Bluel0la/Set_1 · Datasets at Hugging Face

Instruction stringclasses 5 values	Input stringlengths 43 704	response stringlengths 1 1.77k
Provide a detailed solution to the given mathematical problem.	what is value of x if 32 - y = 16 x - y ? a) 1 , b) 3 , c) 4 , d) 2 , e) 6	"32 - y = 16 x - y 32 = 16 x - y + y 16 x = 32 x = 2 answer is option d"
Solve the problem systematically, showing each step clearly.	15.06 * 0.00000001 = ? a) 15060000 , b) 0.001506 , c) 0.01506 , d) 1.506 e - 07 , e) none of these	"explanation : clearly after decimal 10 digits should be there . option d"
Work through the equation methodically, ensuring clarity at each step.	city x has a population 7 times as great as the population of city y , which has a population twice as great as the population of city z . what is the ratio of the population of city x to the population of city z ? a) 1 : 8 , b) 1 : 4 , c) 2 : 1 , d) 14 : 1 , e) 18 : 1	x = 7 y , y = 2 * z x : y , y : z 7 : 1 , 2 : 1 14 : 2 , 2 : 1 so , x : z = 14 : 1 (d)
Work through the equation methodically, ensuring clarity at each step.	seed mixture x is 40 % ryegrass and 60 % bluegrass by weight ; seed mixture y is 25 % ryegrass and 75 % fescue . if a mixture of x and y contains 30 % ryegrass , what percent of the weight of the mixture is x ? a) 10 % , b) 33.33 % , c) 40 % , d) 50 % , e) 66.67 %	"x - wt of mix x added y - wt of mix y added 0.4 x + 0.25 y = 0.3 (x + y) 0.1 x = 0.05 y x / y = 5 / 10 = 1 / 2 y / x = 2 (y + x) / x = 3 = > x / (y + x) * 100 = 100 / 3 = 33.33 % answer : b"
Solve the problem systematically, showing each step clearly.	a certain machine produces 550 units of product p per hour . working continuously at this constant rate , this machine will produce how many units of product p in 5 days ? a) 7,000 , b) 24,000 , c) 66,000 , d) 100,000 , e) 168,000	"since 5 days consist of 24 * 5 hours the total is 120 hours . since every hour the machine produces 550 units of product p the total product during 120 hours is 120 * 550 = 66,000 . correct option : c"
Explain the solution process for the given equation.	if 2 ^ 4 , 3 ^ 3 , and 10 ^ 3 are factors of the product of 1,452 and w , where w is a positive integer , what is the smallest possible value of w ? a) 198 , b) 288 , c) 363 , d) 360 , e) 484	i will go with d (pending elements to match is 2 ^ 2 * 3 ^ 2 * 10 ^ 1 = 360
Break down and solve the equation step by step.	xy = 2 then what is (2 ^ (x + y) ^ 2) / (2 ^ (x - y) ^ 2) a) 2 , b) 4 , c) 8 , d) 16 , e) 256	(x + y) ^ 2 - (x - y) ^ 2 (x + y + x - y) (x + y - x + y) (2 x) (2 y) 4 xy 8 2 ^ 8 = 256 answer e
Break down and solve the equation step by step.	how many 3 - digit even numbers are possible such that if one of the digits is 4 , the next / succeeding digit to it should be 7 ? a) 305 , b) 5 , c) 365 , d) 405 , e) 495	470 , 472 , 474 , 476 , and 478 , so total 5 . hence option b .
Provide a detailed solution to the given mathematical problem.	a cycle is bought for rs . 900 and sold for rs . 1120 , find the gain percent ? a) 11 , b) 24 , c) 99 , d) 77 , e) 18	"900 - - - - 220 100 - - - - ? = > 24 % answer : b"
Work through the equation methodically, ensuring clarity at each step.	in what time will a train 100 m long cross an electric pole , it its speed be 72 km / hr ? a) 2.5 , b) 5 , c) 2.4 , d) 2.8 , e) 2.1	"speed = 72 * 5 / 18 = 20 m / sec time taken = 100 / 20 = 5 sec . answer : b"
Explain the solution process for the given equation.	3 business people wish to invest in a new company . each person is willing to pay one third of the total investment . after careful calculations , they realize that each of them would pay $ 2200 less if they could find two more equal investors . how much is the total investment in the new business ? a) $ 11,000 , b) $ 50,000 , c) $ 16,500 , d) $ 6,600 , e) $ 3,600	total investment between 5 : (x / 5) total investment including 2200 less between 3 people (x - (2200 * 3)) / 3 set both eq . equal to each other . 16,500 answer c)
Break down and solve the equation step by step.	if you keep rolling a pair of dice together till a sum of 5 or 7 is obtained , then what is the probability that a sum of 5 comes before a sum of 7 ? a) 5 / 5 , b) 4 / 5 , c) 3 / 5 , d) 2 / 5 , e) 1 / 5	this can be solved in a generic and complex way but let us not go into all that . there can be four ways through which the pair of dice results in a sum of 5 . there can be six ways through which the pair of dice can result in a sum of 7 . now , we want the probability of the pair of dice resulting in a sum of 5 before a sum of 7 . thus probability = 4 / (4 + 6) = 4 / 10 or 2 / 5 . correct answer is d) 2 / 5
Break down and solve the equation step by step.	a train sets off at 9 : 00 am at the speed of 70 km / h . another train starts at 10 : 30 am in the same direction at the rate of 84 km / h . at what time will the second train catch the first train ? a) 4 : 30 , b) 5 : 00 , c) 5 : 30 , d) 6 : 00 , e) 6 : 30	"in one hour and thirty minutes the first train travels 105 km . the second train catches the first train at a rate of 84 km / h - 70 km / h = 14 km / h . the second train will catch the first train in 105 / 14 = 7.5 hours , so at 6 : 00 pm . the answer is d ."
Work through the equation methodically, ensuring clarity at each step.	a man buys a cycle for rs . 2300 and sells it at a loss of 30 % . what is the selling price of the cycle ? a) s . 1690 , b) s . 1610 , c) s . 1890 , d) s . 1602 , e) s . 1092	"s . p . = 70 % of rs . 2300 = rs . 70 x 2300 / 100 = rs . 1610 answer : option b"
Solve the problem systematically, showing each step clearly.	for any integer k greater than 1 , the symbol k * denotes the product of all integers between 1 and k , inclusive . if k * is a multiple of 315 what is the least possible value of k ? a) 5 , b) 7 , c) 9 , d) 11 , e) 13	315 = 3 * 3 * 5 * 7 thus k must include numbers at least up to the number 7 . the answer is b .
Solve the problem systematically, showing each step clearly.	if integer k is equal to the sum of all even multiples of 25 between 200 and 600 , what is the greatest prime factor of k ? a) 5 , b) 7 , c) 11 , d) 13 , e) 17	"if we break down what the stem is asking what is the sum of all mult of 50 between 200 and 600 . using arithmetic progression to find n : 600 = 200 + (n - 1) 50 400 + 50 = 50 n 450 = 50 n = > n = 9 the sum would be : 9 * mean mean = [ 600 + 200 ] / 2 = 400 9 * 400 = 3600 d"
Work through the equation methodically, ensuring clarity at each step.	36 welders work at a constant rate they complete an order in 3 days . if after the first day , 12 welders start to work on the other project , how many more days the remaining welders will need to complete the rest of the order ? a) 3 , b) 2 , c) 8 , d) 4 , e) 6	"1 . we need to find out the time taken by 24 workers after day 1 . 2 . total no . of wokers * total time taken = time taken by 1 worker 3 . time taken by 1 worker = 36 * 3 = 108 days 4 . but on day 1 thirty - six workers had already worked finishing 1 / 3 of the job . so 24 workers have to finish only 2 / 3 of the job . 5 . total time taken by 24 workers can be got from formula used at (2) . i . e . , 24 * total time taken = 108 . total time taken by 6 workers to finish the complete job is 108 / 24 = 4.5 days . 6 . time taken by 24 workers to finish 2 / 3 of the job is 2 / 3 * 4.5 = 3 days . the answer is choice a"
Break down and solve the equation step by step.	on the first of the year , james invested x dollars at proudstar bank in an account that yields 3 % in interest every quarter year . at the end of the year , during which he made no additional deposits or withdrawals , he had y dollars in the account . if james had invested the same amount in an account which pays interest on a yearly basis , what must the interest rate be for james to have y dollars at the end of the year ? a) 22.04 % , b) 16.12 % , c) 12.5 % , d) 18.25 % , e) 10 %	per quarter , interest = 3 % so for a year , interest = 12 % due to quarter cumulation , effective yield (ytm) would be slight higher than 12 % answer = 12.5 % = answer : c
Provide a detailed solution to the given mathematical problem.	how many positive integers will divide evenly into 190 ? a) 4 , b) 6 , c) 8 , d) 12 , e) 16	the question is asking how many factors 190 has . 190 = 2 * 5 * 19 the number of factors is 2 ^ 3 = 8 the answer is c .
Break down and solve the equation step by step.	two trains , one from howrah to patna and the other from patna to howrah , start simultaneously . after they meet , the trains reach their destinations after 4 hours and 10 hours respectively . the ratio of their speeds is a) 4 : 9 , b) 4 : 3 , c) 4 : 12 , d) 4 : 8 , e) 5 : 2	let us name the trains as a and b . then , (a' s speed) : (b' s speed) = b : a = 10 : 4 = 5 : 2 . answer : e
Break down and solve the equation step by step.	find number which is 60 % less than 120 . a) 18 , b) 22 , c) 28 , d) 48 , e) 98	"explanation : 60 % less is 40 % of the given number therefore , 40 % of 120 is 48 . answer : d"
Provide a detailed solution to the given mathematical problem.	if 7 spiders make 7 webs in 7 days , then 1 spider will make 1 web in how many days ? a) 7 , b) 6 , c) 5 , d) 4 , e) 3	let the required number days be x . less spiders , more days (indirect proportion) less webs , less days (direct proportion) spiders 1 : 7 webs 7 : 1 1 x 7 x x = 7 x 1 x 7 = > x = 7 answer is 7
Break down and solve the equation step by step.	two cars start from the opposite places of a main road , 150 km apart . first car runs for 30 km and takes a right turn and then runs 15 km . it then turns left and then runs for another 25 km and then takes the direction back to reach the main road . in the mean time , due to minor break down the other car has run only 35 km along the main road . what would be the distance between two cars at this point ? a) 65 , b) 38 , c) 60 , d) 28 , e) 21	answer : c) 60 km
Solve the problem systematically, showing each step clearly.	difference between the length and breadth of a rectangle is 23 m . ifits perimeter is 206 m , then its area is ? ? ['a) 1220 m ^ 2','b) 1520 m ^ 2','c) 2520 m ^ 2','d) 2600 m ^ 2','e) 3420 m ^ 2']	we have : (l - b) = 23 and 2 (l + b) = 206 or (l + b) = 103 . solving the two equations , we get : l = 63 and b = 40 . area = (l x b) = (63 x 40) m 2 = 2520 m ^ 2 c
Work through the equation methodically, ensuring clarity at each step.	if 30 honey bees make 30 grams of honey in 30 days , then 1 honey bee will make 1 gram of honey in how many days ? a) 1 , b) 3.5 , c) 30 , d) 49 , e) 30	"explanation : let the required number days be x . less honey bees , more days (indirect proportion) less honey , less days (direct proportion) honey bees 1 : 30 : : 30 : x honey 30 : 1 = > 1 x 30 x x = 30 x 1 x 30 = > x = 30 . answer : c"
Explain the solution process for the given equation.	find the principle on a certain sum of money at 5 % per annum for 3 1 / 5 years if the amount being rs . 2030 ? a) rs . 2000 , b) rs . 1750 , c) rs . 2010 , d) rs . 2005 , e) none of these	"explanation : 2030 = p [ 1 + (5 * 16 / 5) / 100 ] p = 1750 answer : option b"
