euclaise/mathqa_programs huggingface dataset:
Problem: the average age of students of a class is 15.8 years . the average age of boys in the class is 16.4 years and that of the girls is 15.4 years , the ratio of the number of boys to the number of girls in the class is :
Options: a ) 2 : 9 , b ) 2 : 3 , c ) 2 : 1 , d ) 2 : 2 , e ) 2 : 6
Rationale: "explanation : let the ratio be k : 1 . then , k * 16.4 + 1 * 15.4 = ( k + 1 ) * 15.8 < = > ( 16.4 - 15.8 ) k = ( 15.8 - 15.4 ) < = > k = 0.4 / 0.6 = 2 / 3 . required ratio = 2 / 3 : 1 = 2 : 3 . answer : b"
Answer: b
Python program:
a = 15 - 8
b = 16 - 4
c = a / b
Problem: at what rate percent on simple interest will rs . 850 amount to rs . 950 in 5 years ?
Options: a ) 2.35 % , b ) 5.95 % , c ) 4.35 % , d ) 5.33 % , e ) 6.33 %
Rationale: "100 = ( 850 * 5 * r ) / 100 r = 2.35 % answer : a"
Answer: a
Python program:
a = 950 - 850
b = a / 850
c = b / 5
d = c * 100
Problem: if 4 people are selected from a group of 9 married couples , what is the probability that none of them would be married to each other ?
Options: a ) 1 / 33 , b ) 2 / 85 , c ) 1 / 3 , d ) 56 / 85 , e ) 11 / 85
Rationale: if we are to select 4 people from 9 couples without any restriction , how many ways can we make the selection ? 18 ! / 4 ! 14 ! = 3060 if we are to select 4 people from 9 couples with restriction that no married couple can both make it to the group , only a representative ? 9 ! / 4 ! 5 ! = 126 but we know that to select a person from each couple , take 2 possibilities 126 * 2 * 2 * 2 * 2 = 2016 probability = desired / all possibilities = 2016 / 3060 = 56 / 85 answer : d
Answer: d
Python program:
a = 9 * 2
b = 9 * 2
c = a / b
d = 4 * 4
e = 9 * 2
f = e - 1
g = d / f
h = c * g
i = 4 * 4
j = i - 2
k = 4 * 4
l = j / k
m = h * l
n = 4 * 4
o = n - 2
p = o - 2
q = 4 * 4
r = q - 1
s = p / r
t = m * s
Problem: joe went on a diet 4 months ago when he weighed 222 pounds . if he now weighs 198 pounds and continues to lose at the same average monthly rate , in approximately how many months will he weigh 180 pounds ?
Options: a ) 3 , b ) 3.5 , c ) 3 , d ) 4.5 , e ) 5
Rationale: 222 - 198 = 24 pounds lost in 4 months 24 / 4 = 6 , so joe is losing weight at a rate of 6 pounds per month . . . . in approximately how many months will he weigh 180 pounds ? a simple approach is to just list the weights . now : 198 lbs in 1 month : 192 lbs in 2 months : 186 lbs in 3 months : 180 lbs answer : c
Answer: c
Python program:
a = 198 - 180
b = 222 - 198
c = b / 4
d = a / c
Problem: how many integers are divisible by 3 between 10 ! and 10 ! + 40 inclusive ?
Options: a ) 6 , b ) 7 , c ) 8 , d ) 13 , e ) 10
Rationale: "d - 7 10 ! is divisible by 3 there are 12 numbers between 10 ! and 10 ! + 40 that are divisible by 3 . hence 13"
Answer: d
Python program:
a = 40 / 3
b = a + 1
Problem: if the perimeter of a rectangular stadium is 800 m , its length when its breadth is 300 m is ?
Options: a ) 500 , b ) 400 , c ) 300 , d ) 200 , e ) 100
Rationale: 2 ( l + 300 ) = 800 = > l = 100 m answer : e
Answer: e
Python program:
a = 800 / 2
b = a - 300
Problem: the ratio between the current age of a and b is 6 : 7 . if b is 4 years old than a . what will be the ratio of the ages of a and b after 4 years ?
