regisss/math_qa huggingface dataset:
Problem: mid intended to type a 7 - digit number , but the two 3 ' s he meant to type did not appear . what appeared instead was the 5 - digit number 52115 . how many different 7 - digit numbers could mid have meant to type ?
Options: a ) 10 , b ) 16 , c ) 21 , d ) 24 , e ) 27
Rationale: should be 21 . mid intended to type a seven - digit number there are two possibilities for placing 2 3 s . case 1 : two 3 s were missed consecutively . i . e . he typed 33 and it came blank on screen . - 5 - 2 - 1 - 1 - 5 - in this arrangement we can fit 33 in 6 ways . ( six dashes , each dash represent one possible place for placing 33 ) case 2 : two 3 s are not together , i . e . they have one or more digits between them . - 5 - 2 - 1 - 1 - 5 - , in this arrangement if we place first 3 at first dash i . e . 35 - 2 - 1 - 1 - 5 - then the other 3 can fit into 5 places . if we place first 3 at second dash i . e . - 532 - 1 - 1 - 5 - then the other 3 can fit into 4 places . if we place first 3 at third dash i . e . - 5 - 231 - 1 - 5 - then the other 3 can fit into 3 places . if we place first 3 at fourth dash i . e . - 5 - 2 - 131 - 5 - then the other 3 can fit into 2 places . if we place first 3 at fifth dash i . e . - 5 - 2 - 1 - 135 - then the other 3 can fit into 1 place . so total 15 ways . case 2 + case 1 = 6 + 15 = 21 ways answer c
Answer: c
Problem: in a clothing store , there are 6 different colored neckties ( red , orange , yellow , green , blue , and indigo ) and 4 different colored shirts ( red , orange , yellow , green , blue , and indigo ) that must be packed into boxes for gifts . if each box can only fit one necktie and one shirt , what is the probability that all of the boxes will contain a necktie and a shirt of the same color ?
Options: a ) 719 / 720 , b ) 1 / 24 , c ) 2 / 233 , d ) 3 / 543 , e ) 1 / 720
Rationale: 4 ties and 4 shirts . . . red tie can take any of 4 shirts . . orange can take any of the remaining 3 shirts yellow any of remaining 2 . . and so on till last indigo chooses the 1 remaining . . total ways = 4 * 3 * 2 * 1 = 24 out of this 720 , only 1 way will have same colour tie and shirt . . prob = 1 / 720 b
Answer: b
Problem: a , b , c subscribe rs . 50000 for a business , a subscribes rs . 4000 more than b and b rs . 5000 more than c . out of a total profit of rs . 70000 , a receives :
Options: a ) 14700 , b ) 24703 , c ) 29400 , d ) 33777 , e ) 34778
Rationale: explanation : let c = x . then , b = x + 5000 and a = x + 5000 + 4000 = x + 9000 . so , x + x + 5000 + x + 9000 = 50000 < = > 3 x = 36000 < = > x = 12000 . a : b : c = 21000 : 17000 : 12000 = 21 : 17 : 12 . a ' s share = rs . ( 70000 * 21 / 50 ) = rs . 29,400 . answer : c )
Answer: c
Problem: a canoe has two oars , left and right . each oar either works or breaks . the failure or non - failure of each oar is independent of the failure or non - failure of the other . you can still row the canoe with one oar . the probability that the left oar works is 3 / 5 . the probability that the right oar works is also 3 / 5 . what is the probability q that you can still row the canoe ?
Options: a ) 9 / 25 , b ) 10 / 25 , c ) 6 / 10 , d ) 2 / 3 , e ) 21 / 25
Rationale: "simply look at the question from the other side . what is the probability that you can ’ t row the canoe ? this would be 2 / 5 x 2 / 5 = 4 / 25 . using the idea that the probability q of something happening is 1 – the probability that it doesn ’ t happen , you can use the following equation to reach the right answer : 1 – 4 / 25 = 21 / 25 . answer choice e ."
Answer: e
Problem: john purchased a grinder and a mobile for rs . 15000 & rs . 8000 respectively . he sold the grinder at a loss of 4 % and the mobile phone at a profit of 10 % . overall how much he make aprofit .
Options: a ) rs . 90 , b ) rs . 120 , c ) rs . 200 , d ) rs . 250 , e ) rs . 290
Rationale: "let the sp of the refrigerator and the mobile phone be rs . r and rs . m respectively . r = 15000 ( 1 - 4 / 100 ) = 15000 - 600 m = 8000 ( 1 + 10 / 100 ) = 8000 + 800 total sp - total cp = r + m - ( 15000 + 8000 ) = - 600 + 800 = rs . 200 as this is positive , an overall profit of rs . 200 was made . c"
Answer: c
Problem: 0.03 x 0.5 = ?
Options: a ) 0.0001 , b ) 0.001 , c ) 0.015 , d ) 0.1 , e ) none of these
Rationale: "explanation : 3 x 5 = 15 . sum of decimal places = 3 0.02 x 0.5 = 0.015 answer - c"
Answer: c
Problem: how much interest can a person get on rs . 7200 at 17.5 % p . a . simple interest for a period of two years and 6 months ?
Options: a ) rs . 4150 , b ) rs . 3850 , c ) rs . 3770 , d ) rs . 3150 , e ) none of these
Rationale: i = ( 7200 * 2.5 * 17.5 ) / 100 = ( 7200 * 5 * 35 ) / ( 100 * 2 * 2 ) = rs . 3150 answer : d
Answer: d
Problem: what is the units digit of 13 ^ 4 * 17 ^ 2 * 29 ^ 3 ?
Options: a ) 9 , b ) 7 , c ) 5 , d ) 3 , e ) 1
Rationale: "3 ^ 1 = 3 3 ^ 2 = 9 3 ^ 3 = 7 3 ^ 4 = 1 , it is first digit 7 ^ 2 = 9 , it is second digit 9 ^ 3 = 9 , it is third digit 1 * 9 * 9 = 1 answer : e"
Answer: e
Problem: donovan took a math test and got 35 correct and 13 incorrect answers . what was the percentage of correct answers ? ( round to the nearest hundredth )
Options: a ) 57.78 % , b ) 67.78 % , c ) 72.92 % , d ) 47.78 % , e ) 97.78 %
Rationale: 35 correct our of 48 total ( 35 + 13 ) 35 / 48 correct answer c
Answer: c
Problem: the slant height of a cone is 14 cm and radius of the base is 12 cm , find the curved surface of the cone .
Options: a ) 150 , b ) 170 , c ) 48 , d ) 528 , e ) 600
Rationale: "π * 12 * 14 = 528 answer : d"
Answer: d
Problem: triangle xyz is an isosceles right triangle . if side xy is longer than side yz , and the area of the triangle is 49 , what is the measure of side xy ?
