Cheukting/mathqa huggingface dataset:
Question (general): a company wants to spend equal amounts of money for the purchase of two types of computer printers costing $ 350 and $ 200 per unit , respectively . what is the fewest number of computer printers that the company can purchase ?
Options: a ) 9 , b ) 10 , c ) 11 , d ) 12 , e ) 13
Rationale: "the smallest amount that the company can spend is the lcm of 350 and 200 , which is 1400 for each , which is total 2800 . the number of 1 st type of computers which costing $ 350 = 1400 / 350 = 4 . the number of 2 nd type of computers which costing $ 200 = 1400 / 200 = 7 . total = 4 + 7 = 11 answer is c ."
Answer: c
Question (geometry): if the height of a cone is increased by 120 % then its volume is increased by ?
Options: a ) 100 , b ) 777 , c ) 998 , d ) 729 , e ) 120
Rationale: "120 % answer : e"
Answer: e
Question (gain): after decreasing 40 % in the price of an article costs rs . 1050 . find the actual cost of an article ?
Options: a ) 1200 , b ) 1750 , c ) 1050 , d ) 1570 , e ) 1005
Rationale: "cp * ( 60 / 100 ) = 1050 cp = 17.5 * 100 = > cp = 1750 answer : b"
Answer: b
Question (general): a fair sided dice labeled 1 to 6 is tossed 2 times . what is the probability the sum of the 2 throws is 5 ?
Options: a ) 1 / 6 , b ) 7 / 216 , c ) 4 / 36 , d ) 9 / 216 , e ) 11 / 216
Rationale: the total possible number of combination if the dice is thrown 2 times is 36 . there are 4 possible outcomes that the sum of the number is 5 which is ( 2 + 3 ) ( 3 + 2 ) ( 1 + 4 ) and ( 4 + 1 ) answer : c
Answer: c
Question (gain): a merchant marks his goods up by 50 % and then offers a discount of 10 % on the marked price . what % profit does the merchant make after the discount ?
Options: a ) 21 % , b ) 25 % , c ) 69 % , d ) 35 % , e ) 19 %
Rationale: "let the price be 100 . the price becomes 150 after a 50 % markup . now a discount of 10 % on 150 . profit = 135 - 100 35 % answer d"
Answer: d
Question (gain): i bought two books ; for rs . 460 . i sold one at a loss of 15 % and other at a gain of 19 % and then i found each book was sold at the same price . find the cost of the book sold at a loss ?
Options: a ) 337.33 , b ) 280.33 , c ) 299.33 , d ) 268.33 , e ) 289.33
Rationale: "x * ( 85 / 100 ) = ( 460 - x ) 119 / 100 x = 280 answer : d"
Answer: d
Question (physics): what is the least number of squares tiles required to pave the floor of a room 6 m 75 cm long and 4 m 5 cm broad ?
Options: a ) 15 , b ) 20 , c ) 40 , d ) 44 , e ) 54
Rationale: "length of largest tile = h . c . f . of 675 cm and 405 cm = 135 cm . area of each tile = ( 135 x 135 ) cm 2 . required number of tiles = 675 x 405 / ( 135 ^ 2 ) = 15 answer : a"
Answer: a
Question (gain): in how many years will a sum of money doubles itself at 5 % per annum simple interest
Options: a ) 25 years , b ) 28 years , c ) 10 years , d ) 20 years , e ) 21 years
Rationale: "let the initial sum be x then , amount = 2 x ( because sum will be double ) amount = principal + s . i 2 x = x + s . i s . i = x s . i = ( p * r * t ) / 100 x = ( x * 5 * t ) / 100 therefore , t = 20 years answer : d"
Answer: d
Question (general): for any positive integer n , the sum of the first n positive integers equals n ( n + 1 ) / 2 . what is the sum of odd all the integers between 1 and 50 ?
