Bluel0la/Set_1 · Datasets at Hugging Face

Instruction stringclasses 5 values	Input stringlengths 43 704	response stringlengths 1 1.77k
Solve the problem systematically, showing each step clearly.	a and b can together finish a work in 30 days . they worked together for 20 days and then b left . after another 20 days , a finished the remaining work . in how many days a alone can finish the job ? a) 40 , b) 50 , c) 54 , d) 60 , e) none of these	"(a + b)' s 20 days work = (1 / 30 * 20) = 2 / 3 . remaining work = (1 - 2 / 3) = 1 / 3 . now , 1 / 3 work is done by a in 20 days . whole work will be done by a in (20 * 3) = 60 days . correct option : d"
Work through the equation methodically, ensuring clarity at each step.	if 20 men can build a water fountain 56 metres long in 6 days , what length of a similar water fountain can be built by 35 men in 3 days ? a) 40 m , b) 44 m , c) 47 m , d) 49 m , e) 50 m	explanation : let the required length be x metres more men , more length built (direct proportion) less days , less length built (direct proportion) men 20 : 35 days 6 : 3 : : 56 : x therefore (20 x 6 x x) = (35 x 3 x 56) x = (35 x 3 x 56) / 120 = 49 hence , the required length is 49 m . answer : d
Provide a detailed solution to the given mathematical problem.	a can complete a project in 20 days and b can complete the same project in 30 days . if a and b start working on the project together and a quits 15 days before the project is completed , in how many days total will the project be completed ? a) 18 , b) 21 , c) 24 , d) 27 , e) 30	a' s rate is 1 / 20 of the project per day . b' s rate is 1 / 30 of the project per day . the combined rate is 1 / 12 of the project per day . in the last 15 days , b can do 1 / 2 of the project . thus a and b must complete 1 / 2 of the project , which takes 6 days . the total number of days is 6 + 15 = 21 . the answer is b .
Solve the problem systematically, showing each step clearly.	a rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered . if the area of the field is 60 sq . feet , how many feet of fencing will be required ? a) 34 , b) 40 , c) 68 , d) 88 , e) 26	"we have : l = 20 ft and lb = 60 sq . ft . so , b = 3 ft . length of fencing = (l + 2 b) = (20 + 6) ft = 26 ft . answer : e"
Provide a detailed solution to the given mathematical problem.	define a = the sum of digits of the number 7 ^ 100 and b = the sum of the digits of a . what is the sum of the digits of b ? a) 7 , b) 16 , c) 10 , d) 11 , e) can not be determined	following approach will take you only couple of sec . \ (7 ^ { 100 } = (7 ^ { 50 }) ^ 2 \) the sum of digits (digital root) of a perfect square can only be 1 , 4 , 7 or 9 . for example \ (7 ^ 2 = 49 \) = > dr = 4 + 9 = 13 = > 1 + 3 = 4 and etc . checking available options . answer a .
Provide a detailed solution to the given mathematical problem.	10 machines , each working at the same constant rate , together can complete a certain job in 12 days . how many additional machines , each working at the same constant rate , will be needed to complete the job in 8 days ? a) 2 , b) 3 , c) 4 , d) 5 , e) 8	"another solution which is faster is since each machine works at a constant rate . the time needs to bought down from 12 to 8 . so the new time is 2 / 3 of the original time . thus to achieve this we need the rate to be 3 / 2 of original . so 3 / 2 * 10 = 15 so we need 15 - 10 = 5 more machines . answer : d"
Solve the problem systematically, showing each step clearly.	2 a 33 + b 9 ____ 99 if a and b represent positive single digits in the correctly worked computation above , what is the value of a + 3 b ? a) 2 , b) 7 , c) 9 , d) 11 , e) 12	adding the digits in unit' s place , a + 3 + 9 = 19 (since a and b are positive single digits) = > a = 7 now adding the digits in ten' s place , 2 + 3 + b + 1 = 9 (1 has been carried over from nit' s place addition) = > b = 3 a + 3 b = 2 + 2 * 3 = 11 answer d
Solve the problem systematically, showing each step clearly.	a car started running at a speed of 30 km / hr and the speed of the car was increased by 2 km / hr at the end of every hour . find the total distance covered by the car in the first 12 hours of the journey . a) 342 km , b) 352 km , c) 642 km , d) 742 km , e) 492 km	"the total distance covered by the car in the first 12 hours = 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 + 52 = sum of 12 terms in ap whose first term is 30 and last term is 52 = 12 / 2 [ 30 + 52 ] = 492 km . answer : e"
Provide a detailed solution to the given mathematical problem.	the effective annual rate of interest corresponding to a nominal rate of 12 % per annum payable half - yearly is ? a) 16.06 % , b) 16.07 % , c) 12.36 % , d) 6.09 % , e) 6.19 %	"amount of rs . 100 for 1 year when compounded half - yearly = [ 100 * (1 + 6 / 100) 2 ] = rs . 112.36 effective rate = (112.36 - 100) = 12.36 % answer : c"
Explain the solution process for the given equation.	in a school 10 % of the boys are same in number as 1 / 4 th of the girls . what is the ratio of boys to the girls in the school ? a) 5 : 2 , b) 2 : 3 , c) 1 : 4 , d) 3 : 7 , e) 2 : 5	"10 % of b = 1 / 4 g 10 b / 100 = g / 4 b = 5 g / 2 b / g = 5 / 2 b : g = 5 : 2 answer is a"
Break down and solve the equation step by step.	two tests had the same maximum mark . the pass percentages in the first and the second test were 40 % and 45 % respectively . a candidate scored 216 marks in the second test and failed by 75 marks in that test . find the pass mark in the first test ? a) 768 , b) 243 , c) 246 , d) 288 , e) 224	let the maximum mark in each test be m . the candidate failed by 75 marks in the second test . pass mark in the second test = 249 + 75 = 324 45 / 100 m = 324 pass mark in the first test = 40 / 100 m = 40 / 45 * 324 = 288 . answer : d
Solve the problem systematically, showing each step clearly.	10 play kabadi , 40 play kho kho only , 5 play both gmaes . then how many in total ? a) 30 , b) 35 , c) 38 , d) 40 , e) 50	"10 play kabadi = > n (a) = 10 , 5 play both gmaes . = > n (anb) = 5 40 play kho kho only , = > n (b) = n (b only) + n (anb) = 40 + 5 = 45 total = > n (aub) = n (a) + n (b) - n (anb) = 10 + 45 - 5 = 50 answer : e"
Provide a detailed solution to the given mathematical problem.	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) 76 , c) 98 , d) 27 , e) 24	"(3 x - 9) : (5 x - 9) = 9 : 17 x = 12 = > 3 x = 36 answer : a"
Solve the problem systematically, showing each step clearly.	if a person walks at 14 km / hr instead of 10 km / hr , he would have walked 20 km more . the actual distance travelled by him is a) 50 km , b) 56 km , c) 70 km , d) 80 km , e) 90 km	let the actual distance travelled be x km . then x / 10 = (x + 20) / 14 - - > 14 x = 10 x + 200 - - > 4 x = 200 - - > x = 50 km . answer : a .
Explain the solution process for the given equation.	a retailer buys 40 pens at the market price of 36 pens from a wholesaler , if he sells these pens giving a discount of 1 % , what is the profit % ? a) 5 , b) 10 , c) 15 , d) 20 , e) none of them	"let the market price of each pen be rs 1 then , c . p of 40 pens = rs 36 s . p of 40 pens = 99 % of rs 40 = rs 39.60 profit % = ((3.60 * 100) / 36) % = 10 % answer is b ."
Provide a detailed solution to the given mathematical problem.	a bullet train 130 m long is running with a speed of 50 kmph . in what time will it pass a man who is running at 10 kmph in the direction opposite to that in which the bullet train is going ? a) 23 sec , b) 13 sec , c) 12 sec , d) 11 sec , e) 16 sec	"b 13 sec speed of the bullet train relative to man = (50 + 10) kmph = 60 * 5 / 18 m / sec = 30 / 3 m / sec . time taken by the bullet train to cross the man = time taken by it to cover 130 m at (30 / 3) m / sec = (130 * 3 / 30) sec = 13 sec"
Solve the problem systematically, showing each step clearly.	what is the sum of all even numbers from 1 to 501 ? a) 122821 , b) 281228 , c) 281199 , d) 62750 , e) 128111	"explanation : 500 / 2 = 250 250 * 251 = 62750 answer : d"
Explain the solution process for the given equation.	a straight line in the xy - plane has y - intercept of 12 and a slope of 4 / 7 . given the x - coordinate of the point is 49 , find it' s y - coordinate . a) 27 , b) 16 , c) 29.7 , d) 28 , e) 18	"eq of line = y = mx + c m = 4 / 7 c = 12 x = 49 substitute the givens : y = (4 / 7 * 49) + 12 , y = 16 + 12 = 28 correct option is d"
Explain the solution process for the given equation.	if the ratio of a to b is 7 to 3 and the ratio of b to c is 1 to 5 , what is the ratio of a to c ? a) 4 / 15 , b) 1 / 3 , c) 2 / 5 , d) 4 / 5 , e) 7 / 15	"a : b = 7 : 3 - - 1 b : c = 1 : 5 = > b : c = 3 : 15 - - 2 from 1 and 2 , we get a : c = 7 : 15 answer e"
Explain the solution process for the given equation.	a sum fetched a total simple interest of rs . 4016.25 at the rate of 5 % p . a . in 5 years . what is the sum ? a) 12021 , b) 14520 , c) 16065 , d) 18925 , e) 12887	"principal = (100 * 4016.25) / (5 * 5) = rs . 16065 . answer : c"
Explain the solution process for the given equation.	find the greatest number that will divide 256 , 193 and 210 leaving 6 , 18 and 100 as remainders respectively a) 19 , b) 25 , c) 23 , d) 20 , e) 26	"explanation : answer will be hcf of (256 - 6 , 193 - 18 , 210 - 10) hcf of (250 , 175 , 200) = 25 option b"
Provide a detailed solution to the given mathematical problem.	there were totally 100 men . 83 are married . 75 have t . v , 85 have radio , 70 have a . c . how many men have t . v , radio , a . c and also married ? a) 11 , b) 12 , c) 13 , d) 14 , e) 15	100 - (100 - 83) - (100 - 75) - (100 - 85) - (100 - 70) = 100 - 17 - 25 - 15 - 30 = 100 - 87 = 13 answer : c
Solve the problem systematically, showing each step clearly.	if a rectangular billboard has an area of 91 square feet and a perimeter of 40 feet , what is the length of each of the shorter sides ? a) 4 , b) 7 , c) 8 , d) 13 , e) 26	"this question can be solved algebraically or by testing the answers . we' re told that a rectangle has an area of 91 and a perimeter of 40 . we' re asked for the length of one of the shorter sides of the rectangle . since the answers are all integers , and the area is 91 , the shorter side will almost certainly be less than 10 (since 10 x 10 = 100 , but we' re not dealing with a square) . let' s test answer b : 7 if . . . the shorter side = 7 . . . the area = 91 . . . . 91 / 7 = 13 = the longer side perimeter = 7 + 7 + 13 + 13 = 40 b"
Break down and solve the equation step by step.	6) a marketing firm determined that , of 160 households surveyed , 80 used neither brand a nor brand b soap . 60 used only brand a soap and for every household that used both brands of soap , 3 used only brand b soap . how many of the 200 household surveyed used both brands of soap ? a) a) 15 , b) b) 20 , c) c) 30 , d) d) 40 , e) e) 5	"80 = at least one of soap a or b both brands = x brand b = 3 x = > 60 + x + 3 x = 120 = > 4 x = 20 = > x = 5 answer - e"
Break down and solve the equation step by step.	what is the value of (p + q) / (p - q) if p / q is 7 ? a) 4 / 3 , b) 2 / 3 , c) 2 / 6 , d) 7 / 8 , e) 8 / 7	"(p + q) / (p - q) = [ (p / q) + 1 ] / [ (p / q) - 1 ] = (7 + 1) / (7 - 1) = 8 / 6 = 4 / 3 answer : a"
Solve the problem systematically, showing each step clearly.	how many different positive integers exist between 10 ^ 6 and 10 ^ 7 , the sum of whose digits is equal to 2 ? a) 5 , b) 6 , c) 7 , d) 8 , e) 18	"total cases = > 1000000 = > 6 cases for 1 being present at any of the 6 zero and last case 2000000 hence & cases answer : c"
