Cheukting/mathqa huggingface dataset:
Question (general): how many diagonals does a 62 - sided convex polygon have ?
Options: a ) 1829 , b ) 2356 , c ) 3458 , d ) 3843 , e ) 4820
Rationale: "a 62 - sided convex polygon has 62 vertices . if we examine a single vertex , we can see that we can connect it with 59 other vertices to create a diagonal . note that we ca n ' t connect the vertex to itself and we ca n ' t connect it to its adjacent vertices , since this would not create a diagonal . if each of the 62 vertices can be connected with 59 vertices to create a diagonal then the total number of diagonals would be ( 62 ) ( 59 ) = 3658 however , we must recognize that we have counted every diagonal twice . to account for counting each diagonal twice , we must divide 3658 by 2 to get 1829 . the answer is a ."
Answer: a
Question (general): if y > 0 , ( 8 y ) / 20 + ( 3 y ) / 10 is what percent of y ?
Options: a ) 40 % , b ) 50 % , c ) 60 % , d ) 70 % , e ) 80 %
Rationale: "can be reduced to 4 y / 10 + 3 y / 10 = 7 y / 10 = 70 % answer d"
Answer: d
Question (general): how many kg of pure salt must be added to 30 kg of 2 % solution of salt and water to increase it to a 10 % solution ?
Options: a ) 8 / 3 , b ) 9 / 7 , c ) 7 / 4 , d ) 5 / 2 , e ) 6 / 7
Rationale: "amount salt in 30 kg solution = 2 * 30 / 100 = 0.6 kg let x kg of pure salt be added then ( 0.6 + x ) / ( 30 + x ) = 10 / 100 60 + 100 x = 300 + 10 x 90 x = 240 x = 8 / 3 answer is a"
Answer: a
Question (general): on dividing 14698 by a certain number , we get 89 as quotient and 14 as remainder . what is the divisor ?
Options: a ) 743 , b ) 154 , c ) 852 , d ) 741 , e ) 165
Rationale: "divisor * quotient + remainder = dividend divisor = ( dividend ) - ( remainder ) / quotient ( 14698 - 14 ) / 89 = 165 answer ( e )"
Answer: e
Question (physics): how long does a train 110 m long traveling at 60 kmph takes to cross a bridge of 190 m in length ?
Options: a ) 18.9 sec , b ) 88.9 sec , c ) 22.9 sec , d ) 18.0 sec , e ) 72.0 sec
Rationale: "d = 110 + 190 = 300 m s = 60 * 5 / 18 = 50 / 3 t = 300 * 3 / 50 = 18.0 sec answer : d"
Answer: d
Question (general): paulson spends 75 % of his income . his income is increased by 20 % and he increased his expenditure by 10 % . find the percentage increase in his savings ?
Options: a ) 20 % , b ) 25 % , c ) 50 % , d ) 75 % , e ) 60 %
Rationale: "let original income = $ 100 expenditure = $ 75 savings = $ 25 new income = $ 120 new expenditure = $ 110 / 100 * 75 = $ 165 / 2 new savings = 120 - 165 / 2 = $ 75 / 2 increase in savings = 75 / 2 - 25 = $ 25 / 2 increase % = 25 / 2 * 1 / 25 * 100 = 50 % answer is c"
Answer: c
Question (general): rice weighing 33 / 4 pounds was divided equally and placed in 4 containers . how many ounces of rice were in each container ? ( note that 1 pound = 16 ounces )
Options: a ) 27 , b ) 30 , c ) 33 , d ) 36 , e ) 39
Rationale: "33 / 4 ÷ 4 = 33 / 16 pounds in each container 33 / 16 pounds * 16 ounces / pound = 33 ounces in each container the answer is c ."
Answer: c
Question (other): ratio between rahul and deepak is 4 : 3 , after 6 years rahul age will be 22 years . what is deepak present age ?
Options: a ) 22 , b ) 15 , c ) 77 , d ) 266 , e ) 12
Rationale: "present age is 4 x and 3 x , = > 4 x + 6 = 22 = > x = 4 so deepak age is = 3 ( 4 ) = 12 answer : e"
Answer: e
Question (gain): in a certain company , a third of the workers do not have a retirement plan . 20 % of the workers who do not have a retirement plan are women , and 40 % of the workers who do have a retirement plan are men . if 112 of the workers of that company are men , how many of the workers are women ?