Options: a ) 7 : 8 , b ) 8 : 9 , c ) 4 : 6 , d ) 4 : 8 , e ) 5 : 8
Rationale: a 7 : 8 if a ’ s age and b ’ s age will be 6 a years and 7 a years correspondingly then , 7 a – 6 a = 4 a = 4 needed ratio = ( 6 a + 4 ) : ( 7 a + 4 ) = 28 : 32 = 7 : 8
Answer: a
Python program:
a = 6 * 4
b = a + 4 
c = 6 * 4
d = c + 4 
e = d + 4 
f = b / e
Problem: the milk level in a rectangular box measuring 56 feet by 25 feet is to be lowered by 6 inches . how many gallons of milk must be removed ? ( 1 cu ft = 7.5 gallons )
Options: a ) 100 , b ) 250 , c ) 750 , d ) 5250 , e ) 5635
Rationale: "6 inches = 1 / 2 feet ( there are 12 inches in a foot . ) , so 56 * 25 * 1 / 2 = 700 feet ^ 3 of milk must be removed , which equals to 700 * 7.5 = 5250 gallons . answer : d ."
Answer: d
Python program:
a = 56 * 25
b = 1 / 2
c = a * b
d = c * 7
Problem: what is the sum of all even numbers from 1 to 501 ?
Options: a ) 122821 , b ) 281228 , c ) 281199 , d ) 62750 , e ) 128111
Rationale: "explanation : 500 / 2 = 250 250 * 251 = 62750 answer : d"
Answer: d
Python program:
a = 1 * 501
b = a / 4
Problem: during 2005 , a company produced an average of 2,500 products per month . how many products will the company need to produce from 2006 through 2008 in order to increase its monthly average for the period from 2005 through 2008 by 300 % over its 2005 average ?
Options: a ) 450,000 , b ) 475,000 , c ) 500,000 , d ) 525,000 , e ) 550,000
Rationale: "company produced 12 * 2500 = 30,000 products in 2005 . if company produces x products from 2006 to 2008 , then total amount of product produced in 4 years ( 2005 through 2008 ) is x + 30,000 . the gives the average of ( x + 30,000 ) / 4 . this average needs to be 200 % higher than that in 2005 . in math terms , 30,000 + 300 % ( 30,000 ) = 120,000 . so : ( x + 30,000 ) / 4 = 120,000 x + 30,000 = 480,000 x = 450,000 the answer is a ."
Answer: a
Python program:
a = 2008 - 2005
b = a / 2
c = 300 * b
Problem: how many quarters are equal to 3 dollars ?
Options: a ) 1 , b ) 8 , c ) 12 , d ) 9 , e ) 7
Rationale: "3 * 4 = 12 quarters answer : c"
Answer: c
Python program:
a = 3 * 4
Problem: if k ^ 3 is divisible by 60 , what is the least possible value of integer k ?
Options: a ) 12 , b ) 30 , c ) 60 , d ) 90 , e ) 120
Rationale: "60 = 2 ^ 2 * 3 * 5 therefore k must include at least 2 * 3 * 5 = 30 . the answer is b ."
Answer: b
Python program:
a = 60 / 2
b = a / 2
Problem: the purchase price of an article is $ 48 . in order to include 20 % of cost for overhead and to provide $ 12 of net profit , the markup should be
Options: a ) 15 % , b ) 25 % , c ) 35 % , d ) 40 % , e ) 45 %
Rationale: "cost price of article = 48 $ % of overhead cost = 20 net profit = 12 $ we need to calculate % markup net profit as % of cost price = ( 12 / 48 ) * 100 = 25 % total markup should be = 25 + 20 = 45 % answer e"
Answer: e
Python program:
a = 12 / 48
b = a * 100
c = b + 20
Problem: pipe a can fill a tank in 10 hrs and pipe b can fill it in 8 hrs . if both the pipes are opened in the empty tank . there is an outlet pipe in 3 / 4 th of the tank . in how many hours will it be fill 3 / 4 th of that tank ?
Options: a ) 3 hr , b ) 3.10 hr , c ) 3.15 hr , d ) 3.20 hr , e ) 3.30 hr
Rationale: part filled a in 1 hr = ( 1 / 10 ) part filled b in 1 hr = ( 1 / 8 ) part filled by ( a + b ) together in 1 hr = ( 1 / 10 ) + ( 1 / 8 ) = 18 / 80 so , the tank will be full in 80 / 18 hrs . time taken to fill exact 3 / 4 th of the tank = ( 80 / 18 ) * ( 3 / 4 ) = 3.20 hrs answer : d
Answer: d
Python program:
a = 3 / 4
b = 1 / 10
c = 1 / 8
d = b + c 
e = 1/(d)
f = a * e
Problem: a no . when divided by the sum of 555 and 445 gives 2 times their difference as quotient & 30 as remainder . find the no . is ?
