euclaise/mathqa_programs huggingface dataset:
Problem: the number of positive integers n in the range 12 ≤ n ≤ 40 such that the product ( n − 1 ) ( n − 2 ) . . . 3 . 2.1 is not divisible by n is
Options: a ) 5 , b ) 7 , c ) 13 , d ) 14 , e ) 16
Rationale: explanation : ( n - 1 ) ! is always divisible by n unless and untill n is a prime number e . g . ( 5 - 1 ) ! = 4 ! which is not divisible by 5 bcuz 5 is a prime number therefore , find all prime numbers in the range of 12 to 40 i . e 13 , 17 , 19 , 23 , 29 , 31 and 37 hence , the number of positive integers n is 7 . answer : b
Answer: b
Python program:
a = 3 + 2 
b = a + 2
Problem: 79 laboratories raise the bacterium , the laboratory have 48 culture dishes on average , which has about 25,073 bacteria each . how many bacteria are there approximately ?
Options: a ) 10 ^ 7 , b ) 10 ^ 9 , c ) 10 ^ 8 , d ) 10 ^ 10 , e ) 10 ^ 11
Rationale: 79 laboratories raise the bacterium , the laboratory have 48 culture dishes on average , which has about 25,073 bacteria each . how many bacteria are there approximately ? a . 10 ^ 7 b . 10 ^ 8 c . 10 ^ 9 d . 10 ^ 10 e . 10 ^ 11 - > due to approximately , 79 = 80 , 48 = 50 , 25,073 = 25,000 are derived , which makes ( 79 ) ( 48 ) ( 25,075 ) = ( 80 ) ( 50 ) ( 25,000 ) = 10 ^ 8 . the answer is c .
Answer: c
Python program:
a = 79 * 48
b = const_0_25 * 1000
c = b * 100
d = 3 + 4 
e = d * 10
f = c + e 
g = f + 3 
h = a * g
Problem: a set consists of 15 numbers , all are even or multiple of 5 . if 8 numbers are even and 10 numbers are multiple of 5 , how many numbers is multiple of 10 ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 5
Rationale: "{ total } = { even } + { multiple of 5 } - { both } + { nether } . since { neither } = 0 ( allare even or multiple of 5 ) then : 15 = 8 + 10 - { both } + 0 ; { both } = 3 ( so 1 number is both even and multiple of 5 , so it must be a multiple of 10 ) . answer : d ."
Answer: d
Python program:
a = 15 - 10
Problem: the greatest number that divides 2928 and 3240 leaving remainders 5 and 1 respectively is :
Options: a ) 74 , b ) 78 , c ) 79 , d ) 81 , e ) 85
Rationale: "explanation : 2928 - 5 = 2923 , 3240 - 1 = 3239 highest number that can divide 2923 and 3239 is hcf of numbers . hcf of 2923 and 3239 = 79 answer : c"
Answer: c
Python program:
a = 3240 - 1
b = math.gcd(2928, 3240)
c = a / b
Problem: find large number from below question the difference of 2 no ' s is 1365 . on dividing the larger number by the smaller , we get 6 as quotient and the 15 as remainder
Options: a ) 1236 , b ) 1356 , c ) 1470 , d ) 1556 , e ) 1635
Rationale: "let the smaller number be x . then larger number = ( x + 1365 ) . x + 1365 = 6 x + 15 5 x = 1350 x = 270 large number = 270 + 1365 = 1635 e"
Answer: e
Python program:
a = 2 - 6
b = 1365 - 1
c = a / b
d = c * 1365
Problem: a sum was put at simple interest at certain rate for 3 years . had it been put at 1 % higher rate it would have fetched rs . 72 more . the sum is : a . rs . 2,400 b . rs . 2,100 c . rs . 2,200 d . rs . 2,480
Options: a ) 2000 , b ) 2100 , c ) 2200 , d ) 2300 , e ) 2400
Rationale: 1 percent for 3 years = 72 1 percent for 1 year = 24 = > 100 percent = 2400 answer : e
Answer: e
Python program:
a = 72 / 3
b = a * 100
Problem: a and b together can do a work in 3 days . if a alone can do it in 6 days . in how many days can b alone do it ?
