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.	if a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster , he would have covered 70 more miles than he actually did . how many more miles would he have covered than he actually did if he had driven 1 hours longer and at an average rate of 10 miles per hour faster on that day ? a ) 100 , b ) 120 , c ) 130 , d ) 150 , e ) 160	"case 1 : let rate = r , time = t , and distance = d so d = rt case 2 : ( d + 70 ) = ( r + 5 ) ( t + 1 ) case 3 : ( d + x ) = ( r + 10 ) ( t + 1 ) x = 130 ans c"
Explain the solution process for the given equation.	what is the ratio g of the surface area of a cube to the surface area of a rectangular solid identical to the cube in all ways except that its length has been doubled ? ['a ) 1 / 4','b ) 3 / 8','c ) 1 / 2','d ) 3 / 5','e ) 2']	one side surface area of a cube = x * x = x ^ 2 total 6 sides = 6 x ^ 2 as for the rectangular , height ( h ) and width ( w ) are same as cube , x . only length = 2 x . l x h = 2 x * x = 2 x ^ 2 - - - - > 4 sides = 2 x ^ 2 * 4 = 8 x ^ 2 w * h = x * x = x ^ 2 - - - - - - > 2 sides = x ^ 2 * 2 = 2 x ^ 2 total 6 sides = 8 x ^ 2 + 2 x ^ 2 = 10 x ^ 2 ratio of cube area to rectangular area g = 6 x ^ 2 / 10 x ^ 2 - - - - > 6 / 10 - - - - > 3 / 5 ( d )
Work through the equation methodically, ensuring clarity at each step.	the minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time . how much does the clock gain or lose in 12 hours ? a ) 27.273 seconds , b ) 29.273 seconds , c ) 37.273 seconds , d ) 47.273 seconds , e ) 17.273 seconds	after 60 minute rotation the minute hand will make an angle of 30 degree with hour hand . after that let the time required to make an overtake is t minute in this time angle covered by minute hand = ( 360 / 60 ) t = 6 t similarly angle covered by hour hand = ( 30 / 60 ) t = t / 2 so for overtake 6 t = 30 + t / 2 or t = 60 / 11 min = 5 min and 5 / 11 min = 5 min and 300 / 11 sec = 5 min and 27.273 sec so clock loses 27.273 seconds answer : a
Provide a detailed solution to the given mathematical problem.	in a hostel , the number of students decreased by 8 % and the price of food increased by 20 % over the previous year . if each student consumes the same amount of food then by how much should the consumption of food be cut short by every student , so that the total cost of the food remains the same as that of the previous year ? a ) 19 % , b ) 15 % , c ) 25 % , d ) 40 % , e ) 9.4 %	"cost of food ( c ) = food consumed per student ( f ) * number of students ( n ) * price of food ( p ) originally , c = fnp when number of students decrease by 8 % , and the price of food increases by 20 % , c = f ( new ) * ( 0.92 n ) * ( 1.2 p ) = > f ( new ) = f / ( 0.92 * 1.2 ) = > f ( new ) = 0.906 f therefore the new cost of food must be 90.6 % of the old cost , or the cost of food must decrease by 9.4 % answer : e"
Explain the solution process for the given equation.	the value of a machine depreciates at 25 % per annum . if its present value is $ 1 , 50,000 , at what price should it be sold after two years such that a profit of $ 24,000 is made ? a ) $ 258375 , b ) $ 438375 , c ) $ 128375 , d ) $ 108375 , e ) $ 158375	"the value of the machine after two years = 0.75 * 0.75 * 1 , 50,000 = $ 84,375 sp such that a profit of $ 24,000 is made = 84,375 + 24,000 = $ 1 , 08,375 d"
Break down and solve the equation step by step.	a milk man has 15 liters of milk . if he mixes 5 liters of water , which is freely available , in 20 liters of pure milk . if the cost of pure milk is rs . 18 per liter , then the profit of the milkman , when he sells all the mixture at cost price is : a ) 20 % , b ) 25 % , c ) 33.33 % , d ) 18 % , e ) none of these	"explanation : when the water is freely available and all the water is sold at the price of the milk , then the water gives the profit on the cost of 20 liters of milk . therefore , profit percentage = 20 % . answer : a"
Solve the problem systematically, showing each step clearly.	how many cubes of 2 cm edge can be cut out of a cube of 24 cm edge a ) 1136 , b ) 1728 , c ) 1216 , d ) 1484 , e ) none of these	"explanation : number of cubes = ( 24 x 24 x 24 ) / ( 2 x 2 x 2 ) = 1728 answer : b"
Break down and solve the equation step by step.	how many integers from 10 to 180 , inclusive , are divisible by 3 but not divisible by 7 ? a ) 45 , b ) 49 , c ) 50 , d ) 52 , e ) 56	"we should find # of integers divisible by 3 but not by 3 * 7 = 21 . # of multiples of 21 in the range from 10 to 180 , inclusive is ( 168 - 21 ) / 21 + 1 = 8 ; 57 - 8 = 49 . answer : b ."