Options: a ) 122443 , b ) 154546 , c ) 165454 , d ) 186545 , e ) 220030
Rationale: "( 555 + 445 ) * 2 * 110 + 30 = 220000 + 30 = 220030 e"
Answer: e
Python program:
a = 555 + 445 
b = a * 2
c = 555 - 445
d = b * c
e = d + 30
Problem: there are some rabbits and peacocks in a zoo . the total number of their heads is 60 and total number of their legs is 192 . find the number of total rabbits ?
Options: a ) 36 , b ) 77 , c ) 88 , d ) 99 , e ) 22
Rationale: let the number of rabbits and peacocks be ' r ' and ' p ' respectively . as each animal has only one head , so r + p = 60 - - - ( 1 ) each rabbit has 4 legs and each peacock has 2 legs . total number of legs of rabbits and peacocks , 4 r + 2 p = 192 - - - ( 2 ) multiplying equation ( 1 ) by 2 and subtracting it from equation ( 2 ) , we get = > 2 r = 72 = > r = 36 . answer : a
Answer: a
Python program:
a = 2 * 60
b = 192 - a
c = 4 - 2
d = b / c
Problem: when positive integer n is divided by 5 , the remainder is 2 . when n is divided by 11 , the remainder is 8 . what is the smallest positive integer k such that k + n is a multiple of 55 ?
Options: a ) 2 , b ) 3 , c ) 6 , d ) 7 , e ) 10
Rationale: "n = 5 p + 2 = 11 q + 8 n + 3 = 5 p + 5 = 11 q + 11 n + 3 is a multiple of 5 and 11 , so it is a multiple of 55 . the answer is b ."
Answer: b
Python program:
a = 55 - reminder
Problem: a person can row at 10 kmph in still water . if the velocity of the current is 2 kmph and it takes him 10 hour to row to a place and come back , how far is the place ?
Options: a ) 24 km , b ) 30 km , c ) 48 km , d ) 12 km , e ) 48 km
Rationale: "speed of down stream = 10 + 2 = 12 kmph speed of upstream = 10 - 2 = 8 kmph let the required distance be xkm x / 12 + x / 8 = 10 2 x + 3 x = 240 x = 48 km answer is e"
Answer: e
Python program:
a = 10 - 2
b = 10 + 2 
c = a * b
d = c * 10
e = 10 - 2
f = 10 + 2 
g = e + f 
h = d / g
Problem: what is the least number of square tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad ?
Options: a ) 724 , b ) 804 , c ) 814 , d ) 844 , e ) none
Rationale: "solution length of largest tile = h . c . f . of 1517 cm & 902 cm = 41 cm . area of each tile = ( 41 x 41 ) cm 2 ∴ required number of tiles = [ 1517 x 902 / 41 x 41 ] = 814 . answer c"
Answer: c
Python program:
a = 9 * 100
b = a + 2 
c = 15 * 100
d = b * c
e = 2 ** 2
f = d / e
Problem: the average of first 8 prime numbers is ?
Options: a ) 10.11 , b ) 9.625 , c ) 12.11 , d ) 13.11 , e ) 14.11
Rationale: "sum of first 8 prime nos . = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77 average = 77 / 8 = 9.625 answer : b"
Answer: b
Python program:
a = 8 + 1
Problem: from a group of 3 boys and 3 girls , 2 children are to be randomly selected . what is the probability that 1 boy and 1 girl will be selected ?
Options: a ) 1 / 10 , b ) 4 / 9 , c ) 1 / 2 , d ) 9 / 15 , e ) 2 / 3
Rationale: "the total number of ways to choose 2 children from 6 is 6 c 2 = 15 the number of ways to choose 1 boy and 1 girl is 3 * 3 = 9 p ( 1 boy and 1 girl ) = 9 / 15 the answer is d ."
Answer: d
Python program:
a = math.comb(3, 2)
b = math.comb(3, 2)
c = a * b
d = 3 + 3 
e = math.comb(d, 2)
f = c / e
Problem: the average of first five multiples of 7 is ?
