euclaise/mathqa_programs huggingface dataset:
Problem: the average of first seven prime numbers which are between 30 and 70 is
Options: a ) 35.4 , b ) 42 , c ) 44.4 , d ) 57 , e ) 67
Rationale: "explanation : first seven prime numbers which are between 30 and 70 = 31 , 37 , 41 , 43 , 47 , 53 , 59 average = ( 31 + 37 + 41 + 43 + 47 + 53 + 59 ) / 7 = 44.4 answer : c"
Answer: c
Python program:
a = 30 + 1
Problem: a batsman makes a score of 90 runs in the 17 th inning and thus increases his averages by 3 . what is his average after 17 th inning ?
Options: a ) 25 , b ) 31 , c ) 27 , d ) 29 , e ) 42
Rationale: "let the average after 17 innings = x total runs scored in 17 innings = 17 x average after 16 innings = ( x - 3 ) total runs scored in 16 innings = 16 ( x - 3 ) total runs scored in 16 innings + 90 = total runs scored in 17 innings = > 16 ( x - 3 ) + 90 = 17 x = > 16 x - 48 + 90 = 17 x = > x = 42 answer is e ."
Answer: e
Python program:
a = 17 * 3
b = 90 - a
c = b + 3
Problem: if x / y = 3 / z , then 4 x ^ 2 =
Options: a ) y / z , b ) xy , c ) y ² / z ² , d ) 36 y ² / z ² , e ) 15 y ² / z ²
Rationale: "this question is most easily solved by isolating y in the equation and substituting into the expression 4 x ² : x / y = 3 / z x = 3 y / z if we substitute 3 y / z into the expression for x , we get : 4 ( 3 y / z ) ² = 4 ( 9 y ² / z ² ) = 36 y ² / z ² . the correct answer is choice ( d ) ."
Answer: d
Python program:
a = 3 + 4 
b = 3 - 4
c = a / b
Problem: the captain of a cricket team of 11 members is 25 years old and the wicket keeper is 3 years older . if the ages of these two are excluded , the average age of the remaining players is one year less than the average age of the whole team . what is the average age of the team ?
Options: a ) 20 years , b ) 21 years , c ) 22 years , d ) 23 years , e ) 24 years
Rationale: "explanation : let the average age of the whole team by x years . 11 x â € “ ( 25 + 28 ) = 9 ( x - 1 ) 11 x â € “ 9 x = 44 2 x = 44 x = 22 . so , average age of the team is 22 years . answer c"
Answer: c
Python program:
a = 25 + 3 
b = 25 + a 
c = 3 * 3
d = b - c
e = d / 2
Problem: on a certain day , orangeade was made by mixing a certain amount of orange juice with an equal amount of water . on the next day , orangeade was made by mixing the same amount of orange juice with twice the amount of water . on both days , all the orangeade that was made was sold . if the revenue from selling the orangeade was the same for both days and if the orangeade was sold at $ 0.90 per glass on the first day , what was the price per glass on the second day ?
Options: a ) $ 0.15 , b ) $ 0.20 , c ) $ 0.30 , d ) $ 0.40 , e ) $ 0.6
Rationale: "on the first day 1 unit of orange juice and 1 unit of water was used to make 2 units of orangeade ; on the second day 1 unit of orange juice and 2 units of water was used to make 3 units of orangeade ; so , the ratio of the amount of orangeade made on the first day to the amount of orangeade made on the second day is 2 to 3 . naturally the ratio of the # of glasses of orangeade made on the first day to the # of glasses of orangeade made on the second day is 2 to 3 . we are told thatthe revenue from selling the orangeade was the same for both daysso the revenue from 2 glasses on the first day equals to the revenue from 3 glasses on the second day . say the price of the glass of the orangeade on the second day was $ x then 2 * 0.9 = 3 * x - - > x = $ 0.6 . answer : e ."
Answer: e
Python program:
a = 1 + 1 
b = a * 0
c = 1 + 2 
d = b / c
Problem: if a person walks at 25 km / hr instead of 10 km / hr , he would have walked 31 km more . the actual distance traveled by him is ?