Break down and solve the equation step by step.	what is the compound interest paid on a sum of rs . 1500 for the period of 2 years at 10 % per annum . a ) 230 , b ) 175 , c ) 225 , d ) 315 , e ) none of these	"solution = interest % for 1 st year = 10 interest % for 2 nd year = 10 + 10 % of 10 = 10 + 10 * 10 / 100 = 11 total % of interest = 10 + 11 = 21 total interest = 21 % 1500 = 1500 * ( 21 / 100 ) = 315 answer d"
Provide a detailed solution to the given mathematical problem.	pipe a fills a tank of capacity 950 liters at the rate of 40 liters a minute . another pipe b fills the same tank at the rate of 30 liters a minute . a pipe at the bottom of the tank drains the tank at the rate of 20 liters a minute . if pipe a is kept open for a minute and then closed and pipe b is open for a minute and then closed and then pipe c is open for a minute and then closed and the cycle is repeated , when will the tank be full ? a ) 57 minutes , b ) 14 minutes , c ) 39 minutes , d ) 40 minutes 20 seconds , e ) none of these	in one cycle they fill 40 + 30 - 20 = 50 liters 950 = 50 * n = > n = 19 here n = number of cycles . total time = 19 * 3 = 57 as in one cycle there are 3 minutes . thus 57 minutes answer : a
Break down and solve the equation step by step.	the cost of registration at a professional association meeting was $ 50 per person ; a lunch for registrants only was available for an additional $ 22 per person . if the number of registrants who paid for lunch was 20 more than the number who did not , and if receipts for registration and lunch totaled $ 50,240 , how many people paid just for registration at the meeting ? a ) 700 , b ) 800 , c ) 1,300 , d ) 1,500 , e ) 400	hope this might be useful to you . let the number of people who have opted only to register = x now since the registration cost is 50 $ per person , the total amount sums to = 50 x $ as per the information given in the question , the number of registrants who paid for lunch was 20 more than the number who did not . that means , total number of people who registered and paid for lunch = 20 + x . for the people who registered for lunch the cost is 50 $ ( for the event registration ) + 22 $ ( for lunch ) = 72 $ . total amount in this case sums to = 72 ( 20 + x ) = 1440 + 72 x now , total amount received was 50240 . thus , from the above data , 50 x + 1440 + 72 x = 50240 122 x = 50240 - 1440 122 x = 48800 x = 400 . hence the correct ans is e
Work through the equation methodically, ensuring clarity at each step.	at what rate percent of simple interest will a sum of money double itself in 66 years ? a ) 6.54 % , b ) 4.54 % , c ) 1.51 % , d ) 1.94 % , e ) 5.54 %	"let sum = x . then , s . i . = x . rate = ( 100 * s . i . ) / ( p * t ) = ( 100 * x ) / ( x * 66 ) = 100 / 66 = 1.51 % answer : c"
Provide a detailed solution to the given mathematical problem.	in a garden , 26 trees are planted at equal distances along a yard 300 metres long , one tree being at each end of the yard . what is the distance between two consecutive trees ? a ) 10 metres , b ) 20 metres , c ) 14 metres , d ) 12 metres , e ) 13 metres	"between 26 trees , there are 25 gaps length of each gap = 300 / 25 = 12 i . e . , distance between two consecutive trees = 12 answer : d"
Solve the problem systematically, showing each step clearly.	if two integers x , y ( x > y ) are selected from - 5 to 5 ( inclusive ) , how many cases are there ? a ) 55 , b ) 60 , c ) 65 , d ) 70 , e ) 75	"there are 11 integers from - 5 to 5 inclusive . 11 c 2 = 55 . the answer is a ."