Options: a ) 6 , b ) 8 , c ) 9 , d ) 5 , e ) 21
Rationale: "average = 7 ( 1 + 2 + 3 + 4 + 5 ) / 5 = 105 / 5 = 21 . answer : e"
Answer: e
Python program:
a = 7 + 1
Problem: two pipes p and q can fill a cistern in 12 and 10 minutes respectively . both are opened together , but at the end of 3 minutes the first is turned off . how much longer will the cistern take to fill ?
Options: a ) 1 / 8 , b ) 1 / 2 , c ) 2 / 4 , d ) 1 / 4 , e ) 1 / 4
Rationale: "3 / 12 + x / 10 = 1 x = 8 1 / 2 answer : b"
Answer: b
Python program:
a = 1 / 12
b = 1 / 10
c = a + b 
d = c * 3
e = 1 - d
f = e * 10
Problem: ( 169 ) ^ 2 - ( 168 ) ^ 2 =
Options: a ) 1 , b ) 100 , c ) 229 , d ) 329 , e ) 337
Rationale: using the formula : ( a + 1 ) ^ 2 - a ^ 2 = 2 a + 1 so , answer = 168 * 2 + 1 = 336 + 1 = 337 = answer = e
Answer: e
Python program:
a = 169 ** 2
b = 168 ** 2
c = a - b
Problem: a meal cost $ 33.50 and there was no tax . if the tip was more than 10 pc but less than 15 pc of the price , then the total amount paid should be :
Options: a ) 40 - 42 , b ) 39 - 41 , c ) 38 - 40 , d ) 37 - 39 , e ) 36 - 37
Rationale: "10 % ( 33.5 ) = 3.35 15 % ( 33.5 ) = 5.025 total amount could have been 33.5 + 3.35 and 33.5 + 5.025 = > could have been between 36.85 and 38.525 = > approximately between 37 and 39 answer is d ."
Answer: d
Python program:
a = 15 / 100
b = 33 * 50
c = b + 33
Problem: what least number should be added to 1054 , so that the sum is completely divisible by 23
Options: a ) a ) 4 , b ) b ) 1 , c ) c ) 2 , d ) d ) 3 , e ) e ) 5
Rationale: "explanation : ( 1054 / 23 ) gives remainder 19 19 + 4 = 23 , so we need to add 4 answer : option a"
Answer: a
Python program:
a = 23 - reminder
Problem: in a recent head - to - head run - off election , 15,000 absentee ballets were cast . 1 / 5 of the absentee ballets were thrown out and 1 / 4 of the remaining absentee ballets were cast for candidate a . how many absentee votes did candidate b receive ?
Options: a ) 2,000 , b ) 3,000 , c ) 6,000 , d ) 8,000 , e ) 9,000
Rationale: "4 / 5 * 3 / 4 ( total absentee votes ) = 3 / 5 ( total votes ) = 3 / 5 * 15000 = 9000 answer is e"
Answer: e
Python program:
a = 1 * 4
b = a * 1000
c = 1 * 4
d = c * 1000
e = 1 / 5
f = d * e
g = b - f
h = 1 * 4
i = h * 1000
j = 1 * 4
k = j * 1000
l = 1 / 5
m = k * l
n = i - m
o = 1 / 4
p = n * o
q = g - p
r = q / 1000
s = r / 4
t = s * 1
Problem: the operation is defined for all integers a and b by the equation ab = ( a - 1 ) ( b - 1 ) . if y 10 = 90 , what is the value of y ?
Options: a ) 10 , b ) 12 , c ) 15 , d ) 11 , e ) 20
Rationale: "ab = ( a - 1 ) ( b - 1 ) y 10 = ( y - 1 ) ( 10 - 1 ) = 90 - - > y - 1 = 10 - - > y = 11 answer : d"
Answer: d
Python program:
a = 10 - 1
b = 90 / a
c = b + 1
Problem: the distance between two cities a and b is 330 km . a train starts from a at 8 a . m . and travel towards b at 60 km / hr . another train starts from b at 9 a . m . and travels towards a at 75 km / hr . at what time will they meet ?