Options: a ) 30 , b ) 25 , c ) 20.67 , d ) 12.33 , e ) 19.48
Rationale: "let the actual distance traveled is ' x ' km then by given conditions , we have x / 10 = ( x + 31 ) / 25 25 x = 10 x + 310 x = 20.67 km ans - c"
Answer: c
Python program:
a = 25 - 10
b = 31 / a
c = 10 * b
Problem: the speed of a boat in still water is 8 kmph . if it can travel 1 km upstream in 1 hr , what time it would take to travel the same distance downstream ?
Options: a ) 14 minutes , b ) 4 minutes , c ) 24 minutes , d ) 28 minutes , e ) 23 minutes
Rationale: "speed of the boat in still water = 8 km / hr speed upstream = 1 ⁄ 1 = 1 km / hr speed of the stream = 8 - 1 = 7 km / hr speed downstream = ( 8 + 7 ) = 15 km / hr answer is b time taken to travel 1 km downstream = 1 / 15 hr = ( 1 × 60 ) / 15 = 4 minutes"
Answer: b
Python program:
a = 8 - 1
Problem: # p is defined as 2 p - 20 for any number p . what is p , if # ( # ( # p ) ) = 12 ?
Options: a ) – 108 , b ) 19 , c ) 10 , d ) 16 , e ) 18
Rationale: # p = 2 p - 20 - - - > # ( # p ) = 2 ( 2 p - 20 ) - 20 = 4 p - 60 and thus # ( 4 p - 60 ) = 2 ( 4 p - 60 ) - 20 = 8 p - 140 = 12 - - - > 8 p = 152 - - - > p = 19 , b is the correct answer .
Answer: b
Python program:
a = 20 * 12
b = 20 + a 
c = b + 12 
d = 2 * 4
e = c / d
Problem: if 90 percent of 600 is 50 percent of x , then x = ?
Options: a ) 100 , b ) 1000 , c ) 1080 , d ) 1020 , e ) 1200
Rationale: "0.9 * 600 = 0.5 * x x = 9 / 5 * 600 = 1080"
Answer: c
Python program:
a = 90 * 600
b = a / 50
Problem: a number increased by 20 % gives 1080 . the number is ?
Options: a ) 800 , b ) 700 , c ) 500 , d ) 900 , e ) 600
Rationale: "formula = total = 100 % , increase = ` ` + ' ' decrease = ` ` - ' ' a number means = 100 % that same number increased by 20 % = 120 % 120 % - - - - - - - > 1080 ( 120 ã — 9 = 1080 ) 100 % - - - - - - - > 900 ( 100 ã — 9 = 900 ) option ' d '"
Answer: d
Python program:
a = 20 / 100
b = 1 + a 
c = 1080 / b
Problem: a train 500 m long can cross an electric pole in 20 sec and then find the speed of the train ?
Options: a ) 87 kmph , b ) 97 kmph , c ) 72 kmph , d ) 90 kmph , e ) 19 kmph
Rationale: "length = speed * time speed = l / t s = 500 / 20 s = 25 m / sec speed = 25 * 18 / 5 ( to convert m / sec in to kmph multiply by 18 / 5 ) speed = 90 kmph answer : d"
Answer: d
Python program:
a = 500 / 1000
b = 20 / 3600
c = a / b
Problem: if 5 < x < 12 and y = x + 3 , what is the greatest possible integer value of x + y ?
Options: a ) 23 , b ) 24 , c ) 25 , d ) 26 , e ) 27
Rationale: "x + y = x + x + 3 = 2 x + 3 we need to maximize this value and it needs to be an integer . 2 x is an integer when the decimal of x is . 0 or . 5 the largest such value is 11.5 then x + y = 11.5 + 14.5 = 26 . the answer is d ."
Answer: d
Python program:
a = 3 + 10 
b = a + 10
Problem: on dividing a number by 357 , we get 42 as remainder . on dividing the same number by 17 , what will be the remainder ?
Options: a ) 4 , b ) 5 , c ) 8 , d ) 7 , e ) 2
Rationale: "let x be the number and y be the quotient . then , x = 357 * y + 42 = ( 17 * 21 * y ) + ( 17 * 2 ) + 8 = 17 * ( 21 y + 2 ) + 8 . required number = 8 . answer is c"
Answer: c
Python program:
a = 3 ** 0
b = a / 357
c = 42 ** 2
d = c / 357
e = math.floor(d)
f = b - e
g = f * 357
Problem: evaluate : 20 - 16 ÷ 4 × 3 =
Options: a ) 16 , b ) 10 , c ) 4 , d ) 8 , e ) 6
Rationale: "according to order of operations , 16 ÷ 4 × 3 ( division and multiplication ) is done first from left to right 16 ÷ 4 × 3 = 4 × 3 = 12 hence 20 - 16 ÷ 4 × 3 = 20 - 12 = 8 correct answer d ) 8"
Answer: d
Python program:
a = 16 * 4
b = a * 3
c = 20 - b
Problem: a train 1020 m long running at 102 kmph crosses a platform in 50 sec . what is the length of the platform ?