Provide a detailed solution to the given mathematical problem.	john and david can finish a job together in 1 hours . if john can do the job by himself in 2 hours , what percent of the job does david do ? a ) 40 % , b ) 45 % , c ) 50 % , d ) 55 % , e ) 60 %	you can also plug in numbers . for example , bob and alice work at a donut factory and make 10 donuts which is the job ( i picked this as a smart number ) . john on his own works 10 / 2 = 5 donuts per hour . john and david work 10 / 1 = 10 donuts per hour so david works 5 donuts / hour to find out the percentage , david works 5 donuts / hr x 1 hours = 5 donuts per hour . therefore 5 donuts / 10 donuts = 1 / 2 = 50 % answer : c
Explain the solution process for the given equation.	a number consists of 3 digits whose sum is 10 . the middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed . the number is a ) 145 , b ) 185 , c ) 253 , d ) 370 , e ) none	solution let the number be x . then 2 x = 10 or x = 5 . so , the number is either 253 or 352 . since the number increases on reversing the digits , so the hundred' s digit is smaller than the units digit . hence , required number = 253 . answer c
Explain the solution process for the given equation.	excluding stoppages , the average speed of a bus is 60 km / hr and including stoppages , the average speed of the bus is 30 km / hr . for how many minutes does the bus stop per hour ? a ) 22 , b ) 30 , c ) 77 , d ) 20 , e ) 99	"in 1 hr , the bus covers 60 km without stoppages and 30 km with stoppages . stoppage time = time take to travel ( 60 - 30 ) km i . e 30 km at 60 km / hr . stoppage time = 30 / 60 hrs = 30 min . answer : b"
Break down and solve the equation step by step.	( 26.3 * 12 * 20 ) / 3 + 125 = ? a ) 2339 , b ) 2229 , c ) 4429 , d ) 5529 , e ) 2669	b 2229 ? = [ ( 26.3 * 12 * 20 ) / 3 ] + 125 ? = 2104 + 125 = 2229
Explain the solution process for the given equation.	a test has 180 questions . each question has 5 options , but only 1 option is correct . if test - takers mark the correct option , they are awarded 1 point . however , if an answer is incorrectly marked , the test - taker loses 0.25 points . no points are awarded or deducted if a question is not attempted . a certain group of test - takers attempted different numbers of questions , but each test - taker still received the same net score of 40 . what is the maximum possible number of such test - takers ? a ) 29 , b ) 33 , c ) 35 , d ) 40 , e ) 42	"a correct answers get you 1 point , an incorrect answer gets you minus 1 / 4 point and a skipped question gets you 0 points . since there are 200 total questions , there are a variety of ways to get a total of 40 points . let c be the number of correct answers and let i be the number of incorrect answers . to get 40 points , a test taker must have at least 40 correct answers . then c = > 40 . for every correct question above 40 , the test taker has 4 incorrect answers . then , the i = 4 * ( c - 40 ) . also , i + c < = 180 . thus 5 c < = 340 and so c < = 68 . then 40 < = c < = 68 and c can have 29 possible values . the answer is a ."