Options: a ) 10.30 a . m , b ) 11 a . m , c ) 10 a . m , d ) 12 noon , e ) none of these
Rationale: "explanation : assume that they meet x hours after 8 a . m . then , train 1 , starting from a , travelling towards b , travels x hours till the trains meet ⇒ distance travelled by train 1 in x hours = speed × time = 60 x then , train 2 , starting from b , travelling towards a , travels ( x - 1 ) hours till the trains meet ⇒ distance travelled by train 2 in ( x - 1 ) hours = speed × time = 75 ( x - 1 ) total distance travelled = distance travelled by train 1 + distance travelled by train 2 = > 330 = 60 x + 75 ( x - 1 ) = > 12 x + 15 ( x - 1 ) = 66 = > 12 x + 15 x - 15 = 66 = > 27 x = 66 + 15 = 81 = > 3 x = 9 = > x = 3 hence the trains meet 3 hours after 8 a . m . , i . e . at 11 a . m . answer : option b"
Answer: b
Python program:
a = 330 + 75 
b = 60 + 75 
c = a / b
d = c + 8
Problem: a card game called “ high - low ” divides a deck of 52 playing cards into 2 types , “ high ” cards and “ low ” cards . there are an equal number of “ high ” cards and “ low ” cards in the deck and “ high ” cards are worth 2 points , while “ low ” cards are worth 1 point . if you draw cards one at a time , how many ways can you draw “ high ” and “ low ” cards to earn 5 points if you must draw exactly 2 “ low ” cards ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 12
Rationale: "great question ravih . this is a permutations problem ( order matters ) with repeating elements . given thatlowcards are worth 1 pt andhigh cards 2 pts , and you must draw 3 low cards , we know that you must also draw 1 high card . the formula for permutations problems with repeating elements isn ! / a ! b ! . . . where n represents the number of elements in the group and a , b , etc . represent the number of times that repeating elements are repeated . here there are 4 elements and thelowcard is repeated 2 times . as a result , the formula is : 4 ! / 2 ! which represents ( 4 * 3 * 2 * 1 ) / ( 2 * 1 ) which simplifies to just 4 , giving you answer e ."
Answer: e
Python program:
a = 4 * 2
b = a * 2
c = b * 1
d = 2 * 2
e = d * 1
f = c / e
Problem: one pump drains one - half of a pond in 2.5 hours , and then a second pump starts draining the pond . the two pumps working together finish emptying the pond in one - half hour . how long would it take the second pump to drain the pond if it had to do the job alone ?
Options: a ) 1 hour , b ) 1.2 hour , c ) 3 hours , d ) 1.25 hours , e ) 6 hours
Rationale: "the tricky part here , i believed is one half hour = 1 / 2 . then everything would be easy . we have the 1 st pump working rate / hour = 1 / 2 : 5 / 2 = 1 / 5 working rate of 2 pumps : 1 / 2 : 1 / 2 = 1 . working rate of 2 nd pump : 1 - 1 / 5 = 4 / 5 - - > time taken for the 2 nd pump to finish : 1 : 4 / 5 = 5 / 4 = 1.25 hours . d"
Answer: d
Python program:
a = 2 * 5
b = 1 / a
c = 1 - b
d = 1 / c
Problem: a high school has 400 students 1 / 2 attend the arithmetic club , 5 / 8 attend the biology club and 3 / 4 attend the chemistry club . 3 / 8 attend all 3 clubs . if every student attends at least one club how many students attend exactly 2 clubs .
Options: a ) 35 , b ) 45 , c ) 55 , d ) 60 , e ) 50
Rationale: a - club has 200 members ( 1 / 2 of 400 ) b - club has 250 members ( 5 / 8 of 400 ) c - club has 300 members ( 3 / 4 of 400 ) we can create an equation to solve this : 200 + 250 + 150 = n + x + 2 y where n is the number of students , x is the number of students in two clubs , and y is the number of students in three clubs . the question provides y for us ( 150 ) . 750 = 400 + x + 300 x = 50 e
Answer: e
Python program:
a = 400 * 1
b = a / 2
c = 400 * 5
d = c / 8
e = b + d 
f = 400 * 3
g = f / 4
h = e + g 
i = 400 * 3
j = i / 8
k = j * 2
l = h - k
m = l - 400
Problem: in a group of cows and hens , the number of legs are 30 more than twice the number of heads . the number of cows is
Options: a ) 15 , b ) 7 , c ) 10 , d ) 12 , e ) 14
Rationale: "explanation : let the number of cows be x and the number of hens be y . then , 4 x + 2 y = 2 ( x + y ) + 30 4 x + 2 y = 2 x + 2 y + 30 2 x = 30 x = 15 answer : a"
Answer: a
Python program:
a = 30 - 4
Problem: a train , 800 meter long is running with a speed of 78 km / hr . it crosses a tunnel in 1 minute . what is the length of the tunnel ?