Options: a ) 500 , b ) 625 , c ) 750 , d ) 550 , e ) 600
Rationale: "numbers are 1 , 3,5 - - - - - - - , 47,49 average of the set : ( largest + smallest ) / 2 = ( 49 + 1 ) / 2 = 25 ; # of terms : ( largest - smallest ) / 2 + 1 = ( 49 - 1 ) / 2 + 1 = 25 the sum = 25 * 25 = 625 answer : b"
Answer: b
Question (general): on dividing a number by 357 , we get 41 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 + 41 = ( 17 * 21 * y ) + ( 17 * 2 ) + 7 = 17 * ( 21 y + 2 ) + 7 . required number = 7 . answer is d"
Answer: d
Question (physics): machines x and y produce bottles at their respective constant rates . machine x produces k bottles in 6 hours and machine y produces k bottles in 12 hours . how many hours does it take machines x and y , working simultaneously , to produce 12 k bottles ?
Options: a ) 8 , b ) 12 , c ) 15 , d ) 48 , e ) 24
Rationale: "x rate = k / 6 y rate = k / 12 k / 6 + k / 12 = 12 k / t solving t = 48 answer d"
Answer: d
Question (other): a certain music store stocks 800 cellos and 600 violas . of these instruments , there are 80 cello - viola pairs , such that a cello and a viola were both made with wood from the same tree ( each tree can make at most one viola and one cello , so there are no pairs other than these 90 ) . if one viola and one cello are chosen at random , what is the probability that the two instruments are made with wood from the same tree ?
Options: a ) 3 / 16,000 , b ) 1 / 8,100 , c ) 1 / 600 , d ) 1 / 90 , e ) 2 / 45
Rationale: "solution provided by stanford 2012 is correct : 80 / 800 choosing one of the cellos which has a pair viola , 1 / 600 choosing the viola which is the pair of chosen cello - - > p = 80 / 800 * 1 / 600 = 1 / 6,000 . answer : c ."
Answer: c
Question (general): in a race where 8 cars are running , the chance that car x will win is 1 / 2 , that y will win is 1 / 4 and that z will win is 1 / 3 . assuming that a dead heat is impossible , find the chance that one of them will win .
Options: a ) 1 / 140 , b ) 11 / 12 , c ) 14 / 8 , d ) 13 / 12 , e ) 57 / 120
Rationale: required probability = p ( x ) + p ( y ) + p ( z ) ( all the events are mutually exclusive ) . = 1 / 2 + 1 / 4 + 1 / 3 = 13 / 12 answer : d
Answer: d
Question (physics): at a tanning salon , customers are charged $ 10 for their first visit in a calendar month and $ 4 for each visit after that in the same calendar month . in the last calendar month , 100 customers visited the salon , of which 30 made a second visit , and 10 made a third visit . all other customers made only one visit . if those visits were the only source of revenue for the salon , what was the revenue for the last calendar month at the salon ?
Options: a ) $ 1220 , b ) $ 1160 , c ) $ 1300 , d ) $ 1340 , e ) $ 1880
Rationale: "i get b . this question seems too straightforward for 600 + . am i missing something ? 100 first - time visits - - > 100 ( 10 ) = $ 1000 30 + 10 = 40 subsequent visits - - > 40 ( 4 ) = $ 160 total revenue : 1000 + 160 = $ 1160 the answer is b ."
Answer: b
Question (physics): 3 pumps , working 8 hours a day , can empty a tank in 2 days . how many hours a day should 4 pumps work in order to empty the tank in 1 day ?
Options: a ) 10 , b ) 12 , c ) 8 , d ) 15 , e ) 18
Rationale: "explanation : let the required hours needed be x more pumps , less hours ( indirect proportion ) more days , less hours ( indirect proportion ) hence we can write as ( pumps ) 3 : 4 ( days ) 2 : 1 } : : x : 8 ⇒ 3 × 2 × 8 = 4 × 1 × x ⇒ 3 × 2 × 2 = x ⇒ x = 12 answer : option b"
Answer: b
Question (other): in what ratio must rice of rs . 54 per kg be mixed with rice of rs . 35 per kg so that cost of mixture is rs . 42 per kg ?
Options: a ) 9 : 7 , b ) 7 : 12 , c ) 12 : 7 , d ) 5 : 6 , e ) 7 : 9
Rationale: ( 42 - 35 ) / ( 54 - 42 ) = 7 / 12 = 7 : 12 answer : b
Answer: b
Question (general): the sum of the numbers is 264 . if the first number be twice the second and third number be one - third of the first , then the second number is :
Options: a ) 48 , b ) 54 , c ) 72 , d ) 84 , e ) 27
Rationale: let the second number be x . then , first number = 2 x and third number = 2 x / 3 . 2 x + x + 2 x / 3 = 264 11 x / 3 = 264 x = 72 answer : c
Answer: c
Question (gain): if rs . 64 amount to rs . 83.20 in 2 years , what will rs . 86 amount to in 4 years at the same rate percent per annum ?