Options: a ) 287 , b ) 298 , c ) 396 , d ) 726 , e ) 267
Rationale: "d = 102 * 5 / 18 = 50 = 1416 â € “ 1020 = 396 answer : c"
Answer: c
Python program:
a = 102 * const_0_2778
b = 50 * a
c = b - 1020
Problem: a seller of used cars has 15 cars to sell and each of his clients selected 3 cars that he liked most . if each car was selected exactly thrice , how many clients visited the garage ?
Options: a ) 8 , b ) 10 , c ) 12 , d ) 14 , e ) 15
Rationale: "ifno caris selected more than once then the number of clients = 15 / 3 = 5 but since every car is being selected three times so no . of clients must be thrice as well = 5 * 3 = 15 answer : option e"
Answer: e
Python program:
a = 15 / 3
b = a * 3
Problem: if a rectangular room measures 11 meters by 6 meters by 4 meters , what is the volume of the room in cubic centimeters ? ( 1 meter = 100 centimeters )
Options: a ) 24,000 , b ) 240,000 , c ) 2 , 400,000 , d ) 24 , 000,000 , e ) 264 , 000,000
Rationale: "e . 264 , 000,000 11 * 100 * 6 * 100 * 4 * 100 = 264 , 000,000"
Answer: e
Python program:
a = 4 * 100
b = 1 / 10
c = a * b
d = 6 * 100
e = c * d
f = 11 * 100
g = e * f
Problem: a rock is dropped into a well and the distance traveled is 16 t 2 feet , where t is the time . if the water splash is heard 3 seconds after the rock was dropped , and that the speed of sound is 1100 ft / sec , approximate the height of the well
Options: a ) 134.8 feet , b ) 136.2 feet , c ) 132.7 feet , d ) 132.2 feet , e ) 115.3 feet
Rationale: let t 1 be the time it takes the rock to reach the bottom of the well . if h is the height of the well , we can write h = 16 t 1 2 let t 2 be the time it takes sound wave to reach the top of the well . we can write h = 1100 t 2 the relationship between t 1 and t 2 is t 1 + t 2 = 3 eliminate h and combine the equations h = 16 t 1 2 and h = 1100 t 2 to obtain 16 t 1 2 = 1100 t 2 we now substitute t 2 by 3 - t 1 in the above equation 16 t 1 2 = 1100 ( 3 - t 1 ) the above is a quadratic equation that may be written as 16 t 1 2 + 1100 t 1 - 3300 = 0 the above equations has two solutions and only one of them is positive and is given by t 1 = 2.88 seconds ( 2 decimal places ) we now calculate the height h of the well h = 16 t 1 2 = 16 ( 2.88 ) 2 = 132.7 feet answer c
Answer: c
Python program:
a = 1100 * 1100
b = 1100 * 3
c = 4 * 16
d = b * c
e = a + d 
f = math.sqrt(e)
g = f - 1100
h = 16 * 2
i = g / h
j = 1100 * 1100
k = 1100 * 3
l = 4 * 16
m = k * l
n = j + m 
o = math.sqrt(n)
p = o - 1100
q = 16 * 2
r = p / q
s = i * r
t = 16 * s
Problem: a man can row his boat with the stream at 18 km / h and against the stream in 4 km / h . the man ' s rate is ?
Options: a ) 1 kmph , b ) 3 kmph , c ) 8 kmph , d ) 7 kmph , e ) 5 kmph
Rationale: "ds = 18 us = 4 s = ? s = ( 18 - 4 ) / 2 = 7 kmph answer : d"
Answer: d
Python program:
a = 18 - 4
b = a / 2
Problem: how is 2 % expressed as a decimal fraction ?