Work through the equation methodically, ensuring clarity at each step.	24 * 22 = ? / 8 a) 4224 , b) 2224 , c) 5374 , d) 2434 , e) 4254	"a 4224 ? / 8 = 24 * 22 ? = 24 * 22 * 8 ? = 4224"
Solve the problem systematically, showing each step clearly.	john was 43 years old when he married betty . they just celebrated their fifth wedding anniversary , and betty' s age is now 7 / 8 of john' s . how old is betty ? a) 24 , b) 26 , c) 42 , d) 30 , e) 32	"assume betty' s age on marriage = x years . john' s age on marriage = 43 john' s age after 5 years = 48 years . betty' s age after 5 years = x + 5 given : x + 5 = 7 / 8 (48) = 42 therefore betty' s current age = 42 option c"
Explain the solution process for the given equation.	in a class of 50 students , 3 / 5 went away do painting , 1 / 5 went to play in the field . how many students are left in the classroom ? a) 20 , b) 30 , c) 5 , d) 8 , e) 10	3 / 5 + 1 / 5 are the students who have left the classroom 3 / 5 + 1 / 5 = (3 + 1) / 5 , since denominator is same for both fractions = 4 / 5 remaining students in the classroom = 1 - 4 / 5 = 5 / 5 - 4 / 5 = (5 - 4) / 5 = 1 / 5 1 / 5 of 50 = 10 ans : e
Work through the equation methodically, ensuring clarity at each step.	the average of 8 numbers is 25 , that of the first 2 is 20 and of the next 3 is 26 . the sixth number is less than the seventh by 4 , and less than the eighth by 6 . the last number is a) 30 , b) 32 , c) 40 , d) 36 , e) 38	explanation : sum of all the numbers = 25 * 8 = 200 sum of first two numbers = 40 sum of next three numbers (i . e . , 3 rd , 4 th and 5 th) = 78 so , sum of first five numbers = 118 and sum of 6 th , 7 th and 8 th number = 82 assuming sixth number to be x , we get 7 th number = x + 4 and 8 th number = x + 6 . so , x + (x + 4) + (x + 6) = 82 = > x = 24 . hence , last number (8 th number) = 30 . answer : a
Explain the solution process for the given equation.	if the numbers 1 to 96 are written on 96 pieces of paper , (one on each) and one piece is picked at random , then what is the probability that the number drawn is neither prime nor composite ? a) 1 / 50 , b) 1 / 25 , c) 1 / 96 , d) 1 , e) 2	there are 25 primes , 70 composite numbers from 1 to 96 . the number which is neither prime nor composite is 1 . therefore , required probability = 1 / 96 . answer : c
Solve the problem systematically, showing each step clearly.	jim drove 642 miles of a 1200 miles journey . how many more miles does he need to drive to finish his journey ? a) 113 miles , b) 432 miles , c) 558 miles , d) 887 miles , e) 767 miles	"the number of miles to drive to finish his journey is given by 1200 - 642 = 558 miles correct answer c"
Explain the solution process for the given equation.	the ages of patrick and michael are in the ratio of 3 : 5 and that of michael and monica are in the ratio of 3 : 5 . if the sum of their ages is 145 , what is the difference between the ages of patrick and monica ? a) 27 , b) 48 , c) 45 , d) 72 , e) 18	"ages of p and mi = 3 x + 5 x ages of mi and mo = 3 x : 5 x rationalizing their ages . ratio of their ages will be 9 x : 15 x : 25 x sum = 47 x = 145 x = 3 difference if ages of pa and mo = 25 x - 9 x = 16 x = 16 * 3 = 48 answer b"
Work through the equation methodically, ensuring clarity at each step.	evaluate : 1234562 - 12 * 3 * 2 = ? a) 32435453 , b) 1234554 , c) 76768786 , d) 97979797 , e) 75868656	"according to order of operations , 12 ? 3 ? 2 (division and multiplication) is done first from left to right 12 * * 2 = 4 * 2 = 8 hence 1234562 - 12 * 3 * 2 = 1234562 - 8 = 1234554 correct answer b"
Explain the solution process for the given equation.	two stations p and q are 200 km apart on a straight track . one train starts from p at 7 a . m . and travels towards q at 20 kmph . another train starts from q at 8 a . m . and travels towards p at a speed of 25 kmph . at what time will they meet ? a) 10 am , b) 12 noon , c) 10.30 am , d) 12.30 am , e) 11 am	"assume both trains meet after x hours after 7 am distance covered by train starting from p in x hours = 20 x km distance covered by train starting from q in (x - 1) hours = 25 (x - 1) total distance = 200 = > 20 x + 25 (x - 1) = 200 = > 45 x = 225 = > x = 5 means , they meet after 5 hours after 7 am , ie , they meet at 12 noon answer is b ."
Solve the problem systematically, showing each step clearly.	a bottle contains a certain solution . in the bottled solution , the ratio of water to soap is 3 : 2 , and the ratio of soap to salt is five times this ratio . the solution is poured into an open container , and after some time , the ratio of water to soap in the open container is halved by water evaporation . at that time , what is the ratio of water to salt in the solution ? a) 1 : 1 , b) 2 : 3 , c) 3 : 2 , d) 9 : 4 , e) 45 : 8	"water : soap = 3 : 2 soap : salt = 15 : 2 = > for 15 soap , salt = 2 = > for 2 soap , salt = (2 / 15) * 2 = 4 / 15 so , water : soap : salt = 3 : 2 : 4 / 15 = 45 : 30 : 4 after open container , water : soap : salt = 22.5 : 30 : 4 so , water : salt = 22.5 : 4 = 45 : 8 e"
Explain the solution process for the given equation.	four machines , each working at the same constant rate , together can complete a certain job in 20 days . how many additional machines , each working at the same constant rate , will be needed to complete the job in 10 days ? a) 5 , b) 3 , c) 4 , d) 6 , e) 1	"rate of one machine = 1 job / (20 * 4) days let x = number of machines needed to complete the job in 10 days 1 / (4 * 20) * 10 * x = 1 job x = 8 8 - 4 = 4 answer : c"
Explain the solution process for the given equation.	what could be the range of a set consisting of odd multiples of (6 + 1) ? a) 70 , b) 24 , c) 35 , d) 62 , e) 21	"range = the difference between the greatest and the smallest numbers in the sequence . our sequence is odd and is a multiple of 7 . every number in that sequence can be represented like this : 7 * (2 n + 1) where n is any positive integer . range = 7 * (2 m + 1) - 7 * (2 n + 1) = 7 * 2 * (m - n) = 14 * (m - n) . m , n - any positive integers the answer must be divisible by 14 , which is only 70 . the correct answer is a ."
Provide a detailed solution to the given mathematical problem.	a manufacturer sells a pair of glasses to a wholesale dealer at a profit of 18 % . the wholesaler sells the same to retailer at a profit of 20 % . the retailer in turn sells them to a customer for rs . 30.09 , there by earning a profit of 25 % . the cost price for the manufacturer is ? a) 33 , b) 29 , c) 17 , d) 10 , e) 16	"explanation : let the cost price for the manufacturer be rs . x . then , 125 % of 120 % of 118 % of x = 30.09 . 125 / 100 * 120 / 100 * 118 / 100 * x = 3009 / 100 177 / 100 x = 3009 / 100 = > x = 17 answer : c"
Work through the equation methodically, ensuring clarity at each step.	two numbers are in the ratio 3 : 5 . if 9 be subtracted from each , they are in the ratio of 9 : 17 . the first number is : a) 36 , b) 77 , c) 88 , d) 55 , e) 221	"(3 x - 9) : (5 x - 9) = 9 : 17 x = 12 = > 3 x = 36 answer : a"
Provide a detailed solution to the given mathematical problem.	a shopkeeper buys two articles for rs . 1000 each and then sells them , making 40 % profit on the first article and 40 % loss on second article . find the net profit or loss percent ? a) 200 , b) 278 , c) 282 , d) 400 , e) 270	"profit on first article = 40 % of 1000 = 400 . this is equal to the loss he makes on the second article . that , is he makes neither profit nor loss . answer : d"
Solve the problem systematically, showing each step clearly.	of the 65 cars on a car lot , 45 have air - bag , 30 have power windows , and 12 have both air - bag and power windows . how many of the cars on the lot have neither air - bag nor power windows ? a) 8 , b) 2 , c) 10 , d) 15 , e) 18	total - neither = all air bag + all power windows - both or 65 - neither = 45 + 30 - 12 = 63 . = > neither = 2 , hence b . answer : b
Explain the solution process for the given equation.	two numbers are respectively 20 % and 50 % more than a third number . what percentage is the first of the second ? a) 25 % , b) 60 % , c) 80 % , d) 55 % , e) 74 %	"required answer = (100 + x) / (100 + y) * 100 = 120 * 100 / 150 = 80 % answer is c"
Explain the solution process for the given equation.	here is a simple arithmetical puzzle in his own flowery , poetical language . if one - fifth of a hive of bees flew to the badamba flower , one - third flew to the slandbara , 3 times the difference of these two numbers flew to an arbour , and one bee continued to fly about , attracted on each side by the fragrant ketaki and malati , what was the total number of bees ? a) 11 , b) 15 , c) 18 , d) 17 , e) 19	b 15 the number of bees was 15
Solve the problem systematically, showing each step clearly.	the weight of every type a widget is the same , the weight of every type b widget is the same , and the weight of every type c widget is the same . if the weight of 7 type a widgets is equal to the weight of 2 type b widgets , and the weight of 5 type b widgets is equal to the weight of 7 type c widgets . what is the ratio of the total weight of 1 type a widget and 1 type b widget , to the total weight of 1 type b widget and 1 type c widget ? a) 1 : 2 , b) 2 : 3 , c) 3 : 4 , d) 4 : 5 , e) 5 : 6	"5 b = 7 c and so b = 7 c / 5 7 a = 2 b and so a = 2 b / 7 = 2 c / 5 a + b = 2 c / 5 + 7 c / 5 = 9 c / 5 b + c = 7 c / 5 + c = 12 c / 5 the ratio of a + b : b + c = 9 : 12 = 3 : 4 the answer is c ."
Provide a detailed solution to the given mathematical problem.	the area of rectangular field of length 90 meters and breadth 80 meters is equal to the area of a square plot . what will be the length of the diagonal of the square plot ? ['a) 100 m','b) 120 m','c) 95 m','d) 80 m','e) 150 m']	given that the length of the rectangular field l = 90 m and breadth = b = 80 m . then its area lb = 90 x 80 sq . m we know that the area of square = (1 / 2) x (diagonal) 2 from the given question , we have area of square plot = area of rectangular field that is , 90 x 80 = 1 / 2 x diagonal 2 diagonal ^ 2 = 2 x 7200 = 14400 = (120) 2 diagonal = 120 m . hence b
Work through the equation methodically, ensuring clarity at each step.	if' a' and' b' are non - zero numbers such that their sum is 6 times the product , what is the value of 1 / a + 1 / b ? a) 5 , b) 6 , c) 7 , d) 8 , e) 12	a + b = 6 ab 1 / a + 1 / b = (b + a) / (ab) = 6 . the answer is b .
Work through the equation methodically, ensuring clarity at each step.	in the game of dubblefud , red balls , blue balls and green balls are each worth 2 , 4 and 5 points respectively . in a certain selection of balls , the product of the point values of the balls is 16000 . if the number of blue balls in this selection equals the number of green balls , how many red balls are in the selection ? a) 1 , b) 2 , c) 3 , d) 4 , e) 5	this is equivalent to : - 2 x * 4 y * 5 z = 16000 y = z (given) 2 x * 4 y * 5 y = 16000 2 x * y ^ 2 = 16000 / 20 2 x * y ^ 2 = 800 now from options given we will figure out which number will divide 800 and gives us a perfect square : - which gives us x = 4 as 2 * 4 * y ^ 2 = 800 y ^ 2 = 100 y = 10 number of red balls = 4 hence d