Work through the equation methodically, ensuring clarity at each step.	a thief is spotted by a policeman from a distance of 225 meters . when the policeman starts the chase , the thief also starts running . if the speed of the thief be 8 km / hr and that of the policeman 10 km / hr , how far the thief will have run before he is overtaken ? a) 350 m , b) 200 m , c) 400 m , d) 900 m , e) none of them	"relative speed of the policeman = (10 - 8) km / hr = 2 km / hr . time taken by police man to cover (225 m / 1000) x 1 / 2 hr = 9 / 80 hr . in 9 / 80 hrs , the thief covers a distance of 8 x 9 / 80 km = 9 / 10 km = 900 m answer is d ."
Solve the problem systematically, showing each step clearly.	a shopkeeper sells 450 metres of cloth for rs . 18000 at a loss of rs . 5 per metre . find his cost price for one metre of cloth ? a) 12 , b) 27 , c) 29 , d) 45 , e) 21	sp per metre = 18000 / 450 = rs . 40 loss per metre = rs . 5 cp per metre = 40 + 5 = rs . 45 . answer : d
Solve the problem systematically, showing each step clearly.	if one and a half women , drink one and a half tea in one and a half minutes . how many tea can 9 women drink in 3 minutes ? a) 11 , b) 8 , c) 37 , d) 15 , e) 18	e 18 explanation : more minutes implies more tea . more women implies more tea . the time become twice (3 minutes / 1.5 minutes) . women become six times (9 / 1.5) . number of tea = 2 * 6 * original (1.5)
Solve the problem systematically, showing each step clearly.	a car traveled 462 miles per tankful of gasoline on the highway and 336 miles per tankful of gasoline in the city . if the car traveled 15 fewer miles per gallon in the city than on the highway , how many miles per gallon did the car travel in the city ? a) 14 , b) 16 , c) 21 , d) 22 , e) 40	"i treat such problems as work ones . work = rate * time mileage (m) = rate (mpg) * gallons (g) x gallons is a full tank { 462 = rx { 336 = (r - 15) x solve for r , r = 55 55 - 15 = 40 mpg e"
Explain the solution process for the given equation.	what is x if x + 2 y = 10 and y = 4 ? a) a) 10 , b) b) 8 , c) c) 6 , d) d) 4 , e) e) 2	"x = 10 - 2 y x = 10 - 8 . x = 2 answer : e"
Work through the equation methodically, ensuring clarity at each step.	after decreasing 24 % in the price of an article costs rs . 320 . find the actual cost of an article ? a) 400 , b) 520 , c) 651 , d) 525 , e) 421	"cp * (76 / 100) = 320 cp = 4.21 * 100 = > cp = 421 answer : e"
Work through the equation methodically, ensuring clarity at each step.	a business finds that their ratio of spending to income is 5 to 9 . their total profit is $ 48000 . what is their total income ? a) $ 90000 , b) $ 108000 , c) $ 119000 , d) $ 126000 , e) $ 140000	let x equal the amount of spending , and y equal the amount of income . we know that x = 5 / 9 y we also know that x = y - 48000 therefore we can simplify to : y - 48000 = 5 / 9 y we isolate y , and get : 4 / 9 y = 48000 and to solve for y , we get : y = 48000 answer : b
Solve the problem systematically, showing each step clearly.	when x is multiplied by 3 , the result is 26 more than the result of subtracting x from 62 . what is the value of x ? a) - 4 , b) - 2 , c) 11 , d) 13 , e) 22	"the equation that can be formed is : 3 x - 26 = 62 - x or , 4 x = 88 or , x = 22 . answer : e"
Break down and solve the equation step by step.	two trains start from p and q respectively and travel towards each other at a speed of 50 km / hr and 40 km / hr respectively . by the time they meet , the first train has travelled 1 ookm more than the second . the distance between p and q is a) 500 km , b) 600 km , c) 700 km , d) 900 km , e) 400 km	"explanation : at the time of meeting , let the distance travelled by the second train be x km then , distance covered by the first train is (x + 100) km = > x / 40 = (x + 100) / 50 = > 50 x = 40 x + 4000 = > x = 400 km so , distance between p and q = (x + x + 100) km = 900 km answer : option d"
Work through the equation methodically, ensuring clarity at each step.	the effective annual rate of interest corresponding to a nominal rate of 6 % per annum payable half yearly is a) 6.06 % , b) 6.07 % , c) 6.08 % , d) 6.09 % , e) none	"solution amount of rs . 100 for 1 year when compounded half yearly = rs . [ 100 x (1 + 3 / 100) 2 ] = rs . 106.09 % . effective rate = (106.09 - 100) % = rs . 6.09 % . answer d"
Solve the problem systematically, showing each step clearly.	if rs . 480 amount to rs . 540 in 4 years , what will it amount to in 6 years at the same rate % per annum ? a) s . 575 , b) s . 595 , c) s . 590 , d) s . 570 , e) s . 585	"80 = (480 * 4 * r) / 100 r = 3.125 % i = (480 * 6 * 3.125) / 100 = 90 480 + 90 = 570 answer : d"
Work through the equation methodically, ensuring clarity at each step.	a sum of money lent out at s . i . amounts to a total of $ 590 after 2 years and to $ 815 after a further period of 5 years . what was the initial sum of money that was invested ? a) $ 490 , b) $ 500 , c) $ 510 , d) $ 520 , e) $ 530	"s . i for 5 years = $ 815 - $ 590 = $ 225 the s . i . is $ 45 / year s . i . for 2 years = $ 90 principal = $ 590 - $ 90 = $ 500 the answer is b ."
Work through the equation methodically, ensuring clarity at each step.	what least number should be added to 1052 , so that the sum is completely divisible by 23 a) a) 4 , b) b) 1 , c) c) 6 , d) d) 3 , e) e) 5	"explanation : (1052 / 23) gives remainder 17 17 + 6 = 23 , so we need to add 6 answer : option c"
Provide a detailed solution to the given mathematical problem.	kavi had a stock of 600 bags in his bookshop . he sold 25 on monday , 70 on tuesday , 100 on wednesday , 110 on thursday and 145 on friday . what percentage of the bags were not sold ? a) 10 % , b) 25 % , c) 64 % , d) 42 % , e) 17 %	let n be the total number of bags sold . hence n = 25 + 70 + 100 + 110 + 145 = 450 let m be the bags not sold m = 600 - n = 600 - 450 = 150 percentage bags not sold / total number of bags = 150 / 600 = 0.25 = 25 % correct answer b
Explain the solution process for the given equation.	4 mat - weavers can weave 4 mats in 4 days . at the same rate , how many mats would be woven by 16 mat - weavers in 16 days ? a) 25 , b) 19 , c) 39 , d) 61 , e) 64	"let the required number of bottles be x . more weavers , more mats (direct proportion) more days , more mats (direct proportion) wavers 4 : 16 : : 4 : x days 4 : 16 4 * 4 * x = 16 * 16 * 4 x = (16 * 16 * 4) / (4 x 4) x = 64 . answer is e ."
Break down and solve the equation step by step.	on selling 17 balls at rs . 720 , there is a loss equal to the cost price of 5 balls . the cost price of a ball is a) 20 , b) 30 , c) 40 , d) 50 , e) 60	"(c . p . of 17 balls) - (s . p . of 17 balls) = (c . p . of 5 balls) c . p . of 12 balls = s . p . of 17 balls = rs . 720 . c . p . of 1 ball = rs . 720 / 12 = rs . 60 . answer : e"
Provide a detailed solution to the given mathematical problem.	cereal a is 10 % sugar by weight , whereas healthier but less delicious cereal b is 2 % sugar by weight . to make a delicious and healthy mixture that is 5 % sugar , what should be the ratio of cereal a to cereal b , by weight ? a) 2 : 5 , b) 3 : 5 , c) 4 : 7 , d) 2 : 6 , e) 1 : 4	"2 % is 3 % - points below 5 % and 10 % is 5 % - points above 5 % . the ratio of a : b should be 3 : 5 . the answer is b ."
Work through the equation methodically, ensuring clarity at each step.	square abcd is the base of the cube while square efgh is the cube' s top face such that point e is above point a , point f is above point b etc . what is the distance between the midpoint of edge ab and the midpoint of edge eh if the area of square abcd is 2 ? ['a) 1 / sqrt 2','b) 1','c) sqrt 2','d) sqrt 3','e) 2 sqrt 3']	distance from mid point of ab to ad = sqrt [ (1 / sqrt 2) ^ 2 + (1 / sqrt 2) ^ 2 ] = 1 the distance between the midpoint of edge ab and the midpoint of edge eh = sqrt [ 1 ^ 2 + (sqrt 2) ^ 2 ] = sqrt 3 . answer : d
Explain the solution process for the given equation.	what is the value of n if the sum of the consecutive odd integers from 1 to n equals 169 ? a) 47 , b) 25 , c) 37 , d) 33 , e) 29	# of terms = (n - 1 / 2) + 1 { (last term - first term) / 2 + 1 \| sum = (1 + n) / 2 * # of terms = (n + 1) ^ 2 / 4 = 169 n + 1 = 13 * 2 n + 1 = 26 n = 25 . answer : b
Break down and solve the equation step by step.	a small college reduced its faculty by approximately 23 percent to 195 professors . what was the original number of faculty members ? a) 253 , b) 208 , c) 224 , d) 254 , e) 302	"if x is the original number of faculty members , then after 23 % reduction in faculty members number is . 77 x but we are given . 77 x = 195 x = 253 so the original number of faculty members is 253 correct answer - a"
Solve the problem systematically, showing each step clearly.	if 3 cats can kill 3 rats in 3 minutes , how long will it take 100 cats to kill 100 rats ? a) 1 minutes , b) 2 minutes , c) 3 minutes , d) 4 minutes , e) 5 minutes	"it will take 3 minutes for 100 cats to kill 100 rats . 1 cat can kill 1 rat in 3 minutes , so 100 cats can kill 100 rats in 3 minutes answer c"
Work through the equation methodically, ensuring clarity at each step.	pipe a can fill a tank in 6 hours . due to a leak at the bottom , it takes 8 hours for the pipe a to fill the tank . in what time can the leak alone empty the full tank ? a) 13 , b) 24 , c) 18 , d) 19 , e) 12	"let the leak can empty the full tank in x hours 1 / 6 - 1 / x = 1 / 8 = > 1 / x = 1 / 6 - 1 / 8 = (4 - 3) / 24 = 1 / 24 = > x = 24 . answer : b"
Solve the problem systematically, showing each step clearly.	{ - 10 , - 6 , - 5 , - 4 , - 2.5 , - 1 , 0 , 2.5 , 4 , 6 , 7 , 10 } a number is to be selected at random from the set above . what is the probability that the number will be a solution to the equation (x 3) (x + 14) (2 x + 5) = 0 ? a) 1 / 12 , b) 1 / 6 , c) 1 / 4 , d) 1 / 3 , e) 1 / 2	x = - 4 prob = 1 / 12 answer - a
Explain the solution process for the given equation.	two pipes a and b can fill a tank in 4 and 5 hours respectively . if they are turned up alternately for one hour each , the time taken to fill the tank is ? a) 4 hrs 29 min , b) 4 hrs 24 min , c) 8 hrs 24 min , d) 4 hrs 44 min , e) 5 hrs 24 min	1 / 4 + 1 / 5 = 9 / 20 20 / 9 = 2 2 / 9 9 / 20 * 2 = 9 / 10 - - - - 4 hours wr = 1 - 9 / 10 = 1 / 10 1 h - - - - 1 / 4 ? - - - - - 1 / 10 2 / 5 * 60 = 24 = 4 hrs 24 min answer : b
Solve the problem systematically, showing each step clearly.	a diagonal of a polygon is an segment between two non - adjacent vertices of the polygon . how many diagonals does a regular 30 - sided polygon have ? a) 875 , b) 405 , c) 1425 , d) 2025 , e) 2500	"there' s a direct formula for this . number of diagonals in a regular polygon = [ n * (n - 3) ] / 2 , n = number of sides of the regular polygon . here , n = 30 . plugging it in , we get 405 diagonals ! answer (b) ."
Solve the problem systematically, showing each step clearly.	a corporation that had $ 10 billion in profits for the year paid out $ 500 million in employee benefits . approximately what percent of the profits were the employee benefits ? (note : 1 billion = 10 ^ 9) a) 50 % , b) 20 % , c) 5 % , d) 2 % , e) 0.2 %	"required answer = [ employee benefit / profit ] * 100 = [ (500 million) / (10 billion) ] * 100 = [ (500 * 10 ^ 6) / (10 * 10 ^ 9) ] * 100 = (50 / 1000) * 100 = 5 % so answer is (c)"