Options: a ) rs . 114.80 , b ) rs . 86.70 , c ) rs . 55.40 , d ) rs . 51.60 , e ) none
Rationale: "solution p = rs . 64 , s . i . = rs . ( 83.20 - 64 ) = rs . 19.20 , t = 2 years . so , rate = ( 100 x 19.20 / 64 x 2 ) = 15 % . now , p = rs . 86 , r = 15 % , t = 4 years . ∴ s . i . = rs . ( 86 x 15 x 4 / 100 ) rs . 51.60 . answer d"
Answer: d
Question (physics): nd the area of trapezium whose parallel sides are 20 cm and 18 cm long , and the distance between them is 12 cm ?
Options: a ) 243 cm 2 , b ) 239 cm 2 , c ) 267 cm 2 , d ) 230 cm 2 , e ) 228 cm 2
Rationale: "area of a trapezium = 1 / 2 ( sum of parallel sides ) * ( perpendicular distance between them ) = 1 / 2 ( 20 + 18 ) * ( 12 ) = 228 cm 2 answer : e"
Answer: e
Question (general): if x = 6 ^ 36 and x ^ x = 6 ^ k , what is k ?
Options: a ) 6 ^ 36 , b ) 6 ^ 38 , c ) 6 ^ 72 , d ) 6 ^ 216 , e ) 6 ^ 432
Rationale: "solution : we know that x = 6 ^ 36 which implies x ^ x = ( 6 ^ 36 ) ^ ( 6 ^ 36 ) = 6 ^ ( 36 * 6 ^ 36 ) [ because ( x ^ y ) ^ z = x ^ ( y * z ) ) ] so 6 ^ ( 6 ^ 2 * 6 ^ 36 ) = 6 ^ ( 6 ^ ( 2 + 36 ) ) [ because x ^ a * x ^ b = x ^ ( a + b ) ] therefore x ^ x = 6 ^ ( 6 ^ 38 ) given that x ^ x = 6 ^ k so 6 ^ ( 6 ^ 38 ) = 6 ^ k since the base is same the exponent will also be same therefore k = 6 ^ 38 answer : b"
Answer: b
Question (geometry): a cylindrical can has a radius of 6 centimeters and a height of 12 centimeters . what is the area , in square centimeters , of a rectangular label that completely covers the curved surface of the can without over - lapping ?
Options: a ) 16 pi , b ) 64 pi , c ) 96 pi , d ) 144 pi , e ) 576 pi
Rationale: "they are asking the lateral surface area of the cylinder . the lateral surface area = 2 * pi * r * h = 2 * pi * 6 * 12 = 144 pi answer is d ."
Answer: d
Question (physics): a digital wristwatch was set accurately at 8.30 a . m and then lost 2 seconds every 5 minutes . what time was indicated on the watch at 6.30 p . m of the same day if the watch operated continuously that time ?
Options: a ) 5.56 , b ) 5.58 , c ) 6.0 , d ) 6.23 , e ) 6.26
Rationale: for every 5 min watch looses 2 secs , so for 60 ms it looses 24 sec , i . e . , for 1 hr it looses 24 secs 8.30 amto 6.30 pm = 10 hr it looses 240 secs i . e . , 4 mins , hence d watch will show 6.26 answer : e
Answer: e
Question (general): what is the remainder of w = 3 ^ 19 when divided by 10 ?
Options: a ) 0 , b ) 1 , c ) 5 , d ) 7 , e ) 9
Rationale: "i looked for patterns : ^ 2 - units digit 9 ^ 3 - units digit 7 ^ 4 - units digit 1 ^ 5 - units digit 3 hence , we can see that when raised to a power which is multiple of 4 , the units digit is 1 , and when to an even power not multiple of 4 , the units digit is 9 and we can then see : ^ 16 - units digit 1 , or ^ 18 - units digit 9 and ^ 19 - units digit 7 therefore , w = when divided by 10 , the remainder must be 7 . d"
Answer: d
Question (geometry): if each edge of cube increased by 10 % , the percentage increase in surface area is :
Options: a ) 33.1
b ) 33.5
c ) 33.7
d ) 33.2
e ) 33.9
Rationale: 100 × ( 110 ) / 100 × ( 110 ) / 100 × ( 110 ) / 100 = > 1331 / 100 = 33.1 % answer is a .