Solve the problem systematically, showing each step clearly.	what number has a 5 : 1 ratio to the number 11 ? a) 22 , b) 50 , c) 55 , d) 52 , e) 12	"5 : 1 = x : 10 x = 55 answer : c"
Break down and solve the equation step by step.	a man can do a piece of work in 7 days , but with the help of his son , he can do it in 3 days . in what time can the son do it alone ? a) 5 , b) 7 , c) 15 / 2 , d) 21 / 4 , e) 18 / 5	"son' s 1 day' s work = (1 / 3) - (1 / 7) = 4 / 21 the son alone can do the work in 21 / 4 days answer is d"
Break down and solve the equation step by step.	the average age of a group of 10 persons was decreased by 3 years when one person , whose age was 44 years , was replaced by a new person . find the age of the new person ? a) 18 , b) 56 , c) 12 , d) 17 , e) 14	"initial average age of the 10 persons be p . age of the new person q . sum of the ages of the initial 10 persons = 10 p new average = (p - 3) 10 (p - 3) = 10 p - 44 + q = > q = 14 answer : e"
Explain the solution process for the given equation.	a sports equipment store sold ping pong rackets for a total of $ 735 . if the average (arithmetic mean) price of a pair of rackets is $ 9.8 , how many pairs were sold ? a) 50 , b) 100 , c) 490 , d) 500 , e) 75	"average price for a pair of rackets = $ 9.8 total cost = $ 9.8 * x = $ 735 x = 75 pairs were sold . answer : e"
Solve the problem systematically, showing each step clearly.	when tom works alone he chops 4 lb . salad in 3 minutes , and when tammy works alone she chops 3 lb . salad in 4 minutes . they start working together , and after some time finish chopping 65 lb . of salad . of those 80 lb . , the salad quantity chopped by tammy is what percent greater than the quantifying chopped by tom ? . a) 125 % , b) 100 % , c) 25 % , d) 225 % , e) 400 %	"tom chops 8 lbs in 12 minutes tammy chopswhen tom works alone he chops 2 lb . salad in 3 minutes , and when tammy works alone she chops 3 lb . salad in 2 minutes . they start working together , and after some time finish chopping 65 lb . of salad . of those 80 lb . , the salad quantity chopped by tammy is what percent greater than the quantifying chopped by tom ? . 9 lbs in 6 minutes so in the same amount of time , tammy chops 125 % more than tom , since 9 is 125 % greater than 4 . so 125 % is the answer . note that the actual time does n' t matter . if you multiply the time each work by x , you' ll multiply the work each does by x , and 9 x is still 125 % greater than 4 x . ans : a"
Work through the equation methodically, ensuring clarity at each step.	bella is taking a trip in her car from new york (point a) chicago , illinois (point b) . she is traveling at an average speed of 50 miles per hour , if the total distance from point a to point b is 790 miles , in approximately how long will bella reach her destination ? a) 10 hours , b) 12.50 hours , c) 15.80 hours , d) 25 hours , e) 1 day	answer is (c) . if she is traveling at a speed of 50 miles per hour and her trip is a total of 790 miles , 790 miles divided by 50 miles per hours would equal 15.80 hours , just shy of 16 hours .
Break down and solve the equation step by step.	the least number which when divided by 2 , 3 , 5 and 6 leaves a remainder 3 , but when divided by 9 leaves no remainder , is a) 16 , b) 63 , c) 73 , d) 83 , e) none of these	"explanation : l . c . m of 2 , 3 , 5 , 6 = 30 therefore required number is of the form 30 k + 3 . least value of k for which (30 k + 3) is divisible by 9 is k = 2 therefore required number = (30 x 2 + 3) = 63 . answer : b"
Explain the solution process for the given equation.	john bought a shirt on sale for 25 % off the original price and another 25 % off the discounted price . if the final price was $ 19 , what was the price before the first discount ? a) $ 45.10 , b) $ 34.31 , c) $ 28.44 , d) $ 67.54 , e) $ 33.77	"let x be the price before the first discount . the price after the first discount is x - 25 % x (price after first discount) a second discount of 25 % of the discounted price after which the final price is 19 (x - 25 % x) - 25 % (x - 25 % x) = 19 solve for x x = $ 33.77 correct answer e"
Solve the problem systematically, showing each step clearly.	in the biology lab of ` ` jefferson'' high school there are 5.4 * 10 ^ 6 germs , equally divided among 10800 petri dishes . how many germs live happily in a single dish ? a) 100 , b) 200 , c) 500 , d) 1000 , e) 5000	5.4 * 10 ^ 6 / 10800 = = > 54 * 10 ^ 3 / 108 = = > 54000 / 108 = 500 answer : c
Work through the equation methodically, ensuring clarity at each step.	by selling 8 pencils for a rupee a man loses 60 % . how many for a rupee should he sell in order to gain 60 % ? a) 8 , b) 9 , c) 2 , d) 6 , e) 4	"40 % - - - 8 160 % - - - ? 40 / 160 * 12 = 2 answer : c"
Solve the problem systematically, showing each step clearly.	village x has a population of 68000 , which is decreasing at the rate of 1200 per year . village y has a population of 42000 , which is increasing at the rate of 800 per year . in how many years will the population of the two villages be equal ? a) 15 , b) 19 , c) 11 , d) 18 , e) 13	let the population of two villages be equal after p years then , 68000 - 1200 p = 42000 + 800 p 2000 p = 26000 p = 13 answer is e .
Explain the solution process for the given equation.	if 125 % of j is equal to 25 % of k , 150 % of k is equal to 50 % of l , and 175 % of l is equal to 75 % of m , then 20 % of m is equal to what percent of 200 % of j ? a) 0.35 , b) 3.5 , c) 35 , d) 350 , e) 3500	first of all , let us write the information in form of equations and numbers 125 j = 5 k or 5 j = k (i) 150 k = 50 l or 3 k = l (ii) 175 l = 75 m or 7 l = 3 m (iii) we need to find a relation between j and m from (i) and (ii) , 15 j = l multiplying this by 7 , 105 j = 7 l = 3 m hence , 35 j = m now , we are asked : 20 % of m is equal to what percent of 200 % of j assume j = 1 , hence m = 35 20 % of m = 7 and 200 % of j = 2 20 % of m = 3.5 (200 % of j) = (350 / 100) * (200 % of j) hence 20 % of m = 350 % of 200 % of j . option d
Solve the problem systematically, showing each step clearly.	if [ [ x ] ] = x ^ 2 + 2 x + 4 , what is the value of [ [ 2 ] ] ? a) 3 , b) 9 , c) 12 , d) 19 , e) 25	these functions questions might look intimidating , but they just test your knowledge about how well you can substitute values [ [ x ] ] = x ^ 2 + 2 x + 4 [ [ 2 ] ] = 2 ^ 2 + 2 * 2 + 4 = 12 . option c
Solve the problem systematically, showing each step clearly.	the compound ratio of 1 : 4 , 3 : 2 and 4 : 5 ? a) 1 : 9 , b) 1 : 7 , c) 1 : 2 , d) 3 : 10 , e) 1 : 4	"1 / 4 * 3 / 2 * 4 / 5 = 3 / 10 = 3 : 10 answer : d"
Work through the equation methodically, ensuring clarity at each step.	a pair of articles was bought for $ 720 at a discount of 10 % . what must be the marked price of each of the article ? a) $ 300 , b) $ 500 , c) $ 350 , d) $ 400 , e) $ 600	"s . p . of each of the article = 720 / 2 = $ 360 let m . p = $ x 90 % of x = 360 x = 360 * 100 / 90 = $ 400 answer is d"
Provide a detailed solution to the given mathematical problem.	two trains are moving in opposite directions at 60 km / hr and 90 km / hr . their lengths are 1.10 km and 0.15 km respectively . the time taken by the slower train to cross the faster train in seconds is ? a) 65 sec , b) 30 sec , c) 48 sec , d) 33 sec , e) 12 sec	"relative speed = 60 + 90 = 150 km / hr . = 150 * 5 / 18 = 125 / 3 m / sec . distance covered = 1.10 + 0.15 = 1.25 km = 1250 m . required time = 1250 * 3 / 125 = 30 sec . answer : b"
Provide a detailed solution to the given mathematical problem.	how many two - element subsets of < 1 , 23 , 45 , 67 > are there that do not contain the pair of elements 2 and 4 ? a) 11 , b) 12 , c) 14 , d) 15 , e) 20	7 c 2 - 1 = 20 answer is e
Solve the problem systematically, showing each step clearly.	the diameter of the driving wheel of a bus in 140 cm . how many revolutions per minute must the wheel make in order to keep a speed of 20 kmph ? a) 21.21 , b) 20.25 , c) 32.25 , d) 75.75 , e) 62.61	distance covered in 1 min = (20 * 1000) / 60 = 333.33 m circumference of the wheel = (2 * (22 / 7) * . 70) = 4.4 m no of revolution per min = 333.33 / 4.4 = 75.75 answer : d
Break down and solve the equation step by step.	what is the least common multiple of 5,40 , 36,50 a) 3698 , b) 874 , c) 3692 , d) 1800 , e) 3214	"the given numbers are 5,40 , 36,50 thus the l . c . m is 5 * 2 * 2 * 1 * 2 * 9 * 5 = 1800 the answer is d"
Solve the problem systematically, showing each step clearly.	if 13 = 13 w / (1 - w) , then (5 w) 2 = a) 1 / 4 , b) 5 , c) 1 , d) 2 , e) 3	"13 - 13 w = 13 w 26 w = 13 w = 1 / 2 5 w = 5 / 2 5 w * 2 = 5 / 2 * 2 = 5 answer : b"
Explain the solution process for the given equation.	what is the total number of integers between 2 and 38 that are divisible by 4 ? a) 8 , b) 9 , c) 7 , d) 6 , e) 11	"4 , 6 , 8 , . . . , 32,36 this is an equally spaced list ; you can use the formula : n = (largest - smallest) / (' space') + 1 = (36 - 4) / (4) + 1 = 32 / 4 + 1 = 8 + 1 = 9 answer is b"
Solve the problem systematically, showing each step clearly.	a and b complete a work in 8 days . a alone can do it in 12 days . if both together can do the work in how many days ? a) 4.8 days , b) 3.8 days , c) 3.65 days , d) 3.77 days , e) 5.75 days	"1 / 8 + 1 / 12 = 5 / 24 24 / 5 = 4.8 days answer : a"
Work through the equation methodically, ensuring clarity at each step.	there has been successive increases of 25 % and then 10 % in the price of gas from the previous month . by what percentage should a driver reduce gas consumption so that the expenditure does not change ? a) 23 % , b) 27 % , c) 31 % , d) 35 % , e) 39 %	"let p be the original price per unit of gas . let x be the original gas consumption . let y be the reduced gas consumption . y * 1.1 * 1.25 * p = x * p y = x / (1.1 * 1.25) which is about 0.73 x which is a decrease of about 27 % . the answer is b ."
Break down and solve the equation step by step.	65 % of x = 20 % of 487.50 . find the value of x ? a) 100 , b) 150 , c) 160 , d) 180 , e) 199	65 % of x = 20 % of 487.50 then , 65 / 100 * x = 20 / 100 * 4875 / 10 x = 150 answer is b
Provide a detailed solution to the given mathematical problem.	a number , x is chosen at random from the set of positive integers less than 10 . what is the probability that (7 / x) > x ? a) 1 / 5 , b) 7 / 9 , c) 1 / 3 , d) 2 / 3 , e) 2 / 9	"number x has to be chosen from numbers 1 - 9 (7 / x) > x = > 7 > x ^ 2 = > x ^ 2 - 7 < 0 x can have 2 values only 1 , 2 therefore , probability = 2 / 9 answer e"
Provide a detailed solution to the given mathematical problem.	what is the perimeter of a square with area p ^ 2 / 16 ? ['a) 3 p / 4','b) 3 p ^ 2 / 4','c) p','d) 3 p ^ 2','e) 4 p / 3']	area of square , (side) ^ 2 = (p / 4) ^ 2 therefore side of the square = p / 4 perimeter of square = 4 * side = 4 * (p / 4) = p answer is c .