Options: a ) 0.2 , b ) 0.02 , c ) 0.002 , d ) 0.0002 , e ) 2
Rationale: 2 / 100 = 0.02 answer : b
Answer: b
Python program:
a = 2 / 100
Problem: for any integer n greater than 1 , # n denotes the product of all the integers from 1 to n , inclusive . how many prime numbers v are there between # 6 + 2 and # 6 + 6 , inclusive ?
Options: a ) none , b ) one , c ) two , d ) three , e ) four
Rationale: "none is the answer . a . because for every k 6 ! + k : : k , because 6 ! : : k , since k is between 2 and 6 . a"
Answer: a
Python program:
a = 1 + 1
Problem: 25.25 / 3000 is equal to :
Options: a ) 0.008416667 , b ) 0.110773333 , c ) 0.12526234 , d ) 0.01072333 , e ) 0.12725002
Rationale: "25.25 / 3000 = 2525 / 300000 = 0.008416667 answer : a"
Answer: a
Python program:
a = 25 / 25
Problem: a man can row downstream at 28 kmph and upstream at 10 kmph . find the speed of the man in still water and the speed of stream respectively ?
Options: a ) 2 , 9 , b ) 4 , 9 , c ) 8 , 9 , d ) 19 , 9 , e ) 7 , 9
Rationale: "let the speed of the man in still water and speed of stream be x kmph and y kmph respectively . given x + y = 28 - - - ( 1 ) and x - y = 10 - - - ( 2 ) from ( 1 ) & ( 2 ) 2 x = 38 = > x = 19 , y = 9 . answer : d"
Answer: d
Python program:
a = 28 + 10 
b = a / 2
c = b / 2
Problem: two bullet trains of equal lengths take 8 seconds and 15 seconds respectively to cross a telegraph post . if the length of each bullet train be 120 metres , in what time ( in seconds ) will they cross each other travelling in opposite direction ?
Options: a ) 13 sec . , b ) 10.4 sec . , c ) 12 sec . , d ) 17 sec . , e ) 19 sec .
Rationale: speed of the first bullet train = 120 / 8 m / sec = 15 m / sec . speed of the second bullet train = 120 / 15 m / sec = 8 m / sec . relative speed = ( 15 + 8 ) = 23 m / sec . required time = ( 120 + 120 ) / 23 sec = 10.4 sec . b
Answer: b
Python program:
a = 120 + 120 
b = speed + ( 
c = a / b
Problem: how many even number in the range between 10 to 170 inclusive are not divisible by 3
Options: a ) 15 , b ) 30 , c ) 31 , d ) 33 , e ) 54
Rationale: "we have to find the number of terms that are divisible by 2 but not by 6 ( as the question asks for the even numbers only which are not divisible by 3 ) for 2 , 10 , 12,14 . . . 170 using ap formula , we can say 170 = 10 + ( n - 1 ) * 2 or n = 81 . for 6 , 12,18 , . . . 168 using ap formula , we can say 168 = 12 + ( n - 1 ) * 6 or n = 27 . hence , only divisible by 2 but not 3 = 81 - 27 = 54 . hence , answer e"
Answer: e
Python program:
a = 170 - 10
b = a - 2
c = b / 2
d = 170 - 2
e = 3 * 4
f = d - e
g = f - 3
h = g - 3
i = h / 3
j = i / 2
k = c - j
Problem: a rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered . if the area of the field is 600 sq . ft , how many feet of fencing will be required ?
Options: a ) 80 feet , b ) 70 feet , c ) 60 feet , d ) 50 feet , e ) 20 feet
Rationale: "explanation : we are given with length and area , so we can find the breadth . as length * breadth = area = > 20 * breadth = 600 = > breadth = 30 feet area to be fenced = 2 b + l = 2 * 30 + 20 = 80 feet answer : option a"
Answer: a
Python program:
a = 600 / 20
b = a * 2
c = b + 20
Problem: when positive integer x is divided by positive integer y , the remainder is 8 . if x / y = 96.16 , what is the value of y ?
Options: a ) 96 , b ) 75 , c ) 50 , d ) 25 , e ) 12
Rationale: "by the definition of a remainder , the remainder here is equal to 8 / y . the remainder in decimal form is given as . 16 therefore , 8 / y = . 16 solve for y and get 50 . c"
Answer: c
Python program:
a = math.floor(96, 16)
b = 96 - 16
c = 8 / b
Problem: the mean of 30 values was 140 . it was detected on rechecking that one value 145 was wrongly copied as 135 for the computation of the mean . find the correct mean .