Work through the equation methodically, ensuring clarity at each step.	to produce an annual income of rs . 1200 from a 12 % stock at 90 , the amount of stock needed is : a) rs . 10,000 , b) rs . 10,800 , c) rs . 14,400 , d) rs . 16,000 , e) none of these	solution for an income of rs . 12 , stock needed = rs . 100 . for an income of rs . 1200 , stock needed = rs . 100 / 12 x 1200 = 10000 answer a
Solve the problem systematically, showing each step clearly.	a cubical block of metal weighs 4 pounds . how much will another cube of the same metal weigh if its sides are twice as long ? a) 48 , b) 32 , c) 24 , d) 18 , e) 12	"for example our cube have a side 1 meter , so we have 1 cubical meter in this cube and this cubical meter weigth 4 pounds if we take cube with side 2 meters we will have 8 cubical meters in this cube 8 meters * 4 pounds = 32 pounds so answer is b and similar but more theoretical approach : if we have sides a and b than they have equal ration with their areas : a / b = a ^ 2 / b ^ 2 and they have equal ration with their volumes : a / b = a ^ 3 / b ^ 3 we have two sides 1 / 2 so their volume will be in ratio 1 / 8 weight of one cube * volume of another cube 4 * 8 = 32 so answer is b"
Explain the solution process for the given equation.	a boat running downstream covers a distance of 12 km in 2 hours while for covering the same distance upstream , it takes 3 hours . what is the speed of the boat in still water ? a) 8 kmph , b) 5 kmph , c) 4 kmph , d) 6 kmph , e) 7 kmph	"speed downstream = 12 / 2 = 6 kmph speed upstream = 12 / 3 = 4 kmph speed of the boat in still water = 6 + 4 / 2 = 5 kmph answer : b"
Work through the equation methodically, ensuring clarity at each step.	a student scored an average of 75 marks in 3 subjects : physics , chemistry and mathematics . if the average marks in physics and mathematics is 90 and that in physics and chemistry is 70 , what are the marks in physics ? a) 86 , b) 16 , c) 76 , d) 95 , e) 26	"given m + p + c = 75 * 3 = 225 - - - (1) m + p = 90 * 2 = 180 - - - (2) p + c = 70 * 2 = 140 - - - (3) where m , p and c are marks obtained by the student in mathematics , physics and chemistry . p = (2) + (3) - (1) = 180 + 140 - 225 = 95 answer : d"
Explain the solution process for the given equation.	in a certain pond , 30 fish were caught , tagged , and returned to the pond . a few days later , 50 fish were caught again , of which 2 were found to have been tagged . if the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond , what is the approximate number of fish in the pond ? a) 750 , b) 625 , c) 1,250 , d) 2,500 , e) 10,000	"this is a rather straight forward ratio problem . 1 . 30 fish tagged 2 . 2 out of the 50 fish caught were tagged thus 2 / 50 2 / 50 = 30 / x thus , x = 750 think of the analogy : 2 fish is to 50 fish as 50 fish is to . . . ? you' ve tagged 50 fish and you need to find what that comprises as a percentage of the total fish population - we have that information with the ratio of the second catch . a"
Solve the problem systematically, showing each step clearly.	if 1 / 2 of the air in a tank is removed with each stroke of a vacuum pump , what fraction of the original amount of air has been removed after 1 strokes ? a) 1 / 2 , b) 7 / 8 , c) 1 / 4 , d) 1 / 8 , e) 1 / 16	"left after 1 st stroke = 1 / 2 so removed = 1 - 1 / 2 = 1 / 2"
Work through the equation methodically, ensuring clarity at each step.	if money is invested at r percent interest , compounded annually , the amount of the investment will double in approximately 50 / r years . if luke' s parents invested $ 13,500 in a long term bond that pays 12 percent interest compounded annually , what will be the approximate total amount of the investment 12 years later , when luke is ready for college ? a) 62,000 , b) 85,500 , c) 95,500 , d) 100,500 , e) 100,000	answer equals d in 48 years . i thought by 50 th year it would reach 100,500 . options should have been separated more widely for clarity .
Provide a detailed solution to the given mathematical problem.	there are two inlets and one outlet to a cistern . one of the inlets takes 4 hours to fill up the cistern and the other inlet takes twice as much time to fill up the same cistern . both of the inlets are turned on at 8 : 30 am with the cistern completely empty , and at 10 : 30 am , the outlet is turned on and it takes 1 more hour to fill the cistern completely . how much time does the outlet working alone takes to empty the cistern when the cistern is full ? a) 2 hours , b) 2.5 hours , c) 3 hours , d) 3.5 hours , e) 8 hours	"the combined inflow rate of the two inlets is 1 / 4 + 1 / 8 = 3 / 8 cistern / hour . thus , working together , it takes 8 / 3 hours (time is reciprocal of rate) to fill the cistern . from 8 : 30 am to 10 : 30 am , so in 2 hours , the inlet pipes will fill (time) * (rate) = 2 * 3 / 8 = 3 / 4 th of the cistern . then the outlet is turned on and the remaining 1 / 4 th of the cistern is filled in 1 hour . letting x to be the rate of the outlet , we would have : 3 / 8 - x = 1 / 4 - - > x = 1 / 8 cistern / hour , which means that it takes 8 hours the outlet working alone to empty the cistern . answer : e ."
Work through the equation methodically, ensuring clarity at each step.	40 men shake hands with each other . maximum no of handshakes without cyclic handshakes . a) 780 , b) 200 , c) 210 , d) 220 , e) 230	"or , if there are n persons then no . of shakehands = nc 2 = 40 c 2 = 780 answer : a"
Explain the solution process for the given equation.	if the volume of two cubes are in the ratio 125 : 1 , the ratio of their edges is : a) 3 : 1 , b) 5 : 1 , c) 3 : 5 , d) 3 : 7 , e) none of these	"explanation : let the edges be a and b of two cubes , then a 3 / b 3 = 125 / 1 = > (a / b) 3 = (5 / 1) 3 a / b = 5 / 1 = > a : b = 5 : 1 option b"
Work through the equation methodically, ensuring clarity at each step.	in x game of billiards , x can give y 20 points in 60 and he can give z 50 points in 60 . how many points can y give z in x game of 100 ? a) 30 , b) 20 , c) 25 , d) 40 , e) 75	"x scores 60 while y score 40 and z scores 10 . the number of points that z scores when y scores 100 = (100 * 50) / 40 = 25 . in x game of 100 points , y gives (100 - 25) = 75 points to c . e"
Break down and solve the equation step by step.	the ratio of three numbers is 1 : 2 : 4 and the sum of their squares is 1701 . the sum of the numbers is ? a) a) 10 , b) b) 12 , c) c) 15 , d) d) 14 , e) e) 9	"let the numbers be x , 2 x , 4 x then , x ^ 2 + 4 x ^ 2 + 16 x ^ 2 = 1701 21 x ^ 2 = 1701 x ^ 2 = 81 x = 9 answer is e"
Provide a detailed solution to the given mathematical problem.	there is 60 % increase in an amount in 6 years at si . what will be the ci of rs . 16,000 after 3 years at the same rate ? a) 2372 , b) 2572 , c) 4542 , d) 2343 , e) 5296	"let p = rs . 100 . then , s . i . rs . 60 and t = 6 years . r = 100 x 60 = 10 % p . a . 100 x 6 now , p = rs . 16000 . t = 3 years and r = 10 % p . a . c . i . = rs . 16000 x 1 + 10 3 - 1 100 = rs . 16000 x 331 1000 = 5296 . e"
Work through the equation methodically, ensuring clarity at each step.	a prize of $ 600 is to be distributed among 15 winners , each of whom must be awarded at least $ 15 . if 2 / 5 of the prize will be distributed to 3 / 5 of the winners , what is the greatest possible individual award ? a) $ 200 , b) $ 220 , c) $ 285 , d) $ 300 , e) $ 360	"total value of the prize = $ 600 number of people = 15 2 / 5 of 600 (= $ 240) should be distributed among 3 / 5 of 15 (= 9 people) with each getting $ 15 each . remaining money = 600 - 240 = $ 360 . now in order to' maximize' 1 prize , we need to minimise the others and we have been given that each should get $ 15 . thus , minimising the remaining 5 people (= 15 - 9 - 1 .' - 1' to exclude 1 that needs to be maximised) = 5 * 15 = 75 . thus the maximum award can be = 360 - 75 = $ 285 , hence c is the correct answer ."
Provide a detailed solution to the given mathematical problem.	a trader cheats both his supplier and customer by using faulty weights . when he buys from the supplier , he takes 10 % more than the indicated weight . when he sells to his customer , he gives the customer a weight such that 30 % of that is added to the weight , the weight claimed by the trader is obtained . if he charges the cost price of the weight that he claims , find his profit percentage . a) 20 % , b) 43 % , c) 99.33 % , d) 75.109 % , e) 47 %	anyways , one can infer that he' steals' 10 % from suppliers and then charges 30 % extra to customers so basically 1.1 * 1.3 = 1.43 given that 1 is start point , we get 43 % more hence answer is b
Work through the equation methodically, ensuring clarity at each step.	walking 6 / 7 of his usual rate , a boy reaches his school 4 min early . find his usual time to reach the school ? a) 28 m , b) 20 m , c) 22 m , d) 23 m , e) 24 m	speed ratio = 1 : 6 / 7 = 7 : 6 time ratio = 6 : 4 1 - - - - - - - - 6 4 - - - - - - - - - ? = > 24 m answer : e
Work through the equation methodically, ensuring clarity at each step.	machine p and machine q are each used to manufacture 330 sprockets . it takes machine p 10 hours longer to produce 330 sprockets than machine q . machine q produces 10 % more sprockets per hour than machine a . how many sprockets per hour does machine a produce ? a) 5 , b) 15 , c) 3 , d) 95 , e) 125	"p makes x sprockets per hour . then q makes 1.1 x sprockets per hour . 330 / x = 330 / 1.1 x + 10 1.1 (330) = 330 + 11 x 11 x = 33 x = 3 the answer is c ."
Solve the problem systematically, showing each step clearly.	0.0003 x 0.3 = ? a) 0.0009 , b) 9.0 e - 05 , c) 0.09 , d) 0.009 , e) 9.0	"3 x 3 = 9 . sum of decimal places = 5 0.0003 x 0.3 = 0.00009 answer : option b"
Solve the problem systematically, showing each step clearly.	when 2 / 9 of the votes on a certain resolution have been counted , 3 / 4 of those counted are in favor of the resolution . what fraction e of the remaining votes must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution ? a) 11 / 14 , b) 13 / 18 , c) 4 / 7 , d) 3 / 7 , e) 3 / 14	if we use variable for total votes there will be too many fractions to manipulate with , so pick some smart # : let set total # of votes is 18 . 2 / 9 of the votes on a certain resolution have been counted - - > 4 counted and 18 - 4 = 14 votes left to be counted ; 3 / 4 of those counted are in favor of the resolution - - > 3 in favor and 1 against ; ratio of those who voted against to those who voted for to be 2 to 1 there should be total of 18 * 2 / 3 = 12 people who voted against , so in the remaining 14 votes there should be 12 - 1 = 11 people who voted against . thus e = 11 / 14 of the remaining votes must be against . answer : a .
Work through the equation methodically, ensuring clarity at each step.	raman mixed 54 kg of butter at rs . 150 per kg with 36 kg butter at the rate of rs . 125 per kg . at what price per kg should he sell the mixture to make a profit of 40 % in the transaction ? a) 129 , b) 287 , c) 196 , d) 188 , e) 112	"explanation : cp per kg of mixture = [ 54 (150) + 36 (125) ] / (54 + 36) = rs . 140 sp = cp [ (100 + profit %) / 100 ] = 140 * [ (100 + 40) / 100 ] = rs . 196 . answer : c"
Solve the problem systematically, showing each step clearly.	a man whose bowling average is 12.4 , takes 3 wickets for 26 runs and there by decreases his average by 0.4 . the number of wickets taken by him before his last match is ? a) 23 , b) 24 , c) 25 , d) 26 , e) 27	"12.4 * x + 26 = (3 + x) 12 solve equation x = 25 answer : c"