Answer: a
Question (physics): a train 110 m long is running with a speed of 56 km / hr . in what time will it pass a man who is running at 6 km / hr in the direction opposite to that in which the train is going ?
Options: a ) 5 , b ) 6.4 , c ) 7 , d ) 8 , e ) 9
Rationale: "speed of train relative to man = 56 + 6 = 62 km / hr . = 62 * 5 / 18 = 155 / 9 m / sec . time taken to pass the men = 110 * 9 / 155 = 6.4 sec . answer : option b"
Answer: b
Question (general): if a and b are integers and ( a * b ) ^ 5 = 48 y , y could be :
Options: a ) 8 , b ) 16 , c ) 162 , d ) 144 , e ) 100
Rationale: "distribute the exponent . a ^ 5 * b ^ 5 = 48 y find the prime factorization of 48 . this is 2 ^ 4 * 3 ^ 1 . we need 2 ^ 1 * 3 ^ 4 ( or some other power of 3 that will give us a multiple of 3 ^ 5 as our second term ) . 3 ^ 4 * 2 = 81 * 2 = 162 the answer is c ."
Answer: c
Question (general): 5 men are equal to as many women as are equal to 8 boys . all of them earn rs . 105 only . men â € ™ s wages are ?
Options: a ) 6 rs , b ) 7 rs , c ) 8 rs , d ) 4 rs , e ) 1 rs
Rationale: "5 m = xw = 8 b 5 m + xw + 8 b - - - - - 105 rs . 5 m + 5 m + 5 m - - - - - 105 rs . 15 m - - - - - - 105 rs . = > 1 m = 7 rs . answer : b"
Answer: b
Question (physics): the pilot of a small aircraft with a 40 - gallon fuel tank wants to fly to cleveland , which is 520 miles away . the pilot recognizes that the current engine , which can fly only 8 miles per gallon , will not get him there . by how many miles per gallon must the aircraft ’ s fuel efficiency be improved to make the flight to cleveland possible ?
Options: a ) 5 , b ) 4 , c ) 12 , d ) 40 , e ) 160
Rationale: "actual miles / gallon is = 520 / 40 = 13 miles / gallon . current engine miles / gallon is 8 miles / gallon . additional 5 miles / gallon is required to match the actual mileage . imo option a ."
Answer: a
Question (physics): jerry travels 8 miles at an average speed of 40 miles per hour , stops for 11 minutes , and then travels another 20 miles at an average speed of 60 miles per hour . what is jerry ’ s average speed , in miles per hour , for this trip ?
Options: a ) 37 , b ) 42.5 , c ) 44 , d ) 50 , e ) 52.5
Rationale: "total time taken by jerry = ( 8 / 40 ) * 60 minutes + 11 minutes + ( 20 / 60 ) * 60 minutes = 45 minutes average speed = total distance / total time = ( 8 + 20 ) miles / ( 45 / 60 ) hours = 28 * 60 / 45 = 37 miles per hour answer : option a"
Answer: a
Question (physics): car a runs at the speed of 65 km / hr & reaches its destination in 8 hr . car b runs at the speed of 75 km / h & reaches its destination in 4 h . what is the respective ratio of distances covered by car a & car b ?
Options: a ) 26 : 15 , b ) 12 : 7 , c ) 13 : 7 , d ) 15 : 6 , e ) 13 : 6
Rationale: "sol . distance travelled by car a = 65 ã — 8 = 520 km distance travelled by car b = 75 ã — 4 = 300 km ratio = 520 / 300 = 26 : 15 a"
Answer: a
Question (general): 30 ^ 10 / 90 ^ 5 = ?