Options: a ) 80 , b ) 98 , c ) 105 , d ) 120 , e ) 210
Rationale: "set up equation : x = total number of workers 112 = 0,4 * 2 / 3 * x + 0,8 * 1 / 3 * x 112 = 16 / 30 x x = 210 210 - 112 = 98 answer b"
Answer: b
Question (physics): if an object travels 70 feet in 2 seconds , what is the object ’ s approximate speed in miles per hour ? ( note : 1 mile = 5280 feet )
Options: a ) 15.95 , b ) 17.28 , c ) 19.54 , d ) 21.73 , e ) 23.86
Rationale: "70 feet / 2 seconds = 35 feet / second ( 35 feet / second ) * ( 3600 seconds / hour ) * ( 1 mile / 5280 feet ) = 23.86 miles / hour ( approximately ) the answer is e ."
Answer: e
Question (physics): 85 litres of diesel is required to travel 600 km using a 800 cc engine . if the volume of diesel required to cover a distance varies directly as the capacity of the engine , then how many litres of diesel is required to travel 800 km using 1200 cc engine ?
Options: a ) 80 litres , b ) 90 litres , c ) 120 litres , d ) 170 litres , e ) none of these
Rationale: "explanatory answer to cover a distance of 800 kms using a 800 cc engine , the amount of diesel required = 800 / 600 * 85 = 113.33 litres . however , the vehicle uses a 1200 cc engine and the question states that the amount of diesel required varies directly as the engine capacity . i . e . , for instance , if the capacity of engine doubles , the diesel requirement will double too . therefore , with a 1200 cc engine , quantity of diesel required = 1200 / 800 * 113.33 = 170 litres . answer d"
Answer: d
Question (gain): evaluate 28 % of 650 + 45 % of 280
Options: a ) 232 , b ) 308 , c ) 252 , d ) 262 , e ) 282
Rationale: "explanation : = ( 28 / 100 ) * 650 + ( 45 / 100 ) * 280 = 182 + 126 = 308 answer : option b"
Answer: b
Question (physics): a can give b 300 meters start and c 600 meters start in a kilometer race . how much start can b give c in a kilometer race ?
Options: a ) 111.12 , b ) 111.67 , c ) 428.57 , d ) 111.11 , e ) 101.12
Rationale: "a runs 1000 m while b runs 700 m and c runs 400 m . the number of meters that c runs when b runs 1000 m , = ( 1000 * 400 ) / 700 = 571.43 m . b can give c = 1000 - 571.43 = 428.57 m . answer : c"
Answer: c
Question (gain): there are 2000 students in a school and among them 10 % of them attends chess class . 50 % of the students who are in the chess class are also enrolled for swimming . no other students are interested in swimming so how many will attend the swimming class if all enrolled attends ?
Options: a ) 1 , b ) 10 , c ) 100 , d ) 50 , e ) 20
Rationale: "10 % of 2000 gives 200 . so 200 attends chess and 50 % of 200 gives 100 so 100 enrolled for swimming answer : c"
Answer: c
Question (general): a new tower has just been built at the verbico military hospital ; the number of beds available for patients at the hospital is now 6 times the number available before the new tower was built . currently , 1 / 3 of the hospital ' s original beds , as well as 1 / 5 of the beds in the new tower , are occupied . for the purposes of renovating the hospital ' s original wing , all of the patients in the hospital ' s original beds must be transferred to beds in the new tower . if patients are neither admitted nor discharged during the transfer , what fraction of the beds in the new tower will be unoccupied once the transfer is complete ?