Provide a detailed solution to the given mathematical problem.	if 80 % of a number is equal to four - third of another number , what is the ratio of first number to the second number ? a) 5 : 3 , b) 1 : 4 , c) 3 : 7 , d) 6 : 11 , e) 2 : 3	"let 80 % of a = 4 / 3 b then 80 a / 100 = 4 b / 3 4 a / 5 = 4 b / 3 a / b = 5 / 3 a : b = 5 : 3 answer is a"
Break down and solve the equation step by step.	a searchlight on top of the watchtower makes 2 revolutions per minute . what is the probability that a man appearing near the tower will stay in the dark for at least 5 seconds ? a) 5 / 6 , b) 5 / 12 , c) 7 / 12 , d) 11 / 12 , e) 11 / 15	"the searchlight completes one revolution every 30 seconds . the probability that the man' s area will be lit up is 5 / 30 = 1 / 6 . the probability that he will stay in the dark is 1 - 1 / 6 = 5 / 6 the answer is a ."
Break down and solve the equation step by step.	a certain no . when divided by 110 leaves a remainder 25 , what is the remainder if the same no . be divided by 15 ? a) 2 , b) 4 , c) 7 , d) 8 , e) 9	"explanation : 110 + 25 = 135 / 15 = 9 (remainder) e"
Solve the problem systematically, showing each step clearly.	when a = x + (3 / x) and b = x - (3 / x) , (2 ^ a ^ 2) / (2 ^ b ^ 2) = ? a) 2 , b) 4 , c) 8 , d) 16 , e) 4096	"- - > (2 ^ a ^ 2) / (2 ^ b ^ 2) = { (2) ^ (a ^ 2 - b ^ 2) } = 2 ^ (a - b) (a + b) . since a - b = 6 / x and a + b = 2 x , 2 ^ (a - b) (a + b) = 2 ^ (6 / x) (2 x) = 2 ^ 12 = 4096 therefore , the answer is e"
Work through the equation methodically, ensuring clarity at each step.	a cube of edge 15 cm is immersed completely in a rectangular vessel containing water . if the dimensions of the base of vessel are 20 cm * 15 cm , find the rise in water level ? a) 6 cm , b) 8.25 cm , c) 11.25 cm , d) 15.12 cm , e) 20.62 cm	"increase in volume = volume of the cube = 15 * 15 * 15 cm ^ 3 rise in water level = volume / area = 15 * 15 * 15 / 20 * 15 = 11.25 cm answer is c"
Break down and solve the equation step by step.	what is the smallest positive integer x such that (x + 1) ^ 2 is divisible by 28 , 98 , 242 , and 308 ? a) 153 , b) 308 , c) 121 , d) 96 , e) 511	"28 = 2 * 2 * 7 98 = 2 * 7 * 7 242 = 2 * 11 * 11 308 = 2 * 2 * 7 * 11 so (x + 1) ^ 2 = 2 * 2 * 7 * 7 * 11 * 11 , which means (x + 1) = 2 * 7 * 11 = 154 , which means x = 153 , which is option a"
Provide a detailed solution to the given mathematical problem.	what is the area of a triangle with the following vertices l (2 , 3) , m (5 , 1) , and n (3 , 5) ? ['a) 3','b) 4','c) 5','d) 6','e) 7']	there is a direct formula to calculate the are of a triangle based on coordinates of its vertices and one could use it to solve this problem . though if you make a diagram minimum simple calculations will be needed : answer : b .
Solve the problem systematically, showing each step clearly.	pipe a can fill a tank in 32 minutes . pipe b can fill the same tank 5 times faster than pipe a . if both the pipes are connected to the tank so that they fill the tank simultaneously , how long will it take for the empty tank to overflow ? a) 4 minutes , b) 2 / 7 minutes , c) 32 / 7 minutes , d) 224 / 7 minutes , e) 28 minutes	"pipe a can fill a tank in 32 minutes - - > the rate of a = 1 / 32 tank / minute ; pipe b can fill the same tank 5 times fasterthan pipe a - - > the rate of b = 1 / 32 + 5 / 32 = 6 / 32 tank / minute . their combined rate = 1 / 32 + 6 / 32 = 7 / 32 tank / minute ; therefore , a and b can fill the tank in 32 / 7 minutes . answer : c"
Explain the solution process for the given equation.	find number which is 70 % less than 60 . a) 18 , b) 22 , c) 28 , d) 26 , e) 98	"explanation : 70 % less is 30 % of the given number therefore , 30 % of 60 is 18 . answer : a"
Provide a detailed solution to the given mathematical problem.	x can give y 100 meters start and z 200 meters start in x kilometer race . how much start can y give z in x kilometer race ? a) 112.2 m , b) 216.3 m , c) 111.12 m , d) 213.27 m , e) 222.40 m	c 111.12 m x runs 1000 m while y runs 900 m and z runs 800 m . the number of meters that z runs when y runs 1000 m , = (1000 * 800) / 900 = 8000 / 9 = 888.88 m . y can give z = 1000 - 888.88 = 111.12 m .
Break down and solve the equation step by step.	if 0.75 of a number is equal to 0.025 of another number , the ratio of the numbers is : a) 2 : 3 , b) 3 : 4 , c) 1 : 30 , d) 20 : 3 , e) 30 : 7	"0.75 a = 0.025 b - > a / b = 0.025 / 0.75 = 25 / 750 = 1 / 30 : . a : b = 1 : 30 answer : c"
Explain the solution process for the given equation.	how many different values of positive integer x , for which \| x + 5 \| < x , are there ? a) 0 , b) 2 , c) 3 , d) 8 , e) 16	"answer e i opted to put the random value option . i used 0 , 5 , - 5 and the the extreme of 40 and - 40 . . i was able to solve it in 1 : 09 e"
Explain the solution process for the given equation.	a tank is filled by 3 pipes a , b , c in 10 hours . pipe c is twice as fast as b and b is twice as fast as a . how much will pipe a alone take to fill the tank ? a) 20 hr , b) 52 hr , c) 70 hr , d) 66 hr , e) 48 hr	"suppose pipe a alone take x hours to fill the tank then pipe b and c will take x / 2 and x / 4 hours respectively to fill the tank . 1 / x + 2 / x + 4 / x = 1 / 10 7 / x = 1 / 10 x = 70 hours answer is c"
Work through the equation methodically, ensuring clarity at each step.	due to construction , the speed limit along an 10 - mile section of highway is reduced from 55 miles per hour to 25 miles per hour . approximately how many minutes more will it take to travel along this section of highway at the new speed limit than it would have taken at the old speed limit ? a) a) 6.24 , b) b) 13.1 , c) c) 10 , d) d) 15 , e) e) 24	"old time in minutes to cross 10 miles stretch = 10 * 60 / 55 = 10 * 12 / 11 = 10.9 new time in minutes to cross 10 miles stretch = 10 * 60 / 25 = 10 * 12 / 5 = 24 time difference = 13.1 ans : b"
Work through the equation methodically, ensuring clarity at each step.	two vessels p and q contain 62.5 % and 87.5 % of alcohol respectively . if 12 litres from vessel p is mixed with 4 litres from vessel q , the ratio of alcohol and water in the resulting mixture is ? a) 19 : 2 , b) 18 : 3 , c) 11 : 1 , d) 11 : 5 , e) 11 : 4	"quantity of alcohol in vessel p = 62.5 / 100 * 2 = 15 / 2 litres quantity of alcohol in vessel q = 87.5 / 100 * 4 = 7 / 2 litres quantity of alcohol in the mixture formed = 15 / 2 + 7 / 2 = 11 litres as 16 litres of mixture is formed , ratio of alcohol and water in the mixture formed = 11 : 5 answer : d"
Break down and solve the equation step by step.	if p (a) = 4 / 5 and p (b) = 2 / 5 , find p (a n b) if a and b are independent events . a) 8 / 25 , b) 1 / 25 , c) 2 / 25 , d) 3 / 14 , e) 25 / 8	"p (a n b) = p (a) . p (b) p (a n b) = 4 / 5 . 2 / 5 p (a n b) = 8 / 25 . a"
Explain the solution process for the given equation.	find the greatest number of 5 digits which is exactly divisible by 12 , 15 and 18 ? a) a) 13050 , b) b) 12960 , c) c) 10025 , d) d) 11080 , e) e) 12080	the largest five digit numbers are 13050 , 12960,12080 13050 is not divisible by 12 12960 is divisible by 18 , 12 and 15 answer : b
Work through the equation methodically, ensuring clarity at each step.	solve the equation for x : 6 x - 87 + 3 x = 4 + 9 - x a) 12 , b) 5 , c) 10 , d) 9 , e) 3	"c 10 9 x + x = 13 + 87 10 x = 100 = > x = 10"
Solve the problem systematically, showing each step clearly.	the average (arithmetic mean) of 16 students first quiz scores in a difficult english class is 60.5 . when one student dropped the class , the average of the remaining scores increased to 64.0 . what is the quiz score of the student who dropped the class ? a) a) 8 , b) b) 25 , c) c) 40 , d) d) 55 , e) e) 70	"total score of 16 students is 16 * 60.50 = 968 total score of 15 students is 15 * 64 = 960 so , the score of the person who left is 8 (968 - 960) answer will be (a"
Break down and solve the equation step by step.	how many integers between 0 and 1000 do not contain the integer 1 a) 727 , b) 728 , c) 758 , d) 748 , e) 792	from 1 to 99 = 19 integers having 1 from 101 to 199 = all are having 1 similarly 201 - 299 , 301 - 399 , 401 - 499 , . . . . . so on , will have 19 integers having 1 answer = 1000 - (19 * 9 + 99 + 2) / / 2 is added for 100 and 1000 = 728 answer : b
Solve the problem systematically, showing each step clearly.	if 8 people take an hour to complete a piece of work , then how long should 16 people will take to complete the same piece of work ? a) 15 min , b) 30 min , c) 26 min , d) 28 min , e) 18 min	if 8 people take an hour to complete a piece of work , then 16 people will take 8 * 60 / 16 = 30 mins to complete the same piece of work . answer : b
Solve the problem systematically, showing each step clearly.	if grapes are 90 % water and raisins are 17 % water , then how many kilograms did a quantity of raisins , which currently weighs 14 kilograms , weigh when all the raisins were grapes ? (assume that the only difference between their raisin - weight and their grape - weight is water that evaporated during their transformation .) a) 108.6 , b) 110.4 , c) 112.6 , d) 114.8 , e) 116.2	"let x be the original weight of the grapes . the weight of the grape pulp was 0.1 x . since the grape pulp is 83 % of the raisins , 0.1 x = 0.83 (14 kg) . then x = 8.3 * 14 = 116.2 kg . the answer is e ."
Solve the problem systematically, showing each step clearly.	what is the smallest integer that is multiple of 3 , 5,9 a) a) 70 , b) b) 45 , c) c) 200 , d) d) 280 , e) e) 140	"it is the lcm of 3 , 5 and 9 which is 45 . the answer is b ."
Solve the problem systematically, showing each step clearly.	mixture contains alcohol and water in the ratio 4 : 3 . if 8 liters of water is added to the mixture , the ratio becomes 4 : 5 . find the quantity of alcohol in the given mixture . a) 10 , b) 99 , c) 27 , d) 4 , e) 29	"let the quantity of alcohol and water be 4 x litres and 3 x litres respectively 32 x = 4 (3 x + 5) 20 x = 20 x = 1 quantity of alcohol = (4 x 1) litres = 4 litres . answer : d"
Provide a detailed solution to the given mathematical problem.	what is the units digit of (493) (915) (381) (756) (22) a) 0 , b) 1 , c) 4 , d) 5 , e) 6	"just multiply the digits in the units place for each term and you will get the answer . it should be 0 . you got a 5 as a unit digit and an even number term . so the multiplication of this will definitely yield a 0 . answer has to be 0 . i also tried it using the calculator and the answer is 1 . imo b ."
Work through the equation methodically, ensuring clarity at each step.	the dimensions of a rectangular solid are 4 inches , 5 inches , and 9 inches . if a cube , a side of which is equal to one of the dimensions of the rectangular solid , is placed entirely within thespherejust large enough to hold the cube , what the ratio of the volume of the cube to the volume within thespherethat is not occupied by the cube ? a) 2 : 5 , b) 5 : 9 , c) 5 : 16 , d) 25 : 7 , e) 32 : 25	answer : b .
Provide a detailed solution to the given mathematical problem.	a number a is squared and then multiplied by negative 6 . the result of this operation is equal to 3 times the sum of fourth times a and two . what is one possible value of a ? a) 2 , b) 3 , c) 4 , d) 1 , e) 5	- 6 * a ^ 2 = 3 (4 a + 2) a = - 1 or - 1 a = - 1 = b answer : d