Options: a ) 4
b ) 4 √ 2
c ) 14
d ) 8 √ 2
e ) can not be determined from the information provided
Rationale: ans c . . 14 . . xy being larger means it is the hyp . . area = ( 1 / 2 ) * ( yz ) ^ 2 = 4 or yz = 7 * \ sqrt { 2 } . . therefore hyp = xy = 14
Answer: c
Problem: if the price of sugar rises from rs . 6 per kg to rs . 7.50 per kg , a person , to have no increase in his expenditure on sugar , will have to reduce his consumption of sugar by
Options: a ) 10 % , b ) 20 % , c ) 25 % , d ) 30 % , e ) none
Rationale: "solution let the original consumption = 100 kg and new consumption = x kg . so , 100 x 6 = x × 7.50 x = 80 kg . reduction in consumption = 20 % . answer b"
Answer: b
Problem: in a lottery there are 10 prizes and 25 blanks . a lottery is drawn at random . what is the probability of getting a blank ?
Options: a ) 2 / 3 , b ) 6 / 7 , c ) 5 / 7 , d ) 8 / 7 , e ) 4 / 3
Rationale: total draws = prizes + blanks = 10 + 25 = 35 probability of getting a blank = 25 / 35 = 5 / 7 correct option is c
Answer: c
Problem: what is the maximum number of pieces that a circular pizza can be cut into with 6 linear cuts ?
Options: a ) 12
b ) 13
c ) 16
d ) 20
e ) 22
Rationale: 1 cut makes 2 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 a fifth cut can make 16 pieces by cutting through 5 of the pieces a sixth cut can make 22 pieces by cutting through 6 of the pieces e
Answer: e
Problem: walking at the rate of 4 kmph a man cover certain distance in 2 hr 45 min . running at a speed of 16.5 kmph the man will cover the same distance in .
Options: a ) 12 min , b ) 25 min , c ) 40 min , d ) 48 min , e ) 52 min
Rationale: distance = speed * time 4 * 11 / 4 = 11 km new speed = 16.5 kmph therefore time = d / s = 11 / 16.5 = 40 min answer : c
Answer: c
Problem: how many positive factors do 120 and 96 have in common ?
Options: a ) 8 , b ) 12 , c ) 16 , d ) 18 , e ) 24
Rationale: "the number of common factors will be same as number of factors of the highest common factor ( hcf ) hcf of 120 and 96 is 24 number of factors of 24 = 8 answer : a"
Answer: a
Problem: if p and q are positive integers each greater than 1 , and 15 ( p + 1 ) = 29 ( q + 1 ) , what is the least possible value of p + q ?
Options: a ) 36 , b ) 42 , c ) 44 , d ) 46 , e ) none
Rationale: 17 ( p + 1 ) = 29 ( q + 1 ) - - > ( p + 1 ) / ( q + 1 ) = 29 / 17 - - > the least positive value of p + 1 is 29 , so the least value of p is 28 and the least positive value of q + 1 is 15 , so the least value of q is 14 - - > the least value of p + q is 28 + 14 = 42 . answer : b .
Answer: b
Problem: how many numbers from 10 to 50 are exactly divisible by 3 ?
Options: a ) 13 , b ) 15 , c ) 16 , d ) 17 , e ) 18
Rationale: "12 , 15 , 18 , 21 , 24 , 27 , 30 , 33 , 36 , 39 , 42 , 45,48 . 13 numbers . 10 / 3 = 3 and 50 / 3 = 16 = = > 16 - 3 = 13 . therefore 13 digits a )"
Answer: a
Problem: the percentage profit earned by selling an article for rs . 1820 is equal to the percentage loss incurred by selling the same article for rs . 1280 . at what price should the article be sold to make 25 % profit ?
Options: a ) 3000 , b ) 1230 , c ) 2000 , d ) 5600 , e ) 1937.5
Rationale: "c . p . be rs . x . then , ( 1820 - x ) / x * 100 = ( x - 1280 ) / x * 100 1820 - x = x - 1280 2 x = 3100 = > x = 1550 required s . p . = 125 % of rs . 1550 = 125 / 100 * 1550 = rs . 1937.5 . answer e"
Answer: e
Problem: if the remainder is 3 when positive integer n is divided by 18 , what is the remainder when n is divided by 6 ?
Options: a ) 3 , b ) 1 , c ) 2 , d ) 3 , e ) 4
Rationale: assume x is quotient here , n = 18 x + 3 - - - - - - - - - - ( 1 ) and n = 6 x + ? we can also write first term as n = ( 18 x + 6 ) = 6 ( 3 x + 1 ) + 3 ie 6 ( 3 x + 1 ) + 3 ie the first term is perfectly divisible by 6 . so , the remainder left is 3 . so , answer ( a ) is right choice .
Answer: a
Problem: how many 4 digit numbers are there , if it is known that the first digit is even , the second is odd , the third is prime , the fourth ( units digit ) is divisible by 3 , and the digit 7 can be used only once ?
Options: a ) 20 , b ) 150 , c ) 304 , d ) 310 , e ) 320
Rationale: "4 options for the first digit : 2 , 4 , 6 , 8 ; 5 options for the second digit : 1 , 3 , 5 , 7 , 9 ; 4 options for the third digit : 2 , 3 , 5 , 7 ; 4 options for the fourth digit : 0 , 3 , 6 , 9 . four digit # possible without the restriction ( about the digit 2 ) : 4 * 5 * 4 * 4 = 320 numbers with seven 7 - s , 7 x 7 x 4 * 1 * 1 * 4 = 16 . thus there are 320 - 16 = 304 such numbers . answer : c ."
Answer: c
Problem: two kinds of vodka are mixed in the ratio 1 : 2 and 2 : 1 and they are sold fetching the profit 10 % and 20 % respectively . if the vodkas are mixed in equal ratio and the individual profit percent on them are increased by 4 / 3 and 5 / 3 times respectively , then the mixture will fetch the profit of
Options: a ) 18 % , b ) 20 % , c ) 21 % , d ) 23 % , e ) can not be determined
Rationale: "mixture 1 and 2 having individual profit % of 10 and 20 respectively . individual profits of vodka a for mixture 1 = 1 / 3 * 10 = 3.33 individual profits of vodka b for mixture 1 = 2 / 3 * 10 = 6.77 individual profits for a and b are increased 4 / 3 and 5 / 3 times . means profits of mixtures 1 and 2 will increase too . increased profits of vodka a for mixture 1 = 4 / 3 * 3.33 = 4.44 increased profits of vodka b for mixture 2 = 5 / 3 * 6.77 = 10.55 total increased profits for mixture 1 = 14.99 similarly total increased profits for mixture 2 = 24.42 using allegation formula va : vb = 1 : 1 = [ 24.42 - ( profit of mixture ) ] / [ ( profit of mixture ) - 14.99 ] hence profit of mixture = 19.705 approx = 20 % answer : b"
Answer: b
Problem: farm tax is levied on the 40 % of the cultivated land . the tax department collected total $ 3840 through the farm tax from the village of mr . willam . mr . willam paid only $ 480 as farm tax . the percentage of total land of mr . willam over the total taxable land of the village is :
Options: a ) 31.25 % , b ) 25 % , c ) 12.5 % , d ) 0.2083 % , e ) none
Rationale: "only trick n this question is to ignore 40 % information as farm tax is levied uniformly in the village and that includes mr william ' s land . what percentage of tax mr william paid ? this will be equal to the percentage of total cultivated land he holds over the total cultivated land in the village . that leads to ( 480 / 3840 ) x 100 = 12.5 % in percentage terms . but the question asks ratio between his total land to total cultivated land . hence the answer is 12.5 % x ( 100 / 40 ) = 31.25 % and the answer is not there in the options . the correct answer is ( a ) ."