Options: a ) 11 / 30 , b ) 29 / 60 , c ) 17 / 30 , d ) 19 / 30 , e ) 11 / 15
Rationale: "i think e - 11 / 15 is the correct answer . here goes : lets assume originally the number of beds = x after the new tower , the total combined no of beds = 6 x so old = x , new = 5 x now 1 / 3 of x are occupied and 1 / 5 of 5 x are occupied which simplifies to ( 5 / 5 ) x we are shifting 1 / 3 of x to the new ward so there will now be : 1 / 3 of x plus 5 / 5 of x occupied in the new ward . add them up to get 4 / 3 of x there are 5 x beds in new tower so ratio is : ( 4 / 3 ) x / 5 x = 4 / 15 of x subtract that from 15 / 15 of x and you get the number of un - occupied beds to total capacity of new tower = 11 / 15 . e"
Answer: e
Question (general): the sum of two numbers is 40 and their difference is 4 . the ratio of the numbers is
Options: a ) 11 : 9 , b ) 11 : 18 , c ) 21 : 19 , d ) 21 : 19 , e ) none
Rationale: sol . let the numbers be x and y . then , x + y / x - y = 40 / 4 = 10 ⇔ ( x + y ) = 10 ( x - y ) ⇔ 9 x = 11 y ⇔ x / y = 11 / 9 . answer a
Answer: a
Question (general): on a race track a maximum of 5 horses can race together at a time . there are a total of 25 horses . there is no way of timing the races . what is the minimum number q of races we need to conduct to get the top 3 fastest horses ?
Options: a ) 5 , b ) 7 , c ) 8 , d ) 10 , e ) 11
Rationale: q = 7 is the correct answer . good solution buneul . b
Answer: b
Question (general): the ages of 2 persons differ by 21 years . if 15 years ago the elder one be 8 times as old as the younger one , find the present age of elder person .
Options: a ) 30 , b ) 48.5 , c ) 50.4 , d ) 62.6 , e ) 47.9
Rationale: "age of the younger person = x age of the elder person = x + 21 8 ( x - 15 ) = x + 21 - 15 x = 18 age of elder person = 18 + 21 = 39 answer is e"
Answer: e
Question (physics): in a 100 m race , a covers the distance in 30 seconds and b in 45 second . in this race a beats b by :
Options: a ) 33.33 m , b ) 25 m , c ) 45 m , d ) 30 m , e ) none of these
Rationale: solution distance covered by b in 15 sec . = ( 100 / 45 x 15 ) m = 33.33 m . ∴ a beats b by 33.33 metres . answer a
Answer: a
Question (general): if ( 2 to the x ) - ( 2 to the ( x - 2 ) ) = 3 ( 2 to the 12 ) , what is the value of x ?
Options: a ) 9 , b ) 11 , c ) 13 , d ) 15 , e ) 14
Rationale: i am guessing the question is : ( 2 to the power x ) - ( 2 to the power ( x - 2 ) ) = 3 ( 2 to the power 12 ) 2 ^ x - 2 ^ ( x - 2 ) = 3 . 2 ^ 12 hence x = 14 . answer is e
Answer: e
Question (physics): a , b , c can do a work in 15 , 20,45 days respectively . they get rs 500 for their work . what is the share of a ?
Options: a ) rs . 240 , b ) rs . 340 , c ) rs . 260 , d ) rs . 280 , e ) rs . 440
Rationale: lcm = 180 share of a = ( lcm / a x total amount ) / lcm / a + lcm / b + lcm / c = ( 180 / 15 ) / ( 180 / 15 + 180 / 20 + 180 / 45 ) = ( 12 / 25 ) * 500 = rs . 240 answer : a
Answer: a
Question (physics): express a speed of 252 kmph in meters per second ?
Options: a ) 10 mps , b ) 05 mps , c ) 09 mps , d ) 12 mps , e ) 70 mps
Rationale: "e 70 mps 252 * 5 / 18 = 70 mps"
Answer: e
Question (general): the temperature of a certain cup of coffee 10 minutes after it was poured was 120 degrees fahrenheit . if the temperature f of the coffee t minutes after it was poured can be determined by the formula f = 120 ( 2 ^ - at ) + 60 , where f is in degrees fahrenheit and a is a constant . then the temperature of the coffee 40 minutes after it was poured was how many degrees fahrenheit ?