Options: a ) 10 , b ) 99 , c ) 77 , d ) 55 , e ) 6
Rationale: "1 / 3 – 1 / 6 = 1 / 6 = > 6 answer : e"
Answer: e
Python program:
a = 1 / 3
b = 1 / 6
c = a - b
d = 1/(c)
e = 2 + 3 
f = 2 / e
g = d + f
Problem: a man is 26 years older than his studen . in two years , his age will be twice the age of his student . the present age of his student is :
Options: a ) 20 , b ) 21 , c ) 22 , d ) 23 , e ) 24
Rationale: let the student ' s present age be x years . then , man ' s present age = ( x + 26 ) years . ( x + 26 ) + 2 = 2 ( x + 2 ) x + 28 = 2 x + 4 x = 24 . answer : e
Answer: e
Python program:
a = 26 - 2
Problem: a = { 2 , 3 , 4 , 5 } b = { 4 , 5 , 6 , 7 , 8 } two integers will be randomly selected from the sets above , one integer from set a and one integer from set b . what is the probability t that the sum of the two integers will equal 9 ?
Options: a ) 0.15 , b ) 0.20 , c ) 0.25 , d ) 0.30 , e ) 0.33
Rationale: rearrange the first set : a = { 54 , 32 } b = { 45 , 67 , 8 } as you can see numbers in each column ( the numbers of the same color ) give the sum of 9 . so there are 4 such pares possible , total # of pairs is 4 * 5 = 20 . t = favorable / total = 4 / 20 = 0.2 . answer : b . or : we can select any number from set a ( 4 / 4 = 1 ) but in this case we must select its matching pair from set b ( the number of the same color ) and since there are only one matching pair of this particular number in b then the probability of this is 1 / 5 . so , overall : t = 1 * 1 / 5 . answer : b .
Answer: b
Python program:
a = 4 * 5
b = 4 / a
Problem: the population of a bacteria culture doubles every 4 minutes . approximately how many minutes will it take for the population to grow from 1,000 to 500,000 bacteria
Options: a ) 10 , b ) 12 , c ) 14 , d ) 36 , e ) 18
Rationale: "this one ' s easy . 1000 * 2 ^ t = 500,000 2 ^ t = 500 now gauging , since 2 ^ 8 = 256 , then 2 ^ 9 = 512 so t = 9 but be careful , ' t ' is in time intervals of 4 minutes so answer is 9 * 4 = 36 minutes answer ( d )"
Answer: d
Python program:
a = 4 + 1 
b = a * 1
c = b * 100
d = c / 1
e = math.log(d)
f = e * 4
Problem: how many pieces can a square pizza be cut into by making 4 linear cuts ?
Options: a ) 10
b ) 11
c ) 12
d ) 13
e ) 14
Rationale: 1 cut makes two pieces a second cut can make 4 pieces by cutting through 2 of the pieces a third cut can make 7 pieces by cutting through 3 of the pieces a fourth cut can make 11 pieces by cutting through 4 of the pieces b
Answer: b
Python program:
a = 4 * 3
b = a - 1
Problem: two trains of length 220 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively . in what time will they be clear of each other from the moment they meet ?
Options: a ) 28 , b ) 266 , c ) 990 , d ) 25 , e ) 11
Rationale: "relative speed = ( 42 + 30 ) * 5 / 18 = 4 * 5 = 20 mps . distance covered in passing each other = 220 + 280 = 500 m . the time required = d / s = 500 / 20 = 25 sec . answer : d"
Answer: d
Python program:
a = 220 + 280 
b = 42 + 30 
c = b * const_0_2778
d = a / c
Problem: we bought orange juice and apple juice at the store . a bottle of orange juice costs 70 cents and a bottle of apple juice costs 60 cents . we bought a total of 70 bottles for $ 46.20 . how many bottles of orange juice did we buy ?
Options: a ) 32 , b ) 36 , c ) 38 , d ) 42 , e ) 48
Rationale: let oj be the bottles of orange juice and let aj be the bottles of apple juice . ( oj ) + aj = 70 . aj = 70 - ( oj ) . 70 ( oj ) + 60 aj = 4620 . 70 ( oj ) + 60 ( 70 - oj ) = 4620 . 10 ( oj ) + 4200 = 4620 . oj = 42 . the answer is d .