Options: a ) 650 meter , b ) 555 meter , c ) 500 meter , d ) 458 meter , e ) none of these
Rationale: explanation : let length of tunnel is x meter distance = 800 + x meter time = 1 minute = 60 seconds speed = 78 km / hr = 78 * 5 / 18 m / s = 65 / 3 m / s distance = speed * time = > 800 + x = 65 / 3 ∗ 60 = > 800 + x = 20 ∗ 65 = 1300 = > x = 1300 − 800 = 500 so the length of the tunnel is 500 meters . option c
Answer: c
Python program:
a = 78 * 1000
b = a / 3600
c = b * const_60
d = c - 800
Problem: how many digits are in ( 8 × 10 ^ 14 ) ( 10 × 10 ^ 10 ) ?
Options: a ) 24 , b ) 25 , c ) 26 , d ) 27 , e ) 28
Rationale: "he question simplifies to ( 8 × 10 ^ 14 ) ( 10 ^ 11 ) = > 8 * 10 ^ 25 = > will contain 25 zeros + 1 digit 8 = > 26 ans c"
Answer: c
Python program:
a = 3 + 4 
b = 10 + a
Problem: richard walks along sunrise boulevard daily . he starts walking at 07 : 00 from block 10 and walks to block 90 where he turns around and walks back to block 70 , where he stops at 07 : 30 . the blocks along the boulevard are numbered sequentially ( 1 , 2,3 ) , and each block measures 40 meters . what is richard ' s speed in meters per minute ?
Options: a ) 75 , b ) 180 , c ) 198 , d ) 216 , e ) 252
Rationale: total distance from 10 to 90 = 80 + from 90 to 70 = 20 so the dist is 100 × 30 ( per block dist ) speed = 3000 mts / 40 min = 75 m / min a is the answer
Answer: a
Python program:
a = 90 - 10
b = 90 - 70
c = a + b 
d = c * 30
e = d / 40
Problem: when n is divided by 32 , the remainder is 5 . what is the remainder when 4 n is divided by 8 ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
Rationale: "let n = 5 ( leaves a remainder of 5 when divided by 32 ) 4 n = 4 ( 5 ) = 20 , which leaves a remainder of 4 when divided by 8 . answer b"
Answer: b
Python program:
a = 5 - reminder
Problem: a person ' s present age is two - fifth of the age of his mother . after 10 years , he will be one - half of the age of his mother . how old is the mother at present ?
Options: a ) a ) 25 , b ) b ) 40 , c ) c ) 32 , d ) d ) 50 , e ) e ) 28
Rationale: let the mother ' s present age be x years then the person ' s present age = 2 x / 5 ( 2 x / 5 ) + 10 = 1 / 2 ( x + 10 ) 2 ( 2 x + 50 ) = 5 ( x + 10 ) x = 50 answer is d
Answer: d
Python program:
a = 2 + 3 
b = 10 - a
c = 1 / 2
d = 2 + 3 
e = 2 / d
f = c - e
g = b / f
Problem: | x + 3 | – | 4 - x | = | 5 + x | how many solutions will this equation have ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: "you have | x + 3 | - | 4 - x | = | 8 + x | first , look at the three values independently of their absolute value sign , in other words : | x + 3 | - | 4 - x | = | 8 + x | ( x + 3 ) - ( 4 - x ) = ( 8 + x ) now , you ' re looking at x < - 8 , so x is a number less than - 8 . let ' s pretend x = - 10 here to make things a bit easier to understand . when x = - 10 i . ) ( x + 3 ) ( - 10 + 3 ) ( - 7 ) ii . ) ( 4 - x ) ( 4 - [ - 10 ] ) ( double negative , so it becomes positive ) ( 4 + 10 ) ( 14 ) iii . ) ( 8 + x ) ( 8 + - 10 ) ( - 2 ) in other words , when x < - 8 , ( x + 3 ) and ( 8 + x ) are negative . to solve problems like this , we need to check for the sign change . here is how i do it step by step . i . ) | x + 3 | - | 4 - x | = | 8 + x | ii . ) ignore absolute value signs ( for now ) and find the values of x which make ( x + 3 ) , ( 4 - x ) and ( 8 + x ) = to zero as follows : ( x + 3 ) x = - 3 ( - 3 + 3 ) = 0 ( 4 - x ) x = 4 ( 4 - 4 ) = 0 ( 8 + x ) x = - 8 ( 8 + - 8 ) = 4 e"
Answer: e
Python program:
a = 4 + 3 
b = a * 2
c = b / 5
Problem: if the population of a certain country increases at the rate of one person every 40 seconds , by how many persons does the population increase in 1 hour ?