Answer: a
Problem: the area of a circular field is 13.86 hectares . find the cost of fencing it at the rate of rs . 4.50 per metre .
Options: a ) 2399 , b ) 3888 , c ) 2999 , d ) 5940 , e ) 2888
Rationale: "explanation : area = ( 13.86 x 10000 ) sq . m = 138600 sq . m circumference = cost of fencing = rs . ( 1320 x 4.50 ) = rs . 5940 . answer : d ) 5940"
Answer: d
Problem: a mixture contains alcohol and water in the ratio 4 : 3 . if 6 litres of water is added to the mixture , the ratio becomes 4 : 5 . find the quantity of alcohol in the given mixture
Options: a ) 15 litres , b ) 10 litres , c ) 30 litres , d ) 12 litres , e ) 8 litres
Rationale: "let the quantity of alcohol and water be 4 x litres and 3 x litres respectively 4 x / ( 3 x + 6 ) = 4 / 5 20 x = 4 ( 3 x + 6 ) 8 x = 24 x = 3 quantity of alcohol = ( 4 x 3 ) litres = 12 litres . answer is d"
Answer: d
Problem: if the sides of a triangle are 26 cm , 25 cm and 10 cm , what is its area ?
Options: a ) 120 , b ) 110 , c ) 130 , d ) 140 , e ) 125
Rationale: "the triangle with sides 26 cm , 25 cm and 10 cm is right angled , where the hypotenuse is 26 cm . area of the triangle = 1 / 2 * 25 * 10 = 125 cm 2 answer : option e"
Answer: e
Problem: ( 7.5 × 7.5 + 37.5 + 2.5 × 2.5 ) is equal to :
Options: a ) 30 , b ) 60 , c ) 80 , d ) 100 , e ) none of these
Rationale: solution given expression = ( 7.5 × 7.5 + 2 × 7.5 × 2.5 + 2.5 × 2.5 ) = ( a 2 + 2 ab + b 2 ) = ( a + b ) 2 = ( 7.5 + 2.5 ) 2 = 102 = 100 . answer d
Answer: d
Problem: a theater charges $ 12 for seats in the orchestra and $ 8 for seats in the balcony . on a certain night , a total of 350 tickets were sold for a total cost of $ 3,320 . how many more tickets were sold that night for seats in the balcony than for seats in the orchestra ?
Options: a ) 90 , b ) 110 , c ) 120 , d ) 130 , e ) 220
Rationale: "orchestra seats - a balcony seats - b a + b = 350 and 12 a + 8 b = 3320 solving equations simultaneously ( multiply equation 1 with 8 and subtract from second equation ) 4 a = 3320 - 8 * 350 = 3320 - 2800 = 520 i . e . a = 130 and b = 350 - 130 = 220 more seats in balcony than orchestra = b - a = 220 - 130 = 90 answer : option a"
Answer: a
Problem: if a - b = 3 and a 2 + b 2 = 29 , find the value of ab .
Options: a ) 10 , b ) 12 , c ) 13 , d ) 15 , e ) 18
Rationale: "2 ab = ( a 2 + b 2 ) - ( a - b ) 2 = 29 - 9 = 20 ab = 10 . answer : a"
Answer: a
Problem: each child has 4 pencils and 13 skittles . if there are 8 children , how many pencils are there in total ?
Options: a ) 16 , b ) 12 , c ) 18 , d ) 22 , e ) 32
Rationale: 4 * 8 = 32 . answer is e .
Answer: e
Problem: sandy had $ 140 left after spending 30 % of the money she took for shopping . how much money did sandy take along with her ?
Options: a ) $ 180 , b ) $ 200 , c ) $ 220 , d ) $ 240 , e ) $ 260
Rationale: let the money sandy took for shopping be x . 0.7 x = 140 x = 200 the answer is b .
Answer: b
Problem: three pipes , a , b , & c are attached to a tank . a & b can fill it in 20 & 30 minutes respectively while c can empty it in 15 minutes . if a , b & c are kept open successively for 1 minute each , how soon will the tank be filled ?
Options: a ) 2 hours , b ) 4 hours , c ) 3 hours , d ) 5 hours , e ) 6 hours
Rationale: "in three minute 1 / 20 + 1 / 30 - 1 / 15 = 1 / 60 part is filled 3 min - - - - - - - - 1 / 60 parts x min - - - - - - - - - 1 part ( full ) x = 180 min = 3 hours answer : c"
Answer: c
Problem: a total of $ 20,000 was invested in two certificates of deposit at simple annual interest rates of 6 percent and 8 percent , respectively . if the total interest on the two certificates was $ 1,440 at the end of one year , what fractional part of the $ 20.000 was invested at the higher rate ?
Options: a ) 3 / 8 , b ) 2 / 5 , c ) 1 / 2 , d ) 3 / 5 , e ) 3 / 4
Rationale: "let ` ` $ x ' ' be invested with the higher rate of 8 % thus , ` ` 20000 - x ' ' must have been invested with lower rate of 6 % simple interest = principal * rate * time for the amount invested at 8 % principal = x rate = 0.08 time = 1 simple interest = x * 0.08 * 1 = 0.08 x for the amount invested at 6 % principal = 20000 - x rate = 0.06 time = 1 simple interest = ( 20000 - x ) * 0.06 * 1 = 1200 - 0.06 x total interest = $ 1440 0.08 x + 1200 - 0.06 x = 1440 0.02 x = 240 x = 240 / 0.02 = 12000 fraction of x per total amount = 12000 / 20000 = 12 / 20 = 3 / 5 ans : ` ` d ' '"
Answer: d
Problem: the average of 5 quantities is 6 . the average of 3 of them is 4 . what is the average of remaining 2 numbers ?
Options: a ) 9 , b ) 10 , c ) 8 , d ) 9.5 , e ) none of these
Rationale: answer : a
Answer: a
Problem: having received his weekly allowance , john spent 3 / 5 of his allowance at the arcade . the next day he spent one third of his remaining allowance at the toy store , and then spent his last $ 0.40 at the candy store . what is john ’ s weekly allowance ?
Options: a ) $ 1.50 , b ) $ 2.00 , c ) $ 2.50 , d ) $ 3.00 , e ) $ 3.50
Rationale: "x = 3 x / 5 + 1 / 3 * 2 x / 5 + 40 4 x / 15 = 40 x = 150 = $ 1.50 the answer is a ."
Answer: a
Problem: how much space , in cubic units , is left vacant when maximum number of 2 x 2 x 2 cubes are fitted in a rectangular box measuring 8 x 10 x 9 ?
Options: a ) 60 , b ) 65 , c ) 70 , d ) 75 , e ) 80
Rationale: "no of cubes that can be accommodated in box = ( 8 * 10 * 9 ) / ( 2 * 2 * 2 ) 8 * 10 in numerator can be perfectly divided by 2 * 2 in denominator . side with length 9 ca n ' t be perfectly divided by 2 and hence is the limiting factor . closet multiple of 2 less that 9 is 8 . so vacant area in cube = = 10 * 8 ( 9 - 8 ) = 10 * 8 * 1 = 80 ans - e"
Answer: e
Problem: how many 4 digit numbers are there , if it is known that the first digit is even , the second is odd , the third is prime , the fourth ( units digit ) is divisible by 3 , and the digit 3 can be used only once ?