Answer: d
Python program:
a = 46 * 2
b = 60 * 70
c = a - b
d = 70 - 60
e = c / d
Problem: if x is to be chosen at random from the integers between 1 to 7 , inclusive , and y is to be chosen at random from the integers between 7 and 11 , inclusive , what is the probability that x + y will be even ?
Options: a ) 3 / 5 , b ) 4 / 7 , c ) 5 / 7 , d ) 18 / 35 , e ) 23 / 35
Rationale: "x + y will be even if x and y are both even or both odd . p ( x and y are both even ) = 3 / 7 * 2 / 5 = 6 / 35 p ( x and y are both odd ) = 4 / 7 * 3 / 5 = 12 / 35 p ( x + y is even ) = 6 / 35 + 12 / 35 = 18 / 35 the answer is d ."
Answer: d
Python program:
a = 2 * 2
b = 3 * 2
c = a + b 
d = 7 - 1
e = 7 * d
f = c / e
Problem: a glucose molecule contains 6 carbon , 12 hydrogen and 6 oxygen atoms . during photosynthesis , green plants create glucose daily . if , over a period of 12 days , an eucalyptus plant ( a green plant ) creates 12000 molecules of glucose then approximately how many more atoms of hydrogen than carbon are created on average per day ?
Options: a ) 60,000 , b ) 6,000 , c ) 2,500 , d ) 250 , e ) 0
Rationale: solution : we know that 12,000 glucose molecules are created over a period of 12 days . therefore 12,000 / 12 = 1,000 glucose molecules are created on average per day . each glucose molecule contains 6 carbon atoms and 12 hydrogen atoms . so 1,000 glucose molecules contain 6 × 1,000 = 6,000 carbon atoms and 12 × 1,000 = 12,000 hydrogen atoms . the difference is 12,000 – 6,000 = 6,000 . the correct answer is b .
Answer: b
Python program:
a = 12000 / 12
b = a * 12
Problem: a basketball player practices 2 hours a day monday through friday . how many hours must he practice saturday and sunday combined if he wants to practice an average ( arithmetic mean ) of 3 hours a day for the 7 day week ?
Options: a ) 8 , b ) 9 , c ) 10 , d ) 11 , e ) 12
Rationale: total number of hours practiced mon through fri : 5 * 2 = 10 total number of hours that will be practiced in a 7 day week considering 3 hours a day average : 7 * 3 = 21 thus 21 - 10 = 11 hours still need to be practiced ( on sat and sun combined ) d
Answer: d
Python program:
a = 3 * 7
b = 2 + 3 
c = b * 2
d = a - c
Problem: when the positive integer k is divided by the positive integer n , the remainder is 11 . if k / n = 83.2 , what is the value of n ?
Options: a ) 9 , b ) 20 , c ) 55 , d ) 70 , e ) 81
Rationale: here ' s an approach that ' s based on number properties and a bit ofbrute forcemath : we ' re told that k and n are both integers . since k / n = 81.2 , we can say that k = 81.2 ( n ) n has tomultiply outthe . 2 so that k becomes an integer . with the answers that we have to work with , n has to be a multiple of 5 . eliminate a and e . with the remaining answers , we can test the answers and find the one that fits the rest of the info ( k / n = 81.2 and k / n has a remainder of 11 ) answer b : if n = 20 , then k = 1624 ; 1624 / 20 has a remainder of 4 not a match answer c : if n = 55 , then k = 4466 ; 4466 / 55 has a remainder of 11 match . final answer : e
Answer: e
Python program:
a = 83 * 2
b = a - 2
c = b / 10
d = 2 / 10
e = 11 / d
f = c * e
g = f + 11 
h = 2 / 10
i = 11 / h
j = g / i
Problem: what sum of money will produce rs . 90 as simple interest in 4 years at 4 1 / 2 percent ?
Options: a ) rs . 550 , b ) rs . 650 , c ) rs . 600 , d ) rs . 500 , e ) rs . 700
Rationale: "explanation : 90 = ( p * 4 * 9 / 2 ) / 100 90 = 36 p / 100 p = 500 answer : option d"
Answer: d
Python program:
a = 1 / 2
b = 4 + a 
c = 4 * b
d = c / 100
e = 90 / d
Problem: two trains are moving in the same direction at 72 kmph and 36 kmph . the faster train crosses a man in the slower train in 7 seconds . find the length of the faster train ?