Explain the solution process for the given equation.	a and b invests rs . 6000 and rs . 9000 in a business . after 4 months , a withdraws half of his capital and 2 months later , b withdraws one - third of his capital . in what ratio should they share the profits at the end of the year ? a ) 32 : 99 , b ) 82 : 31 , c ) 24 : 45 , d ) 34 : 89 , e ) 35 : 21	"a : b ( 6000 * 4 ) + ( 3000 * 8 ) : ( 9000 * 6 ) + ( 6000 * 6 ) 48000 : 90000 24 : 45 answer : c"
Explain the solution process for the given equation.	a candidate appearing for an examination has to secure 42 % marks to pass paper i . but he secured only 42 marks and failed by 22 marks . what is the maximum mark for paper i ? a ) 110 , b ) 120 , c ) 130 , d ) 152 , e ) 150	"he secured 42 marks nd fail by 22 marks so total marks for pass the examinatn = 64 let toal marks x x * 42 / 100 = 64 x = 152 answer : d"
Solve the problem systematically, showing each step clearly.	b completes a work in 8 days . a alone can do it in 10 days . if both work together , the work can be completed in how many days ? a ) 3.75 days , b ) 4.44 days , c ) 5 days , d ) 6.44 days , e ) 7 days	"1 / 8 + 1 / 10 = 9 / 40 40 / 9 = 4.44 days answer : b"
Work through the equation methodically, ensuring clarity at each step.	if 21 lines are drawn in a plane such that no two of them are parallel and no three are concurrent , then in how many points do they intersect ? a ) 150 , b ) 170 , c ) 190 , d ) 210 , e ) 230	"if two lines are not parallel , then they will intersect in exactly one point . lines can be extended infinitely on both ends so somewhere they will intersect with each other if they are not parallel . we are also given that no three lines are concurrent . this means that no three lines intersect at the same point . so every pair of two lines we select will have a unique point of intersection which they will not share with any third line . the number of ways to select 2 lines from 21 lines is 21 c 2 = 210 the answer is d ."
Solve the problem systematically, showing each step clearly.	what is the remainder when 43 ^ 91 is divided by 5 ? a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4	"the units digit of the exponents of 3 cycle in a group of 4 : { 3 , 9 , 7 , 1 } 91 has the form 4 k + 3 so the units digit of 43 ^ 91 is 7 . the remainder when dividing by 5 is 2 . the answer is c ."
Explain the solution process for the given equation.	how many integerskgreater than 100 and less than 1200 are there such that if the hundreds and the units digits ofkare reversed , the resulting integer is k + 99 ? a ) 50 , b ) 60 , c ) 70 , d ) 80 , e ) 100	"numbers will be like 102 = > 201 = 102 + 99 203 = > 302 = 103 + 99 so the hundereth digit and units digit are consecutive where unit digit is bigger than hundred digit . there will be ten pairs of such numbers for every pair there will 10 numbers like for 12 = > 102 , 112,132 , 142,152 , 162,172 , 182,192 . total = 10 * 10 = 100 hence e"
Solve the problem systematically, showing each step clearly.	the total price of a basic computer and printer are $ 2,500 . if the same printer had been purchased with an enhanced computer whose price was $ 500 more than the price of the basic computer , then the price of the printer would have been 1 / 3 of that total . what was the price of the basic computer ? a ) 1500 , b ) 1600 , c ) 1750 , d ) 1900 , e ) 2000	"let the price of basic computer be c and the price of the printer be p : c + p = $ 2,500 . the price of the enhanced computer will be c + 500 and total price for that computer and the printer will be 2,500 + 500 = $ 3,000 . now , we are told that the price of the printer is 1 / 3 of that new total price : p = 1 / 3 * $ 3,000 = $ 1000 plug this value in the first equation : c + 1000 = $ 2,500 - - > c = $ 1,500 . answer : a ."