Options: a ) 20 , b ) 150 , c ) 225 , d ) 250 , e ) 316
Rationale: "4 options for the first digit : 2 , 4 , 6 , 8 ; 5 options for the second digit : 1 , 3 , 5 , 7 , 9 ; 4 options for the third digit : 2 , 3 , 5 , 7 ; 4 options for the fourth digit : 0 , 3 , 6 , 9 . four digit # possible without the restriction ( about the digit 6 ) : 4 * 5 * 4 * 4 = 320 numbers with three 3 - s , 3 x 3 x 3 4 * 1 * 1 * 1 = 4 thus there are 320 - 4 = 316 such numbers . answer : e ."
Answer: e
Problem: in an examination , there were 2,000 candidates , out of which 900 candidates were girls and rest were boys . if 32 % of the boys and 32 % of the girls passed , then the total percentage of failed candidates is ?
Options: a ) 35.67 % , b ) 64.75 % , c ) 68.00 % , d ) 69.57 % , e ) none of these
Rationale: "girls = 900 , boys = 1100 passed = ( 32 % of 1100 ) + ( 32 % of 900 ) = 352 + 288 = 640 failed = 2000 - 640 = 1360 failed % = [ ( 1360 / 2000 ) x 100 ] % = 68 % . answer : c"
Answer: c
Problem: find ( 7 x + 5 y ) / ( x - 2 y ) if x / 2 y = 3 / 2 ?
Options: a ) 24 , b ) 25 , c ) 26 , d ) 27 , e ) 29
Rationale: "x / 2 y = 3 / 2 = > x = 6 y / 2 = 3 y = > ( 7 x + 5 y ) / ( x - 2 y ) = ( ( 7 * ( 3 y ) ) + 5 y ) / ( 3 y - 2 y ) = > 26 y / y = 26 answer : c"
Answer: c
Problem: when a student weighing 45 kgs left a class , the average weight of the remaining 59 students increased by 200 g . what is the average weight of the remaining 59 students
Options: a ) 55 , b ) 56 , c ) 57 , d ) 58 , e ) 59
Rationale: explanation : let the average weight of the 59 students be a . so the total weight of the 59 of them will be 59 * a . the questions states that when the weight of this student who left is added , the total weight of the class = 59 a + 45 when this student is also included , the average weight decreases by 0.2 kgs 59 a + 45 / 60 = a − 0.2 = > 59 a + 45 = 60 a - 12 = > 45 + 12 = 60 a - 59 a = > a = 57 answer : option c
Answer: c
Problem: mike drives his new corvette from san francisco to las vegas , a journey of 640 miles . he drives the first half of the trip at an average rate of 80 miles per hour , but has to slow down for the second half of his journey . if the second half of the trip takes him 200 percent longer than the first half , what is his average rate j in miles per hour for the entire trip ?
Options: a ) 26.7 , b ) j = 30.0 , c ) j = 40.0 , d ) j = 53.3 , e ) 60.0
Rationale: veritas prepofficial solution correct answer : c using the formula : time = distance / rate , we find that mike takes 4 hours to cover the first 320 miles of his trip . since the 2 nd 320 miles take 200 % longer than the first , it takes mike 8 hours longer , or 12 hours . ( note : 200 % longer than the first half is not 200 % of the first half . ) the overall time is 4 hours + 12 hours or 16 hours . since the definition of average rate = total distance traveled / total time of travel , mike ' s average rate = 640 / 16 or 40 miles per hour . answer choice c is correct .
Answer: c
Problem: if f ( x ) = 12 - x ^ 2 / 2 and f ( 2 k ) = 3 k , what is one possible value for k ?
Options: a ) 2 , b ) 3 , c ) 4 , d ) 6 , e ) 8
Rationale: first of all , see thisgmat blog postand check the related lesson linked below for some background on function notation . we can plug anything in for x and get a result . you can find f ( 1 ) , for example , by plugging in 1 where x is , and you would get 12 - 1 / 2 = 11.5 . or we could find f ( 2 ) , which would be 12 - 4 / 2 = 10 . so the notation f ( 2 k ) means that we are going to plug a 2 k in for x everywhere in the formula for f ( x ) . that would be : f ( 2 k ) = 12 - ( 2 k ) ^ 2 / 2 = 12 - 2 k ^ 2 . remember that we have to square both the 2 and the k , to get 4 k 2 . now , this expression , the output , we will set equal to 2 k . 12 - 2 k ^ 2 = 2 k - - > k = - 4 or k = 3 . all the answers are positive , so we choose k = 3 . answer = b
Answer: b
Problem: two numbers n and 14 have lcm = 56 and gcf = 12 . find n .
Options: a ) 48 , b ) 34 , c ) 44 , d ) 54 , e ) 64
Rationale: "the product of two integers is equal to the product of their lcm and gcf . hence . 14 × n = 56 × 12 n = 56 × 12 / 14 = 48 correct answer a"
Answer: a
Problem: express a speed of 414 kmph in meters per second ?
Options: a ) 201 mps , b ) 114 mps , c ) 118 mps , d ) 115 mps , e ) 111 mps
Rationale: "414 * 5 / 18 = 115 mps answer : d"
Answer: d
Problem: two pipes p and q can fill a cistern in 10 and 15 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 / 2 , b ) 1 / 4 , c ) 2 / 4 , d ) 1 / 4 , e ) 1 / 4
Rationale: "3 / 10 + x / 15 = 1 x = 10 1 / 2 answer : a"
Answer: a
Problem: the malibu country club needs to drain its pool for refinishing . the hose they use to drain it can remove 60 cubic feet of water per minute . if the pool is 80 feet wide by 150 feet long by 10 feet deep and is currently at 100 % capacity , how long will it take to drain the pool ?
Options: a ) 1400 , b ) 1600 , c ) 1800 , d ) 1500 , e ) 2000
Rationale: "volume of pool = 80 * 150 * 10 cu . ft , 100 % full = 60 * 100 * 10 * 1 cu . ft water is available to drain . draining capacity = 60 cu . ft / min therefore time taken = 80 * 150 * 10 * 1 / 60 min = 2000 min e"
Answer: e
Problem: tom and john traveled in the same direction along the equal route at their constant speed rates of 15 km per hour and 10 km per hour , respectively . after 15 minutes tom passed john , tom reaches a certain gas station , how many r minutes it takes john to reach the station ?
Options: a ) 5 min , b ) 6 min , c ) 7 and 1 / 2 min , d ) 8 min , e ) 10 min
Rationale: since the question states “ after 15 minutes ” , we can say tom traveled 15 / 4 km for 15 minutes as he can travel 15 km per hour . hence , using the same logic , we can say john traveled 10 / 4 km as he travels 10 km per hour . so , john has to travel ( 15 / 4 ) - ( 10 / 4 ) km = 5 / 4 km more . since john ’ s speed is 10 km / hour , which means 1 km / 6 minutes . as he has to travel 5 / 4 km more , it is going to take him 6 ( 5 / 4 ) minutes . hence , r = 6 ( 5 / 4 ) = 15 / 2 minutes . the correct answer is c .