Options: a ) 70 , b ) 20 , c ) 30 , d ) 75 , e ) 15
Rationale: "relative speed = ( 72 - 36 ) * 5 / 18 = 2 * 5 = 10 mps . distance covered in 27 sec = 7 * 10 = 70 m . the length of the faster train = 70 m . answer : a"
Answer: a
Python program:
a = 72 - 36
b = a / const_3_6
c = b * 7
Problem: if the average ( arithmetic mean ) of a and b is 80 , and the average of b and c is 180 , what is the value of a − c ?
Options: a ) − 220 , b ) − 200 , c ) 100 , d ) 135 , e ) it can not be determined from the information given
Rationale: "a + b = 160 b + c = 360 a - c = - 200 . imo option b ."
Answer: b
Python program:
a = 180 * 2
b = 80 * 2
c = a - b
Problem: exactly 3 / 7 of the people in the room are under the age of 21 , and exactly 5 / 8 of the people in the room are over the age of 65 . if the total number of the people in the room is greater than 50 and less than 100 , how many people in the room are under the age of 21 ?
Options: a ) 24 , b ) 35 , c ) 39 , d ) 60 , e ) 65
Rationale: "the total number of the people in the room must be a multiple of both 7 and 8 ( in order 3 / 7 and 5 / 8 of the number to be an integer ) , thus the total number of the people must be a multiple of lcm of 7 and 8 , which is 56 . since , the total number of the people in the room is greater than 50 and less than 100 , then there are 56 people in the room . therefore there are 3 / 7 * 56 = 24 people in the room under the age of 21 . answer : a ."
Answer: a
Python program:
a = 7 * 8
b = a * 3
c = b / 7
Problem: alice picked a number of pears , each of which she sold , canned , or poached . she canned 20 percent more than she poached , and she poached 50 percent less than she sold . if there were a total of 42 pears , then how many pears did she sell ?
Options: a ) 20 , b ) 21 , c ) 22 , d ) 23 , e ) 24
Rationale: let x be the amount of pears that she sold . then she poached 0.5 x . then she canned 1.2 * 0.5 x = 0.6 x . the total amount of pears is x + 0.5 x + 0.6 x = 2.1 x the fraction of pears that she sold is 1 / 2.1 = 10 / 21 . the number of pears she sold is ( 10 / 21 ) * 42 = 20 . the answer is a .
Answer: a
Python program:
a = 50 / 100
b = 50 / 100
c = 20 / 100
d = 1 + c 
e = b * d
f = a + e 
g = 1 + f 
h = 42 / g
Problem: the value of x + x ( x ^ x ) when x = 10 is
Options: a ) 100000000010 , b ) 10000000010 , c ) 10000000000 , d ) 1200000000 , e ) 12000000010
Rationale: solution : x + x ( x ^ x ) put the value of x = 10 in the above expression we get , 10 + 10 ( 10 ^ 10 ) = 10 + 10 ( 10 ã — 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 ) = 10 + 10 ( 10000000000 ) = 10 + 100000000000 = 100000000010 answer : ( a )
Answer: a
Python program:
a = 10 ** 10
b = 10 * a
c = b + 10
Problem: if k is a non - negative integer and 30 ^ k is a divisor of 929260 then 3 ^ k - k ^ 3 =
Options: a ) 0 , b ) 1 , c ) 45 , d ) 130 , e ) 520
Rationale: 9 + 2 + 9 + 2 + 6 + 0 = 28 , so this number is not divisible by 3 and thus not divisible by 30 . therefore , k = 0 3 ^ k - k ^ 3 = 1 - 0 = 1 the answer is b .
Answer: b
Python program:
a = 3 / 3
Problem: a sum was put at simple interest at a certain rate for 4 years had it been put at 2 % higher rate , it would have fetched 56 more . find the sum .