Options: a ) 6 ^ 5 , b ) 5 ^ 6 , c ) 3 ^ 6 , d ) 10 ^ 5 , e ) 15 ^ 3
Rationale: "30 ^ 10 / 90 ^ 5 = ? a . 6 ^ 5 b . 5 ^ 6 c . 3 ^ 6 d . 10 ^ 5 e . 15 ^ 3 - > 30 ^ 10 / 90 ^ 5 = ( 30 ^ 10 ) / ( 3 ^ 5 ) ( 30 ^ 5 ) = ( 30 ^ 5 ) / ( 3 ^ 5 ) = ( 3 ^ 5 ) ( 10 ^ 5 ) / ( 3 ^ 5 ) = 10 ^ 5 . thus , d is the answer ."
Answer: d
Question (geometry): the ratio of the areas of a square and rhombus whose base is same is :
Options: a ) 1 : 2 , b ) 2 : 1 , c ) 1 : 1 , d ) 3 : 1 , e ) 2 : 3
Rationale: "solution : the square and the rhombus are equal in area if they have a common base . answer is c"
Answer: c
Question (physics): a boat can travel with a speed of 25 km / hr in still water . if the speed of the stream is 5 km / hr , find the time taken by the boat to go 120 km downstream .
Options: a ) 1 hr , b ) 2 hrs , c ) 3 hrs , d ) 4 hrs , e ) 5 hrs
Rationale: "speed downstream = ( 25 + 5 ) km / hr = 30 km / hr . time taken to travel 120 km downstream = 120 / 30 hrs = 4 hrs . answer : d"
Answer: d
Question (gain): the banker â € ™ s gain on a sum due 3 years hence at 17 % per annum is rs . 100 . the banker â € ™ s discount is
Options: a ) rs . 960 , b ) rs . 840 , c ) rs . 296 , d ) rs . 760 , e ) none
Rationale: "solution t . d = ( b . g x 100 / r x t ) = rs . ( 100 x 100 / 17 x 3 ) = rs . 196 . b . d = rs ( 196 + 100 ) = rs . 296 . answer c"
Answer: c
Question (gain): an error 4 % in excess is made while measuring the side of a square . the percentage of error in the calculated area of the square is :
Options: a ) 4.16 , b ) 5.16 , c ) 7.16 , d ) 8.16 , e ) 9.16
Rationale: "explanation : 100 cm is read as 104 cm . a 1 = ( 100 × 100 ) cm 2 = 10000 and a 2 = ( 104 × 104 ) cm 2 = 10816 ( a 2 - a 1 ) = 10816 - 10000 = 816 = > 816 / 10000 * 100 = 8.16 answer : d"
Answer: d
Question (gain): a 90 - liter solution of cool - drink is made from 5 % jasmine water . if 8 liters of jasmine and 2 liters of water were added to the solution , what percent of the solution is jasmine ?
Options: a ) 6.5 % , b ) 8.5 % , c ) 10.5 % , d ) 12.5 % , e ) 14.5 %
Rationale: the percent of jasmine in the resulting solution is : ( amount of jasmine ) / ( total volume ) ( 0.05 ( 90 ) + 8 ) / 100 = 12.5 / 100 = 12.5 % the answer is d .
Answer: d
Question (other): if the ratio of two number is 4 : 6 and lcm of the number is 280 then what is the number .
Options: a ) 12 , b ) 20 , c ) 25 , d ) 30 , e ) 35
Rationale: product of two no = lcm * hcf 4 x * 6 x = 280 * x x = 12 answer : a
Answer: a
Question (general): by how much does the larger root of the equation 2 b ^ 2 + 5 b = 12 exceed the smaller root ?
Options: a ) 5 / 2 , b ) 10 / 3 , c ) 7 / 2 , d ) 14 / 3 , e ) 11 / 2
Rationale: "for 2 b ^ 2 + 5 b = 12 roots are [ - 5 + sqrt ( 25 + 96 ) ] / 4 or [ - 5 - sqrt ( 25 + 96 ) ] / 4 = 1.5 or - 4 hence larger root 1.5 is 1.5 - ( - 4 ) = 5.5 = 11 / 2 greater than smaller root ( - 4 ) . hence option ( e ) ."
Answer: e
Question (other): between 100 to 200 how many no are divisible by both 3 and 2 and 100,200 are inclusive ?
Options: a ) 77 , b ) 78 , c ) 79 , d ) 80 , e ) 81
Rationale: prime numbers are 21 so 100 - 21 = 79 answer : c
Answer: c
Question (general): ifaequals the sum of the even integers from 2 to 20 , inclusive , andbequals the sum of the odd integers from 1 to 19 , inclusive , what is the value of ( a + b ) / 2 ?