Options: a ) 67.5 , b ) 75 , c ) 80 , d ) 85 , e ) 90
Rationale: "answer : b the temperature of coffee 10 minutes after it was poured ( 120 f ) will help in solving the constant “ a ” . 120 = 120 ( 2 ^ 10 a ) + 60 2 ^ - 1 = 2 ^ 10 a a = - 1 / 10 the temperature of coffee 40 minutes after it was poured is : f = 120 ( 2 ^ - 40 / 10 ) + 60 f = 120 * 1 / 16 + 60 f = 15 / 2 + 60 f = 135 / 2 = 67.5 a"
Answer: a
Question (geometry): a circular garden is surrounded by a fence of negligible width along the boundary . if the length of the fence is 1 / 3 of th area of the garden . what is the radius of the circular garden ?
Options: a ) 6 , b ) 2 , c ) 4 , d ) 8 , e ) 16
Rationale: "as per the question - - width is negligible now , let l be the length of the fence = 2 pir l = 1 / 3 ( pir ^ 2 ) pir ^ 2 = 6 pir r = 6 answer : a"
Answer: a
Question (general): jean drew a gumball at random from a jar of pink and blue gumballs . since the gumball she selected was blue and she wanted a pink one , she replaced it and drew another . the second gumball also happened to be blue and she replaced it as well . if the probability of her drawing the two blue gumballs was 25 / 49 , what is the probability that the next one she draws will be pink ?
Options: a ) 1 / 49 , b ) 2 / 7 , c ) 3 / 7 , d ) 16 / 49 , e ) 40 / 49
Rationale: "the probability of drawing a pink gumball both times is the same . the probability that she drew two blue gumballs = 25 / 49 = ( 5 / 7 ) * ( 5 / 7 ) therefore probability that the next one she draws is pink = 2 / 7 option ( b )"
Answer: b
Question (geometry): the areas of the two spheres are in the ratio 1 : 4 . the ratio of their volume is ?
Options: a ) 1 : 4
b ) 1 : 16
c ) 1 : 8
d ) 1 : 64
e ) 2 : 3
Rationale: 4 π r 12 : 4 π r 22 = 1 : 4 r 1 : r 2 = 1 : 2 4 / 3 π r 13 : 4 / 3 π r 23 r 13 : r 23 = 1 : 8 answer : c
Answer: c
Question (gain): a hostel had provisions for 250 men for 40 days . if 50 men left the hostel , how long will the food last at the same rate ?
Options: a ) 30 , b ) 40 , c ) 50 , d ) 60 , e ) 65
Rationale: "a hostel had provisions for 250 men for 40 days if 50 men leaves the hostel , remaining men = 250 - 50 = 200 we need to find out how long the food will last for these 200 men . let the required number of days = x days more men , less days ( indirect proportion ) ( men ) 250 : 200 : : x : 40 250 × 40 = 200 x 5 × 40 = 4 x x = 5 × 10 = 50 answer c"
Answer: c
Question (physics): the distance between parallel sides of the hexagon is 9 . determine the length of the hexagon
Options: a ) 18 * sqrt 3
b ) 18 * sqrt 2
c ) 18 * sqrt 4
d ) 19 * sqrt 3
e ) 17 * sqrt 3
Rationale: length of each side of regular hexagon will be 3 * sqrt 3 . total length ( perimeter ) of regular hexagon will be 18 * sqrt 3 answer : a
Answer: a
Question (physics): a train covers a distance of 12 km in 10 min . if it takes 3 sec to pass a telegraph post , then the length of the train is ?
Options: a ) 70 m , b ) 16 m , c ) 60 m , d ) 80 m , e ) 88 m
Rationale: "speed = ( 12 / 10 * 60 ) km / hr = ( 72 * 5 / 18 ) m / sec = 20 m / sec . length of the train = 20 * 3 = 60 m . answer : c"
Answer: c
Question (other): if g and f are both odd prime numbers andg < f , then how many different positive integer factors does 2 gfhave ?
Options: a ) 3 , b ) 4 , c ) 6 , d ) 8 , e ) 12
Rationale: g and f are both odd prime numbers - it means either g or f is not 2 and since prime numbers have only two factors - 1 and the number itself g and f each will have ( 1 + 1 ) = 2 factors hence 2 gf will have ( 1 + 1 ) ( 1 + 1 ) ( 1 + 1 ) = 8 factors d is the answer
Answer: d
Question (physics): a and b go around a circular track of length 600 m on a cycle at speeds of 30 kmph and 60 kmph . after how much time will they meet for the first time at the starting point ?