Options: a ) 151 , b ) 140.33 , c ) 152 , d ) 148 , e ) none of the above
Rationale: "corrected mean = 140 × 30 − 135 + 145 / 30 = 4200 − 135 + 145 / 30 = 4210 / 30 = 140.33 answer b"
Answer: b
Python program:
a = 30 * 140
b = 145 - 135
c = a + b 
d = c / 30
Problem: two trains 110 meters and 180 meters in length respectively are running in opposite directions , one at the rate of 80 km and the other at the rate of 65 kmph . in what time will they be completely clear of each other from the moment they meet ?
Options: a ) 7.2 , b ) 7.85 , c ) 6.85 , d ) 5.85 , e ) 6.15
Rationale: "t = ( 110 + 180 ) / ( 80 + 65 ) * 18 / 5 t = 7.20 answer : a"
Answer: a
Python program:
a = 110 + 180 
b = 80 + 65 
c = b * const_0_2778
d = a / c
Problem: you buy a piece of land with an area of √ 900 , how long is one side of the land plot ?
Options: a ) 28
b ) 29
c ) 30
d ) 31
e ) 32
Rationale: try filling the numbers into the answer y x y = find the closest to 900 . answer c
Answer: c
Python program:
a = math.sqrt(900)
Problem: a large box contains 21 small boxes and each small box contains 25 chocolate bars . how many chocolate bars are in the large box ?
Options: a ) 250 , b ) 350 , c ) 450 , d ) 550 , e ) 525
Rationale: "the number of chocolate bars is equal to 21 * 25 = 525 correct answer e"
Answer: e
Python program:
a = 21 * 25
Problem: a typist uses a sheet measuring 20 cm by 30 cm lenghtwise . if a margin of 2 cm is left on each side and a 3 cm margin on the top and bottom , then what is the percentage of page used by the typist ?
Options: a ) 62 % , b ) 63 % , c ) 64 % , d ) 65 % , e ) 66 %
Rationale: area of the sheet = ( 20 * 30 ) cm 2 = 600 cm 2 area used for typing = ( [ 20 - 4 ] * [ 30 - 6 ] ) cm 2 = 384 cm 2 therefore required percentage = ( 384 / 600 ) * 100 = 64 % answer c
Answer: c
Python program:
a = 2 * 2
b = 20 - a
c = 3 * 2
d = 30 - c
e = b * d
f = 20 * 30
g = e / f
h = g * 100
Problem: an association of mathematics teachers has 1,260 members . only 525 of these members cast votes in the election for president of the association . what percent of the total membership voted for the winning candidate if the winning candidate received 62 percent of the votes cast ?
Options: a ) 75 % , b ) 58 % , c ) 42 % , d ) 34 % , e ) 25.83 %
Rationale: "total number of members = 1260 number of members that cast votes = 525 since , winning candidate received 62 percent of the votes cast number of votes for winning candidate = ( 62 / 100 ) * 525 = 325.5 percent of total membership that voted for winning candidate = ( 325.5 / 1260 ) * 100 = 25.83 % answer e"
Answer: e
Python program:
a = 62 / 100
b = a * 525
c = 4 ** 2
d = 100 * c
e = b / d
f = e * 100
Problem: for any positive integer n , the sum of the first n positive integers equals n ( n + 1 ) / 2 . what is the sum of all the even integers between 99 and 161 ?
Options: a ) 2670 , b ) 2980 , c ) 3550 , d ) 4030 , e ) 4540
Rationale: "100 + 102 + . . . + 160 = 100 * 31 + ( 2 + 4 + . . . + 60 ) = 100 * 31 + 2 * ( 1 + 2 + . . . + 30 ) = 100 * 31 + 2 ( 30 ) ( 31 ) / 2 = 100 * 31 + 30 * 31 = 130 ( 31 ) = 4030 the answer is d ."
Answer: d
Python program:
a = 161 - 1
b = 99 + 1 
c = a - b
d = c / 2
e = d + 1
Problem: a clock shows the time as 11 a . m . if the minute hand gains 5 minutes every hour , how many minutes will the clock gain by 6 p . m . ?