Options: a ) 100 , b ) 110 , c ) 105 , d ) 20 , e ) 215
Rationale: "yes ! there is really a faster way to solve it . sum of consecutive odd or even integers = ( no . of odd or even ints ) * ( first int + last int ) / 2 here a = sum of even ints from 2 to 20 , inclusive number of even ints = 10 , first int + last int = 2 + 20 = 22 a = 10 * 22 / 2 = 110 b = sum of odd ints from 1 to 19 , inclusive number of odd ints = 10 , first int + last int = 1 + 19 = 20 a = 10 * 20 / 2 = 100 ( a + b ) / 2 = 110 + 100 = 210 / 2 = 105 ans : c"
Answer: c
Question (gain): martin has to secure 80 % marks to clear his exam of class 9 th . he got 200 marks and failed by 200 marks . what is the maximum marks ?
Options: a ) 500 , b ) 505 , c ) 510 , d ) 515 , e ) 525
Rationale: a 500 to pass the exam ravish needs 200 + 200 = 400 marks . = > ( 400 / 80 ) * 100 = 500
Answer: a
Question (geometry): the length of each side of square a is increased by 100 percent to make square b . if the length of the side of square b is increased by 60 percent to make square c , by what percent is the area of square c greater than the sum of the areas of squares a and b ?
Options: a ) 75 % , b ) 80 % , c ) 104.8 % , d ) 150.5 % , e ) 180 %
Rationale: "let length of each side of square a be 10 area of a = 10 ^ 2 = 100 since , length of each side of square a is increased by 100 percent to make square b length of each side of square b = 2 * 10 = 20 area of b = 20 ^ 2 = 400 since , length of the side of square b is increased by 60 percent to make square c length of each side of square c = 1.6 * 20 = 32 area of c = 32 ^ 2 = 1024 difference in areas of c and cummulative areas of a and b = 1024 - ( 400 + 100 ) = 524 percent is the area of square c greater than the sum of the areas of squares a and b = ( 524 / 500 ) * 100 % = 104.8 % answer c"
Answer: c
Question (geometry): find the perimeter and area of a square of side 15 cm .
Options: a ) 144 , b ) 169 , c ) 196 , d ) 121 , e ) 225
Rationale: "we know that the perimeter of square = 4 ã — side side = 15 cm therefore , perimeter = 4 ã — 15 cm = 60 cm now , area of the square = ( side ã — side ) sq . units = 15 ã — 15 cm â ² = 225 cm â ² answer : e"
Answer: e
Question (general): what is the remainder when 980 * 1050 * 880 * 1010 is divided by 22 ?
Options: a ) 15 , b ) 10 , c ) 12 , d ) 8 , e ) 3
Rationale: "oa is ' c ' . oe : take the remainder from each of 980 / 22 , 1050 / 22 and so on . . 980 / 22 gives remainder = 12 1050 / 22 gives remainder = 16 880 / 22 gives remainder = 0 1010 / 22 gives remainder = 20 the net remainder is the product of above individual remainders . i . e = 12 * 16 * 0 * 20 break them into pairs 12 * 16 / 22 gives remainder 16 and 0 * 20 / 22 gives remainder 20 so 16 * 20 / 22 gives remainder 12 . answer : c"
Answer: c
Question (other): a pizzeria in castroville creates its own special blend of shredded cheese at a cost of $ 696.05 per kilogram . the blend includes mozzarella , at $ 504.35 per kilogram , and romano cheese , at $ 887.75 per kilogram . if the employees have already shredded 19 kilograms of mozzarella cheese , how many kilograms of romano cheese will they need to shred now ?
Options: a ) 19 kg , b ) 13 kg , c ) 10 kg , d ) 23 kg , e ) 15 kg
Rationale: 504.35 ( 19 ) + 887.75 x = 696.05 ( 19 + x ) solve the equation . 504.35 ( 19 ) + 887.75 x = 696.05 ( 19 + x ) 9 , 582.65 + 887.75 x = 13 , 224.95 + 696.05 x 9 , 582.65 + 191.7 x = 13 , 224.95 191.7 x = 3 , 642.3 x = 19 . answer is a .