Answer: c
Problem: a professional athlete was offered a 3 - year contract to play with team k that provided for an annual salary of $ 100000 in the first year , an increase in annual salary of 20 % over the previous year for the next two years , and a bonus of $ 50000 on signing . team l offered a 3 - year contract providing for an annual salary of $ 150000 in the first year , an increase in annual salary of 20 % over the previous year for the next two years , and no signing bonus . if he accepts the offer of team l and fulfills the 3 - year contract terms , the athlete will receive how much more money by choosing team l over team k ?
Options: a ) $ 132,000 , b ) $ 50,000 , c ) $ 82,500 , d ) $ 92,000 , e ) $ 100,000
Rationale: team k ' s contract = $ 100,000 + $ 100,000 * 1.2 + $ 100,000 * 1.2 * 1.2 + $ 50,000 = $ 414,000 team l ' s contract = $ 150,000 + $ 150,000 * 1.2 + $ 150,000 * 1.2 * 1.2 = $ 546,000 the difference = $ 132,000 . answer : a .
Answer: a
Problem: there are 19 teams in the hockey league , and each team faces all the other teams 10 times each . how many games are played in the season ?
Options: a ) 1710 , b ) 1920 , c ) 2440 , d ) 2860 , e ) 3260
Rationale: "the number of ways to choose two teams is 19 c 2 = 19 * 18 / 2 = 171 the total number of games in the season is 10 * 171 = 1710 . the answer is a ."
Answer: a
Problem: in a family 11 people eat only vegetarian , 6 people eat only non veg . , 9 people eat both veg and non veg . . how many people eat veg in the family ?
Options: a ) 20 , b ) 11 , c ) 9 , d ) 31 , e ) 21
Rationale: "total people eat veg = only veg + both veg and non veg total = 11 + 9 = 20 answer = a"
Answer: a
Problem: a cyclist climbs a hill with a length of 400 m with a speed of 7.2 km / h . when descending , the speed is two times greater . how much time is necessary for the cyclist to climb and descend the hill ?
Options: a ) 30 seconds , b ) 20 seconds , c ) 40 seconds , d ) 10 seconds , e ) none
Rationale: 7.2 km = 7200 m . 1 h = 3600 s . the speed will be 7200 m / 3600 s = 2 m / s . the necessary time to climb will be 20 seconds . when descending , if the speed is two times greater , the necessary time will be two times smaller , so 10 seconds . the total necessary time is 20 s + 10 s = 30 seconds . answer a
Answer: a
Problem: the radius of a semi circle is 6.8 cm then its perimeter is ?
Options: a ) 32.7 , b ) 32.4 , c ) 22.4 , d ) 34.9 , e ) 35.1
Rationale: "36 / 7 r = 6.8 = 34.9 answer : d"
Answer: d
Problem: 6.513 / 10.02 =
Options: a ) 0.35 , b ) 0.3505 , c ) 0.3509 , d ) 0.65 , e ) 0.3527
Rationale: "6.513 / 10.02 6.513 / 1002 = 6.5 move the comma two places to the left as we have 2 decimal places from the divisor : 0.65 . answer : d"
Answer: d
Problem: the ratio between the number of sheep and the number of horses at the stewart farm is 6 to 7 , if each horse is fed 230 ounces of horse food per day and the farm needs a total 12,880 ounces of horse food per day , what is the number of sheep in the farm ?
Options: a ) 18 , b ) 28 , c ) 48 , d ) 56 , e ) 60
Rationale: "let the number of sheeps and horses be 4 x and 7 x . now total number of horses = total consumption of horse food / consumption per horse = 12880 / 230 = 56 , which is equal to 7 x . = > x = 8 sheeps = 6 x = 6 * 8 = 48 . hence c ."
Answer: c
Problem: what is the 23 th digit to the right of the decimal point in the decimal form of 1 / 11 ?
Options: a ) 1 , b ) 6 , c ) 3 , d ) 5 , e ) 0
Rationale: "to solve , we first have to use some long division . long division allows us to get 1 / 11 in decimal form , which is 0.090909 … where “ 09 ” is repeating . we see that the 1 st , 3 rd , 5 th digit to the right of the decimal point is a 0 and that the 2 nd , 4 th , 6 th digit to the right of the decimal point is a 9 . in other words , each odd - positioned digit is a 0 , and each even - positioned digit is a 9 . then the 23 th digit to the right of the decimal point and we see that 23 is odd , then 23 th digit is a 0 . answer e ."
Answer: e
Problem: the present age of the father is 4 years more than 4 times the age of his son . 4 years hence , father ' s age will be 20 years more than twice the age of the son . find the present age of the father ?
Options: a ) 25 , b ) 56 , c ) 45 , d ) 36 , e ) 44
Rationale: let the son ' s present age be x years father ' s present age = ( 4 x + 4 ) 4 x + 4 + 4 = 2 ( x + 4 ) + 10 4 x + 8 = 2 x + 18 x = 5 father ' s present age = 4 x + 4 = 4 * 10 + 4 = 44 years answer is e
Answer: e
Problem: find the slope of the line perpendicular to the line y = ( 1 / 3 ) x - 7
Options: a ) - 1 , b ) - 4 , c ) - 3 , d ) 0 , e ) - 5
Rationale: "two lines are perpendicular if the product of their slopes is equal to - 1 . the slope of the given line is equal to 1 / 3 . if m is the slope of the line perpendicular to the given line , then m × ( 1 / 3 ) = - 1 solve for m m = - 3 correct answer c ) - 3"
Answer: c
Problem: a scuba diver descends at a rate of 60 feet per minute . a diver dive from a ship to search for a lost ship at the depth of 3600 feet below the sea level . . how long will he take to reach the ship ?
Options: a ) 60 minutes , b ) 72 minutes , c ) 75 minutes , d ) 76 minutes , e ) 77 minutes
Rationale: time taken to reach = 3600 / 60 = 60 minutes answer : a
Answer: a
Problem: in a certain company , the ratio of the number of managers to the number of non - managers in any department must always be greater than 7 : 32 . in the company , what is the maximum number of non - managers in a department that has 8 managers ?
Options: a ) 35 , b ) 36 , c ) 37 , d ) 38 , e ) 39
Rationale: "8 / 7 * 32 = 36.6 the answer is b ."
Answer: b
Problem: two letters from the word khantkar are selected at random . what is the probability that at least one of the selected letters is a consonant ?
Options: a ) 25 / 28 , b ) 27 / 28 , c ) 30 / 28 , d ) 40 / 28 , e ) 50 / 28
Rationale: there are 6 consonants and 2 vowels in khantkar . probability that at least one of the selected letters is a consonant = 1 - ( probability of both vowels ) probability of both vowels = 2 c 2 / 8 c 2 = 1 / 28 so , answer is 1 - 1 / 28 = 27 / 28 answer : b
Answer: b
Problem: a paint store mixes 3 / 4 pint of red paint and 2 / 3 pint of white paint to make a new paint color called perfect pink . how many pints of red paint would be needed to make 30 pints of perfect pink paint ?