Solve the problem systematically, showing each step clearly.

a and b can together finish a work in 30 days . they worked together for 20 days and then b left . after another 20 days , a finished the remaining work . in how many days a alone can finish the job ? a) 40 , b) 50 , c) 54 , d) 60 , e) none of these

"(a + b)' s 20 days work = (1 / 30 * 20) = 2 / 3 . remaining work = (1 - 2 / 3) = 1 / 3 . now , 1 / 3 work is done by a in 20 days . whole work will be done by a in (20 * 3) = 60 days . correct option : d"

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

if 20 men can build a water fountain 56 metres long in 6 days , what length of a similar water fountain can be built by 35 men in 3 days ? a) 40 m , b) 44 m , c) 47 m , d) 49 m , e) 50 m

explanation : let the required length be x metres more men , more length built (direct proportion) less days , less length built (direct proportion) men 20 : 35 days 6 : 3 : : 56 : x therefore (20 x 6 x x) = (35 x 3 x 56) x = (35 x 3 x 56) / 120 = 49 hence , the required length is 49 m . answer : d

Provide a detailed solution to the given mathematical problem.

a can complete a project in 20 days and b can complete the same project in 30 days . if a and b start working on the project together and a quits 15 days before the project is completed , in how many days total will the project be completed ? a) 18 , b) 21 , c) 24 , d) 27 , e) 30

a' s rate is 1 / 20 of the project per day . b' s rate is 1 / 30 of the project per day . the combined rate is 1 / 12 of the project per day . in the last 15 days , b can do 1 / 2 of the project . thus a and b must complete 1 / 2 of the project , which takes 6 days . the total number of days is 6 + 15 = 21 . the answer is b .

Solve the problem systematically, showing each step clearly.

a rectangular field is to be fenced on three sides leaving a side of 20 feet uncovered . if the area of the field is 60 sq . feet , how many feet of fencing will be required ? a) 34 , b) 40 , c) 68 , d) 88 , e) 26

"we have : l = 20 ft and lb = 60 sq . ft . so , b = 3 ft . length of fencing = (l + 2 b) = (20 + 6) ft = 26 ft . answer : e"

Provide a detailed solution to the given mathematical problem.

define a = the sum of digits of the number 7 ^ 100 and b = the sum of the digits of a . what is the sum of the digits of b ? a) 7 , b) 16 , c) 10 , d) 11 , e) can not be determined

following approach will take you only couple of sec . \ (7 ^ { 100 } = (7 ^ { 50 }) ^ 2 \) the sum of digits (digital root) of a perfect square can only be 1 , 4 , 7 or 9 . for example \ (7 ^ 2 = 49 \) = > dr = 4 + 9 = 13 = > 1 + 3 = 4 and etc . checking available options . answer a .

Provide a detailed solution to the given mathematical problem.

10 machines , each working at the same constant rate , together can complete a certain job in 12 days . how many additional machines , each working at the same constant rate , will be needed to complete the job in 8 days ? a) 2 , b) 3 , c) 4 , d) 5 , e) 8

"another solution which is faster is since each machine works at a constant rate . the time needs to bought down from 12 to 8 . so the new time is 2 / 3 of the original time . thus to achieve this we need the rate to be 3 / 2 of original . so 3 / 2 * 10 = 15 so we need 15 - 10 = 5 more machines . answer : d"

Solve the problem systematically, showing each step clearly.

2 a 33 + b 9 ____ 99 if a and b represent positive single digits in the correctly worked computation above , what is the value of a + 3 b ? a) 2 , b) 7 , c) 9 , d) 11 , e) 12

adding the digits in unit' s place , a + 3 + 9 = 19 (since a and b are positive single digits) = > a = 7 now adding the digits in ten' s place , 2 + 3 + b + 1 = 9 (1 has been carried over from nit' s place addition) = > b = 3 a + 3 b = 2 + 2 * 3 = 11 answer d

Solve the problem systematically, showing each step clearly.

a car started running at a speed of 30 km / hr and the speed of the car was increased by 2 km / hr at the end of every hour . find the total distance covered by the car in the first 12 hours of the journey . a) 342 km , b) 352 km , c) 642 km , d) 742 km , e) 492 km

"the total distance covered by the car in the first 12 hours = 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 + 52 = sum of 12 terms in ap whose first term is 30 and last term is 52 = 12 / 2 [ 30 + 52 ] = 492 km . answer : e"

Provide a detailed solution to the given mathematical problem.

the effective annual rate of interest corresponding to a nominal rate of 12 % per annum payable half - yearly is ? a) 16.06 % , b) 16.07 % , c) 12.36 % , d) 6.09 % , e) 6.19 %

"amount of rs . 100 for 1 year when compounded half - yearly = [ 100 * (1 + 6 / 100) 2 ] = rs . 112.36 effective rate = (112.36 - 100) = 12.36 % answer : c"

Explain the solution process for the given equation.

in a school 10 % of the boys are same in number as 1 / 4 th of the girls . what is the ratio of boys to the girls in the school ? a) 5 : 2 , b) 2 : 3 , c) 1 : 4 , d) 3 : 7 , e) 2 : 5

"10 % of b = 1 / 4 g 10 b / 100 = g / 4 b = 5 g / 2 b / g = 5 / 2 b : g = 5 : 2 answer is a"

Break down and solve the equation step by step.

two tests had the same maximum mark . the pass percentages in the first and the second test were 40 % and 45 % respectively . a candidate scored 216 marks in the second test and failed by 75 marks in that test . find the pass mark in the first test ? a) 768 , b) 243 , c) 246 , d) 288 , e) 224

let the maximum mark in each test be m . the candidate failed by 75 marks in the second test . pass mark in the second test = 249 + 75 = 324 45 / 100 m = 324 pass mark in the first test = 40 / 100 m = 40 / 45 * 324 = 288 . answer : d

Solve the problem systematically, showing each step clearly.

10 play kabadi , 40 play kho kho only , 5 play both gmaes . then how many in total ? a) 30 , b) 35 , c) 38 , d) 40 , e) 50

"10 play kabadi = > n (a) = 10 , 5 play both gmaes . = > n (anb) = 5 40 play kho kho only , = > n (b) = n (b only) + n (anb) = 40 + 5 = 45 total = > n (aub) = n (a) + n (b) - n (anb) = 10 + 45 - 5 = 50 answer : e"

Provide a detailed solution to the given mathematical problem.

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) 76 , c) 98 , d) 27 , e) 24

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

Solve the problem systematically, showing each step clearly.

if a person walks at 14 km / hr instead of 10 km / hr , he would have walked 20 km more . the actual distance travelled by him is a) 50 km , b) 56 km , c) 70 km , d) 80 km , e) 90 km

let the actual distance travelled be x km . then x / 10 = (x + 20) / 14 - - > 14 x = 10 x + 200 - - > 4 x = 200 - - > x = 50 km . answer : a .

Explain the solution process for the given equation.

a retailer buys 40 pens at the market price of 36 pens from a wholesaler , if he sells these pens giving a discount of 1 % , what is the profit % ? a) 5 , b) 10 , c) 15 , d) 20 , e) none of them

"let the market price of each pen be rs 1 then , c . p of 40 pens = rs 36 s . p of 40 pens = 99 % of rs 40 = rs 39.60 profit % = ((3.60 * 100) / 36) % = 10 % answer is b ."

Provide a detailed solution to the given mathematical problem.

a bullet train 130 m long is running with a speed of 50 kmph . in what time will it pass a man who is running at 10 kmph in the direction opposite to that in which the bullet train is going ? a) 23 sec , b) 13 sec , c) 12 sec , d) 11 sec , e) 16 sec

"b 13 sec speed of the bullet train relative to man = (50 + 10) kmph = 60 * 5 / 18 m / sec = 30 / 3 m / sec . time taken by the bullet train to cross the man = time taken by it to cover 130 m at (30 / 3) m / sec = (130 * 3 / 30) sec = 13 sec"

Solve the problem systematically, showing each step clearly.

what is the sum of all even numbers from 1 to 501 ? a) 122821 , b) 281228 , c) 281199 , d) 62750 , e) 128111

"explanation : 500 / 2 = 250 250 * 251 = 62750 answer : d"

Explain the solution process for the given equation.

a straight line in the xy - plane has y - intercept of 12 and a slope of 4 / 7 . given the x - coordinate of the point is 49 , find it' s y - coordinate . a) 27 , b) 16 , c) 29.7 , d) 28 , e) 18

"eq of line = y = mx + c m = 4 / 7 c = 12 x = 49 substitute the givens : y = (4 / 7 * 49) + 12 , y = 16 + 12 = 28 correct option is d"

Explain the solution process for the given equation.

if the ratio of a to b is 7 to 3 and the ratio of b to c is 1 to 5 , what is the ratio of a to c ? a) 4 / 15 , b) 1 / 3 , c) 2 / 5 , d) 4 / 5 , e) 7 / 15

"a : b = 7 : 3 - - 1 b : c = 1 : 5 = > b : c = 3 : 15 - - 2 from 1 and 2 , we get a : c = 7 : 15 answer e"

Explain the solution process for the given equation.

a sum fetched a total simple interest of rs . 4016.25 at the rate of 5 % p . a . in 5 years . what is the sum ? a) 12021 , b) 14520 , c) 16065 , d) 18925 , e) 12887

"principal = (100 * 4016.25) / (5 * 5) = rs . 16065 . answer : c"

Explain the solution process for the given equation.

find the greatest number that will divide 256 , 193 and 210 leaving 6 , 18 and 100 as remainders respectively a) 19 , b) 25 , c) 23 , d) 20 , e) 26

"explanation : answer will be hcf of (256 - 6 , 193 - 18 , 210 - 10) hcf of (250 , 175 , 200) = 25 option b"

Provide a detailed solution to the given mathematical problem.

there were totally 100 men . 83 are married . 75 have t . v , 85 have radio , 70 have a . c . how many men have t . v , radio , a . c and also married ? a) 11 , b) 12 , c) 13 , d) 14 , e) 15

100 - (100 - 83) - (100 - 75) - (100 - 85) - (100 - 70) = 100 - 17 - 25 - 15 - 30 = 100 - 87 = 13 answer : c

Solve the problem systematically, showing each step clearly.

if a rectangular billboard has an area of 91 square feet and a perimeter of 40 feet , what is the length of each of the shorter sides ? a) 4 , b) 7 , c) 8 , d) 13 , e) 26

"this question can be solved algebraically or by testing the answers . we' re told that a rectangle has an area of 91 and a perimeter of 40 . we' re asked for the length of one of the shorter sides of the rectangle . since the answers are all integers , and the area is 91 , the shorter side will almost certainly be less than 10 (since 10 x 10 = 100 , but we' re not dealing with a square) . let' s test answer b : 7 if . . . the shorter side = 7 . . . the area = 91 . . . . 91 / 7 = 13 = the longer side perimeter = 7 + 7 + 13 + 13 = 40 b"

Break down and solve the equation step by step.