Solve the problem systematically, showing each step clearly.	if a 2 + b 2 + c 2 = 941 and ab + bc + ca = 10 , then a + b + c is a ) 44 , b ) 56 , c ) 16 , d ) 32 , e ) 31	"by formula , ( a + b + c ) ^ 2 = a ^ 2 + b ^ 2 + c ^ 2 + 2 ( ab + bc + ca ) , since , a ^ 2 + b ^ 2 + c ^ 2 = 941 and ab + bc + ca = 10 , ( a + b + c ) ^ 2 = 941 + 2 ( 10 ) = 961 = 31 ^ 2 therefore : a + b + c = 31 answer : e"
Provide a detailed solution to the given mathematical problem.	a merchant has 1000 kg of sugar part of which he sells at 8 % profit and the rest at 18 % profit . he gains 14 % on the whole . the quantity sold at 18 % profit is ? a ) 700 kg , b ) 500 kg , c ) 600 kg , d ) 800 kg , e ) 900 kg	"by the rule of alligation : profit of first part profit of second part so , ratio of 1 st and 2 nd parts = 4 : 6 = 2 : 3 . quantity of 2 nd kind = ( 3 / 5 x 1000 ) kg = 600 kg answer : c"
Provide a detailed solution to the given mathematical problem.	the total marks obtained by a student in mathematics and physics is 20 and his score in chemistry is 20 marks more than that in physics . find the average marks scored in mathamatics and chemistry together . a ) 40 , b ) 26 , c ) 20 , d ) 28 , e ) 22	"let the marks obtained by the student in mathematics , physics and chemistry be m , p and c respectively . given , m + c = 20 and c - p = 20 m + c / 2 = [ ( m + p ) + ( c - p ) ] / 2 = ( 20 + 20 ) / 2 = 20 . answer : c"
Provide a detailed solution to the given mathematical problem.	the price of a t . v . set worth rs . 8000 is to be paid in 20 installments of rs . 750 each . if the rate of interest be 6 % per annum , and the first installment be paid at the time of purchase , then the value of the last installment covering the interest as well will be ? a ) 7250 , b ) 7200 , c ) 7820 , d ) 6000 , e ) 2799	"money paid in cash = rs . 750 balance payment = ( 8000 - 1000 ) = rs . 7250 answer : a"
Provide a detailed solution to the given mathematical problem.	a shopkeeper sold an article offering a discount of 10 % and earned a profit of 25 % . what would have been the percentage of profit earned if no discount was offered ? a ) 39 % , b ) 52 % , c ) 15 % , d ) 21 % , e ) 28 %	"let c . p . be $ 100 . then , s . p . = $ 125 let marked price be $ x . then , 90 / 100 * x = 125 x = 12500 / 90 = $ 139 now , s . p . = $ 139 , c . p . = rs . 100 profit % = 39 % . answer : a"
Work through the equation methodically, ensuring clarity at each step.	one side of the bottom layer of a triangular pyramid has 12 balls . how many are there in the whole pyramid ? note that the pyramid is equilateral and solid . ['a ) 286','b ) 364','c ) 504','d ) 164','e ) 165']	as there are 12 balls along one side , it means that there are 12 layers of balls . the top most layer has 1 ball . the second layer has 3 ( 1 + 2 ) balls . the third layer has 6 ( 1 + 2 + 3 ) balls . the fourth layer has 10 ( 1 + 2 + 3 + 4 ) balls . the fifth layer has 15 ( 1 + 2 + 3 + 4 + 5 ) balls . similarly , there are 21 , 28 , 36 , 45 , 55 , 66 and 78 balls in the remaining layers . hence , the total number of balls are = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 balls answer : b
Break down and solve the equation step by step.	if n = 2 ^ 0.25 and n ^ b = 8 , b must equal a ) 3 / 80 , b ) 3 / 5 , c ) 12 , d ) 5 / 3 , e ) 80 / 3	"25 / 100 = 1 / 4 n = 2 ^ 1 / 4 n ^ b = 2 ^ 3 ( 2 ^ 1 / 4 ) ^ b = 2 ^ 3 b = 12 answer : c"
Solve the problem systematically, showing each step clearly.	susan finds that she spends her free time during summer vacation between swimming , reading , and hanging out with friends in a ratio of 1 : 4 : 10 . how many hours did she spend reading if she hung out with her friends for 20 hours ? a ) 2 , b ) 4 , c ) 5 , d ) 8 , e ) 10	the ratio is 1 hour swimming : 4 hours reading : 10 hours hanging out with friends . divide 20 by 10 to find 1 ` ` part'' of the ratio . 20 / 10 = 2 multiply this by 4 to find the hours spent reading . 2 * 4 = 8 the answer is d .