Options: a ) 9 , b ) 16 , c ) 15 , d ) 25 1 / 3 , e ) 28 1 / 2
Rationale: 3 / 4 pint is required to make 3 / 4 + 2 / 3 = 17 / 12 pint of perfect pink so 17 / 12 pint requires 3 / 4 pint of red . . 1 pint will require 3 / 4 * 12 / 17 = 9 / 17 . . 30 pints will require 9 / 17 * 30 = 15 pints . . c
Answer: c
Problem: a certain elevator has a safe weight limit of 2,500 pounds . what is the greatest possible number of people who can safely ride on the elevator at one time with the average ( arithmetic mean ) weight of half the riders being 180 pounds and the average weight of the others being 230 pounds ?
Options: a ) 7 , b ) 8 , c ) 9 , d ) 10 , e ) 12
Rationale: "lets assume there are 2 x people . half of them have average weight of 180 and other half has 230 . maximum weight is = 2500 so 180 * x + 230 * x = 2500 = > 410 x = 2500 = > x is approximately equal to 6 . so total people is 2 * 6 = 12 answer e ."
Answer: e
Problem: there are 3 vessels of equal capacity . vessel a contains milk and water in the ratio 4 : 3 ; vessel b contains milk and water in the ratio 2 : 1 and vessel c contains milk and water in the ratio 3 : 2 . if the mixture in all the 3 vessels is mixed up . what will be the ratio of water to milk ?
Options: a ) 9 : 6 , b ) 3 : 2 , c ) 118 : 126 , d ) 122 : 193 , e ) 201 : 132
Rationale: 4 : 3 = > 4 x + 3 x = 7 x 2 : 1 = > 2 y + 1 y = 3 y 3 : 2 = > 3 z + 2 z = 5 z 7 x = 3 y = 5 z take lcm of 7 , 3,5 = 105 or simply ; x = 3 * 5 = 15 y = 7 * 5 = 35 z = 3 * 7 = 21 so , ratio of water : milk = ( 3 x + y + 2 z ) / ( 4 x + 2 y + 3 z ) = 122 / 193 ans : d
Answer: d
Problem: each year for 4 years , a farmer increased the number of trees in a certain orchard by 1 / 4 of the number of trees in the orchard of the preceding year . if all of the trees thrived and there were 5625 trees in the orchard at the end of 4 year period , how many trees were in the orchard at the beginning of the 4 year period .
Options: a ) 2304 , b ) 2305 , c ) 2309 , d ) 2310 , e ) 2311
Rationale: "trees increase by 1 / 4 the number of trees in preceding year . hence , correct answer must be divisible by 4 . based on divisibility rules , if last 2 digits are divisible by 4 then the number is divisible by 4 . thus , we can eliminate b , c , d , e the answer to be a again , trees increase by 1 / 4 the number of trees in preceding year . hence , the number of trees increase by 5 / 4 times the number of trees the preceding year . if x = initial number of trees = 2304 year 1 = 5 / 4 x year 2 = ( 5 / 4 ) ( 5 / 4 ) x year 3 = ( 5 / 4 ) ( 5 / 4 ) ( 5 / 4 ) x year 4 = ( 5 / 4 ) ( 5 / 4 ) ( 5 / 4 ) ( 5 / 4 ) x only for answer d : ( 5 / 4 ) ( 5 / 4 ) ( 5 / 4 ) ( 5 / 4 ) 2304 = 5625 hence , correct answer = a"
Answer: a
Problem: the time it took car p to travel 800 miles was 2 hours less than the time it took car r to travel the same distance . if car p ’ s average speed was 10 miles per hour greater than that of car r , what was car r ’ s average speed , in miles per hour ?
Options: a ) 43 , b ) 58 , c ) 60 , d ) 70 , e ) 80
Rationale: "let speed of car r be = x then speed of car p = x + 10 a / q , ( 800 / x ) - ( 800 / ( x + 10 ) ) = 2 solving for x = 58 miles \ hr . b"
Answer: b
Problem: a flamboyant comic book villain has abducted a bystander ! captain valiant is alerted to the crime and reaches the start of the villain ' s trail 30 minutes later . the villain is in a getaway car going 40 kmph , and captain valiant can fly at 50 kmph . how long will the bystander have to wait for rescue ?
Options: a ) 1 hour , b ) 1 1 / 2 hours , c ) 2 hours , d ) 2 1 / 2 hours , e ) 3 hours
Rationale: distance covered by the villain in 1 / 2 hour = 20 km now , 10 km is compensated in 1 hour therefore 20 km will be compensated in 2 hours added to the half hour head start , the bystander has been kidnapped for 2 1 / 2 hours . correct answer : d
Answer: d
Problem: a man can do a piece of work in 5 days , but with the help of his son , he can do it in 4 days . in what time can the son do it alone ?
Options: a ) 15 , b ) 17 , c ) 20 , d ) 8 , e ) 9
Rationale: "son ' s 1 day ' s work = ( 1 / 4 ) - ( 1 / 5 ) = 1 / 20 the son alone can do the work in 20 days answer is c"
Answer: c
Problem: a train running at the speed of 60 km / hr crosses a pole in 27 seconds . find the length of the train ?
Options: a ) 150 meter , b ) 876 meter , c ) 167 meter , d ) 450 meter , e ) 169 meter
Rationale: "speed = 60 * ( 5 / 18 ) m / sec = 50 / 3 m / sec length of train ( distance ) = speed * time ( 50 / 3 ) * 27 = 450 meter answer : d"
Answer: d
Problem: find the greatest number that will divide 400 , 435 and 541 leaving 9 , 10 and 14 as remainders respectively
Options: a ) 19 , b ) 17 , c ) 13 , d ) 9 , e ) 8
Rationale: "explanation : answer will be hcf of ( 400 - 9 , 435 - 10 , 541 - 14 ) hcf of ( 391 , 425 , 527 ) = 17 option b"
Answer: b
Problem: what is the largest 4 digit number exactly divisible by 88
Options: a ) 9900 , b ) 9999 , c ) 9988 , d ) 9944 , e ) 9966
Rationale: "explanation : largest 4 digit number is 9999 after doing 9999 ÷ 88 we get remainder 55 hence largest 4 digit number exactly divisible by 88 = 9999 - 55 = 9944 answer : option d"
Answer: d
Problem: if the cost price of 15 articles is same as the selling price of 25 articles . find the gain or loss percentage ?
Options: a ) 30 % gain , b ) 30 % loss , c ) 40 % gain , d ) 40 % loss , e ) 50 % loss
Rationale: "explanation : 15 cp = 25 sp cost price cp = 25 selling price sp = 15 formula = ( sp - cp ) / cp * 100 = ( 15 - 25 ) / 25 * 100 = 40 % loss answer : option d"
Answer: d
Problem: a and b complete a work in 16 days . a alone can do it in 40 days . if both together can do the work in how many days ?
Options: a ) 1.0875 days , b ) 0.1875 days , c ) 0.0675 days , d ) 0.0875 days , e ) 0.0775 days
Rationale: "1 / 16 + 1 / 40 = 0.0875 days answer : d"
Answer: d
Problem: find the area of the quadrilateral of one of its diagonals is 22 cm and its off sets 9 cm and 6 cm ?
