 + 10 / 140 ) = 1 / 4 5 women and 10 children will complete the work in 4 days . answer : a"
Answer: a
Python program:
a = 4 * 7
b = 5 / a
c = 4 * 14
d = 4 / c
e = b + d 
f = 1/(e)
Problem: if p is a prime number greater than 3 , find the remainder when p ^ 2 + 13 is divided by 12 .
Options: a ) 6 , b ) 1 , c ) 0 , d ) 2 , e ) 7
Rationale: "every prime number greater than 3 can be written 6 n + 1 or 6 n - 1 . if p = 6 n + 1 , then p ^ 2 + 13 = 36 n ^ 2 + 12 n + 1 + 13 = 36 n ^ 2 + 12 n + 12 + 2 if p = 6 n - 1 , then p ^ 2 + 13 = 36 n ^ 2 - 12 n + 1 + 13 = 36 n ^ 2 - 12 n + 12 + 2 when divided by 12 , it must leave a remainder of 2 . the answer is d ."
Answer: d
Python program:
a = 1 + 4 
b = a ** 2
c = 13 + b 
d = 12 * 3
e = c - d
Problem: yesterday ' s closing prices of 2,640 different stocks listed on a certain stock exchange were all different from today ' s closing prices . the number of stocks that closed at a higher price today than yesterday was 20 percent greater than the number that closed at a lower price . how many of the stocks closed at a higher price today than yesterday ?
Options: a ) 484 , b ) 726 , c ) 1,100 , d ) 1,320 , e ) 1,440
Rationale: "lets consider the below - the number of stocks that closed at a higher price = h the number of stocks that closed at a lower price = l we understand from first statement - > h + l = 2640 - - - - ( 1 ) we understand from second statement - > h = ( 120 / 100 ) l = > h = 1.2 l - - - - ( 2 ) solve eq ( 1 ) ( 2 ) to get h = 1440 . e is my answer ."
Answer: e
Python program:
a = 20 * 100
b = a - 10
c = b - 10
d = 20 / 100
e = 1 + d 
f = e + 1 
g = c / f
h = 20 / 100
i = 1 + h 
j = g * i
Problem: a bag consists of 20 marbles , of which 6 are blue , 9 are red , and the remainder are white . if lisa is to select a marble from the bag at random , what is the probability that the marble will be red or white ?
Options: a ) 7 / 10 , b ) 2 / 4 , c ) 1 / 4 , d ) 1 / 8 , e ) 1 / 16
Rationale: "bag consists of 20 marbles , of which 6 are blue , 9 are red remainder are white . so , white = 20 - 6 - 9 = 5 . probability that the marble will be red or white = probability that the marble will be red + probability that the marble will be white probability that the marble will be red or white = 9 / 20 + 5 / 20 = 14 / 20 = 7 / 10 hence , answer will be a ."
Answer: a
Python program:
a = 6 + 9 
b = 20 - a
c = b + 9 
d = c / 20
Problem: suppose 12 monkeys take 12 minutes to eat 12 bananas . how many monkeys would it take to eat 72 bananas in 72 minutes ?
Options: a ) 9 , b ) 10 , c ) 11 , d ) 12 , e ) 13
Rationale: "one monkey takes 12 min to eat 1 banana , so in 72 mins 1 monkey will eat 6 bananas , so for 72 bananas in 72 min we need 72 / 6 = 12 monkeys answer : d"
Answer: d
Python program:
a = 72 / 12
b = 72 / a
Problem: a farmer with 1,350 acres of land had planted his fields with corn , sugar cane , and tobacco in the ratio of 5 : 2 : 2 , respectively , but he wanted to make more money , so he shifted the ratio to 2 : 2 : 5 , respectively . how many more acres of land were planted with tobacco under the new system ?
Options: a ) 90 , b ) 150 , c ) 270 , d ) 300 , e ) 450
Rationale: originally ( 2 / 9 ) * 1350 = 300 acres were planted with tobacco . in the new system ( 5 / 9 ) * 1350 = 750 acres were planted with tobacco . thus 750 - 300 = 450 more acres were planted with tobacco . the answer is e .
Answer: e
Python program:
a = 100 * 10
b = 3 * 100
c = a + b 
d = 5 * 10
e = c + d 
f = 5 + 2 
g = f + 2 
h = 5 / g
i = e * h
j = 100 * 10
k = 3 * 100
l = j + k 
m = 5 * 10
n = l + m 
o = 5 + 2 
p = o + 2 
q = 2 / p
r = n * q
s = i - r
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 charlie 9 years from now ?