Work through the equation methodically, ensuring clarity at each step.	a car is running at a speed of 120 kmph . what distance will it cover in 12 sec ? a) 396 m , b) 267 m , c) 180 m , d) 200 m , e) 250 m	"speed = 120 kmph = 120 * 5 / 18 = 33 m / s distance covered in 12 sec = 33 * 12 = 396 m answer is a"
Solve the problem systematically, showing each step clearly.	if there are 4 peanuts in a box and mary puts 4 more peanuts inside , how many peanuts are in the box ? a) 8 , b) 9 , c) 10 , d) 11 , e) 12	"4 + 4 = 8 correct answer is a) 8"
Explain the solution process for the given equation.	the speed of a car is 90 km in the first hour and 60 km in the second hour . what is the average speed of the car ? a) 12 , b) 75 , c) 88 , d) 54 , e) 15	"s = (90 + 60) / 2 = 75 kmph answer : b"
Explain the solution process for the given equation.	if 15 people contributed a total of $ 30.00 toward a gift and each of them contributed at least $ 1.00 , then the maximum possible amount any one person could have contributed is a) $ 1.00 , b) $ 16 , c) $ 5.00 , d) $ 6.00 , e) $ 20.00	"b for me 14 people with 1 $ each - > maximum = 16"
Break down and solve the equation step by step.	how many times digit 6 is used while writing numbers from 100 to 1400 ? a) 648 , b) 350 , c) 252 , d) 225 , e) 26	"there are 100 numbers which begin with 600 next , in every 10 numbers such as 100 to 110 , 110 to 120 , 120 to 130 6 comes at least once . number of such intervals = end limit - first no . / interval . our range of numbers is 100 - 1000 1000 - 100 = 900 / 10 = 90 number of 10 s interval in this is 90 . so 90' 6 s' so far we have calculated 190 . the total now comes to 280 . the nearest to which is 350 . hence b ."
Work through the equation methodically, ensuring clarity at each step.	there is enough provisions for 600 men in an army camp for 25 days . if there were 100 men less , how long will the provision last ? a) 10 days , b) 50 days , c) 100 days , d) 150 days , e) 200 days	"exp : we have , m 1 d 1 = m 2 d 2 600 * 25 = 100 * d 2 d 2 = 600 * 25 / 100 = 150 days . answer : d"
Break down and solve the equation step by step.	a walks at 30 kmph and 30 hours after his start , b cycles after him at 50 kmph . how far from the start does b catch up with a ? a) 1200 km , b) 1500 km , c) 2000 km , d) 2250 km , e) 3600 km	"suppose after x km from the start b catches up with a . then , the difference in the time taken by a to cover x km and that taken by b to cover x km is 30 hours . x / 30 - x / 50 = 30 x = 2250 km answer is d"
Work through the equation methodically, ensuring clarity at each step.	every disk in a bag is either blue , yellow or green . the ratio of blue disks to yellow disks to green disks in this bag is 3 : 7 : 8 . if the total number of disks in the bag is 72 , how many more green disks than blue disks are in the bag ? a) 20 , b) 28 , c) 30 , d) 35 , e) 40	"let b : y : g = 3 x : 7 x : 8 x . 3 x + 7 x + 8 x = 18 x = 72 - - > x = 4 . g - b = 8 x - 3 x = 5 x = 20 . the answer is a ."
Solve the problem systematically, showing each step clearly.	a telephone company needs to create a set of 3 - digit area codes . the company is entitled to use only digits 6 , 4 and 3 , which can be repeated . if the product of the digits in the area code must be even , how many different codes can be created ? a) 20 , b) 22 , c) 26 , d) 24 , e) 30	"total # of codes possible is 3 * 3 * 3 = 27 . oit of those 27 codes only the product of 333 and will be odd , the remaining 26 will have either 2 or 4 in them , which ensures that their product will be even . therefore the number of codes where the product of the digits is even = (total) - (restriction) = 27 - 1 = 26 . answer : c"
Work through the equation methodically, ensuring clarity at each step.	each of the products produced yesterday was checked by worker x or worker y . 0.5 % of the products checked by worker x are defective and 0.8 % of the products checked by worker y are defective . if the total defective rate of all the products checked by worker x and worker y is 0.7 % , what fraction of the products was checked by worker y ? a) 2 / 3 , b) 5 / 6 , c) 7 / 8 , d) 4 / 5 , e) 5 / 8	x : 0.5 % is 0.2 % - points from 0.7 % . y : 0.8 % is 0.1 % - points from 0.7 % . therefore the ratio of products checked by y : x is 2 : 1 . thus , worker y checked 2 / 3 of the products . the answer is a .
Break down and solve the equation step by step.	pens , pencils and markers in a jenna' s desk are in the ratio of 2 : 2 : 5 . if there are 10 pens , the number of markers in jenna' s desk is : a) 30 , b) 22 , c) 20 , d) 25 , e) 100	explanation : let pens = 2 x , pencils = 2 x & markers = 5 x . now , 2 x = 10 hence x = 5 . number of markers = 5 x which is 25 . answer : d
Provide a detailed solution to the given mathematical problem.	a goods train runs at the speed of 72 km / hr and crosses a 210 m long platform in 26 sec . what is the length of the goods train ? a) 382 , b) 782 , c) 278 , d) 310 , e) 881	"speed = 72 * 5 / 18 = 20 m / sec . time = 26 sec . let the length of the train be x meters . then , (x + 210) / 26 = 20 x = 310 m . answer : d"
Work through the equation methodically, ensuring clarity at each step.	a library has an average of 660 visitors on sundays and 240 on other days . the average number of visitors per day in a month of 30 days beginning with a sunday is : a) 250 , b) 276 , c) 280 , d) 310 , e) none	"since the month begins with sunday , to there will be five sundays in the month average required = (660 x 5 + 240 x 25) / 30) = 310 answer : option d"
Break down and solve the equation step by step.	find the average of all the numbers between 6 and 34 which are divisible by 6 . a) 18 , b) 20 , c) 24 , d) 30 , e) 21	"solution average = ((12 + 18 + 24 + 30) / 5) = 84 / 4 = 21 . answer e"
Work through the equation methodically, ensuring clarity at each step.	the h . c . f . of two numbers is 38 and the other two factors of their l . c . m . are 12 and 10 . the larger of the two numbers is : a) 678 , b) 456 , c) 234 , d) 476 , e) 432	": explanation : clearly , the numbers are (38 x 12) and (38 x 10) . { \ color { blue } \ therefore } larger number = (38 x 12) = 456 . answer : b) 456"
Solve the problem systematically, showing each step clearly.	if (a - b - c + d = 18) and (a + b - c - d = 6) , what is the value of (b - d) ^ 2 ? a) 4 . , b) 8 . , c) 12 . , d) 16 . , e) 36 .	"eq 1 : a - b - c + d = 18 eq 2 : a + b - c - d = 6 (1) subtract eq 1 from eq 2 a - b - c + d = 18 - a + b - c - d = 6 - - - - - - - - - - - - - - - - - - - - - - - - - 2 b + 2 d = 12 (2) simplify - b + d = 6 b - d = - 6 (b - d) ^ 2 = (- 6) ^ 2 = 36 my answer : e"
Provide a detailed solution to the given mathematical problem.	a 270 meter long train running at the speed of 120 kmph crosses another train running in the opposite direction at the speed of 80 kmph in 9 seconds . what is the lenght of other train . a) 210 m , b) 220 m , c) 230 m , d) 240 m , e) 250 m	"relative speeds = (120 + 80) km / hr = 200 km / hr = (200 * 5 / 18) m / s = (500 / 9) m / s let length of train be xm x + 270 / 9 = 500 / 9 x = 230 ans is 230 m answer : c"
Explain the solution process for the given equation.	if 5 machines can produce 20 units in 10 hours , how long would it take 10 to produce 60 units ? a) 9 , b) 10 , c) 15 , d) 25 , e) 30	"5 machines would produce 60 units in 30 hours . increasing the amount of machines by 2 would mean dividing 30 hours by 2 . 30 / 2 = 15 answer : c"
Break down and solve the equation step by step.	what is the probability of getting atmost one head when two unbiased coin are tossed ? a) 3 / 4 , b) 3 / 2 , c) 4 / 3 , d) 1 / 2 , e) 2 / 6	"here s = { hh , ht , th , tt } let ee = event of getting one head e = { tt , ht , th } p (e) = n (e) / n (s) = 3 / 4 option a"
Break down and solve the equation step by step.	what is the average (arithmetic mean) of all multiples of 10 from 10 to 400 inclusive ? a) 190 , b) 195 , c) 200 , d) 205 , e) 210	"we' re asked for the average of all of the multiples of 10 from 10 to 400 , inclusive . to start , we can figure out the total number of terms rather easily : 1 (10) = 10 2 (10) = 20 . . . 40 (10) = 400 so we know that there are 40 total numbers . we can now figure out the sum of those numbers with' bunching' : 10 + 400 = 410 20 + 390 = 410 30 + 380 = 410 etc . since there are 40 total terms , this pattern will create 20' pairs' of 410 . thus , since the average = (sum of terms) / (number of terms) , we have . . . (20) (410) / (40) = 410 / 2 = 205 final answer : d"
Provide a detailed solution to the given mathematical problem.	a , b and c can do a piece of work in 9 days , 14 days and 28 days respectively . how long will they taken , if all the three work together ? a) 4.12 days , b) 4.52 days , c) 3.52 days , d) 2.52 days , e) 4.92 days	1 / 9 + 1 / 14 + 1 / 28 = 55 / 252 = > 252 / 55 days = > 4.52 days answer : b
Explain the solution process for the given equation.	a sun is divided among x , y and z in such a way that for each rupee x gets , y gets 45 paisa and z gets 30 paisa . if the share of y is rs . 18 , what is the total amount ? a) 70 , b) 105 , c) 178 , d) 177 , e) 169	"x : y : z = 100 : 45 : 30 20 : 9 : 6 9 - - - 18 35 - - - ? = > 70 answer : a"
Explain the solution process for the given equation.	two good train each 100 m long , are running in opposite directions on parallel tracks . their speeds are 45 km / hr and 30 km / hr respectively . find the time taken by the slower train to pass the driver of the faster one . a) 1.2 sec , b) 2.4 sec , c) 4.8 sec , d) 9.6 sec , e) none	"sol . relative speed = (45 + 30) km / hr = (75 x 5 / 18) m / sec = (125 / 6) m / sec . distance covered = (100 + 100) m = 1000 m . required time = (200 x 6 / 125) sec = 9.6 sec . answer d"
Explain the solution process for the given equation.	an article is bought for rs . 600 and sold for rs . 450 , find the loss percent ? a) 16 % , b) 18 % , c) 19 % , d) 20 % , e) 25 %	"600 - - - - 150 100 - - - - ? = > 16 2 / 3 % answer : e"
Solve the problem systematically, showing each step clearly.	a trailer carries 3 , 4 and 5 crates on a trip . each crate weighs no less than 150 kg . what is the maximum weight of the crates on a single trip ? a) 1250 , b) 625 , c) 600 , d) 750 , e) 375	"max no . of crates = 5 . max weight = 150 kg max . weight carried = 5 * 150 = 750 kg = d"
Provide a detailed solution to the given mathematical problem.	mariah has decided to hire three workers . to determine whom she will hire , she has selected a group of 17 candidates . she plans to have one working interview with 3 of the 17 candidates every day to see how well they work together . how many days will it take her to have working interviews with all the different combinations of job candidates ? a) 720 , b) 420 , c) 300 , d) 30 , e) 333	"420 . answer b"