Options: a ) 120 sec , b ) 176 sec , c ) 178 sec , d ) 72 sec , e ) 189 sec
Rationale: "time taken to meet for the first time at the starting point = lcm { length of the track / speed of a , length of the track / speed of b } = lcm { 600 / ( 30 * 5 / 18 ) , 600 / ( 60 * 5 / 18 ) } = 72 sec . answer : d"
Answer: d
Question (gain): if $ 5,000 is invested in an account at a simple annual rate of r percent , the interest is $ 250 . when $ 18,000 is invested at the same interest rate , what is the interest from the investment ?
Options: a ) $ 700 , b ) $ 750 , c ) $ 800 , d ) $ 850 , e ) $ 900
Rationale: "- > 250 / 5,000 = 5 % and 18,000 * 5 % = 900 . thus , e is the answer ."
Answer: e
Question (general): m and n are the x and y coordinates , respectively , of a point in the coordinate plane . if the points ( m , n ) and ( m + p , n + 18 ) both lie on the line defined by the equation x = ( y / 6 ) - ( 2 / 5 ) , what is the value of p ?
Options: a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5
Rationale: "x = ( y / 6 ) - ( 2 / 5 ) , and so y = 6 x + 12 / 5 . the slope is 6 . ( n + 18 - n ) / ( m + p - m ) = 6 p = 3 the answer is c ."
Answer: c
Question (general): if w is the product of the integers from 1 to 30 , inclusive , what is the greatest integer k for which 3 ^ k is a factor of w ?
Options: a ) 10 , b ) 12 , c ) 14 , d ) 16 , e ) 18
Rationale: w = 30 ! 8 w = 30 x 29 x 28 x 27 x 26 x 25 x 24 x 24 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 09 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 out of these 30 , 27 , 24 , 21 , 18 , 15 , 12 , 09 , 06 , 3 are factors of 3 3 x 10 , 3 x 3 x 3 , 3 x 8 , 3 x 3 x 2 , 3 x 5 , 3 x 4 , 3 x 3 x 3 , 3 x 2 , 3 so we have a total of 14 three ' s . . . therefore the maximum value of k can be 14 ( c )
Answer: c
Question (gain): a and b started a business in partnership investing rs . 20000 and rs . 15000 respectively . after 6 months , c joined them with rs . 20000 . what will be b ' s share in the total profit of rs . 25000 earned at the end of 2 years from the starting of the business ?
Options: a ) rs . 7500 , b ) rs . 9000 , c ) rs . 9500 , d ) rs . 10,000 , e ) none
Rationale: solution a : b : c = ( 20000 × 24 ) : ( 15000 × 24 ) : ( 20000 × 18 ) = 4 : 3 : 3 b ' s share = rs . ( 25000 × 3 / 10 ) = rs . 7500 . answer a
Answer: a
Question (physics): length of a rectangular plot is 14 mtr more than its breadth . if the cost of fencing the plot at 26.50 per meter is rs . 5300 , what is the length of the plot in mtr ?
Options: a ) 50 m , b ) 57 m , c ) 80 m , d ) 82 m , e ) 84 m
Rationale: "let breadth = x metres . then , length = ( x + 14 ) metres . perimeter = 5300 m = 200 m . 26.50 2 [ ( x + 14 ) + x ] = 200 2 x + 14 = 100 2 x = 86 x = 43 . hence , length = x + 14 = 57 m b"
Answer: b
Question (other): 9 log 9 ( 7 ) = ?
Options: a ) 1 , b ) 2 , c ) 7 , d ) 4 , e ) 5
Rationale: "exponential and log functions are inverse of each other . hence aloga ( x ) = x , for all x real and positive . and therefore 9 log 9 ( 7 ) = 7 correct answer c"
Answer: c
Question (general): if ( a + b ) = 4 , ( b + c ) = 5 and ( c + d ) = 3 , what is the value of ( a + d ) ?
Options: a ) 16 . , b ) 8 . , c ) 7 . , d ) 2 . , e ) - 2 .
Rationale: "given a + b = 4 b + c = 5 c + d = 3 ( a + b ) - ( b + c ) + ( c + d ) = ( a + d ) = > 4 - 5 + 3 = 2 . option d . . ."