Options: a ) 188 cm 2 , b ) 165 cm 2 , c ) 168 cm 2 , d ) 198 cm 2 , e ) 987 cm 2
Rationale: "1 / 2 * 22 ( 9 + 6 ) = 165 cm 2 answer : b"
Answer: b
Problem: a work which could be finished in 8 days was finished 3 days earlier after 10 more men joined . the number of men employed was ?
Options: a ) 22 , b ) 30 , c ) 88 , d ) 71 , e ) 11
Rationale: "x - - - - - - - 8 ( x + 10 ) - - - - 6 x * 8 = ( x + 10 ) 6 x = 30 \ answer : b"
Answer: b
Problem: a certain protective additive increases from 25 days to 30 days the time between required maintenance checks on an industrial vehicle . by what percent is the time between maintenance checks increased by using the additive ?
Options: a ) 25 % , b ) 20 % , c ) 50 % , d ) 66 2 / 3 % , e ) 75 %
Rationale: "general formula for percent increase or decrease , ( percent change ) : percent = change / original ∗ 100 so , the time between maintenance checks increased by 30 − 25 / 25 ∗ 100 = 20 answer : b ."
Answer: b
Problem: the average of 13 numbers is 58 . average of the first 7 of them is 57 and that of the last 7 is 61 . find the 8 th number ?
Options: a ) 72 , b ) 83 , c ) 45 , d ) 53 , e ) 64
Rationale: "sum of all the 13 numbers = 13 * 58 = 754 sum of the first 7 of them = 7 * 57 = 399 sum of the last 7 of them = 7 * 61 = 427 so , the 8 th number = 427 + 399 - 754 = 72 . answer : a"
Answer: a
Problem: the volumes of two cones are in the ratio 1 : 40 and the radii of the cones are in the ratio of 1 : 2 . what is the length of the wire ?
Options: a ) 8 : 5 , b ) 8 : 9 , c ) 2 : 2 , d ) 8 : 2 , e ) 2 : 8
Rationale: "the volume of the cone = ( 1 / 3 ) π r 2 h only radius ( r ) and height ( h ) are varying . hence , ( 1 / 3 ) π may be ignored . v 1 / v 2 = r 1 ^ 2 . h 1 / r 2 ^ 2 . h 2 = > 1 / 40 = ( 1 ) ^ 2 h 1 / ( 2 ) ^ 2 h 2 = > h 1 / h 2 = 8 / 5 i . e . h 1 : h 2 = 8 : 5 answer : a"
Answer: a
Problem: in the new budget the price of milk rose by 10 % . by how much percent must a person reduce his consumption so that his expenditure on it does not increase ?
Options: a ) 7.5 % , b ) 9.1 % , c ) 10.9 % , d ) 12.6 % , e ) 15 %
Rationale: "reduce in consumption = r / ( 100 + r ) * 100 % = 10 / 110 * 100 = 9.1 % answer is c"
Answer: c
Problem: a copy machine , working at a constant rate , makes 40 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 ) 2,700 , c ) 4,500 , d ) 5,400 , e ) 2,850
Rationale: together the two machines make 40 + 55 = 95 copies per minute . so , in half an hour they will make 95 * 30 = 2,850 copies . answer : e .
Answer: e
Problem: working simultaneously and independently at an identical constant rate , 4 machines of a certain type can produce a total of x units of product p in 6 days . how many of these machines , working simultaneously and independently at this constant rate , can produce a total of 2 x units of product p in 6 days ?
Options: a ) 24 , b ) 18 , c ) 16 , d ) 12 , e ) 8
Rationale: "4 machines do x units in 6 days we have x / 6 = > rate of the 4 machines we know that we need to have 2 x units in 4 days therefore , we need to get to 2 x / 4 rate of the machines . rate of one machine is x / 6 * 1 / 4 = x / 24 . now , we need to know how many machines need to work simultaneously , to get 2 x done in 6 days . 2 x / 6 work needs to be done by machines that work at x / 24 rate . let ' s assign a constant y for the number of machines : ( x / 24 ) * y = 2 x / 6 y = 2 x / 6 * 24 / x cancel 6 with 24 , and x with x and get - > 8 . answer choice e"
Answer: e
Problem: evaluate : 17 + sqrt ( - 7 + 10 ã — 6 ã · 5 ) = ?
Options: a ) 40 , b ) 42 , c ) 44 , d ) 46 , e ) 48
Rationale: "according to order of operations , inner brackets first where 10 x 6 ã · 5 is first calculated since it has a multiplication and a division . 10 x 6 ã · 5 = 60 ã · 5 = 12 hence 17 + sqrt ( - 7 + 10 ã — 6 ã · 5 ) = 17 + sqrt ( - 7 + 12 ) = 17 + sqrt ( 5 ) = 17 + 25 = 42 correct answer b ) 42"
Answer: b
Problem: product of two natural numbers is 19 . then , the sum of reciprocals of their squares is
Options: a ) 290 / 289 , b ) 1 / 289 , c ) 290 / 90 , d ) 362 / 361 , e ) none of these
Rationale: explanation : if the numbers are a , b , then ab = 19 , as 17 is a prime number , so a = 1 , b = 19 . 1 / a 2 + 1 / b 2 = 1 / 1 ( 2 ) + 1 / 19 ( 2 ) = 362 / 361 option d
Answer: d
Problem: if a ( a + 2 ) = 120 and b ( b + 2 ) = 120 , where a ≠ b , then a + b =
Options: a ) - 4 , b ) - 6 , c ) - 2 , d ) - 10 , e ) - 8
Rationale: "i . e . if a = 10 then b = - 12 or if a = - 12 then b = 10 but in each case a + b = - 12 + 10 = - 2 answer : c"
Answer: c
Problem: a rectangular block has the dimensions 5 x 6 x 7 cm it is dropped into a cylindrical vessel of radius 6 cm and height 10 cm . if the level of the fluid in the cylinder rises by 4 cm , what portion of the block is immersed in the fluid ?
Options: a ) 22 / 7 x 24 / 35
b ) 22 / 7 x 36 x 4
c ) 22 / 7 x 36 / 5
d ) 22 / 7 x 37 / 21
e ) 22 / 7 x 32 / 21
Rationale: explanation : since level of water increased in cylinder by height 4 . this is because of the rectangular block . therefore , area of rectangular block immersed in water is = 22 / 7 x ( 6 ) ^ 2 x 4 thats why portion of block immersed in water is = ( 22 / 7 x 36 x 4 ) / total vol . of rectangle = ( 22 / 7 x 36 x 4 ) / ( 7 x 5 x 6 ) = 22 / 7 x 24 / 35 . answer is a
Answer: a
Problem: a train 630 m long , running with a speed of 63 km / hr will pass a tree in ?
Options: a ) 35 , b ) 36 , c ) 37 , d ) 38 , e ) 39
Rationale: "speed = 63 * 5 / 18 = 35 / 2 m / sec time taken = 630 * 2 / 35 = 36 sec answer : option b"
Answer: b
Problem: of the female students at barkely university , 1 / 6 are on the honor roll . of the male students , 2 / 5 are on the honor roll . if 3 / 5 of the students are female , what fraction of all the students are on the honor roll ?