Options: a ) 44 , b ) 52 , c ) 57 , d ) 65 , e ) 80
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 charlie = 7 * 5 = 35 charlie ' s age 9 years hence = 35 + 9 = 44 answer = a"
Answer: a
Python program:
a = 8 * 3
b = a + 76 
c = 5 + 8 
d = c + 7 
e = b / d
f = e * 8
g = f + 9
Problem: the radius of a semi circle is 6.4 cm then its perimeter is ?
Options: a ) 32.9
b ) 32.4
c ) 22.4
d ) 32.8
e ) 32.1
Rationale: 36 / 7 r = 6.4 = 32.9 answer : a
Answer: a
Python program:
a = circumface / (
b = a + 2
Problem: the grade point average of the entire class is 87 . if the average of one third of the class is 95 , what is the average of the rest of the class ?
Options: a ) 81 , b ) 82 , c ) 83 , d ) 84 , e ) 85
Rationale: "let x be the number of students in the class . let p be the average of the rest of the class . 87 x = ( 1 / 3 ) 95 x + ( 2 / 3 ) ( p ) x 261 = 95 + 2 p 2 p = 166 p = 83 . the answer is c ."
Answer: c
Python program:
a = 87 * 4
b = a - 95
c = 4 - 1
d = b / c
Problem: an equilateral triangle t 2 is formed by joining the mid points of the sides of another equilateral triangle t 1 . a third equilateral triangle t 3 is formed by joining the mid - points of t 2 and this process is continued indefinitely . if each side of t 1 is 30 cm , find the sum of the perimeters of all the triangles .
Options: a ) 180 cm , b ) 220 cm , c ) 240 cm , d ) 270 cm , e ) 300 cm
Rationale: "we have 30 for first triangle , when we join mid - points of first triangle we get the second equilateral triangle then the length of second one is 15 and continues . so we have 30,15 , 7.5 , . . . we have ratio = 1 / 2 , and it is gp type . sum of infinite triangle is a / 1 - r = 30 / 1 - ( 1 / 2 ) = 60 equilateral triangle perimeter is 3 a = 3 * 60 = 180 . so option a ."
Answer: a
Python program:
a = triangle_perimeter + (
Problem: john and amanda stand at opposite ends of a straight road and start running towards each other at the same moment . their rates are randomly selected in advance so that john runs at a constant rate of 2 , 3 , 4 , 5 , or 6 miles per hour and amanda runs at a constant rate of 3 , 4 , 5 , 6 , or 7 miles per hour . what is the probability that john has traveled farther than amanda by the time they meet ?
Options: a ) 1 / 3 , b ) 2 / 5 , c ) 3 / 5 , d ) 4 / 25 , e ) 6 / 25
Rationale: "john will run farther if he runs at 6 mph and amanda runs at 5 mph , 4 mph , or 3 mph . in this case , p ( john runs farther ) = 1 / 5 * 3 / 5 = 3 / 25 john will run farther if he runs at 5 mph and amanda runs at 4 mph or 3 mph . in this case , p ( john runs farther ) = 1 / 5 * 2 / 5 = 2 / 25 john will run farther if he runs at 4 mph and amanda runs at 3 mph . in this case , p ( john runs farther ) = 1 / 5 * 1 / 5 = 1 / 25 overall , p ( john runs farther ) = 3 / 25 + 2 / 25 + 1 / 25 = 6 / 25 the answer is e ."
Answer: e
Python program:
a = 4 * 5
b = 6 / a
Problem: the sum of four consecutive even numbers is 292 . what would be the largest number ?
Options: a ) 33 , b ) 88 , c ) 76 , d ) 123 , e ) 12
Rationale: "let the four consecutive even numbers be 2 ( x - 2 ) , 2 ( x - 1 ) , 2 x , 2 ( x + 1 ) their sum = 8 x - 4 = 292 = > x = 37 smallest number is : 2 ( x + 1 ) = 76 . answer : c"
Answer: c
Python program:
a = 292 - 10
b = a - 2
c = b / 4
d = c + 2 
e = d + 2 
f = e ** 2
g = 292 - 10
h = g - 2
i = h / 4
j = i + 2 
k = j + 2 
l = k + 2 
m = l ** 2
n = f + m 
o = 292 - 10
p = o - 2
q = p / 4
r = q ** 2
s = 292 - 10
t = s - 2
u = t / 4
v = u + 2 
w = v ** 2
x = r + w 
y = n + x
Problem: on dividing 127 by a number , the quotient is 5 and the remainder is 2 . find the divisor .