Explain the solution process for the given equation.	20 ^ 10 / 120 ^ 5 = ? a ) 6 ^ 5 , b ) 5 ^ 6 , c ) ( 10 / 3 ) ^ 5 , d ) 6 ^ 3 , e ) 15 ^ 3	"20 ^ 10 / 120 ^ 5 = ? a . 6 ^ 5 b . 5 ^ 6 c . ( 10 / 3 ) ^ 5 d . 6 ^ 3 e . 15 ^ 3 - > 20 ^ 10 / 120 ^ 5 = ( 20 ^ 10 ) / ( 6 ^ 5 ) ( 20 ^ 5 ) = ( 20 ^ 5 ) / ( 6 ^ 5 ) = ( 2 ^ 5 ) ( 10 ^ 5 ) / ( 2 ^ 5 ) ( 3 ^ 5 ) = ( 10 / 3 ) ^ 5 . thus , c is the answer ."
Provide a detailed solution to the given mathematical problem.	the amount of principal rs . 8000 at compound interest at the ratio of 5 % p . a . for 6 years is a ) s . 9621 , b ) s . 6921 , c ) s . 10721 , d ) s . 6261 , e ) s . 6361	"c . i = p ( 1 + r / 100 ) ^ n = 8000 ( 1 + 5 / 100 ) ^ 6 = rs 10721 answer : c"
Explain the solution process for the given equation.	an alloy weighing 48 ounces is 25 % gold . how many ounces of pure gold must be added to create an alloy that is 40 % gold ? a ) 15 , b ) 20 , c ) 12 , d ) 14 , e ) 7	an alloy of 48 oz which is 25 % gold means there is 12 oz of gold . to get to an alloy that is 40 % gold , let' s use this expression : ( 12 + x ) / ( 48 + x ) = 0.40 with x representing the amount of pure gold that must be added to get to 40 % . the expression we are using represents the new total weight of pure gold over the new total weight of the alloy and this fraction should represent 40 % or 0.4 . you will see that 12 is the correct answer , as 24 / 60 = 0.4 choose c
Work through the equation methodically, ensuring clarity at each step.	a and b can do a piece of work in 4 days . b and c can do it in 12 days and a and c in 16 days . working together they will complete the work in how many days ? a ) 4.05 days , b ) 5.05 days , c ) 14.76 days , d ) 8.05 days , e ) 15.05 days	"a + b = 1 / 4 b + c = 1 / 12 c + a = 1 / 16 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2 ( a + b + c ) = 1 / 4 + 1 / 12 + 1 / 16 = 19 / 48 a + b + c = 19 / 96 96 / 19 = 5.05 days answer : b"
Break down and solve the equation step by step.	how many pieces of 75 cm can be cut from a rope 57 meters long ? a ) 30 , b ) 40 , c ) 76 , d ) none , e ) can not be determined	"explanation : total pieces of 75 cm that can be cut from a rope of 57 meters long is = ( 57 meters ) / ( 75 cm ) = ( 57 meters ) / ( 0.75 meters ) = 76 answer : c"
Provide a detailed solution to the given mathematical problem.	the sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 480 m , its area is ? a ) 150 , b ) 240 , c ) 277 , d ) 261 , e ) 281	"5 x + 12 x + 13 x = 480 = > x = 16 a = 80 , b = 192 , c = 208 s = ( 80 + 192 + 208 ) / 2 = 240 answer : b"
Provide a detailed solution to the given mathematical problem.	the area of an isosceles trapezoid with sides of length 5 and bases of length 6 and 12 is ? a ) 44 , b ) 42 , c ) 40 , d ) 38 , e ) 36	"isosceles trapezoid indicates that if we draw perpendicular from two vertex of the smaller side , the longer side with side 12 will be divided into 3 parts = 6 , 3 and 3 ( making one square and two right triangles ) for the right triangle , height will be = ( 5 ^ 2 - 3 ^ 2 ) ^ 1 / 2 = 4 and the area of trapezoid = 1 / 2 ( b 1 + b 2 ) h = 1 / 2 * 18 * 4 = 36 e"