Options: a ) 45 minutes , b ) 55 minutes , c ) 35 minutes , d ) 25 minutes , e ) 40 minutes
Rationale: "there are 7 hours in between 11 a . m . to 6 p . m . 7 * 5 = 35 minutes . answer : c"
Answer: c
Python program:
a = 3 + 4 
b = a * 5
Problem: a man is 26 years older than his son . in two years , his age will be twice the age of his son . what is the present age of his son ?
Options: a ) 23 years , b ) 22 years , c ) 24 years , d ) 20 years , e ) 19 years
Rationale: "let present age of the son = x years then , present age the man = ( x + 26 ) years given that , in 2 years , man ' s age will be twice the age of his son â ‡ ’ ( x + 26 ) + 2 = 2 ( x + 2 ) â ‡ ’ x = 24 answer : c"
Answer: c
Python program:
a = 2 * 2
b = a - 2
c = 26 - b
d = 2 - 1
e = c / d
Problem: a and b can do a piece of work in 6 days . with the help of c they finish the work in 5 days . c alone can do that piece of work in ?
Options: a ) 15.5 days , b ) 19.5 days , c ) 17.5 days , d ) 16.5 days , e ) 30 days
Rationale: "c = 1 / 5 – 1 / 6 = 1 / 30 = > 30 days answer : e"
Answer: e
Python program:
a = 5 / 6
b = 5 - a
c = 1/(b)
Problem: an aeroplane covers a certain distance at a speed of 120 kmph in 5 hours . to cover the same distance in 1 2 / 3 hours , it must travel at a speed of :
Options: a ) 520 , b ) 620 , c ) 820 , d ) 740 , e ) 720
Rationale: "distance = ( 120 x 5 ) = 600 km . speed = distance / time speed = 600 / ( 5 / 3 ) km / hr . [ we can write 1 2 / 3 hours as 5 / 3 hours ] required speed = ( 600 x 3 / 5 ) km / hr = 360 km / hr answer a ) 360 km / hr"
Answer: a
Python program:
a = 120 * 5
b = 2 / 3
c = 1 + b 
d = a / c
e = d / 2
Problem: a man swims downstream 18 km and upstream 12 km taking 3 hours each time , what is the speed of the man in still water ?
Options: a ) 3 , b ) 5 , c ) 6 , d ) 4 , e ) 8
Rationale: "18 - - - 3 ds = 6 ? - - - - 1 12 - - - - 3 us = 4 ? - - - - 1 m = ? m = ( 6 + 4 ) / 2 = 5 answer : b"
Answer: b
Python program:
a = 12 / 3
b = 18 / 3
c = a + b 
d = c / 2
Problem: amar takes as much time in running 18 meters as a car takes in covering 48 meters . what will be the distance covered by amar during the time the car covers 1.3 km ?
Options: a ) 600 m , b ) 325 m , c ) 300 m , d ) 400 m , e ) 100 m
Rationale: "b 325 m distance covered by amar = 18 / 4.8 ( 1.3 km ) = 3 / 8 ( 1300 ) = 325 m answer is b"
Answer: b
Python program:
a = 1 * 3
b = 18 * a
c = b / 48
Problem: a sum of rs . 2678 is lent into two parts so that the interest on the first part for 8 years at 3 % per annum may be equal to the interest on the second part for 3 years at 5 % per annum . find the second sum ?
Options: a ) 1629 , b ) 1648 , c ) 2677 , d ) 2986 , e ) 2679
Rationale: "( x * 8 * 3 ) / 100 = ( ( 2678 - x ) * 3 * 5 ) / 100 24 x / 100 = 40170 / 100 - 15 x / 100 39 x = 40170 = > x = 1030 second sum = 2678 â € “ 1030 = 1648 answer : b"
Answer: b
Python program:
a = 3 * 5
b = a * 2678
c = 3 * 5
d = 8 * 3
e = c + d 
f = b / e
g = 2678 - f
Problem: the sum of two numbers is 24 and their product is 148 . find the sum of the squares of that numbers .
Options: a ) 250 , b ) 280 , c ) 230 , d ) 290 , e ) 250
Rationale: let a and b be the two numbers ( a + b ) ^ 2 = a ^ 2 + 2 ab + b ^ 2 given ( a + b ) = 24 ab = 148 so , 24 ^ 2 = a ^ 2 + b ^ 2 + 2 * 148 576 = a ^ 2 + b ^ 2 + 296 a ^ 2 + b ^ 2 = 280 ans b
Answer: b
Python program:
a = 24 ** 2
b = 148 - a
c = b / 2
Problem: two numbers n and 12 have lcm = 54 and gcf = 8 . find n .