Options: a ) 25 , b ) 30 , c ) 35 , d ) 40 , e ) 45
Rationale: "d = ( d - r ) / q = ( 127 - 2 ) / 5 = 125 / 5 = 25 a"
Answer: a
Python program:
a = 127 / 5
b = math.floor(a)
Problem: two pipes a and b can separately fill a cistern in 80 minutes and 160 minutes respectively . there is a third pipe in the bottom of the cistern to empty it . if all the three pipes are simultaneously opened , then the cistern is full in 40 minutes . in how much time , the third pipe alone can empty the cistern ?
Options: a ) 90 min , b ) 160 min , c ) 110 min , d ) 120 min , e ) 130 min
Rationale: "1 / 40 - ( 1 / 80 + 1 / 160 ) = - 1 / 160 third pipe can empty in 160 minutes answer : b"
Answer: b
Python program:
a = 1/(80)
b = 1/(160)
c = a + b 
d = 1/(40)
e = c - d
f = 1/(e)
Problem: if 15 students in a class average 70 % on an exam and 10 students average 95 % on the same exam , what is the average in percent for all 25 students ?
Options: a ) 77 % , b ) 78 % , c ) 79 % , d ) 80 % , e ) 81 %
Rationale: "( 15 * 70 + 10 * 95 ) / 25 = 80 % the answer is d ."
Answer: d
Python program:
a = 15 * 70
b = 10 * 95
c = a + b 
d = c / 25
Problem: a reduction of 30 % in the price of oil enables a house wife to obtain 9 kgs more for rs . 1800 , what is the reduced price for kg ?
Options: a ) rs . 55 , b ) rs . 60 , c ) rs . 65 , d ) rs . 70 , e ) rs . 75
Rationale: "explanation : 1800 * ( 30 / 100 ) = 540 - - - - 9 ? - - - - 1 = > rs . 60 answer : b"
Answer: b
Python program:
a = 1800 * 30
b = a / 100
c = b / 9
Problem: a train passes a man standing on a platform in 8 seconds and also crosses the platform which is 273 metres long in 20 seconds . the length of the train ( in metres ) is :
Options: a ) 182 , b ) 176 , c ) 175 , d ) 96 , e ) none of these
Rationale: "explanation : let the length of train be l m . acc . to question ( 273 + l ) / 20 = l / 8 2184 + 8 l = 20 l l = 2184 / 12 = 182 m answer a"
Answer: a
Python program:
a = 20 - 8
b = 273 / a
c = b * 8
Problem: a certain manufacturer produces items for which the production costs consist of annual fixed costs totaling $ 130000 and variables costs averaging $ 10 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 $ 150000 ?
Options: a ) 2,858 , b ) 18,667 , c ) 21,429 , d ) 35,000 , e ) 56,000
Rationale: let the items manufactured or sold bex 130000 + 10 x = 15 x - 150000 5 x = 280000 x = 56000 ans : e
Answer: e
Python program:
a = 150000 + 130000 
b = 15 - 10
c = a / b
d = c / 1000
e = d / 3
f = e + 3
Problem: robert is travelling on his cycle and has calculated to reach point a at 2 p . m . if he travels at 10 km / hr ; he will reach there at 12 noon if he travels at 15 km / hr . at what speed must he travel to reach a at 1 p . m . ?
Options: a ) 17 kmph , b ) 19 kmph , c ) 15 kmph , d ) 12 kmph , e ) 16 kmph
Rationale: d 12 kmph let the distance traveled be x km . then , x / 10 - x / 15 = 2 3 x - 2 x = 60 = > x = 60 km . time taken to travel 60 km at 10 km / hr = 60 / 10 = 6 hrs . so , robert started 6 hours before 2 . p . m . i . e . , at 8 a . m . required speed = 60 / 5 = 12 kmph .
Answer: d
Python program:
a = 1 / 10
b = 1 / 15
c = a - b
d = 2 / c
e = 4 + 1 
f = d / e
Problem: seed mixture x is 40 percent ryegrass and 60 percent bluegrass by weight ; seed mixture y is 25 percent ryegrass and 75 percent fescue . if a mixture of x and y contains 32 percent ryegrass , what percent of the weight of this mixture is x ?
Options: a ) 10 % , b ) 33.33 % , c ) 46.66 % , d ) 50 % , e ) 66.66 %
Rationale: "- - - - - - - - - - - - - - - - > ryegrass x - - - - - - - - - - - - - - > 40 % y - - - - - - - - - - - - - - > 25 % m ( mixture ) - - - - > 32 % 0.4 x + ( m - x ) 0.25 = 0.32 m 0.15 x = 0.07 m x = 0.4666 m x = 46.66 % of m c"
Answer: c
Python program:
a = 32 - 25
b = 40 / 100
c = 25 / 100
d = b - c
e = a / d
Problem: during 2003 , a company produced an average of 2000 products per month . how many products will the company need to produce from 2004 through 2007 in order to increase its monthly average for the period from 2003 through 2007 by 150 % over its 2003 average ?