6) a marketing firm determined that , of 160 households surveyed , 80 used neither brand a nor brand b soap . 60 used only brand a soap and for every household that used both brands of soap , 3 used only brand b soap . how many of the 200 household surveyed used both brands of soap ? a) a) 15 , b) b) 20 , c) c) 30 , d) d) 40 , e) e) 5

"80 = at least one of soap a or b both brands = x brand b = 3 x = > 60 + x + 3 x = 120 = > 4 x = 20 = > x = 5 answer - e"

Break down and solve the equation step by step.

what is the value of (p + q) / (p - q) if p / q is 7 ? a) 4 / 3 , b) 2 / 3 , c) 2 / 6 , d) 7 / 8 , e) 8 / 7

"(p + q) / (p - q) = [ (p / q) + 1 ] / [ (p / q) - 1 ] = (7 + 1) / (7 - 1) = 8 / 6 = 4 / 3 answer : a"

Solve the problem systematically, showing each step clearly.

how many different positive integers exist between 10 ^ 6 and 10 ^ 7 , the sum of whose digits is equal to 2 ? a) 5 , b) 6 , c) 7 , d) 8 , e) 18

"total cases = > 1000000 = > 6 cases for 1 being present at any of the 6 zero and last case 2000000 hence & cases answer : c"

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

a thief is spotted by a policeman from a distance of 225 meters . when the policeman starts the chase , the thief also starts running . if the speed of the thief be 8 km / hr and that of the policeman 10 km / hr , how far the thief will have run before he is overtaken ? a) 350 m , b) 200 m , c) 400 m , d) 900 m , e) none of them

"relative speed of the policeman = (10 - 8) km / hr = 2 km / hr . time taken by police man to cover (225 m / 1000) x 1 / 2 hr = 9 / 80 hr . in 9 / 80 hrs , the thief covers a distance of 8 x 9 / 80 km = 9 / 10 km = 900 m answer is d ."

Solve the problem systematically, showing each step clearly.

a shopkeeper sells 450 metres of cloth for rs . 18000 at a loss of rs . 5 per metre . find his cost price for one metre of cloth ? a) 12 , b) 27 , c) 29 , d) 45 , e) 21

sp per metre = 18000 / 450 = rs . 40 loss per metre = rs . 5 cp per metre = 40 + 5 = rs . 45 . answer : d

Solve the problem systematically, showing each step clearly.

if one and a half women , drink one and a half tea in one and a half minutes . how many tea can 9 women drink in 3 minutes ? a) 11 , b) 8 , c) 37 , d) 15 , e) 18

e 18 explanation : more minutes implies more tea . more women implies more tea . the time become twice (3 minutes / 1.5 minutes) . women become six times (9 / 1.5) . number of tea = 2 * 6 * original (1.5)

Solve the problem systematically, showing each step clearly.

a car traveled 462 miles per tankful of gasoline on the highway and 336 miles per tankful of gasoline in the city . if the car traveled 15 fewer miles per gallon in the city than on the highway , how many miles per gallon did the car travel in the city ? a) 14 , b) 16 , c) 21 , d) 22 , e) 40

"i treat such problems as work ones . work = rate * time mileage (m) = rate (mpg) * gallons (g) x gallons is a full tank { 462 = rx { 336 = (r - 15) x solve for r , r = 55 55 - 15 = 40 mpg e"

Explain the solution process for the given equation.

what is x if x + 2 y = 10 and y = 4 ? a) a) 10 , b) b) 8 , c) c) 6 , d) d) 4 , e) e) 2

"x = 10 - 2 y x = 10 - 8 . x = 2 answer : e"

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

after decreasing 24 % in the price of an article costs rs . 320 . find the actual cost of an article ? a) 400 , b) 520 , c) 651 , d) 525 , e) 421

"cp * (76 / 100) = 320 cp = 4.21 * 100 = > cp = 421 answer : e"

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

a business finds that their ratio of spending to income is 5 to 9 . their total profit is $ 48000 . what is their total income ? a) $ 90000 , b) $ 108000 , c) $ 119000 , d) $ 126000 , e) $ 140000

let x equal the amount of spending , and y equal the amount of income . we know that x = 5 / 9 y we also know that x = y - 48000 therefore we can simplify to : y - 48000 = 5 / 9 y we isolate y , and get : 4 / 9 y = 48000 and to solve for y , we get : y = 48000 answer : b

Solve the problem systematically, showing each step clearly.

when x is multiplied by 3 , the result is 26 more than the result of subtracting x from 62 . what is the value of x ? a) - 4 , b) - 2 , c) 11 , d) 13 , e) 22

"the equation that can be formed is : 3 x - 26 = 62 - x or , 4 x = 88 or , x = 22 . answer : e"

Break down and solve the equation step by step.

two trains start from p and q respectively and travel towards each other at a speed of 50 km / hr and 40 km / hr respectively . by the time they meet , the first train has travelled 1 ookm more than the second . the distance between p and q is a) 500 km , b) 600 km , c) 700 km , d) 900 km , e) 400 km

"explanation : at the time of meeting , let the distance travelled by the second train be x km then , distance covered by the first train is (x + 100) km = > x / 40 = (x + 100) / 50 = > 50 x = 40 x + 4000 = > x = 400 km so , distance between p and q = (x + x + 100) km = 900 km answer : option d"

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

the effective annual rate of interest corresponding to a nominal rate of 6 % per annum payable half yearly is a) 6.06 % , b) 6.07 % , c) 6.08 % , d) 6.09 % , e) none

"solution amount of rs . 100 for 1 year when compounded half yearly = rs . [ 100 x (1 + 3 / 100) 2 ] = rs . 106.09 % . effective rate = (106.09 - 100) % = rs . 6.09 % . answer d"

Solve the problem systematically, showing each step clearly.

if rs . 480 amount to rs . 540 in 4 years , what will it amount to in 6 years at the same rate % per annum ? a) s . 575 , b) s . 595 , c) s . 590 , d) s . 570 , e) s . 585

"80 = (480 * 4 * r) / 100 r = 3.125 % i = (480 * 6 * 3.125) / 100 = 90 480 + 90 = 570 answer : d"

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

a sum of money lent out at s . i . amounts to a total of $ 590 after 2 years and to $ 815 after a further period of 5 years . what was the initial sum of money that was invested ? a) $ 490 , b) $ 500 , c) $ 510 , d) $ 520 , e) $ 530

"s . i for 5 years = $ 815 - $ 590 = $ 225 the s . i . is $ 45 / year s . i . for 2 years = $ 90 principal = $ 590 - $ 90 = $ 500 the answer is b ."

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

what least number should be added to 1052 , so that the sum is completely divisible by 23 a) a) 4 , b) b) 1 , c) c) 6 , d) d) 3 , e) e) 5

"explanation : (1052 / 23) gives remainder 17 17 + 6 = 23 , so we need to add 6 answer : option c"

Provide a detailed solution to the given mathematical problem.

kavi had a stock of 600 bags in his bookshop . he sold 25 on monday , 70 on tuesday , 100 on wednesday , 110 on thursday and 145 on friday . what percentage of the bags were not sold ? a) 10 % , b) 25 % , c) 64 % , d) 42 % , e) 17 %

let n be the total number of bags sold . hence n = 25 + 70 + 100 + 110 + 145 = 450 let m be the bags not sold m = 600 - n = 600 - 450 = 150 percentage bags not sold / total number of bags = 150 / 600 = 0.25 = 25 % correct answer b

Explain the solution process for the given equation.

4 mat - weavers can weave 4 mats in 4 days . at the same rate , how many mats would be woven by 16 mat - weavers in 16 days ? a) 25 , b) 19 , c) 39 , d) 61 , e) 64

"let the required number of bottles be x . more weavers , more mats (direct proportion) more days , more mats (direct proportion) wavers 4 : 16 : : 4 : x days 4 : 16 4 * 4 * x = 16 * 16 * 4 x = (16 * 16 * 4) / (4 x 4) x = 64 . answer is e ."

Break down and solve the equation step by step.

on selling 17 balls at rs . 720 , there is a loss equal to the cost price of 5 balls . the cost price of a ball is a) 20 , b) 30 , c) 40 , d) 50 , e) 60

"(c . p . of 17 balls) - (s . p . of 17 balls) = (c . p . of 5 balls) c . p . of 12 balls = s . p . of 17 balls = rs . 720 . c . p . of 1 ball = rs . 720 / 12 = rs . 60 . answer : e"

Provide a detailed solution to the given mathematical problem.

cereal a is 10 % sugar by weight , whereas healthier but less delicious cereal b is 2 % sugar by weight . to make a delicious and healthy mixture that is 5 % sugar , what should be the ratio of cereal a to cereal b , by weight ? a) 2 : 5 , b) 3 : 5 , c) 4 : 7 , d) 2 : 6 , e) 1 : 4

"2 % is 3 % - points below 5 % and 10 % is 5 % - points above 5 % . the ratio of a : b should be 3 : 5 . the answer is b ."

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

square abcd is the base of the cube while square efgh is the cube' s top face such that point e is above point a , point f is above point b etc . what is the distance between the midpoint of edge ab and the midpoint of edge eh if the area of square abcd is 2 ? ['a) 1 / sqrt 2','b) 1','c) sqrt 2','d) sqrt 3','e) 2 sqrt 3']

distance from mid point of ab to ad = sqrt [ (1 / sqrt 2) ^ 2 + (1 / sqrt 2) ^ 2 ] = 1 the distance between the midpoint of edge ab and the midpoint of edge eh = sqrt [ 1 ^ 2 + (sqrt 2) ^ 2 ] = sqrt 3 . answer : d

Explain the solution process for the given equation.

what is the value of n if the sum of the consecutive odd integers from 1 to n equals 169 ? a) 47 , b) 25 , c) 37 , d) 33 , e) 29

# of terms = (n - 1 / 2) + 1 { (last term - first term) / 2 + 1 | sum = (1 + n) / 2 * # of terms = (n + 1) ^ 2 / 4 = 169 n + 1 = 13 * 2 n + 1 = 26 n = 25 . answer : b

Break down and solve the equation step by step.

a small college reduced its faculty by approximately 23 percent to 195 professors . what was the original number of faculty members ? a) 253 , b) 208 , c) 224 , d) 254 , e) 302

"if x is the original number of faculty members , then after 23 % reduction in faculty members number is . 77 x but we are given . 77 x = 195 x = 253 so the original number of faculty members is 253 correct answer - a"

Solve the problem systematically, showing each step clearly.

if 3 cats can kill 3 rats in 3 minutes , how long will it take 100 cats to kill 100 rats ? a) 1 minutes , b) 2 minutes , c) 3 minutes , d) 4 minutes , e) 5 minutes

"it will take 3 minutes for 100 cats to kill 100 rats . 1 cat can kill 1 rat in 3 minutes , so 100 cats can kill 100 rats in 3 minutes answer c"

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

pipe a can fill a tank in 6 hours . due to a leak at the bottom , it takes 8 hours for the pipe a to fill the tank . in what time can the leak alone empty the full tank ? a) 13 , b) 24 , c) 18 , d) 19 , e) 12

"let the leak can empty the full tank in x hours 1 / 6 - 1 / x = 1 / 8 = > 1 / x = 1 / 6 - 1 / 8 = (4 - 3) / 24 = 1 / 24 = > x = 24 . answer : b"

Solve the problem systematically, showing each step clearly.

{ - 10 , - 6 , - 5 , - 4 , - 2.5 , - 1 , 0 , 2.5 , 4 , 6 , 7 , 10 } a number is to be selected at random from the set above . what is the probability that the number will be a solution to the equation (x 3) (x + 14) (2 x + 5) = 0 ? a) 1 / 12 , b) 1 / 6 , c) 1 / 4 , d) 1 / 3 , e) 1 / 2

x = - 4 prob = 1 / 12 answer - a

Explain the solution process for the given equation.

two pipes a and b can fill a tank in 4 and 5 hours respectively . if they are turned up alternately for one hour each , the time taken to fill the tank is ? a) 4 hrs 29 min , b) 4 hrs 24 min , c) 8 hrs 24 min , d) 4 hrs 44 min , e) 5 hrs 24 min

1 / 4 + 1 / 5 = 9 / 20 20 / 9 = 2 2 / 9 9 / 20 * 2 = 9 / 10 - - - - 4 hours wr = 1 - 9 / 10 = 1 / 10 1 h - - - - 1 / 4 ? - - - - - 1 / 10 2 / 5 * 60 = 24 = 4 hrs 24 min answer : b

Solve the problem systematically, showing each step clearly.

a diagonal of a polygon is an segment between two non - adjacent vertices of the polygon . how many diagonals does a regular 30 - sided polygon have ? a) 875 , b) 405 , c) 1425 , d) 2025 , e) 2500

"there' s a direct formula for this . number of diagonals in a regular polygon = [ n * (n - 3) ] / 2 , n = number of sides of the regular polygon . here , n = 30 . plugging it in , we get 405 diagonals ! answer (b) ."