Options: a ) 5 / 9 , b ) 3 / 15 , c ) 26 / 100 , d ) 43 / 60 , e ) 53 / 90
Rationale: "let the total students be 100 given 3 / 5 of the students are females = 60 then males = 2 / 5 = 40 1 / 6 of the females are on honor roll = 10 males on the honor roll = 2 / 5 = 16 total students on honor roll = 10 + 16 = 26 fraction = 26 / 100 c"
Answer: c
Problem: a basket of 1430 apples is divided equally among a group of apple lovers . if 45 people join the group , each apple lover would receive 9 apples less . how many u apples did each person get before 45 people joined the feast ?
Options: a ) 20 . , b ) 21 . , c ) 22 . , d ) 23 . , e ) 24 .
Rationale: "before solving it algebraically , let us prime factorize 1430 = 2 * 5 * 11 * 13 . since number of apples per person * total persons u = 1430 , the answer should be a factor of 1430 . only c is . and that ' s your answer . c"
Answer: c
Problem: what is the smallest number which when increased by 3 is divisible by 18 , 70 , 100 and 21 ?
Options: a ) 5037 , b ) 5447 , c ) 5867 , d ) 6297 , e ) 6617
Rationale: "when increased by 3 , the number must include at least 2 ^ 2 * 3 ^ 2 * 5 ^ 2 * 7 = 6300 the answer is d ."
Answer: d
Problem: if a ( k ) = ( 2 k + 1 ) ^ k , and k = 0 , what is the value of a ( a ( a ( k ) ) ) ?
Options: a ) 343 , b ) 199 , c ) 121 , d ) 69 , e ) 249
Rationale: a ( a ( a ( 0 ) ) ) = a ( a ( 1 ) ) = a ( 3 ) = ( 6 + 1 ) ^ 3 = 343 . thus , the answer is a .
Answer: a
Problem: the cricket team of 11 members is 27 yrs old & the wicket keeper is 3 yrs older . if the ages ofthese 2 are excluded , the average age of theremaining players is 1 year less than the average age of the whole team . what is the average age of the team ?
Options: a ) 21 , b ) 22 , c ) 23 , d ) 25 , e ) 24
Rationale: "let the average age of the whole team be x years . 11 x - ( 27 + 30 ) = 9 ( x - 1 ) = > 11 x - 9 x = 48 = > 2 x = 48 = > x = 24 . so , average age of the team is 24 years . e"
Answer: e
Problem: how many multiples of 5 are there between 1 and 89 ?
Options: a ) 15 , b ) 20 , c ) 13 , d ) 17 , e ) 22
Rationale: "5 multiples are 5 , 10,15 , - - - - - - - - - , 80,85 it should be mentioned whether 1 and 89 are inclusive . the answer is ( 85 - 5 ) / 5 + 1 = 17 answer is d"
Answer: d
Problem: the scoring system in a certain football competition goes as follows : 3 points for victory , 1 point for a draw , and 0 points for defeat . each team plays 20 matches . if a team scored 12 points after 5 games , what is the least number of the remaining matches it has to win to reach the 40 - point mark by the end of the tournament ?
Options: a ) 6 , b ) 7 , c ) 8 , d ) 9 , e ) 10
Rationale: "to get 40 points as end of season we need another 28 points or more from remaining 15 matches : option a = 6 * 3 + 9 * 1 = 27 option b = 7 * 3 + 8 * 1 = 29 hence option b - 7"
Answer: b
Problem: if x is an integer and 3.134 × 10 ^ x is less than 31,000 what is the greatest possible value for x ?
Options: a ) 7 , b ) 6 , c ) 5 , d ) 4 , e ) 3
Rationale: "if x = 4 3.134 × 10 ^ 4 = 31340 > 31000 so , x = 3 answer : e"
Answer: e
Problem: if x and y are integers such that | y + 3 | ≤ 3 and 2 y – 3 x + 6 = 0 , what is the least possible value r of the product xy ?
Options: a ) - 12 , b ) - 3 , c ) 0 , d ) 2 , e ) none of the above
Rationale: "how to deal with inequalities involving absolute values ? first example shows us the so callednumber case in this case we have | y + 3 | ≤ 3 which is generalized | something | ≤ some number . first we solve as if there were no absolute value brackets : y + 3 ≤ 3 y ≤ 0 so y is 0 or negative second scenario - remove the absolute value brackets . put a negative sign around the other side of the inequality , andflip the sign : y + 3 > = - 3 y > = - 6 therefore we have a possible range for y : - 6 = < y < = 0 ok , so far so good , we ' re half way through . what about x ? here ' s the formula : 2 y – 3 x + 6 = 0 , rewrite it as 2 y + 6 = 3 x . you can say that 2 y + 6 is a multiple of 3 ( = 3 x ) . so all values which must be integer must also satisfy this constraint . i ' m just saying that , so it ' s easier to evaluate all the possible numbers ( - 6 , - 3 , 0 ) . if you plug in y = 0 , x will be 2 and xy = 0 as the lowest possible value r . hence , answer choice c is the one to go ."
Answer: c
Problem: in a river flowing at 2 km / hr , a boat travels 40 km upstream and then returns downstream to the starting point . if its speed in still water be 6 km / hr , find the total journey time .
Options: a ) 10 hours , b ) 12 hours , c ) 14 hours , d ) 15 hours , e ) none of these
Rationale: "explanation : speed of the boat = 6 km / hr speed downstream = ( 6 + 2 ) = 8 km / hr speed upstream = ( 6 - 2 ) = 4 km / hr distance travelled downstream = distance travelled upstream = 40 km total time taken = time taken downstream + time taken upstream = ( 40 / 8 ) + ( 40 / 4 ) = 15 d"
Answer: d
Problem: a train covers a distance in 50 min , if it runs at a speed of 48 kmph on an average . the speed at which the train must run to reduce the time of journey to 40 min will be
Options: a ) 45 min , b ) 60 min , c ) 55 min , d ) 70 min , e ) 80 min
Rationale: "time = 50 / 60 hr = 5 / 6 hr speed = 48 mph distance = s * t = 48 * 5 / 6 = 40 km time = 40 / 60 hr = 2 / 3 hr new speed = 40 * 3 / 2 kmph = 60 kmph answer : b ."
Answer: b
Problem: ifaequals the sum of the even integers from 2 to 100 , inclusive , andbequals the sum of the odd integers from 1 to 99 , inclusive , what is the value of a - b ?
Options: a ) 1 , b ) 50 , c ) 19 , d ) 20 , e ) 21
Rationale: "this is a solution from beatthegmat : even numbers : ( 100 - 2 ) / 2 + 1 = 50 even integers . ( 100 + 2 ) / 2 = 51 is the average of the even set . sum = avg * ( # of elements ) = 51 * 50 = 2550 = a odd numbers : ( 99 - 1 ) / 2 + 1 = 50 odd integers . ( 99 + 1 ) / 2 = 50 is the average of the odd set . sum = avg * ( # of elements ) = 50 * 50 = 2500 = b a - b = 2550 - 2500 = 50 . ( b )"
Answer: b