Options: a ) 287,000 , b ) 290,000 , c ) 284,000 , d ) 285,000 , e ) 286,000
Rationale: company produced 12 * 2000 = 24,000 products in 2003 . if company produces x products from 2004 to 2007 , then total amount of product produced in 4 years ( 2003 through 2007 ) is x + 24,000 . the gives the average of ( x + 24,000 ) / 4 . this average needs to be 300 % higher than that in 2003 . in math terms , 36,000 + 150 % ( 36,000 ) = 80,000 . so : ( x + 36,000 ) / 4 = 80,000 x + 36,000 = 320,000 x = 284,000 the answer is c .
Answer: c
Python program:
a = 150 / 100
b = a + 1 
c = b * 2000
d = 2007 - 2003
e = d + 1 
f = c * e
g = f * 12
h = 2000 * 12
i = g - h
j = i / 1000
Problem: in how many ways can a 5 - letter password be chosen , using the letters a , b , c , d , e , f , g , and / or h such that at least one letter is repeated within the password ?
Options: a ) 720 , b ) 864 , c ) 900 , d ) 936 , e ) 26048
Rationale: total number of four letter passwords = 8 * 8 * 8 * 8 * 8 = 32768 - - - - - - ( 1 ) total number of passwords in which no letter repeats = 8 c 5 * 5 ! = 56 * 120 = 6720 - - - - - - ( 2 ) therefore required value = ( 1 ) - ( 2 ) = 32768 - 6720 = 26048 e
Answer: e
Python program:
a = 5 + 3 
b = a ** 5
c = 5 + 3 
d = math.factorial(c)
e = math.factorial(3)
f = d / e
g = b - f
Problem: how many seconds will a 700 metre long train take to cross a man walking with a speed of 3 km / hr in the direction of the moving train if the speed of the train is 63 km / hr ?
Options: a ) 25 , b ) 30 , c ) 42 , d ) 45 , e ) 50
Rationale: "relative speed of the train = 63 - 3 = 60 kmph = 60 * 5 / 18 = 50 / 3 m / sec t = 700 * 3 / 50 = 42 sec answer : c"
Answer: c
Python program:
a = 63 - 3
b = const_0_2778 * a
c = 700 / b
Problem: a man walking at a constant rate of 5 miles per hour is passed by a woman traveling in the same direction along the same path at a constant rate of 15 miles per hour . the woman stops to wait for the man 2 minutes after passing him , while the man continues to walk at his constant rate . how many minutes must the woman wait until the man catches up ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 5 , e ) 4
Rationale: "when the woman passes the man , they are aligned ( m and w ) . they are moving in the same direction . after 5 minutes , the woman ( w ) will be ahead the man ( m ) : m - - - - - - m - - - - - - - - - - - - - - - w w in the 5 minutes , after passing the man , the woman walks the distance mw = ww , which is 2 * 15 / 60 = 1 / 2 miles and the man walks the distance mm , which is 2 * 5 / 60 = 1 / 6 mile . the difference of 1 / 2 - 1 / 6 = 1 / 3 miles ( mw ) will be covered by the man in ( 1 / 3 ) / 5 = 1 / 15 of an hour , which is 4 minutes . answer e ."
Answer: e
Python program:
a = 2 / const_60
b = 15 - 5
c = a * b
d = c / 5
e = const_60 * d
Problem: a certain galaxy is known to comprise approximately 2 x 10 ^ 11 stars . of every 50 million of these stars , one is larger in mass than our sun . approximately how many stars in this galaxy are larger than the sun ?
Options: a ) 800 , b ) 1,250 , c ) 8,000 , d ) 4,000 , e ) 80,000
Rationale: total no . of stars on galaxy = 2 * 10 ^ 11 of every 50 million stars , 1 is larger than sun . 1 million = 10 ^ 6 therofore , 50 million = 50 * 10 ^ 6 total no . of stars larger than sun = 2 * 10 ^ 11 / 50 * 10 ^ 6 = 20 * 10 ^ 3 / 5 = 4000 therefore answer is d
Answer: d
Python program:
a = 2 * 10
b = a / 50
c = 10 ** 4
d = b * c
e = d / 1000
f = e * 2
Problem: a side of beef lost 35 percent of its weight in processing . if the side of beef weighed 550 pounds after processing , how many pounds did it weigh before processing ?