Solve the problem systematically, showing each step clearly.

a corporation that had $ 10 billion in profits for the year paid out $ 500 million in employee benefits . approximately what percent of the profits were the employee benefits ? (note : 1 billion = 10 ^ 9) a) 50 % , b) 20 % , c) 5 % , d) 2 % , e) 0.2 %

"required answer = [ employee benefit / profit ] * 100 = [ (500 million) / (10 billion) ] * 100 = [ (500 * 10 ^ 6) / (10 * 10 ^ 9) ] * 100 = (50 / 1000) * 100 = 5 % so answer is (c)"

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

to produce an annual income of rs . 1200 from a 12 % stock at 90 , the amount of stock needed is : a) rs . 10,000 , b) rs . 10,800 , c) rs . 14,400 , d) rs . 16,000 , e) none of these

solution for an income of rs . 12 , stock needed = rs . 100 . for an income of rs . 1200 , stock needed = rs . 100 / 12 x 1200 = 10000 answer a

Solve the problem systematically, showing each step clearly.

a cubical block of metal weighs 4 pounds . how much will another cube of the same metal weigh if its sides are twice as long ? a) 48 , b) 32 , c) 24 , d) 18 , e) 12

"for example our cube have a side 1 meter , so we have 1 cubical meter in this cube and this cubical meter weigth 4 pounds if we take cube with side 2 meters we will have 8 cubical meters in this cube 8 meters * 4 pounds = 32 pounds so answer is b and similar but more theoretical approach : if we have sides a and b than they have equal ration with their areas : a / b = a ^ 2 / b ^ 2 and they have equal ration with their volumes : a / b = a ^ 3 / b ^ 3 we have two sides 1 / 2 so their volume will be in ratio 1 / 8 weight of one cube * volume of another cube 4 * 8 = 32 so answer is b"

Explain the solution process for the given equation.

a boat running downstream covers a distance of 12 km in 2 hours while for covering the same distance upstream , it takes 3 hours . what is the speed of the boat in still water ? a) 8 kmph , b) 5 kmph , c) 4 kmph , d) 6 kmph , e) 7 kmph

"speed downstream = 12 / 2 = 6 kmph speed upstream = 12 / 3 = 4 kmph speed of the boat in still water = 6 + 4 / 2 = 5 kmph answer : b"

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

a student scored an average of 75 marks in 3 subjects : physics , chemistry and mathematics . if the average marks in physics and mathematics is 90 and that in physics and chemistry is 70 , what are the marks in physics ? a) 86 , b) 16 , c) 76 , d) 95 , e) 26

"given m + p + c = 75 * 3 = 225 - - - (1) m + p = 90 * 2 = 180 - - - (2) p + c = 70 * 2 = 140 - - - (3) where m , p and c are marks obtained by the student in mathematics , physics and chemistry . p = (2) + (3) - (1) = 180 + 140 - 225 = 95 answer : d"

Explain the solution process for the given equation.

in a certain pond , 30 fish were caught , tagged , and returned to the pond . a few days later , 50 fish were caught again , of which 2 were found to have been tagged . if the percent of tagged fish in the second catch approximates the percent of tagged fish in the pond , what is the approximate number of fish in the pond ? a) 750 , b) 625 , c) 1,250 , d) 2,500 , e) 10,000

"this is a rather straight forward ratio problem . 1 . 30 fish tagged 2 . 2 out of the 50 fish caught were tagged thus 2 / 50 2 / 50 = 30 / x thus , x = 750 think of the analogy : 2 fish is to 50 fish as 50 fish is to . . . ? you' ve tagged 50 fish and you need to find what that comprises as a percentage of the total fish population - we have that information with the ratio of the second catch . a"

Solve the problem systematically, showing each step clearly.

if 1 / 2 of the air in a tank is removed with each stroke of a vacuum pump , what fraction of the original amount of air has been removed after 1 strokes ? a) 1 / 2 , b) 7 / 8 , c) 1 / 4 , d) 1 / 8 , e) 1 / 16

"left after 1 st stroke = 1 / 2 so removed = 1 - 1 / 2 = 1 / 2"

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

if money is invested at r percent interest , compounded annually , the amount of the investment will double in approximately 50 / r years . if luke' s parents invested $ 13,500 in a long term bond that pays 12 percent interest compounded annually , what will be the approximate total amount of the investment 12 years later , when luke is ready for college ? a) 62,000 , b) 85,500 , c) 95,500 , d) 100,500 , e) 100,000

answer equals d in 48 years . i thought by 50 th year it would reach 100,500 . options should have been separated more widely for clarity .

Provide a detailed solution to the given mathematical problem.

there are two inlets and one outlet to a cistern . one of the inlets takes 4 hours to fill up the cistern and the other inlet takes twice as much time to fill up the same cistern . both of the inlets are turned on at 8 : 30 am with the cistern completely empty , and at 10 : 30 am , the outlet is turned on and it takes 1 more hour to fill the cistern completely . how much time does the outlet working alone takes to empty the cistern when the cistern is full ? a) 2 hours , b) 2.5 hours , c) 3 hours , d) 3.5 hours , e) 8 hours

"the combined inflow rate of the two inlets is 1 / 4 + 1 / 8 = 3 / 8 cistern / hour . thus , working together , it takes 8 / 3 hours (time is reciprocal of rate) to fill the cistern . from 8 : 30 am to 10 : 30 am , so in 2 hours , the inlet pipes will fill (time) * (rate) = 2 * 3 / 8 = 3 / 4 th of the cistern . then the outlet is turned on and the remaining 1 / 4 th of the cistern is filled in 1 hour . letting x to be the rate of the outlet , we would have : 3 / 8 - x = 1 / 4 - - > x = 1 / 8 cistern / hour , which means that it takes 8 hours the outlet working alone to empty the cistern . answer : e ."

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

40 men shake hands with each other . maximum no of handshakes without cyclic handshakes . a) 780 , b) 200 , c) 210 , d) 220 , e) 230

"or , if there are n persons then no . of shakehands = nc 2 = 40 c 2 = 780 answer : a"

Explain the solution process for the given equation.

if the volume of two cubes are in the ratio 125 : 1 , the ratio of their edges is : a) 3 : 1 , b) 5 : 1 , c) 3 : 5 , d) 3 : 7 , e) none of these

"explanation : let the edges be a and b of two cubes , then a 3 / b 3 = 125 / 1 = > (a / b) 3 = (5 / 1) 3 a / b = 5 / 1 = > a : b = 5 : 1 option b"

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

in x game of billiards , x can give y 20 points in 60 and he can give z 50 points in 60 . how many points can y give z in x game of 100 ? a) 30 , b) 20 , c) 25 , d) 40 , e) 75

"x scores 60 while y score 40 and z scores 10 . the number of points that z scores when y scores 100 = (100 * 50) / 40 = 25 . in x game of 100 points , y gives (100 - 25) = 75 points to c . e"

Break down and solve the equation step by step.

the ratio of three numbers is 1 : 2 : 4 and the sum of their squares is 1701 . the sum of the numbers is ? a) a) 10 , b) b) 12 , c) c) 15 , d) d) 14 , e) e) 9

"let the numbers be x , 2 x , 4 x then , x ^ 2 + 4 x ^ 2 + 16 x ^ 2 = 1701 21 x ^ 2 = 1701 x ^ 2 = 81 x = 9 answer is e"

Provide a detailed solution to the given mathematical problem.

there is 60 % increase in an amount in 6 years at si . what will be the ci of rs . 16,000 after 3 years at the same rate ? a) 2372 , b) 2572 , c) 4542 , d) 2343 , e) 5296

"let p = rs . 100 . then , s . i . rs . 60 and t = 6 years . r = 100 x 60 = 10 % p . a . 100 x 6 now , p = rs . 16000 . t = 3 years and r = 10 % p . a . c . i . = rs . 16000 x 1 + 10 3 - 1 100 = rs . 16000 x 331 1000 = 5296 . e"

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

a prize of $ 600 is to be distributed among 15 winners , each of whom must be awarded at least $ 15 . if 2 / 5 of the prize will be distributed to 3 / 5 of the winners , what is the greatest possible individual award ? a) $ 200 , b) $ 220 , c) $ 285 , d) $ 300 , e) $ 360

"total value of the prize = $ 600 number of people = 15 2 / 5 of 600 (= $ 240) should be distributed among 3 / 5 of 15 (= 9 people) with each getting $ 15 each . remaining money = 600 - 240 = $ 360 . now in order to' maximize' 1 prize , we need to minimise the others and we have been given that each should get $ 15 . thus , minimising the remaining 5 people (= 15 - 9 - 1 .' - 1' to exclude 1 that needs to be maximised) = 5 * 15 = 75 . thus the maximum award can be = 360 - 75 = $ 285 , hence c is the correct answer ."

Provide a detailed solution to the given mathematical problem.

a trader cheats both his supplier and customer by using faulty weights . when he buys from the supplier , he takes 10 % more than the indicated weight . when he sells to his customer , he gives the customer a weight such that 30 % of that is added to the weight , the weight claimed by the trader is obtained . if he charges the cost price of the weight that he claims , find his profit percentage . a) 20 % , b) 43 % , c) 99.33 % , d) 75.109 % , e) 47 %

anyways , one can infer that he' steals' 10 % from suppliers and then charges 30 % extra to customers so basically 1.1 * 1.3 = 1.43 given that 1 is start point , we get 43 % more hence answer is b

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

walking 6 / 7 of his usual rate , a boy reaches his school 4 min early . find his usual time to reach the school ? a) 28 m , b) 20 m , c) 22 m , d) 23 m , e) 24 m

speed ratio = 1 : 6 / 7 = 7 : 6 time ratio = 6 : 4 1 - - - - - - - - 6 4 - - - - - - - - - ? = > 24 m answer : e

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

machine p and machine q are each used to manufacture 330 sprockets . it takes machine p 10 hours longer to produce 330 sprockets than machine q . machine q produces 10 % more sprockets per hour than machine a . how many sprockets per hour does machine a produce ? a) 5 , b) 15 , c) 3 , d) 95 , e) 125

"p makes x sprockets per hour . then q makes 1.1 x sprockets per hour . 330 / x = 330 / 1.1 x + 10 1.1 (330) = 330 + 11 x 11 x = 33 x = 3 the answer is c ."

Solve the problem systematically, showing each step clearly.

0.0003 x 0.3 = ? a) 0.0009 , b) 9.0 e - 05 , c) 0.09 , d) 0.009 , e) 9.0

"3 x 3 = 9 . sum of decimal places = 5 0.0003 x 0.3 = 0.00009 answer : option b"

Solve the problem systematically, showing each step clearly.

when 2 / 9 of the votes on a certain resolution have been counted , 3 / 4 of those counted are in favor of the resolution . what fraction e of the remaining votes must be against the resolution so that the total count will result in a vote of 2 to 1 against the resolution ? a) 11 / 14 , b) 13 / 18 , c) 4 / 7 , d) 3 / 7 , e) 3 / 14

if we use variable for total votes there will be too many fractions to manipulate with , so pick some smart # : let set total # of votes is 18 . 2 / 9 of the votes on a certain resolution have been counted - - > 4 counted and 18 - 4 = 14 votes left to be counted ; 3 / 4 of those counted are in favor of the resolution - - > 3 in favor and 1 against ; ratio of those who voted against to those who voted for to be 2 to 1 there should be total of 18 * 2 / 3 = 12 people who voted against , so in the remaining 14 votes there should be 12 - 1 = 11 people who voted against . thus e = 11 / 14 of the remaining votes must be against . answer : a .