Options: a ) 500 , b ) 600 , c ) 700 , d ) 800 , e ) none of these
Rationale: "difference in s . i . = p × t / 100 ( r 1 − r 2 ) ⇒ 56 = p × 4 × 2 / 100 ( ∵ r 1 - r 2 = 2 ) ⇒ p = 56 × 100 / 4 × 2 = 700 answer c"
Answer: c
Python program:
a = 2 / 100
b = a * 4
c = 56 / b
Problem: a vending machine randomly dispenses 4 different types of fruit candy . there are twice as many apple candies as orange candies , twice as many strawberry candies as grape candies , and twice as many apple candies as strawberry candies . if each candy cost $ 0.1 , and there are exactly 90 candies , what is the minimum amount of money required to guarantee that you would buy at least 3 of each type of candy ?
Options: a ) $ 8.30 , b ) $ 20.75 , c ) $ 22.50 , d ) $ 42.75 , e ) $ 45.00
Rationale: let number of apple , orange , strawberry and grape candies be a , o , s and g respectively . a = 2 o s = 2 g a = 2 s a = 4 g a + o + s + g = 90 = > a + a / 2 + a / 2 + a / 4 = 90 = > 9 a / 4 = 90 = > a = 40 o = 20 s = 20 g = 10 cost of each candy = . 1 $ mininum amount of money required to guarantee that you would buy at least three of each type of candy we can buy 40 apple candies , 20 orange candies , 20 strawberry candies and 3 grape candies to ensure atleast 3 of each type of candies . total = 83 candies amount required = 83 * . 1 = 8.30 $ answer a
Answer: a
Python program:
a = 1 / 2
b = 4 + a 
c = 90 / b
d = c * 2
e = 1 / 2
f = 4 + e 
g = 90 / f
h = d + g 
i = 1 / 2
j = 4 + i 
k = 90 / j
l = h + k 
m = l + 3 
n = 0 * 1
Problem: a can do a work in 6 days , b can do a work in 8 days and c can do it in 12 days . b left work after 4 days . for how many number of days should a and c should work together to complete the remaining work ?
Options: a ) 0.1 , b ) 0.2 , c ) 0.3 , d ) 0.4 , e ) 0.5
Rationale: "b work 1 / 8 * 4 = 1 / 2 remaining work = 1 - 1 / 2 = 1 / 2 a and c work together = 1 / 6 + 1 / 12 = 3 / 12 = 1 / 4 take reciprocal 4 * remaining work = 2 * 1 / 4 = 0.5 answer : e"
Answer: e
Python program:
a = 4 / 6
b = 4 / 8
c = a + b
Problem: the tax on a commodity is diminished by 20 % and its consumption increased by 25 % . the effect on revenue is ?
Options: a ) 2 % decrease , b ) 8 % decrease , c ) 9 % decrease , d ) 0 % no change , e ) 2 % decrease
Rationale: "100 * 100 = 10000 80 * 125 = 10000 - - - - - - - - - - - 10000 - - - - - - - - - - - 0 100 - - - - - - - - - - - ? = > 0 % no change answer : d"
Answer: d
Python program:
a = 25 / 100
b = 1 + a 
c = 20 / 100
d = 1 - c
e = b * d
f = e * 100
g = 100 - f
Problem: if n is a prime number greater than 4 , what is the remainder when n ^ 2 is divided by 12 ?
Options: a ) 0 , b ) 2 , c ) 1 , d ) 3 , e ) 5
Rationale: "there are several algebraic ways to solve this question including the one under the spoiler . but the easiest way is as follows : since we can not have two correct answersjust pick a prime greater than 4 , square it and see what would be the remainder upon division of it by 12 . n = 7 - - > n ^ 2 = 49 - - > remainder upon division 49 by 12 is 1 . answer : c ."
Answer: c
Python program:
a = 4 + 2 
b = a ** 2
c = 12 * 4
d = b - c
Problem: a can do a particular work in 6 days . b can do the same work in 8 days . a and b signed to do it for rs . 3680 . they completed the work in 3 days with the help of c . how much is to be paid to c ?