Options: a ) 191 , b ) 355 , c ) 737 , d ) 846 , e ) 1,560
Rationale: "let weight of side of beef before processing = x ( 65 / 100 ) * x = 550 = > x = ( 550 * 100 ) / 65 = 846 answer d"
Answer: d
Python program:
a = 550 * 100
b = 100 - 35
c = a / b
Problem: right triangle abc is to be drawn in the xy - plane so that the right angle is at a and ab is parallel to the y - axis . if the x - and y - coordinates of a , b , and c are to be integers that are consistent with the inequalities - 5 ≤ x ≤ 2 and 4 ≤ y ≤ 9 , then how many different triangles can be drawn that will meet these conditions ?
Options: a ) 54 , b ) 1440 , c ) 2,160 , d ) 2,916 , e ) 148,824
Rationale: "we have the rectangle with dimensions 9 * 5 ( 9 horizontal dots and 5 vertical ) . ab is parallel to y - axis and ac is parallel to x - axis . choose the ( x , y ) coordinates for vertex a : 9 c 1 * 5 c 1 ; choose the x coordinate for vertex c ( as y coordinate is fixed by a ) : 8 c 1 , ( 9 - 1 = 8 as 1 horizontal dot is already occupied by a ) ; choose the y coordinate for vertex b ( as x coordinate is fixed by a ) : 4 c 1 , ( 5 - 1 = 4 as 1 vertical dot is already occupied by a ) . 9 c 1 * 5 c * 8 c 1 * 4 c 1 = 1440 . answer : b ."
Answer: b
Python program:
a = 5 - 1
b = 5 * a
c = 9 * 5
d = b * c
Problem: one half of a two digit number exceeds its one third by 5 . what is the sum of the digits of the number ?
Options: a ) a ) 3 , b ) b ) 5 , c ) c ) 7 , d ) d ) 9 , e ) e ) 11
Rationale: "x / 2 – x / 3 = 5 = > x = 30 3 + 0 = 3 answer : a"
Answer: a
Python program:
a = 5 * 4
b = reminder + (
Problem: the triplicate ratio of 1 : 8 is ?
Options: a ) 1 : 0 , b ) 1 : 8 , c ) 1 : 7 , d ) 1 : 512 , e ) 1 : 1
Rationale: "13 : 83 = 1 : 512 answer : d"
Answer: d
Python program:
a = 2 ** 0
b = 3 ** 0
c = a / b
Problem: a white paper 2 inches wide is placed around a rectangular black paper with dimensions 8 inches by 12 inches . what is the area of the white paper , in square inches ?
Options: a ) 96
b ) 196
c ) 128
d ) 144
e ) 168
Rationale: this question is an example of a ' punch out ' question - we have to find the area of everything , then ' punch out ' the part that we do n ' t want . we ' re told that a white paper 2 inches wide is placed around a rectangular black paper with dimensions 8 inches by 12 inches . we ' re asked for the area of the frame , in square inches . area of a rectangle = ( length ) ( width ) so the area of the black paper is . . . ( 8 ) ( 12 ) = 96 the white paper ' adds ' 2 inches to the top , bottom , left and right ' sides ' of the picture , so the area of everything is . . . ( 8 + 2 + 2 ) ( 12 + 2 + 2 ) = ( 12 ) ( 16 ) = 196 when we ' punch out ' the area of the black paper , we ' ll be left with the area of the white paper : 192 - 96 = 96 final answer : b
Answer: b
Python program:
a = 8 * 2
b = 12 * a
Problem: find the greatest number which on dividing 4351 and 5161 , leaves a reminder of 8 and 10 respectively
Options: a ) 125 , b ) 111 , c ) 129 , d ) 101 , e ) 141
Rationale: "explanation : in this type of question , its obvious we need to calculate the hcf , trick is hcf of ( 4351 - 8 ) and ( 5161 - 10 ) = hcf ( 4343 , 5151 ) = 101 option d"
Answer: d
Python program:
a = 4351 - 8
b = 5161 - 10
c = math.gcd(a, b)
Problem: the difference of two numbers is 1340 . on dividing the larger number by the smaller , we get 6 as quotient and the 15 as remainder . what is the smaller number ?
Options: a ) a ) 265 , b ) b ) 280 , c ) c ) 290 , d ) d ) 300 , e ) e ) 310
Rationale: "let the smaller number be x . then larger number = ( x + 1340 ) . x + 1340 = 6 x + 15 5 x = 1325 x = 265 smaller number = 265 . a )"
Answer: a
Python program:
a = 1340 + 15 
b = 6 - 1
c = a / b