Options: a ) 90 , b ) 120 , c ) 150 , d ) 180 , e ) 160
Rationale: "answer = 1.5 * 60 = 90 answer is a"
Answer: a
Python program:
a = const_60 / 40
b = a * 1
Problem: when positive integer t is divided by 13 , the remainder is 2 . when n is divided by 8 , the remainder is 5 . how many such values are less than 180 ?
Options: a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: the equation that can be formed t is 13 x + 2 = 8 y + 5 . . 13 x - 3 = 8 y . . . as we can see x can take only odd values as the rhs will always be even . . also x can take values till 13 as 13 * 14 > 180 . . now we have to substitue x as 1 , 35 , 79 , 1113 . . . once we find 7 fitting in , any other value need not be checked as every 4 th value will give us answer so next value will be 15 . . ans 1 . . b
Answer: b
Python program:
a = 5 + 2 
b = 13 * a
c = b + 2 
d = 5 + 2 
e = 13 * d
f = e + 2 
g = c / f
Problem: a room 11 m 47 cm long and 7 m 77 cm broad is to be paved with square tiles . find the least number of square tiles required to cover the floor .
Options: a ) 636 , b ) 640 , c ) 647 , d ) 651 , e ) 675
Rationale: explanation : area of the room = ( 1147 x 777 ) cm 2 . size of largest square tile = h . c . f . of 1147 cm and 777 cm = 37 cm . area of 1 tile = ( 37 x 37 ) cm 2 . number of tiles required = ( 1147 × 777 ) / ( 37 × 37 ) = 651 answer : option d
Answer: d
Python program:
a = 11 * 100
b = a + 47 
c = 7 * 100
d = c + 77 
e = b * d
f = 2 * 4
g = f + 2 
h = 47 - g
i = 2 * 4
j = i + 2 
k = 47 - j
l = h * k
m = e / l
Problem: the distance from city a to city b is 180 miles . while driving from city a to city b , cara drives at a constant speed of 30 miles per hour . dan leaves city a 60 minutes after cara . what is the minimum constant speed in miles per hour that dan must exceed in order to arrive in city b before cara ?
Options: a ) 42 , b ) 44 , c ) 46 , d ) 36 , e ) 50
Rationale: "the time it takes cara to drive to city b is 180 / 30 = 6 hours . dan needs to take less than 5 hours for the trip . dan needs to exceed a constant speed of 180 / 5 = 36 miles per hour . the answer is d ."
Answer: d
Python program:
a = 180 / 30
b = 60 / 60
c = a - b
d = 180 / c
Problem: difference of two parallel sides of atrapezium is 4 cm . perpendicular distance between them is 10 cm . if the area of the trapezium is 250 cm ^ 2 . find the lengths of the parallel side ?
Options: a ) 20 cm and 21 cm
b ) 25 cm and 23 cm
c ) 27 cm and 23 cm
d ) 30 cm and 34 cm
e ) 28 cm and 36 cm
Rationale: let the two parallel sides of the trapezium be a cm and b cm . then , a - b = 4 - - - - - - ( 1 ) and , ( 1 / 2 ) x ( a + b ) x 10 = 475 = > ( a + b ) = ( ( 250 x 2 ) / 10 ) = > a + b = 50 - - - - - - - ( 2 ) solving 1 and 2 , we get : a = 27 , b = 23 so , the two parallel sides are 27 cm and 23 cm . c
Answer: c
Python program:
a = 250 / 10
b = a * 2
c = b + 10 
d = c / 2
Problem: the length of the longest tape in cm which can be used to measure exactly , the length 100 cm ; 2 m 25 cm ; and 7 m 80 cm is :
Options: a ) 30 , b ) 5 , c ) 10 , d ) 36 , e ) 25
Rationale: the three lengths in cm are 100 , 225 & 780 . hcf of 100 , 225 & 780 is 5 . hence , the answer is 5 cm . answer : b
Answer: b
Python program:
a = math.sqrt(25)