Options: a ) s . 380 , b ) s . 600 , c ) s . 460 , d ) s . 400 , e ) s . 480
Rationale: "explanation : amount of work a can do in 1 day = 1 / 6 amount of work b can do in 1 day = 1 / 8 amount of work a + b can do in 1 day = 1 / 6 + 1 / 8 = 7 / 24 amount of work a + b + c can do = 1 / 3 amount of work c can do in 1 day = 1 / 3 - 7 / 24 = 1 / 24 work a can do in 1 day : work b can do in 1 day : work c can do in 1 day = 1 / 6 : 1 / 8 : 1 / 24 = 4 : 3 : 1 amount to be paid to c = 3680 × ( 1 / 8 ) = 460 answer : option c"
Answer: c
Python program:
a = 1 / 3
b = 1 / 6
c = 1 / 8
d = b + c 
e = a - d
f = 3 * e
g = f * 3680
Problem: in what time will a train 100 meters long cross an electric pole , if its speed is 36 km / hr
Options: a ) 8.5 seconds , b ) 10 seconds , c ) 3.5 seconds , d ) 2.5 seconds , e ) 2.6 seconds
Rationale: "first convert speed into m / sec speed = 36 * ( 5 / 18 ) = 10 m / sec time = distance / speed = 100 / 10 = 10 seconds answer : b"
Answer: b
Python program:
a = 36 * const_0_2778
b = 100 / a
Problem: if x / ( 12 p ) is an odd prime number , where x is a positive integer and p is a prime number , what is the least value of x ?
Options: a ) 22 , b ) 33 , c ) 44 , d ) 66 , e ) 72
Rationale: "x / ( 12 p ) = odd prime number x = odd prime number * 11 p least value of x = lowest odd prime number * 11 * lowest value of p = 3 * 12 * 2 = 72 answer e"
Answer: e
Python program:
a = 3 * 12
b = a * 2
Problem: given f ( x ) = 3 x – 5 , for what value of x does 2 * [ f ( x ) ] – 16 = f ( x – 6 ) ?
Options: a ) 1 , b ) 4 , c ) 6 , d ) 7 , e ) 13
Rationale: "2 ( 3 x - 5 ) - 16 = 3 ( x - 6 ) - 5 3 x = 3 x = 1 the answer is a ."
Answer: a
Python program:
a = 2 * 5
b = a + 16 
c = 3 * 6
d = c + 5 
e = b - d
f = 2 * 3
g = 3 * 1
h = f - g
i = e / h
Problem: a batsman in his 12 th innings makes a score of 80 and thereby increases his average by 3 runs . what is his average after the 12 th innings if he had never been ‘ not out ’ ?
Options: a ) 42 , b ) 43 , c ) 44 , d ) 45 , e ) 47
Rationale: let ‘ x ’ be the average score after 12 th innings ⇒ 12 x = 11 × ( x – 3 ) + 80 ∴ x = 47 answer e
Answer: e
Python program:
a = 12 * 3
b = 80 - a
c = b + 3
Problem: each of the three people individually can complete a certain job in 4 , 5 , and 8 hours , respectively . what is the lowest fraction of the job that can be done in 1 hour by 2 of the people working together at their respective rates ?
Options: a ) 9 / 20 , b ) 11 / 20 , c ) 11 / 32 , d ) 13 / 40 , e ) 17 / 40
Rationale: "the two slowest people work at rates of 1 / 5 and 1 / 8 of the job per hour . the sum of these rates is 1 / 5 + 1 / 8 = 13 / 40 of the job per hour . the answer is d ."
Answer: d
Python program:
a = 1 / 5
b = 1 / 8
c = a + b
Problem: a train travels from new york to chicago , a distance of approximately 840 miles , at an average rate of 60 miles per hour and arrives in chicago at 7 : 00 in evening , chicago time . at what hour in the morning , new york time , did the train depart for chicago ? ( note : chicago time is one hour earlier than new york time )
Options: a ) 3 : 00 , b ) 4 : 00 , c ) 5 : 00 , d ) 6 : 00 , e ) 7 : 00
Rationale: "7 : 00 in evening in chicago = 8 : 00 in evening in new york . so , the train was in chicago 8 : 00 in the evening , new york time . the trip took t = d / r = 840 / 60 = 14 hours . therefore , the train depart from new york at 8 : 00 - 14 hours = 6 : 00 in the morning , new york time . answer : d ."
Answer: d
Python program:
a = 7 + 1 
b = a * 3
c = 840 / 60
d = b - c
Problem: if the ratio of a to b is 4 to 3 and the ratio of b to c is 1 to 5 , what is the ratio of a to c ?