Provide a detailed solution to the given mathematical problem.

what is value of x if 32 - y = 16 x - y ? a) 1 , b) 3 , c) 4 , d) 2 , e) 6

"32 - y = 16 x - y 32 = 16 x - y + y 16 x = 32 x = 2 answer is option d"

Solve the problem systematically, showing each step clearly.

15.06 * 0.00000001 = ? a) 15060000 , b) 0.001506 , c) 0.01506 , d) 1.506 e - 07 , e) none of these

"explanation : clearly after decimal 10 digits should be there . option d"

Work through the equation methodically, ensuring clarity at each step.

city x has a population 7 times as great as the population of city y , which has a population twice as great as the population of city z . what is the ratio of the population of city x to the population of city z ? a) 1 : 8 , b) 1 : 4 , c) 2 : 1 , d) 14 : 1 , e) 18 : 1

x = 7 y , y = 2 * z x : y , y : z 7 : 1 , 2 : 1 14 : 2 , 2 : 1 so , x : z = 14 : 1 (d)

Work through the equation methodically, ensuring clarity at each step.

seed mixture x is 40 % ryegrass and 60 % bluegrass by weight ; seed mixture y is 25 % ryegrass and 75 % fescue . if a mixture of x and y contains 30 % ryegrass , what percent of the weight of the mixture is x ? a) 10 % , b) 33.33 % , c) 40 % , d) 50 % , e) 66.67 %

"x - wt of mix x added y - wt of mix y added 0.4 x + 0.25 y = 0.3 (x + y) 0.1 x = 0.05 y x / y = 5 / 10 = 1 / 2 y / x = 2 (y + x) / x = 3 = > x / (y + x) * 100 = 100 / 3 = 33.33 % answer : b"

Solve the problem systematically, showing each step clearly.

a certain machine produces 550 units of product p per hour . working continuously at this constant rate , this machine will produce how many units of product p in 5 days ? a) 7,000 , b) 24,000 , c) 66,000 , d) 100,000 , e) 168,000

"since 5 days consist of 24 * 5 hours the total is 120 hours . since every hour the machine produces 550 units of product p the total product during 120 hours is 120 * 550 = 66,000 . correct option : c"

Explain the solution process for the given equation.

if 2 ^ 4 , 3 ^ 3 , and 10 ^ 3 are factors of the product of 1,452 and w , where w is a positive integer , what is the smallest possible value of w ? a) 198 , b) 288 , c) 363 , d) 360 , e) 484

i will go with d (pending elements to match is 2 ^ 2 * 3 ^ 2 * 10 ^ 1 = 360

Break down and solve the equation step by step.

xy = 2 then what is (2 ^ (x + y) ^ 2) / (2 ^ (x - y) ^ 2) a) 2 , b) 4 , c) 8 , d) 16 , e) 256

(x + y) ^ 2 - (x - y) ^ 2 (x + y + x - y) (x + y - x + y) (2 x) (2 y) 4 xy 8 2 ^ 8 = 256 answer e

Break down and solve the equation step by step.

how many 3 - digit even numbers are possible such that if one of the digits is 4 , the next / succeeding digit to it should be 7 ? a) 305 , b) 5 , c) 365 , d) 405 , e) 495

470 , 472 , 474 , 476 , and 478 , so total 5 . hence option b .

Provide a detailed solution to the given mathematical problem.

a cycle is bought for rs . 900 and sold for rs . 1120 , find the gain percent ? a) 11 , b) 24 , c) 99 , d) 77 , e) 18

"900 - - - - 220 100 - - - - ? = > 24 % answer : b"

Work through the equation methodically, ensuring clarity at each step.

in what time will a train 100 m long cross an electric pole , it its speed be 72 km / hr ? a) 2.5 , b) 5 , c) 2.4 , d) 2.8 , e) 2.1

"speed = 72 * 5 / 18 = 20 m / sec time taken = 100 / 20 = 5 sec . answer : b"

Explain the solution process for the given equation.

3 business people wish to invest in a new company . each person is willing to pay one third of the total investment . after careful calculations , they realize that each of them would pay $ 2200 less if they could find two more equal investors . how much is the total investment in the new business ? a) $ 11,000 , b) $ 50,000 , c) $ 16,500 , d) $ 6,600 , e) $ 3,600

total investment between 5 : (x / 5) total investment including 2200 less between 3 people (x - (2200 * 3)) / 3 set both eq . equal to each other . 16,500 answer c)

Break down and solve the equation step by step.

if you keep rolling a pair of dice together till a sum of 5 or 7 is obtained , then what is the probability that a sum of 5 comes before a sum of 7 ? a) 5 / 5 , b) 4 / 5 , c) 3 / 5 , d) 2 / 5 , e) 1 / 5

this can be solved in a generic and complex way but let us not go into all that . there can be four ways through which the pair of dice results in a sum of 5 . there can be six ways through which the pair of dice can result in a sum of 7 . now , we want the probability of the pair of dice resulting in a sum of 5 before a sum of 7 . thus probability = 4 / (4 + 6) = 4 / 10 or 2 / 5 . correct answer is d) 2 / 5

Break down and solve the equation step by step.

a train sets off at 9 : 00 am at the speed of 70 km / h . another train starts at 10 : 30 am in the same direction at the rate of 84 km / h . at what time will the second train catch the first train ? a) 4 : 30 , b) 5 : 00 , c) 5 : 30 , d) 6 : 00 , e) 6 : 30

"in one hour and thirty minutes the first train travels 105 km . the second train catches the first train at a rate of 84 km / h - 70 km / h = 14 km / h . the second train will catch the first train in 105 / 14 = 7.5 hours , so at 6 : 00 pm . the answer is d ."

Work through the equation methodically, ensuring clarity at each step.

a man buys a cycle for rs . 2300 and sells it at a loss of 30 % . what is the selling price of the cycle ? a) s . 1690 , b) s . 1610 , c) s . 1890 , d) s . 1602 , e) s . 1092

"s . p . = 70 % of rs . 2300 = rs . 70 x 2300 / 100 = rs . 1610 answer : option b"

Solve the problem systematically, showing each step clearly.

for any integer k greater than 1 , the symbol k * denotes the product of all integers between 1 and k , inclusive . if k * is a multiple of 315 what is the least possible value of k ? a) 5 , b) 7 , c) 9 , d) 11 , e) 13

315 = 3 * 3 * 5 * 7 thus k must include numbers at least up to the number 7 . the answer is b .

Solve the problem systematically, showing each step clearly.

if integer k is equal to the sum of all even multiples of 25 between 200 and 600 , what is the greatest prime factor of k ? a) 5 , b) 7 , c) 11 , d) 13 , e) 17

"if we break down what the stem is asking what is the sum of all mult of 50 between 200 and 600 . using arithmetic progression to find n : 600 = 200 + (n - 1) 50 400 + 50 = 50 n 450 = 50 n = > n = 9 the sum would be : 9 * mean mean = [ 600 + 200 ] / 2 = 400 9 * 400 = 3600 d"

Work through the equation methodically, ensuring clarity at each step.

36 welders work at a constant rate they complete an order in 3 days . if after the first day , 12 welders start to work on the other project , how many more days the remaining welders will need to complete the rest of the order ? a) 3 , b) 2 , c) 8 , d) 4 , e) 6

"1 . we need to find out the time taken by 24 workers after day 1 . 2 . total no . of wokers * total time taken = time taken by 1 worker 3 . time taken by 1 worker = 36 * 3 = 108 days 4 . but on day 1 thirty - six workers had already worked finishing 1 / 3 of the job . so 24 workers have to finish only 2 / 3 of the job . 5 . total time taken by 24 workers can be got from formula used at (2) . i . e . , 24 * total time taken = 108 . total time taken by 6 workers to finish the complete job is 108 / 24 = 4.5 days . 6 . time taken by 24 workers to finish 2 / 3 of the job is 2 / 3 * 4.5 = 3 days . the answer is choice a"

Break down and solve the equation step by step.

on the first of the year , james invested x dollars at proudstar bank in an account that yields 3 % in interest every quarter year . at the end of the year , during which he made no additional deposits or withdrawals , he had y dollars in the account . if james had invested the same amount in an account which pays interest on a yearly basis , what must the interest rate be for james to have y dollars at the end of the year ? a) 22.04 % , b) 16.12 % , c) 12.5 % , d) 18.25 % , e) 10 %

per quarter , interest = 3 % so for a year , interest = 12 % due to quarter cumulation , effective yield (ytm) would be slight higher than 12 % answer = 12.5 % = answer : c

Provide a detailed solution to the given mathematical problem.

how many positive integers will divide evenly into 190 ? a) 4 , b) 6 , c) 8 , d) 12 , e) 16

the question is asking how many factors 190 has . 190 = 2 * 5 * 19 the number of factors is 2 ^ 3 = 8 the answer is c .

Break down and solve the equation step by step.

two trains , one from howrah to patna and the other from patna to howrah , start simultaneously . after they meet , the trains reach their destinations after 4 hours and 10 hours respectively . the ratio of their speeds is a) 4 : 9 , b) 4 : 3 , c) 4 : 12 , d) 4 : 8 , e) 5 : 2

let us name the trains as a and b . then , (a' s speed) : (b' s speed) = b : a = 10 : 4 = 5 : 2 . answer : e

Break down and solve the equation step by step.

find number which is 60 % less than 120 . a) 18 , b) 22 , c) 28 , d) 48 , e) 98

"explanation : 60 % less is 40 % of the given number therefore , 40 % of 120 is 48 . answer : d"

Provide a detailed solution to the given mathematical problem.

if 7 spiders make 7 webs in 7 days , then 1 spider will make 1 web in how many days ? a) 7 , b) 6 , c) 5 , d) 4 , e) 3

let the required number days be x . less spiders , more days (indirect proportion) less webs , less days (direct proportion) spiders 1 : 7 webs 7 : 1 1 x 7 x x = 7 x 1 x 7 = > x = 7 answer is 7

Break down and solve the equation step by step.

two cars start from the opposite places of a main road , 150 km apart . first car runs for 30 km and takes a right turn and then runs 15 km . it then turns left and then runs for another 25 km and then takes the direction back to reach the main road . in the mean time , due to minor break down the other car has run only 35 km along the main road . what would be the distance between two cars at this point ? a) 65 , b) 38 , c) 60 , d) 28 , e) 21

answer : c) 60 km

Solve the problem systematically, showing each step clearly.

difference between the length and breadth of a rectangle is 23 m . ifits perimeter is 206 m , then its area is ? ? ['a) 1220 m ^ 2','b) 1520 m ^ 2','c) 2520 m ^ 2','d) 2600 m ^ 2','e) 3420 m ^ 2']

we have : (l - b) = 23 and 2 (l + b) = 206 or (l + b) = 103 . solving the two equations , we get : l = 63 and b = 40 . area = (l x b) = (63 x 40) m 2 = 2520 m ^ 2 c

Work through the equation methodically, ensuring clarity at each step.

if 30 honey bees make 30 grams of honey in 30 days , then 1 honey bee will make 1 gram of honey in how many days ? a) 1 , b) 3.5 , c) 30 , d) 49 , e) 30

"explanation : let the required number days be x . less honey bees , more days (indirect proportion) less honey , less days (direct proportion) honey bees 1 : 30 : : 30 : x honey 30 : 1 = > 1 x 30 x x = 30 x 1 x 30 = > x = 30 . answer : c"

Explain the solution process for the given equation.

find the principle on a certain sum of money at 5 % per annum for 3 1 / 5 years if the amount being rs . 2030 ? a) rs . 2000 , b) rs . 1750 , c) rs . 2010 , d) rs . 2005 , e) none of these

"explanation : 2030 = p [ 1 + (5 * 16 / 5) / 100 ] p = 1750 answer : option b"

Work through the equation methodically, ensuring clarity at each step.