Answer: a
Question (general): product of two natural numbers is 17 . then , the sum of reciprocals of their squares is
Options: a ) 290 / 289 , b ) 1 / 289 , c ) 290 / 90 , d ) 290 / 19 , e ) none of these
Rationale: "explanation : if the numbers are a , b , then ab = 17 , as 17 is a prime number , so a = 1 , b = 17 . 1 / a 2 + 1 / b 2 = 1 / 1 ( 2 ) + 1 / 17 ( 2 ) = 290 / 289 option a"
Answer: a
Question (geometry): the perimeter of an isosceles right triangle is 10 + 10 sq rt 2 . what is the length of the hypotenuse of the triangle ?
Options: a ) 5 , b ) 6 , c ) 7.07 , d ) 8 , e ) 9
Rationale: "side of triangle is a then perimeter = a + a + a . sqrt 2 ( right angle and pythagorus ) = 2 a + a . sqrt 2 = 10 + 10 sqrt 2 or , a . ( 2 + sqrt 2 ) = 10 ( 1 + sqrt 2 ) , a = 10 * ( 1 + sqrt 2 ) / 2 + sqrt 2 = 10 * 2.414 / 3.414 = 0.707 * 10 then hypotenuse = 7.07 c"
Answer: c
Question (general): the average of 10 numbers is 40.2 . later it is found that two numbers have been wrongly copied . the first is 15 greater than the actual number and the second number added is 13 instead of 31 . find the correct average .
Options: a ) 40.2 , b ) 40.4 , c ) 40.5 , d ) 40.8 , e ) none of the above
Rationale: "sum of 10 numbers = 402 corrected sum of 10 numbers = 402 – 13 + 31 – 15 = 405 hence , new average = 405 ⁄ 10 = 40.5 answer c"
Answer: c
Question (general): a straight line in the xy - plane has a slope of 2 and a y - intercept of 2 . on this line , what is the x - coordinate of the point whose y - coordinate is 998 ?
Options: a ) 249 , b ) 498 , c ) 676 , d ) 823 , e ) 1,002
Rationale: "slope of 2 and a y - intercept of 2 y - coordinate is 998 y = 2 x + 2 996 = 2 x x = 498 answer : b . 498"
Answer: b
Question (physics): if two - third of a bucket is filled in 2 minutes then the time taken to fill the bucket completely will be .
Options: a ) 3 minutes , b ) 70 seconds , c ) 60 seconds , d ) 100 seconds , e ) 120 seconds
Rationale: "2 / 3 filled in 2 mint 1 / 3 filled in 1 mint then 2 / 3 + 1 / 3 = 2 + 1 = 3 minutes answer : a"
Answer: a
Question (geometry): marla is hanging a banner diagonally across a rectangular room . the length of the room is 3 feet shorter than its width . if the total area of the room is 130 square feet , what is the length of the banner ( in feet ) ?
Options: a ) between 16 and 17
b ) between 17 and 18
c ) between 18 and 19
d ) between 19 and 20
e ) between 20 and 21
Rationale: to solve this , we first need to set up an equation for the area of the room . if x is the width , then we have x ( x - 3 ) = 130 . by putting the equation in standard form , we get x ^ 2 - 3 x - 130 = 0 . by using the quadratic formula , we get roots of - 10 and 13 . we know that x is the width , and x - 3 is the length , so by using the roots , we get 13 as the width ( x ) , and 10 as the length ( x - 3 ) . once we have this , we can use the pythagorean theorem to solve for the diagonal . plugging in the length and width , we will get d ^ 2 = 13 ^ 2 + 10 ^ 2 = 169 + 100 = 269 the square root of 269 is less than 17 , since 17 ^ 2 = 289 . going down a number , we can compute 16 ^ 2 = 256 < 269 . therefore , the length of the diagonal must be between 16 and 17 . the correct answer is a .
Answer: a
Question (gain): if p gets 25 % more than q , then by what percent does q get less than p ?
Options: a ) 45 , b ) 34 , c ) 20 , d ) 76 , e ) 54
Rationale: let q get 1 unit . 25 % of 1 unit = 1 / 4 = 0.25 p gets ( 1 + 0.25 ) i . e . , 1.25 units . difference = ( 1.25 - 1 ) = 0.25 required percentage = 0.25 / 1.25 * 100 = 1 / 5 * 100 = 20 answer : c
Answer: c