Provide a detailed solution to the given mathematical problem.	a cistern which could be filled in 9 hours takes 1 hour more to be filled owing to a leak in its bottom . if the cistern is full , in what time will the leak empty it ? a ) 50 hours , b ) 62 hours , c ) 90 hours , d ) 75 hours , e ) 100 hours	"let the leak empty the full cistern in x hours 9 x / x - 9 = 9 + 1 x = 90 hours answer is c"
Explain the solution process for the given equation.	a call center has two teams . each member of team a was able to process 2 / 5 calls as compared to each member of team b . if team a has 5 / 8 as many number of call center agents as team b , what fraction of the total calls was processed by team b ? a ) 3 / 2 , b ) 3 / 4 , c ) 4 / 5 , d ) 1 / 2 , e ) 1 / 5	let team b has 8 agents , so team a has 5 agents let each agent of team b picked up 5 calls , so total calls by team b = 40 so , each agent in team a picked up 2 calls , so total calls for team a = 10 fraction for team b = 40 / ( 40 + 10 ) = 4 / 5 = answer = c
Explain the solution process for the given equation.	of the 720 students at a certain university , 25 percent are seniors . if x seniors were to graduate early and leave the university and no additional students entered or left the university , what value of x would reduce the number of seniors at the university to 20 percent ? a ) 30 , b ) 36 , c ) 42 , d ) 45 , e ) 48	sol : 180 - x = 20 % of ( 720 - x ) 4 x = 180 x = 45 answer : d
Work through the equation methodically, ensuring clarity at each step.	if 2 is the remainder when m is divided by 5 , what is the remainder when 3 m is divided by 5 ? a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4	let' s find a number which when divided by 5 leaves remainder 2 = 5 + 2 = 7 i . e . one possible value of m = 7 value of 3 m = 3 * 7 = 21 when em is divided by 5 then remainder = rem ( 21 / 5 ) = 1 answer : option b
Provide a detailed solution to the given mathematical problem.	two sides of a triangle have lengths x and w and meet at a right angle . if the perimeter of the triangle is 4 x , what is the ratio of x to w ? ['a ) a ) 2 : 3','b ) b ) 3 : 4','c ) c ) 4 : 3','d ) d ) 3 : 2','e ) e ) 2 : 1']	ahhhh . . . . my bad . thank you , you' re correct . 8 x = 6 w - - > x / w = 3 / 4 . actually b is the right answer .
Work through the equation methodically, ensuring clarity at each step.	in a two digit number it is known that its unit' s digit exceeds its 10' s digit by 4 and that the product of the given number and the sum of its digits is equal to 208 , what is the number ? a ) 24 , b ) 25 , c ) 26 , d ) 27 , e ) 28	xy is 2 digit no . y - x = 4 ; ( 10 x + y ) * ( 2 x + 4 ) = 208 ; sub y = x + 4 ; we get 26 answer : c
Provide a detailed solution to the given mathematical problem.	if n is a prime number greater than 5 , what is the remainder when n ^ 2 is divided by 23 ? a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 5	"there are several algebraic ways to solve this question including the one under the spoiler . but the easiest way is as follows : since we can not have two correct answersjust pick a prime greater than 5 , square it and see what would be the remainder upon division of it by 23 . n = 5 - - > n ^ 2 = 49 - - > remainder upon division 49 by 23 is 3 . answer : d ."