24 * 22 = ? / 8 a) 4224 , b) 2224 , c) 5374 , d) 2434 , e) 4254

"a 4224 ? / 8 = 24 * 22 ? = 24 * 22 * 8 ? = 4224"

Solve the problem systematically, showing each step clearly.

john was 43 years old when he married betty . they just celebrated their fifth wedding anniversary , and betty' s age is now 7 / 8 of john' s . how old is betty ? a) 24 , b) 26 , c) 42 , d) 30 , e) 32

"assume betty' s age on marriage = x years . john' s age on marriage = 43 john' s age after 5 years = 48 years . betty' s age after 5 years = x + 5 given : x + 5 = 7 / 8 (48) = 42 therefore betty' s current age = 42 option c"

Explain the solution process for the given equation.

in a class of 50 students , 3 / 5 went away do painting , 1 / 5 went to play in the field . how many students are left in the classroom ? a) 20 , b) 30 , c) 5 , d) 8 , e) 10

3 / 5 + 1 / 5 are the students who have left the classroom 3 / 5 + 1 / 5 = (3 + 1) / 5 , since denominator is same for both fractions = 4 / 5 remaining students in the classroom = 1 - 4 / 5 = 5 / 5 - 4 / 5 = (5 - 4) / 5 = 1 / 5 1 / 5 of 50 = 10 ans : e

Work through the equation methodically, ensuring clarity at each step.

the average of 8 numbers is 25 , that of the first 2 is 20 and of the next 3 is 26 . the sixth number is less than the seventh by 4 , and less than the eighth by 6 . the last number is a) 30 , b) 32 , c) 40 , d) 36 , e) 38

explanation : sum of all the numbers = 25 * 8 = 200 sum of first two numbers = 40 sum of next three numbers (i . e . , 3 rd , 4 th and 5 th) = 78 so , sum of first five numbers = 118 and sum of 6 th , 7 th and 8 th number = 82 assuming sixth number to be x , we get 7 th number = x + 4 and 8 th number = x + 6 . so , x + (x + 4) + (x + 6) = 82 = > x = 24 . hence , last number (8 th number) = 30 . answer : a

Explain the solution process for the given equation.

if the numbers 1 to 96 are written on 96 pieces of paper , (one on each) and one piece is picked at random , then what is the probability that the number drawn is neither prime nor composite ? a) 1 / 50 , b) 1 / 25 , c) 1 / 96 , d) 1 , e) 2

there are 25 primes , 70 composite numbers from 1 to 96 . the number which is neither prime nor composite is 1 . therefore , required probability = 1 / 96 . answer : c

Solve the problem systematically, showing each step clearly.

jim drove 642 miles of a 1200 miles journey . how many more miles does he need to drive to finish his journey ? a) 113 miles , b) 432 miles , c) 558 miles , d) 887 miles , e) 767 miles

"the number of miles to drive to finish his journey is given by 1200 - 642 = 558 miles correct answer c"

Explain the solution process for the given equation.

the ages of patrick and michael are in the ratio of 3 : 5 and that of michael and monica are in the ratio of 3 : 5 . if the sum of their ages is 145 , what is the difference between the ages of patrick and monica ? a) 27 , b) 48 , c) 45 , d) 72 , e) 18

"ages of p and mi = 3 x + 5 x ages of mi and mo = 3 x : 5 x rationalizing their ages . ratio of their ages will be 9 x : 15 x : 25 x sum = 47 x = 145 x = 3 difference if ages of pa and mo = 25 x - 9 x = 16 x = 16 * 3 = 48 answer b"

Work through the equation methodically, ensuring clarity at each step.

evaluate : 1234562 - 12 * 3 * 2 = ? a) 32435453 , b) 1234554 , c) 76768786 , d) 97979797 , e) 75868656

"according to order of operations , 12 ? 3 ? 2 (division and multiplication) is done first from left to right 12 * * 2 = 4 * 2 = 8 hence 1234562 - 12 * 3 * 2 = 1234562 - 8 = 1234554 correct answer b"

Explain the solution process for the given equation.

two stations p and q are 200 km apart on a straight track . one train starts from p at 7 a . m . and travels towards q at 20 kmph . another train starts from q at 8 a . m . and travels towards p at a speed of 25 kmph . at what time will they meet ? a) 10 am , b) 12 noon , c) 10.30 am , d) 12.30 am , e) 11 am

"assume both trains meet after x hours after 7 am distance covered by train starting from p in x hours = 20 x km distance covered by train starting from q in (x - 1) hours = 25 (x - 1) total distance = 200 = > 20 x + 25 (x - 1) = 200 = > 45 x = 225 = > x = 5 means , they meet after 5 hours after 7 am , ie , they meet at 12 noon answer is b ."

Solve the problem systematically, showing each step clearly.

a bottle contains a certain solution . in the bottled solution , the ratio of water to soap is 3 : 2 , and the ratio of soap to salt is five times this ratio . the solution is poured into an open container , and after some time , the ratio of water to soap in the open container is halved by water evaporation . at that time , what is the ratio of water to salt in the solution ? a) 1 : 1 , b) 2 : 3 , c) 3 : 2 , d) 9 : 4 , e) 45 : 8

"water : soap = 3 : 2 soap : salt = 15 : 2 = > for 15 soap , salt = 2 = > for 2 soap , salt = (2 / 15) * 2 = 4 / 15 so , water : soap : salt = 3 : 2 : 4 / 15 = 45 : 30 : 4 after open container , water : soap : salt = 22.5 : 30 : 4 so , water : salt = 22.5 : 4 = 45 : 8 e"

Explain the solution process for the given equation.

four machines , each working at the same constant rate , together can complete a certain job in 20 days . how many additional machines , each working at the same constant rate , will be needed to complete the job in 10 days ? a) 5 , b) 3 , c) 4 , d) 6 , e) 1

"rate of one machine = 1 job / (20 * 4) days let x = number of machines needed to complete the job in 10 days 1 / (4 * 20) * 10 * x = 1 job x = 8 8 - 4 = 4 answer : c"

Explain the solution process for the given equation.

what could be the range of a set consisting of odd multiples of (6 + 1) ? a) 70 , b) 24 , c) 35 , d) 62 , e) 21

"range = the difference between the greatest and the smallest numbers in the sequence . our sequence is odd and is a multiple of 7 . every number in that sequence can be represented like this : 7 * (2 n + 1) where n is any positive integer . range = 7 * (2 m + 1) - 7 * (2 n + 1) = 7 * 2 * (m - n) = 14 * (m - n) . m , n - any positive integers the answer must be divisible by 14 , which is only 70 . the correct answer is a ."

Provide a detailed solution to the given mathematical problem.

a manufacturer sells a pair of glasses to a wholesale dealer at a profit of 18 % . the wholesaler sells the same to retailer at a profit of 20 % . the retailer in turn sells them to a customer for rs . 30.09 , there by earning a profit of 25 % . the cost price for the manufacturer is ? a) 33 , b) 29 , c) 17 , d) 10 , e) 16

"explanation : let the cost price for the manufacturer be rs . x . then , 125 % of 120 % of 118 % of x = 30.09 . 125 / 100 * 120 / 100 * 118 / 100 * x = 3009 / 100 177 / 100 x = 3009 / 100 = > x = 17 answer : c"

Work through the equation methodically, ensuring clarity at each step.

two numbers are in the ratio 3 : 5 . if 9 be subtracted from each , they are in the ratio of 9 : 17 . the first number is : a) 36 , b) 77 , c) 88 , d) 55 , e) 221

"(3 x - 9) : (5 x - 9) = 9 : 17 x = 12 = > 3 x = 36 answer : a"

Provide a detailed solution to the given mathematical problem.

a shopkeeper buys two articles for rs . 1000 each and then sells them , making 40 % profit on the first article and 40 % loss on second article . find the net profit or loss percent ? a) 200 , b) 278 , c) 282 , d) 400 , e) 270

"profit on first article = 40 % of 1000 = 400 . this is equal to the loss he makes on the second article . that , is he makes neither profit nor loss . answer : d"

Solve the problem systematically, showing each step clearly.

of the 65 cars on a car lot , 45 have air - bag , 30 have power windows , and 12 have both air - bag and power windows . how many of the cars on the lot have neither air - bag nor power windows ? a) 8 , b) 2 , c) 10 , d) 15 , e) 18

total - neither = all air bag + all power windows - both or 65 - neither = 45 + 30 - 12 = 63 . = > neither = 2 , hence b . answer : b

Explain the solution process for the given equation.

two numbers are respectively 20 % and 50 % more than a third number . what percentage is the first of the second ? a) 25 % , b) 60 % , c) 80 % , d) 55 % , e) 74 %

"required answer = (100 + x) / (100 + y) * 100 = 120 * 100 / 150 = 80 % answer is c"

Explain the solution process for the given equation.

here is a simple arithmetical puzzle in his own flowery , poetical language . if one - fifth of a hive of bees flew to the badamba flower , one - third flew to the slandbara , 3 times the difference of these two numbers flew to an arbour , and one bee continued to fly about , attracted on each side by the fragrant ketaki and malati , what was the total number of bees ? a) 11 , b) 15 , c) 18 , d) 17 , e) 19

b 15 the number of bees was 15

Solve the problem systematically, showing each step clearly.

the weight of every type a widget is the same , the weight of every type b widget is the same , and the weight of every type c widget is the same . if the weight of 7 type a widgets is equal to the weight of 2 type b widgets , and the weight of 5 type b widgets is equal to the weight of 7 type c widgets . what is the ratio of the total weight of 1 type a widget and 1 type b widget , to the total weight of 1 type b widget and 1 type c widget ? a) 1 : 2 , b) 2 : 3 , c) 3 : 4 , d) 4 : 5 , e) 5 : 6

"5 b = 7 c and so b = 7 c / 5 7 a = 2 b and so a = 2 b / 7 = 2 c / 5 a + b = 2 c / 5 + 7 c / 5 = 9 c / 5 b + c = 7 c / 5 + c = 12 c / 5 the ratio of a + b : b + c = 9 : 12 = 3 : 4 the answer is c ."

Provide a detailed solution to the given mathematical problem.

the area of rectangular field of length 90 meters and breadth 80 meters is equal to the area of a square plot . what will be the length of the diagonal of the square plot ? ['a) 100 m','b) 120 m','c) 95 m','d) 80 m','e) 150 m']

given that the length of the rectangular field l = 90 m and breadth = b = 80 m . then its area lb = 90 x 80 sq . m we know that the area of square = (1 / 2) x (diagonal) 2 from the given question , we have area of square plot = area of rectangular field that is , 90 x 80 = 1 / 2 x diagonal 2 diagonal ^ 2 = 2 x 7200 = 14400 = (120) 2 diagonal = 120 m . hence b

Work through the equation methodically, ensuring clarity at each step.

if' a' and' b' are non - zero numbers such that their sum is 6 times the product , what is the value of 1 / a + 1 / b ? a) 5 , b) 6 , c) 7 , d) 8 , e) 12

a + b = 6 ab 1 / a + 1 / b = (b + a) / (ab) = 6 . the answer is b .