Options: a ) 4 / 15 , b ) 1 / 3 , c ) 2 / 5 , d ) 4 / 5 , e ) 7 / 6
Rationale: a : b = 4 : 3 - - 1 b : c = 1 : 5 = > b : c = 3 : 15 - - 2 from 1 and 2 , we get a : c = 4 : 15 answer a
Answer: a
Python program:
a = 4 * 1
b = 3 * 5
c = a / b
Problem: the h . c . f . of two numbers is 63 and the other two factors of their l . c . m . are 11 and 17 . the larger of the two numbers is :
Options: a ) 3421 , b ) 2132 , c ) 1111 , d ) 4323 , e ) 1071
Rationale: ": explanation : clearly , the numbers are ( 63 x 11 ) and ( 63 x 17 ) . { \ color { blue } \ therefore } larger number = ( 63 x 17 ) = 1071 . answer : e ) 1071"
Answer: e
Python program:
a = 63 * 17
Problem: find the probability that a leap year selected at random will not have 53 sundays or mondays .
Options: a ) 6 / 7 , b ) 5 / 7 , c ) 4 / 7 , d ) 3 / 7 , e ) 2 / 7
Rationale: "there are 366 days in a leap year : 52 weeks and 2 more days . so , 52 sundays , 52 mondays and 2 days . these 2 days can be : { mon , tue } , { tue , wed } , { wed , thu } , { thu , fri } , { fri , sat } , { sat , sun } and { sun , mon } ( 7 cases ) . in order to have 53 sundays or 53 mondays we should have either { mon , tue } { sat , sun } or { sun , mon } case . the probability of that is 3 / 7 . so probability of not having 53 sundays or mondays , probability is 1 - 3 / 7 = 4 / 7 answer : c ."
Answer: c
Python program:
a = 3 + 4 
b = 2 / a
Problem: a copy machine , working at a constant rate , makes 25 copies per minute . a second copy machine , working at a constant rate , makes 55 copies per minute . working together at their respective rates , how many copies do the two machines make in half an hour ?
Options: a ) 90 , b ) 2,400 , c ) 4,500 , d ) 5,400 , e ) 324,000
Rationale: "together the two machines make 25 + 55 = 80 copies per minute . so , in half an hour they will make 80 * 30 = 2,400 copies . answer : b ."
Answer: b
Python program:
a = 25 + 55 
b = a * const_60
c = b / 2
Problem: a student multiplied a number by 3 / 5 instead of 5 / 3 . what is the percentage error .
Options: a ) 64 % , b ) 65 % , c ) 66 % , d ) 67 % , e ) none of these
Rationale: "explanation : let the number be x , then , 5 / 3 − 3 / 5 = 16 / 15 x error % = ( 16 / 15 x ∗ 3 / 5 ∗ 100 ) option a"
Answer: a
Python program:
a = 5 / 3
b = 3 / 5
c = a - b
d = 5 / 3
e = c / d
f = e * 100
Problem: solve the given equation 9 ^ y = 3 ^ 12 , what is y ?
Options: a ) 1 , b ) 3 , c ) 4 , d ) 5 , e ) 6
Rationale: 3 ^ 2 y = 3 ^ 12 2 y = 12 , therefore y = 6 e
Answer: e
Python program:
a = math.log(9)
b = math.log(3)
c = a / b
d = 12 / c
Problem: the dimensions of a room are 25 feet * 15 feet * 12 feet . what is the cost of white washing the four walls of the room at rs . 5 per square feet if there is one door of dimensions 6 feet * 3 feet and three windows of dimensions 4 feet * 3 feet each ?
Options: a ) 2277 , b ) 2977 , c ) 2677 , d ) 4530 , e ) 1971
Rationale: "area of the four walls = 2 h ( l + b ) since there are doors and windows , area of the walls = 2 * 12 ( 15 + 25 ) - ( 6 * 3 ) - 3 ( 4 * 3 ) = 906 sq . ft . total cost = 906 * 5 = rs . 4530 answer : d"
Answer: d
Python program:
a = 2 * 12
b = 15 + 25 
c = a * b
d = 6 * 3
e = c - d
f = 4 * 3
g = 3 * f
h = e - g
i = h * 5