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

raman mixed 54 kg of butter at rs . 150 per kg with 36 kg butter at the rate of rs . 125 per kg . at what price per kg should he sell the mixture to make a profit of 40 % in the transaction ? a) 129 , b) 287 , c) 196 , d) 188 , e) 112

"explanation : cp per kg of mixture = [ 54 (150) + 36 (125) ] / (54 + 36) = rs . 140 sp = cp [ (100 + profit %) / 100 ] = 140 * [ (100 + 40) / 100 ] = rs . 196 . answer : c"

Solve the problem systematically, showing each step clearly.

a man whose bowling average is 12.4 , takes 3 wickets for 26 runs and there by decreases his average by 0.4 . the number of wickets taken by him before his last match is ? a) 23 , b) 24 , c) 25 , d) 26 , e) 27

"12.4 * x + 26 = (3 + x) 12 solve equation x = 25 answer : c"

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if 80 % of a number is equal to four - third of another number , what is the ratio of first number to the second number ? a) 5 : 3 , b) 1 : 4 , c) 3 : 7 , d) 6 : 11 , e) 2 : 3

"let 80 % of a = 4 / 3 b then 80 a / 100 = 4 b / 3 4 a / 5 = 4 b / 3 a / b = 5 / 3 a : b = 5 : 3 answer is a"

Break down and solve the equation step by step.

a searchlight on top of the watchtower makes 2 revolutions per minute . what is the probability that a man appearing near the tower will stay in the dark for at least 5 seconds ? a) 5 / 6 , b) 5 / 12 , c) 7 / 12 , d) 11 / 12 , e) 11 / 15

"the searchlight completes one revolution every 30 seconds . the probability that the man' s area will be lit up is 5 / 30 = 1 / 6 . the probability that he will stay in the dark is 1 - 1 / 6 = 5 / 6 the answer is a ."

Break down and solve the equation step by step.

a certain no . when divided by 110 leaves a remainder 25 , what is the remainder if the same no . be divided by 15 ? a) 2 , b) 4 , c) 7 , d) 8 , e) 9

"explanation : 110 + 25 = 135 / 15 = 9 (remainder) e"

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when a = x + (3 / x) and b = x - (3 / x) , (2 ^ a ^ 2) / (2 ^ b ^ 2) = ? a) 2 , b) 4 , c) 8 , d) 16 , e) 4096

"- - > (2 ^ a ^ 2) / (2 ^ b ^ 2) = { (2) ^ (a ^ 2 - b ^ 2) } = 2 ^ (a - b) (a + b) . since a - b = 6 / x and a + b = 2 x , 2 ^ (a - b) (a + b) = 2 ^ (6 / x) (2 x) = 2 ^ 12 = 4096 therefore , the answer is e"

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a cube of edge 15 cm is immersed completely in a rectangular vessel containing water . if the dimensions of the base of vessel are 20 cm * 15 cm , find the rise in water level ? a) 6 cm , b) 8.25 cm , c) 11.25 cm , d) 15.12 cm , e) 20.62 cm

"increase in volume = volume of the cube = 15 * 15 * 15 cm ^ 3 rise in water level = volume / area = 15 * 15 * 15 / 20 * 15 = 11.25 cm answer is c"

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what is the smallest positive integer x such that (x + 1) ^ 2 is divisible by 28 , 98 , 242 , and 308 ? a) 153 , b) 308 , c) 121 , d) 96 , e) 511

"28 = 2 * 2 * 7 98 = 2 * 7 * 7 242 = 2 * 11 * 11 308 = 2 * 2 * 7 * 11 so (x + 1) ^ 2 = 2 * 2 * 7 * 7 * 11 * 11 , which means (x + 1) = 2 * 7 * 11 = 154 , which means x = 153 , which is option a"

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what is the area of a triangle with the following vertices l (2 , 3) , m (5 , 1) , and n (3 , 5) ? ['a) 3','b) 4','c) 5','d) 6','e) 7']

there is a direct formula to calculate the are of a triangle based on coordinates of its vertices and one could use it to solve this problem . though if you make a diagram minimum simple calculations will be needed : answer : b .

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pipe a can fill a tank in 32 minutes . pipe b can fill the same tank 5 times faster than pipe a . if both the pipes are connected to the tank so that they fill the tank simultaneously , how long will it take for the empty tank to overflow ? a) 4 minutes , b) 2 / 7 minutes , c) 32 / 7 minutes , d) 224 / 7 minutes , e) 28 minutes

"pipe a can fill a tank in 32 minutes - - > the rate of a = 1 / 32 tank / minute ; pipe b can fill the same tank 5 times fasterthan pipe a - - > the rate of b = 1 / 32 + 5 / 32 = 6 / 32 tank / minute . their combined rate = 1 / 32 + 6 / 32 = 7 / 32 tank / minute ; therefore , a and b can fill the tank in 32 / 7 minutes . answer : c"

Explain the solution process for the given equation.

find number which is 70 % less than 60 . a) 18 , b) 22 , c) 28 , d) 26 , e) 98

"explanation : 70 % less is 30 % of the given number therefore , 30 % of 60 is 18 . answer : a"

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x can give y 100 meters start and z 200 meters start in x kilometer race . how much start can y give z in x kilometer race ? a) 112.2 m , b) 216.3 m , c) 111.12 m , d) 213.27 m , e) 222.40 m

c 111.12 m x runs 1000 m while y runs 900 m and z runs 800 m . the number of meters that z runs when y runs 1000 m , = (1000 * 800) / 900 = 8000 / 9 = 888.88 m . y can give z = 1000 - 888.88 = 111.12 m .

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if 0.75 of a number is equal to 0.025 of another number , the ratio of the numbers is : a) 2 : 3 , b) 3 : 4 , c) 1 : 30 , d) 20 : 3 , e) 30 : 7

"0.75 a = 0.025 b - > a / b = 0.025 / 0.75 = 25 / 750 = 1 / 30 : . a : b = 1 : 30 answer : c"

Explain the solution process for the given equation.

how many different values of positive integer x , for which | x + 5 | < x , are there ? a) 0 , b) 2 , c) 3 , d) 8 , e) 16

"answer e i opted to put the random value option . i used 0 , 5 , - 5 and the the extreme of 40 and - 40 . . i was able to solve it in 1 : 09 e"

Explain the solution process for the given equation.

a tank is filled by 3 pipes a , b , c in 10 hours . pipe c is twice as fast as b and b is twice as fast as a . how much will pipe a alone take to fill the tank ? a) 20 hr , b) 52 hr , c) 70 hr , d) 66 hr , e) 48 hr

"suppose pipe a alone take x hours to fill the tank then pipe b and c will take x / 2 and x / 4 hours respectively to fill the tank . 1 / x + 2 / x + 4 / x = 1 / 10 7 / x = 1 / 10 x = 70 hours answer is c"

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due to construction , the speed limit along an 10 - mile section of highway is reduced from 55 miles per hour to 25 miles per hour . approximately how many minutes more will it take to travel along this section of highway at the new speed limit than it would have taken at the old speed limit ? a) a) 6.24 , b) b) 13.1 , c) c) 10 , d) d) 15 , e) e) 24

"old time in minutes to cross 10 miles stretch = 10 * 60 / 55 = 10 * 12 / 11 = 10.9 new time in minutes to cross 10 miles stretch = 10 * 60 / 25 = 10 * 12 / 5 = 24 time difference = 13.1 ans : b"

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two vessels p and q contain 62.5 % and 87.5 % of alcohol respectively . if 12 litres from vessel p is mixed with 4 litres from vessel q , the ratio of alcohol and water in the resulting mixture is ? a) 19 : 2 , b) 18 : 3 , c) 11 : 1 , d) 11 : 5 , e) 11 : 4

"quantity of alcohol in vessel p = 62.5 / 100 * 2 = 15 / 2 litres quantity of alcohol in vessel q = 87.5 / 100 * 4 = 7 / 2 litres quantity of alcohol in the mixture formed = 15 / 2 + 7 / 2 = 11 litres as 16 litres of mixture is formed , ratio of alcohol and water in the mixture formed = 11 : 5 answer : d"

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if p (a) = 4 / 5 and p (b) = 2 / 5 , find p (a n b) if a and b are independent events . a) 8 / 25 , b) 1 / 25 , c) 2 / 25 , d) 3 / 14 , e) 25 / 8

"p (a n b) = p (a) . p (b) p (a n b) = 4 / 5 . 2 / 5 p (a n b) = 8 / 25 . a"

Explain the solution process for the given equation.

find the greatest number of 5 digits which is exactly divisible by 12 , 15 and 18 ? a) a) 13050 , b) b) 12960 , c) c) 10025 , d) d) 11080 , e) e) 12080

the largest five digit numbers are 13050 , 12960,12080 13050 is not divisible by 12 12960 is divisible by 18 , 12 and 15 answer : b

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solve the equation for x : 6 x - 87 + 3 x = 4 + 9 - x a) 12 , b) 5 , c) 10 , d) 9 , e) 3

"c 10 9 x + x = 13 + 87 10 x = 100 = > x = 10"

Solve the problem systematically, showing each step clearly.

the average (arithmetic mean) of 16 students first quiz scores in a difficult english class is 60.5 . when one student dropped the class , the average of the remaining scores increased to 64.0 . what is the quiz score of the student who dropped the class ? a) a) 8 , b) b) 25 , c) c) 40 , d) d) 55 , e) e) 70

"total score of 16 students is 16 * 60.50 = 968 total score of 15 students is 15 * 64 = 960 so , the score of the person who left is 8 (968 - 960) answer will be (a"

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how many integers between 0 and 1000 do not contain the integer 1 a) 727 , b) 728 , c) 758 , d) 748 , e) 792

from 1 to 99 = 19 integers having 1 from 101 to 199 = all are having 1 similarly 201 - 299 , 301 - 399 , 401 - 499 , . . . . . so on , will have 19 integers having 1 answer = 1000 - (19 * 9 + 99 + 2) / / 2 is added for 100 and 1000 = 728 answer : b

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if 8 people take an hour to complete a piece of work , then how long should 16 people will take to complete the same piece of work ? a) 15 min , b) 30 min , c) 26 min , d) 28 min , e) 18 min

if 8 people take an hour to complete a piece of work , then 16 people will take 8 * 60 / 16 = 30 mins to complete the same piece of work . answer : b

Solve the problem systematically, showing each step clearly.

if grapes are 90 % water and raisins are 17 % water , then how many kilograms did a quantity of raisins , which currently weighs 14 kilograms , weigh when all the raisins were grapes ? (assume that the only difference between their raisin - weight and their grape - weight is water that evaporated during their transformation .) a) 108.6 , b) 110.4 , c) 112.6 , d) 114.8 , e) 116.2

"let x be the original weight of the grapes . the weight of the grape pulp was 0.1 x . since the grape pulp is 83 % of the raisins , 0.1 x = 0.83 (14 kg) . then x = 8.3 * 14 = 116.2 kg . the answer is e ."

Solve the problem systematically, showing each step clearly.

what is the smallest integer that is multiple of 3 , 5,9 a) a) 70 , b) b) 45 , c) c) 200 , d) d) 280 , e) e) 140

"it is the lcm of 3 , 5 and 9 which is 45 . the answer is b ."

Solve the problem systematically, showing each step clearly.

mixture contains alcohol and water in the ratio 4 : 3 . if 8 liters of water is added to the mixture , the ratio becomes 4 : 5 . find the quantity of alcohol in the given mixture . a) 10 , b) 99 , c) 27 , d) 4 , e) 29

"let the quantity of alcohol and water be 4 x litres and 3 x litres respectively 32 x = 4 (3 x + 5) 20 x = 20 x = 1 quantity of alcohol = (4 x 1) litres = 4 litres . answer : d"

Provide a detailed solution to the given mathematical problem.

what is the units digit of (493) (915) (381) (756) (22) a) 0 , b) 1 , c) 4 , d) 5 , e) 6

"just multiply the digits in the units place for each term and you will get the answer . it should be 0 . you got a 5 as a unit digit and an even number term . so the multiplication of this will definitely yield a 0 . answer has to be 0 . i also tried it using the calculator and the answer is 1 . imo b ."

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the dimensions of a rectangular solid are 4 inches , 5 inches , and 9 inches . if a cube , a side of which is equal to one of the dimensions of the rectangular solid , is placed entirely within thespherejust large enough to hold the cube , what the ratio of the volume of the cube to the volume within thespherethat is not occupied by the cube ? a) 2 : 5 , b) 5 : 9 , c) 5 : 16 , d) 25 : 7 , e) 32 : 25

answer : b .

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a number a is squared and then multiplied by negative 6 . the result of this operation is equal to 3 times the sum of fourth times a and two . what is one possible value of a ? a) 2 , b) 3 , c) 4 , d) 1 , e) 5

- 6 * a ^ 2 = 3 (4 a + 2) a = - 1 or - 1 a = - 1 = b answer : d