Work through the equation methodically, ensuring clarity at each step.

in the game of dubblefud , red balls , blue balls and green balls are each worth 2 , 4 and 5 points respectively . in a certain selection of balls , the product of the point values of the balls is 16000 . if the number of blue balls in this selection equals the number of green balls , how many red balls are in the selection ? a) 1 , b) 2 , c) 3 , d) 4 , e) 5

this is equivalent to : - 2 x * 4 y * 5 z = 16000 y = z (given) 2 x * 4 y * 5 y = 16000 2 x * y ^ 2 = 16000 / 20 2 x * y ^ 2 = 800 now from options given we will figure out which number will divide 800 and gives us a perfect square : - which gives us x = 4 as 2 * 4 * y ^ 2 = 800 y ^ 2 = 100 y = 10 number of red balls = 4 hence d

Solve the problem systematically, showing each step clearly.

what number has a 5 : 1 ratio to the number 11 ? a) 22 , b) 50 , c) 55 , d) 52 , e) 12

"5 : 1 = x : 10 x = 55 answer : c"

Break down and solve the equation step by step.

a man can do a piece of work in 7 days , but with the help of his son , he can do it in 3 days . in what time can the son do it alone ? a) 5 , b) 7 , c) 15 / 2 , d) 21 / 4 , e) 18 / 5

"son' s 1 day' s work = (1 / 3) - (1 / 7) = 4 / 21 the son alone can do the work in 21 / 4 days answer is d"

Break down and solve the equation step by step.

the average age of a group of 10 persons was decreased by 3 years when one person , whose age was 44 years , was replaced by a new person . find the age of the new person ? a) 18 , b) 56 , c) 12 , d) 17 , e) 14

"initial average age of the 10 persons be p . age of the new person q . sum of the ages of the initial 10 persons = 10 p new average = (p - 3) 10 (p - 3) = 10 p - 44 + q = > q = 14 answer : e"

Explain the solution process for the given equation.

a sports equipment store sold ping pong rackets for a total of $ 735 . if the average (arithmetic mean) price of a pair of rackets is $ 9.8 , how many pairs were sold ? a) 50 , b) 100 , c) 490 , d) 500 , e) 75

"average price for a pair of rackets = $ 9.8 total cost = $ 9.8 * x = $ 735 x = 75 pairs were sold . answer : e"

Solve the problem systematically, showing each step clearly.

when tom works alone he chops 4 lb . salad in 3 minutes , and when tammy works alone she chops 3 lb . salad in 4 minutes . they start working together , and after some time finish chopping 65 lb . of salad . of those 80 lb . , the salad quantity chopped by tammy is what percent greater than the quantifying chopped by tom ? . a) 125 % , b) 100 % , c) 25 % , d) 225 % , e) 400 %

"tom chops 8 lbs in 12 minutes tammy chopswhen tom works alone he chops 2 lb . salad in 3 minutes , and when tammy works alone she chops 3 lb . salad in 2 minutes . they start working together , and after some time finish chopping 65 lb . of salad . of those 80 lb . , the salad quantity chopped by tammy is what percent greater than the quantifying chopped by tom ? . 9 lbs in 6 minutes so in the same amount of time , tammy chops 125 % more than tom , since 9 is 125 % greater than 4 . so 125 % is the answer . note that the actual time does n' t matter . if you multiply the time each work by x , you' ll multiply the work each does by x , and 9 x is still 125 % greater than 4 x . ans : a"

Work through the equation methodically, ensuring clarity at each step.

bella is taking a trip in her car from new york (point a) chicago , illinois (point b) . she is traveling at an average speed of 50 miles per hour , if the total distance from point a to point b is 790 miles , in approximately how long will bella reach her destination ? a) 10 hours , b) 12.50 hours , c) 15.80 hours , d) 25 hours , e) 1 day

answer is (c) . if she is traveling at a speed of 50 miles per hour and her trip is a total of 790 miles , 790 miles divided by 50 miles per hours would equal 15.80 hours , just shy of 16 hours .

Break down and solve the equation step by step.

the least number which when divided by 2 , 3 , 5 and 6 leaves a remainder 3 , but when divided by 9 leaves no remainder , is a) 16 , b) 63 , c) 73 , d) 83 , e) none of these

"explanation : l . c . m of 2 , 3 , 5 , 6 = 30 therefore required number is of the form 30 k + 3 . least value of k for which (30 k + 3) is divisible by 9 is k = 2 therefore required number = (30 x 2 + 3) = 63 . answer : b"

Explain the solution process for the given equation.

john bought a shirt on sale for 25 % off the original price and another 25 % off the discounted price . if the final price was $ 19 , what was the price before the first discount ? a) $ 45.10 , b) $ 34.31 , c) $ 28.44 , d) $ 67.54 , e) $ 33.77

"let x be the price before the first discount . the price after the first discount is x - 25 % x (price after first discount) a second discount of 25 % of the discounted price after which the final price is 19 (x - 25 % x) - 25 % (x - 25 % x) = 19 solve for x x = $ 33.77 correct answer e"

Solve the problem systematically, showing each step clearly.

in the biology lab of ` ` jefferson'' high school there are 5.4 * 10 ^ 6 germs , equally divided among 10800 petri dishes . how many germs live happily in a single dish ? a) 100 , b) 200 , c) 500 , d) 1000 , e) 5000

5.4 * 10 ^ 6 / 10800 = = > 54 * 10 ^ 3 / 108 = = > 54000 / 108 = 500 answer : c

Work through the equation methodically, ensuring clarity at each step.

by selling 8 pencils for a rupee a man loses 60 % . how many for a rupee should he sell in order to gain 60 % ? a) 8 , b) 9 , c) 2 , d) 6 , e) 4

"40 % - - - 8 160 % - - - ? 40 / 160 * 12 = 2 answer : c"

Solve the problem systematically, showing each step clearly.

village x has a population of 68000 , which is decreasing at the rate of 1200 per year . village y has a population of 42000 , which is increasing at the rate of 800 per year . in how many years will the population of the two villages be equal ? a) 15 , b) 19 , c) 11 , d) 18 , e) 13

let the population of two villages be equal after p years then , 68000 - 1200 p = 42000 + 800 p 2000 p = 26000 p = 13 answer is e .

Explain the solution process for the given equation.

if 125 % of j is equal to 25 % of k , 150 % of k is equal to 50 % of l , and 175 % of l is equal to 75 % of m , then 20 % of m is equal to what percent of 200 % of j ? a) 0.35 , b) 3.5 , c) 35 , d) 350 , e) 3500

first of all , let us write the information in form of equations and numbers 125 j = 5 k or 5 j = k (i) 150 k = 50 l or 3 k = l (ii) 175 l = 75 m or 7 l = 3 m (iii) we need to find a relation between j and m from (i) and (ii) , 15 j = l multiplying this by 7 , 105 j = 7 l = 3 m hence , 35 j = m now , we are asked : 20 % of m is equal to what percent of 200 % of j assume j = 1 , hence m = 35 20 % of m = 7 and 200 % of j = 2 20 % of m = 3.5 (200 % of j) = (350 / 100) * (200 % of j) hence 20 % of m = 350 % of 200 % of j . option d

Solve the problem systematically, showing each step clearly.

if [ [ x ] ] = x ^ 2 + 2 x + 4 , what is the value of [ [ 2 ] ] ? a) 3 , b) 9 , c) 12 , d) 19 , e) 25

these functions questions might look intimidating , but they just test your knowledge about how well you can substitute values [ [ x ] ] = x ^ 2 + 2 x + 4 [ [ 2 ] ] = 2 ^ 2 + 2 * 2 + 4 = 12 . option c

Solve the problem systematically, showing each step clearly.

the compound ratio of 1 : 4 , 3 : 2 and 4 : 5 ? a) 1 : 9 , b) 1 : 7 , c) 1 : 2 , d) 3 : 10 , e) 1 : 4

"1 / 4 * 3 / 2 * 4 / 5 = 3 / 10 = 3 : 10 answer : d"

Work through the equation methodically, ensuring clarity at each step.

a pair of articles was bought for $ 720 at a discount of 10 % . what must be the marked price of each of the article ? a) $ 300 , b) $ 500 , c) $ 350 , d) $ 400 , e) $ 600

"s . p . of each of the article = 720 / 2 = $ 360 let m . p = $ x 90 % of x = 360 x = 360 * 100 / 90 = $ 400 answer is d"

Provide a detailed solution to the given mathematical problem.

two trains are moving in opposite directions at 60 km / hr and 90 km / hr . their lengths are 1.10 km and 0.15 km respectively . the time taken by the slower train to cross the faster train in seconds is ? a) 65 sec , b) 30 sec , c) 48 sec , d) 33 sec , e) 12 sec

"relative speed = 60 + 90 = 150 km / hr . = 150 * 5 / 18 = 125 / 3 m / sec . distance covered = 1.10 + 0.15 = 1.25 km = 1250 m . required time = 1250 * 3 / 125 = 30 sec . answer : b"

Provide a detailed solution to the given mathematical problem.

how many two - element subsets of < 1 , 23 , 45 , 67 > are there that do not contain the pair of elements 2 and 4 ? a) 11 , b) 12 , c) 14 , d) 15 , e) 20

7 c 2 - 1 = 20 answer is e

Solve the problem systematically, showing each step clearly.

the diameter of the driving wheel of a bus in 140 cm . how many revolutions per minute must the wheel make in order to keep a speed of 20 kmph ? a) 21.21 , b) 20.25 , c) 32.25 , d) 75.75 , e) 62.61

distance covered in 1 min = (20 * 1000) / 60 = 333.33 m circumference of the wheel = (2 * (22 / 7) * . 70) = 4.4 m no of revolution per min = 333.33 / 4.4 = 75.75 answer : d

Break down and solve the equation step by step.

what is the least common multiple of 5,40 , 36,50 a) 3698 , b) 874 , c) 3692 , d) 1800 , e) 3214

"the given numbers are 5,40 , 36,50 thus the l . c . m is 5 * 2 * 2 * 1 * 2 * 9 * 5 = 1800 the answer is d"

Solve the problem systematically, showing each step clearly.

if 13 = 13 w / (1 - w) , then (5 w) 2 = a) 1 / 4 , b) 5 , c) 1 , d) 2 , e) 3

"13 - 13 w = 13 w 26 w = 13 w = 1 / 2 5 w = 5 / 2 5 w * 2 = 5 / 2 * 2 = 5 answer : b"

Explain the solution process for the given equation.

what is the total number of integers between 2 and 38 that are divisible by 4 ? a) 8 , b) 9 , c) 7 , d) 6 , e) 11

"4 , 6 , 8 , . . . , 32,36 this is an equally spaced list ; you can use the formula : n = (largest - smallest) / (' space') + 1 = (36 - 4) / (4) + 1 = 32 / 4 + 1 = 8 + 1 = 9 answer is b"

Solve the problem systematically, showing each step clearly.

a and b complete a work in 8 days . a alone can do it in 12 days . if both together can do the work in how many days ? a) 4.8 days , b) 3.8 days , c) 3.65 days , d) 3.77 days , e) 5.75 days

"1 / 8 + 1 / 12 = 5 / 24 24 / 5 = 4.8 days answer : a"

Work through the equation methodically, ensuring clarity at each step.

there has been successive increases of 25 % and then 10 % in the price of gas from the previous month . by what percentage should a driver reduce gas consumption so that the expenditure does not change ? a) 23 % , b) 27 % , c) 31 % , d) 35 % , e) 39 %

"let p be the original price per unit of gas . let x be the original gas consumption . let y be the reduced gas consumption . y * 1.1 * 1.25 * p = x * p y = x / (1.1 * 1.25) which is about 0.73 x which is a decrease of about 27 % . the answer is b ."

Break down and solve the equation step by step.

65 % of x = 20 % of 487.50 . find the value of x ? a) 100 , b) 150 , c) 160 , d) 180 , e) 199

65 % of x = 20 % of 487.50 then , 65 / 100 * x = 20 / 100 * 4875 / 10 x = 150 answer is b

Provide a detailed solution to the given mathematical problem.

a number , x is chosen at random from the set of positive integers less than 10 . what is the probability that (7 / x) > x ? a) 1 / 5 , b) 7 / 9 , c) 1 / 3 , d) 2 / 3 , e) 2 / 9

"number x has to be chosen from numbers 1 - 9 (7 / x) > x = > 7 > x ^ 2 = > x ^ 2 - 7 < 0 x can have 2 values only 1 , 2 therefore , probability = 2 / 9 answer e"

Provide a detailed solution to the given mathematical problem.

what is the perimeter of a square with area p ^ 2 / 16 ? ['a) 3 p / 4','b) 3 p ^ 2 / 4','c) p','d) 3 p ^ 2','e) 4 p / 3']

area of square , (side) ^ 2 = (p / 4) ^ 2 therefore side of the square = p / 4 perimeter of square = 4 * side = 4 * (p / 4) = p answer is c .