Options: a ) 24 , b ) 34 , c ) 36 , d ) 54 , e ) 64
Rationale: the product of two integers is equal to the product of their lcm and gcf . hence . 12 × n = 54 × 8 n = 54 × 8 / 12 = 36 correct answer c
Answer: c
Python program:
a = 54 * 8
b = a / 12
Problem: the two lines y = x and x = - 6 intersect on the coordinate plane . if z represents the area of the figure formed by the intersecting lines and the x - axis , what is the side length of a cube whose surface area is equal to 6 z ?
Options: a ) 16 , b ) 3 √ 2 , c ) 8 , d ) 2 √ 2 , e ) ( √ 2 ) / 3
Rationale: "800 score official solution : the first step to solving this problem is to actually graph the two lines . the lines intersect at the point ( - 6 , - 6 ) and form a right triangle whose base length and height are both equal to 4 . as you know , the area of a triangle is equal to one half the product of its base length and height : a = ( 1 / 2 ) bh = ( 1 / 2 ) ( 6 × 6 ) = 18 ; so z = 18 . the next step requires us to find the length of a side of a cube that has a face area equal to 18 . as you know the 6 faces of a cube are squares . so , we can reduce the problem to finding the length of the side of a square that has an area of 18 . since the area of a square is equal to s ² , where s is the length of one of its side , we can write and solve the equation s ² = 18 . clearly s = √ 18 = 3 √ 2 , oranswer choice ( b ) ."
Answer: b
Python program:
a = 6 * 6
b = a / 2
c = math.sqrt(b)
Problem: the average age of 35 students in a class is 16 years . the average age of 21 students is 14 . what is the average age of remaining 14 students ?
Options: a ) 14 years , b ) 17 years , c ) 19 years , d ) 21 years , e ) none
Rationale: "solution sum of the ages of 14 students = ( 16 x 35 ) - ( 14 x 21 ) = 560 - 294 . = 266 . ∴ required average = 266 / 14 = 19 years . answer c"
Answer: c
Python program:
a = 35 * 16
b = a + 21 
c = b + 35 
d = 35 * 16
e = c - d
Problem: in the coordinate plane , one of the vertices of a square is the point ( - 6 , - 4 ) . if the diagonals of that square intersect at point ( 3 , 2 ) , what is the area of that square ?
Options: a ) 100
b ) 169
c ) 324
d ) 196
e ) 225
Rationale: one point ( - 6 - 4 ) , intersection ( 3,2 ) so the distance from the first point - 6 - 3 = - 9 is the midpoint of the square - - > whole side 18 , 18 * 18 = 324 c
Answer: c
Python program:
a = 6 * 3
b = 6 * 3
c = a * b
Problem: mr . jones gave 40 % of the money he had to his wife . he also gave 20 % of the remaining amount to his 3 sons . half of the amount now left was spent on miscellaneous items and the remaining amount of rs . 12000 was deposited in the bank . how much money did mr . jones have initially ?
Options: a ) 100000 , b ) 12000 , c ) 15000 , d ) 13000 , e ) 65000
Rationale: "let the initial amount with mr . jones be rs . x then , ( 1 / 2 ) [ 100 - ( 3 * 20 ) ] % of x = 12000  ( 1 / 2 ) * ( 40 / 100 ) * ( 60 / 100 ) * x = 12000  x = ( ( 12000 * 25 ) / 3 ) = 100000 answer a"
Answer: a
Python program:
a = 100 / 2
b = a / 100
c = 40 / 100
d = 1 - c
e = 20 / 100
f = 1 - e
g = d * f
h = b * g
i = 12000 / h
Problem: if n = 3 * 4 * p where p is a prime number greater than 3 , how many different positive non - prime divisors does n have , excluding 1 and n ?
Options: a ) six , b ) seven , c ) eight , d ) nine , e ) ten
Rationale: n = 3 ∗ 2 ^ 2 ∗ p number of divisors = 2 * 3 * 2 = 12 the 12 divisors includes 1 , n , 3 , 2 and p number of different non prime divisors excluding 1 and n = 12 - 5 = 7 answer : b
Answer: b
Python program:
a = 3 * 4
b = 4 + 1 
c = a - b