Answer: d
Question (gain): indu gave bindu rs . 6500 on compound interest for 2 years at 4 % per annum . how much loss would indu has suffered had she given it to bindu for 2 years at 4 % per annum simple interest ?
Options: a ) 12.8 , b ) 20 , c ) 10.4 , d ) 11.6 , e ) 15
Rationale: "6500 = d ( 100 / 4 ) 2 d = 10.4 answer : c"
Answer: c
Question (physics): if the product of two numbers is 84942 and their h . c . f . is 33 , find their l . c . m .
Options: a ) 2574 , b ) 2500 , c ) 1365 , d ) 1574 , e ) none of these
Rationale: "explanation : hcf * lcm = 84942 , because we know product of two numbers = product of hcf and lcm lcm = 84942 / 33 = 2574 option a"
Answer: a
Question (general): if n is a natural number , then ( 6 n ^ 2 + 6 n ) is always divisible by
Options: a ) 6 only , b ) 6 and 12 both , c ) 12 only , d ) 18 only , e ) 12 and 18 both
Rationale: "( 6 n ^ 2 + 6 n ) = 6 n ( n + 1 ) , which is divisible by both 6 and 12 , since n ( n + 1 ) is always even . answer b ."
Answer: b
Question (general): express the ratio 3 2 / 3 : 7 1 / 3 in its simplest form .
Options: a ) 1 : 21 , b ) 1 : 1 , c ) 2 : 1 , d ) 1 : 2 , e ) 2 : 11
Rationale: solution we first convert the mixed numbers 3 2 / 3 and 7 1 / 3 into fractions 3 2 / 3 = 3 * 3 / 3 + 2 / 3 = 11 / 3 7 1 / 3 = 7 * 3 / 3 + 1 / 3 = 22 / 3 the ratio 3 2 / 3 : 7 1 / 3 can be expressed as 11 / 3 ÷ 22 / 3 = 11 / 3 × 3 / 22 simplify = 11 / 22 = 1 / 2 the ratio is 1 / 2 or 1 : 2 answer is d
Answer: d
Question (physics): a runs twice as fast as b and gives b a start of 64 m . how long should the racecourse be so that a and b might reach in the same time ?
Options: a ) 75 m . , b ) 128 m . , c ) 150 m . , d ) 100 m . , e ) none of the above
Rationale: "ratio of speeds of a and b is 2 : 1 b is 64 m away from a but we know that a covers 1 meter ( 2 - 1 ) more in every second than b the time taken for a to cover 64 m is 64 / 1 = 64 m so the total time taken by a and b to reach = 2 * 64 = 128 m answer : b"
Answer: b
Question (physics): on a map , 2.5 inches represent 25 miles . how many miles approximately is the distance if you measured 55 centimeters assuming that 1 - inch is 2.54 centimeters ?
Options: a ) 212.5 , b ) 213.5 , c ) 214.5 , d ) 215.5 , e ) 216.5
Rationale: "1 inch = 2.54 cm 2.5 inch = 2.54 * 2.5 cm 6.35 cm = 25 miles 55 cms = 25 / 6.35 * 55 = 216.5 miles answer : e"
Answer: e
Question (general): a certain bus driver is paid a regular rate of $ 16 per hour for any number of hours that does not exceed 40 hours per week . for any overtime hours worked in excess of 40 hours per week , the bus driver is paid a rate that is 75 % higher than his regular rate . if last week the bus driver earned $ 920 in total compensation , how many total hours did he work that week ?
Options: a ) 36 , b ) 40 , c ) 44 , d ) 50 , e ) 52
Rationale: "for 40 hrs = 40 * 16 = 640 excess = 920 - 640 = 280 for extra hours = . 75 ( 16 ) = 12 + 16 = 28 number of extra hrs = 280 / 28 = 70 / 7 = 10 total hrs = 40 + 10 = 50 answer d 50"
Answer: d
Question (geometry): when magnified 1,000 times by an electron microscope , the image of a certain circular piece of tissue has a diameter of 2 centimeter . the actual diameter of the tissue , in centimeters , is
Options: a ) 0.005
b ) 0.002
c ) 0.001
d ) 0.0005
e ) 0.0002
Rationale: it is very easy if x is the diameter , then the magnified length is 1000 x . ince 1000 x = 2 then x = 2 / 1000 = 0.002 . the answer is b
Answer: b
Question (general): the average age of a class of 20 students is 21 years . the average increased by 1 when the teacher ' s age also included . what is the age of the teacher ?