Work through the equation methodically, ensuring clarity at each step.

a car is running at a speed of 120 kmph . what distance will it cover in 12 sec ? a) 396 m , b) 267 m , c) 180 m , d) 200 m , e) 250 m

"speed = 120 kmph = 120 * 5 / 18 = 33 m / s distance covered in 12 sec = 33 * 12 = 396 m answer is a"

Solve the problem systematically, showing each step clearly.

if there are 4 peanuts in a box and mary puts 4 more peanuts inside , how many peanuts are in the box ? a) 8 , b) 9 , c) 10 , d) 11 , e) 12

"4 + 4 = 8 correct answer is a) 8"

Explain the solution process for the given equation.

the speed of a car is 90 km in the first hour and 60 km in the second hour . what is the average speed of the car ? a) 12 , b) 75 , c) 88 , d) 54 , e) 15

"s = (90 + 60) / 2 = 75 kmph answer : b"

Explain the solution process for the given equation.

if 15 people contributed a total of $ 30.00 toward a gift and each of them contributed at least $ 1.00 , then the maximum possible amount any one person could have contributed is a) $ 1.00 , b) $ 16 , c) $ 5.00 , d) $ 6.00 , e) $ 20.00

"b for me 14 people with 1 $ each - > maximum = 16"

Break down and solve the equation step by step.

how many times digit 6 is used while writing numbers from 100 to 1400 ? a) 648 , b) 350 , c) 252 , d) 225 , e) 26

"there are 100 numbers which begin with 600 next , in every 10 numbers such as 100 to 110 , 110 to 120 , 120 to 130 6 comes at least once . number of such intervals = end limit - first no . / interval . our range of numbers is 100 - 1000 1000 - 100 = 900 / 10 = 90 number of 10 s interval in this is 90 . so 90' 6 s' so far we have calculated 190 . the total now comes to 280 . the nearest to which is 350 . hence b ."

Work through the equation methodically, ensuring clarity at each step.

there is enough provisions for 600 men in an army camp for 25 days . if there were 100 men less , how long will the provision last ? a) 10 days , b) 50 days , c) 100 days , d) 150 days , e) 200 days

"exp : we have , m 1 d 1 = m 2 d 2 600 * 25 = 100 * d 2 d 2 = 600 * 25 / 100 = 150 days . answer : d"

Break down and solve the equation step by step.

a walks at 30 kmph and 30 hours after his start , b cycles after him at 50 kmph . how far from the start does b catch up with a ? a) 1200 km , b) 1500 km , c) 2000 km , d) 2250 km , e) 3600 km

"suppose after x km from the start b catches up with a . then , the difference in the time taken by a to cover x km and that taken by b to cover x km is 30 hours . x / 30 - x / 50 = 30 x = 2250 km answer is d"

Work through the equation methodically, ensuring clarity at each step.

every disk in a bag is either blue , yellow or green . the ratio of blue disks to yellow disks to green disks in this bag is 3 : 7 : 8 . if the total number of disks in the bag is 72 , how many more green disks than blue disks are in the bag ? a) 20 , b) 28 , c) 30 , d) 35 , e) 40

"let b : y : g = 3 x : 7 x : 8 x . 3 x + 7 x + 8 x = 18 x = 72 - - > x = 4 . g - b = 8 x - 3 x = 5 x = 20 . the answer is a ."

Solve the problem systematically, showing each step clearly.

a telephone company needs to create a set of 3 - digit area codes . the company is entitled to use only digits 6 , 4 and 3 , which can be repeated . if the product of the digits in the area code must be even , how many different codes can be created ? a) 20 , b) 22 , c) 26 , d) 24 , e) 30

"total # of codes possible is 3 * 3 * 3 = 27 . oit of those 27 codes only the product of 333 and will be odd , the remaining 26 will have either 2 or 4 in them , which ensures that their product will be even . therefore the number of codes where the product of the digits is even = (total) - (restriction) = 27 - 1 = 26 . answer : c"

Work through the equation methodically, ensuring clarity at each step.

each of the products produced yesterday was checked by worker x or worker y . 0.5 % of the products checked by worker x are defective and 0.8 % of the products checked by worker y are defective . if the total defective rate of all the products checked by worker x and worker y is 0.7 % , what fraction of the products was checked by worker y ? a) 2 / 3 , b) 5 / 6 , c) 7 / 8 , d) 4 / 5 , e) 5 / 8

x : 0.5 % is 0.2 % - points from 0.7 % . y : 0.8 % is 0.1 % - points from 0.7 % . therefore the ratio of products checked by y : x is 2 : 1 . thus , worker y checked 2 / 3 of the products . the answer is a .

Break down and solve the equation step by step.

pens , pencils and markers in a jenna' s desk are in the ratio of 2 : 2 : 5 . if there are 10 pens , the number of markers in jenna' s desk is : a) 30 , b) 22 , c) 20 , d) 25 , e) 100

explanation : let pens = 2 x , pencils = 2 x & markers = 5 x . now , 2 x = 10 hence x = 5 . number of markers = 5 x which is 25 . answer : d

Provide a detailed solution to the given mathematical problem.

a goods train runs at the speed of 72 km / hr and crosses a 210 m long platform in 26 sec . what is the length of the goods train ? a) 382 , b) 782 , c) 278 , d) 310 , e) 881

"speed = 72 * 5 / 18 = 20 m / sec . time = 26 sec . let the length of the train be x meters . then , (x + 210) / 26 = 20 x = 310 m . answer : d"

Work through the equation methodically, ensuring clarity at each step.

a library has an average of 660 visitors on sundays and 240 on other days . the average number of visitors per day in a month of 30 days beginning with a sunday is : a) 250 , b) 276 , c) 280 , d) 310 , e) none

"since the month begins with sunday , to there will be five sundays in the month average required = (660 x 5 + 240 x 25) / 30) = 310 answer : option d"

Break down and solve the equation step by step.

find the average of all the numbers between 6 and 34 which are divisible by 6 . a) 18 , b) 20 , c) 24 , d) 30 , e) 21

"solution average = ((12 + 18 + 24 + 30) / 5) = 84 / 4 = 21 . answer e"

Work through the equation methodically, ensuring clarity at each step.

the h . c . f . of two numbers is 38 and the other two factors of their l . c . m . are 12 and 10 . the larger of the two numbers is : a) 678 , b) 456 , c) 234 , d) 476 , e) 432

": explanation : clearly , the numbers are (38 x 12) and (38 x 10) . { \ color { blue } \ therefore } larger number = (38 x 12) = 456 . answer : b) 456"

Solve the problem systematically, showing each step clearly.

if (a - b - c + d = 18) and (a + b - c - d = 6) , what is the value of (b - d) ^ 2 ? a) 4 . , b) 8 . , c) 12 . , d) 16 . , e) 36 .

"eq 1 : a - b - c + d = 18 eq 2 : a + b - c - d = 6 (1) subtract eq 1 from eq 2 a - b - c + d = 18 - a + b - c - d = 6 - - - - - - - - - - - - - - - - - - - - - - - - - 2 b + 2 d = 12 (2) simplify - b + d = 6 b - d = - 6 (b - d) ^ 2 = (- 6) ^ 2 = 36 my answer : e"

Provide a detailed solution to the given mathematical problem.

a 270 meter long train running at the speed of 120 kmph crosses another train running in the opposite direction at the speed of 80 kmph in 9 seconds . what is the lenght of other train . a) 210 m , b) 220 m , c) 230 m , d) 240 m , e) 250 m

"relative speeds = (120 + 80) km / hr = 200 km / hr = (200 * 5 / 18) m / s = (500 / 9) m / s let length of train be xm x + 270 / 9 = 500 / 9 x = 230 ans is 230 m answer : c"

Explain the solution process for the given equation.

if 5 machines can produce 20 units in 10 hours , how long would it take 10 to produce 60 units ? a) 9 , b) 10 , c) 15 , d) 25 , e) 30

"5 machines would produce 60 units in 30 hours . increasing the amount of machines by 2 would mean dividing 30 hours by 2 . 30 / 2 = 15 answer : c"

Break down and solve the equation step by step.

what is the probability of getting atmost one head when two unbiased coin are tossed ? a) 3 / 4 , b) 3 / 2 , c) 4 / 3 , d) 1 / 2 , e) 2 / 6

"here s = { hh , ht , th , tt } let ee = event of getting one head e = { tt , ht , th } p (e) = n (e) / n (s) = 3 / 4 option a"

Break down and solve the equation step by step.

what is the average (arithmetic mean) of all multiples of 10 from 10 to 400 inclusive ? a) 190 , b) 195 , c) 200 , d) 205 , e) 210

"we' re asked for the average of all of the multiples of 10 from 10 to 400 , inclusive . to start , we can figure out the total number of terms rather easily : 1 (10) = 10 2 (10) = 20 . . . 40 (10) = 400 so we know that there are 40 total numbers . we can now figure out the sum of those numbers with' bunching' : 10 + 400 = 410 20 + 390 = 410 30 + 380 = 410 etc . since there are 40 total terms , this pattern will create 20' pairs' of 410 . thus , since the average = (sum of terms) / (number of terms) , we have . . . (20) (410) / (40) = 410 / 2 = 205 final answer : d"

Provide a detailed solution to the given mathematical problem.

a , b and c can do a piece of work in 9 days , 14 days and 28 days respectively . how long will they taken , if all the three work together ? a) 4.12 days , b) 4.52 days , c) 3.52 days , d) 2.52 days , e) 4.92 days

1 / 9 + 1 / 14 + 1 / 28 = 55 / 252 = > 252 / 55 days = > 4.52 days answer : b

Explain the solution process for the given equation.

a sun is divided among x , y and z in such a way that for each rupee x gets , y gets 45 paisa and z gets 30 paisa . if the share of y is rs . 18 , what is the total amount ? a) 70 , b) 105 , c) 178 , d) 177 , e) 169

"x : y : z = 100 : 45 : 30 20 : 9 : 6 9 - - - 18 35 - - - ? = > 70 answer : a"

Explain the solution process for the given equation.

two good train each 100 m long , are running in opposite directions on parallel tracks . their speeds are 45 km / hr and 30 km / hr respectively . find the time taken by the slower train to pass the driver of the faster one . a) 1.2 sec , b) 2.4 sec , c) 4.8 sec , d) 9.6 sec , e) none

"sol . relative speed = (45 + 30) km / hr = (75 x 5 / 18) m / sec = (125 / 6) m / sec . distance covered = (100 + 100) m = 1000 m . required time = (200 x 6 / 125) sec = 9.6 sec . answer d"

Explain the solution process for the given equation.

an article is bought for rs . 600 and sold for rs . 450 , find the loss percent ? a) 16 % , b) 18 % , c) 19 % , d) 20 % , e) 25 %

"600 - - - - 150 100 - - - - ? = > 16 2 / 3 % answer : e"

Solve the problem systematically, showing each step clearly.

a trailer carries 3 , 4 and 5 crates on a trip . each crate weighs no less than 150 kg . what is the maximum weight of the crates on a single trip ? a) 1250 , b) 625 , c) 600 , d) 750 , e) 375

"max no . of crates = 5 . max weight = 150 kg max . weight carried = 5 * 150 = 750 kg = d"

Provide a detailed solution to the given mathematical problem.

mariah has decided to hire three workers . to determine whom she will hire , she has selected a group of 17 candidates . she plans to have one working interview with 3 of the 17 candidates every day to see how well they work together . how many days will it take her to have working interviews with all the different combinations of job candidates ? a) 720 , b) 420 , c) 300 , d) 30 , e) 333

"420 . answer b"