Options: a ) 40 , b ) 41 , c ) 42 , d ) 45 , e ) 43
Rationale: "total age of all students = 20 ã — 21 total age of all students + age of the teacher = 21 ã — 22 age of the teacher = 21 ã — 22 â ˆ ’ 21 ã — 20 = 21 ( 22 â ˆ ’ 20 ) = 21 ã — 2 = 42 answer is c ."
Answer: c
Question (general): when the positive integer k is divided by the positive integer n , the remainder is 11 . if k / n = 71.2 , what is the value of n ?
Options: a ) 9 , b ) 20 , c ) 55 , d ) 70 , e ) 80
Rationale: "here ' s an approach that ' s based on number properties and a bit ofbrute forcemath : we ' re told that k and n are both integers . since k / n = 81.2 , we can say that k = 81.2 ( n ) n has tomultiply outthe . 2 so that k becomes an integer . with the answers that we have to work with , n has to be a multiple of 5 . eliminate a and e . with the remaining answers , we can test the answers and find the one that fits the rest of the info ( k / n = 81.2 and k / n has a remainder of 11 ) answer b : if n = 20 , then k = 1624 ; 1624 / 20 has a remainder of 4 not a match answer c : if n = 55 , then k = 4466 ; 4466 / 55 has a remainder of 11 match . final answer : e"
Answer: e
Question (general): if a and b are integers and ( a * b ) ^ 5 = 128 y , y could be :
Options: a ) 1 , b ) 2 , c ) 4 , d ) 8 , e ) 16
Rationale: distribute the exponent . a ^ 5 * b ^ 5 = 128 y find the prime factorization of 96 . this is 2 ^ 5 * 2 ^ 2 . we need 2 ^ 3 ( or some other power of 2 that will give us a multiple of 2 ^ 5 as our second term ) . 2 ^ 3 = 8 the answer is d .
Answer: d
Question (physics): a train after traveling for 50 km meets with an accident and then proceeds at 3 / 4 of its former speed and arrives at its destination 35 minutes late . had the accident occurred 24 km farther , it would have reached the destination only 25 minutes late . what is the speed t of the train .
Options: a ) a ) 45 , b ) b ) 33 , c ) c ) 48 , d ) d ) 55 , e ) e ) 61
Rationale: "let y be the balance distance to be covered and x be the former speed . a train after traveling for 50 km meets with an accident and then proceeds at 3 / 4 of its former speed and arrives at its destination 35 minutes late so , y / ( 3 x / 4 ) - y / x = 35 / 60 4 y / 3 x - y / x = 7 / 12 y / x ( 4 / 3 - 1 ) = 7 / 12 y / x * 1 / 3 = 7 / 12 y / x = 7 / 4 4 y - 7 x = 0 . . . . . . . . 1 had the accident occurred 24 km farther , it would have reached the destination only 25 minutes late so , ( y - 24 ) / ( 3 x / 4 ) - ( y - 24 ) / x = 25 / 60 4 ( y - 24 ) / 3 x - ( y - 24 ) / x = 5 / 12 ( y - 24 ) / x ( 4 / 3 - 1 ) = 5 / 12 ( y - 24 ) / x * 1 / 3 = 5 / 12 ( y - 24 ) * 12 = 3 x * 5 ( y - 24 ) * 4 = 5 x 4 y - 5 x = 96 . . . . . . . 2 eq 2 - eq 1 2 x = 96 x = 48 = t ans = c"
Answer: c
Question (general): jacob is now 12 years younger than michael . if 5 years from now michael will be twice as old as jacob , how old will jacob be in 4 years ?
Options: a ) 3 , b ) 7 , c ) 11 , d ) 21 , e ) 25
Rationale: "jacob = x years , mike = x + 12 years 5 years from now , 2 ( x + 5 ) = x + 17 2 x + 10 = x + 17 x = 7 x + 4 = 11 years answer c"
Answer: c