Break down and solve the equation step by step.	a retailer buys a radio for rs 225 . his overhead expenses are rs 28 . he sellis the radio for rs 300 . the profit percent of the retailer is a ) 10 % , b ) 50 % , c ) 25 % , d ) 52 % , e ) 18.5 %	"explanation : cost price = ( 225 + 28 ) = 253 sell price = 300 gain = ( 47 / 253 ) * 100 = 18.5 % . answer : e"
Explain the solution process for the given equation.	a lent rs . 5000 to b for 2 years and rs . 3000 to c for 4 years on simple interest at the same rate of interest and received rs . 3300 in all from both of them as interest . the rate of interest per annum is ? a ) 16 % , b ) 12 % , c ) 74 % , d ) 10 % , e ) 15 %	"let the rate be r % p . a . then , ( 5000 * r * 2 ) / 100 + ( 3000 * r * 4 ) / 100 = 3300 100 r + 120 r = 3300 r = 15 % answer : e"
Solve the problem systematically, showing each step clearly.	an equilateral triangle t 2 is formed by joining the mid points of the sides of another equilateral triangle t 1 . a third equilateral triangle t 3 is formed by joining the mid - points of t 2 and this process is continued indefinitely . if each side of t 1 is 50 cm , find the sum of the perimeters of all the triangles . a ) 180 cm , b ) 220 cm , c ) 240 cm , d ) 270 cm , e ) 300 cm	"we have 50 for first triangle , when we join mid - points of first triangle we get the second equilateral triangle then the length of second one is 25 and continues . so we have 50,25 , 12.5 , . . . we have ratio = 1 / 2 , and it is gp type . sum of infinite triangle is a / 1 - r = 50 / 1 - ( 1 / 2 ) = 100 equilateral triangle perimeter is 3 a = 3 * 100 = 300 . so option e ."
Break down and solve the equation step by step.	when a natural number n is successively divided by 12 , 34 the remainders are 56 , 78 . what will be the sum of the remainders if the order of the division is reversed ? a ) 11 , b ) 12 , c ) 10 , d ) 2 , e ) 7	1 2 3 4 5 6 7 8 leave the top right - most number 8 start with bottom right - most number 5 8 * 3 + 7 = 31 31 * 2 + 6 = 68 68 * 1 + 5 = 73 this is the number required now , do the successive division in the reverse order the sum of the remainders is 10 hence , the correct option is c
Explain the solution process for the given equation.	johnny travels a total of one hour to and from school . on the way there he jogs at 5 miles per hour and on the return trip he gets picked up by the bus and returns home at 25 miles per hour . how far is it to the school ? a ) 2 miles , b ) 4 miles , c ) 6.6 miles , d ) 8 miles , e ) 10 miles	answer : c ) 6.6 miles . average speed for round trip = 2 * a * b / ( a + b ) , where a , b are speeds so , average speed was = 2 * 5 * 25 / ( 5 + 25 ) = 6.6 m / hr the distance between schoolhome should be half of that . ie . 6.6 miles answer c
Break down and solve the equation step by step.	( x + 7 ) is a factor in x ^ 2 - mx - 35 . what is the value of m ? a ) - 3 , b ) - 2 , c ) - 4 , d ) - 5 , e ) - 6	"i solved the second degree equation and found it like this : x ^ 2 - mx - 35 ( x - 5 ) ( x + 7 ) x = 5 or x = - 7 substituting both values for x in the equation we find : x ^ 2 - mx - 35 = > ( - 7 ) ^ 2 - m ( - 7 ) = 35 = > 49 + 7 m = 35 = > 7 m = 35 - 49 = - 14 = > m = - 2 and with 5 , using a similar process we end up with : ( 5 ) ^ 2 - m ( 5 ) = 35 - 10 = 5 m m = - 2 ao , ans b"
Break down and solve the equation step by step.	rounded to 3 decimal places , 1.003 ^ 4 = a ) 1.004 , b ) 1.006 , c ) 1.008 , d ) 1.012 , e ) 1.016	as compared to 1 , 0.003 is a very small quantity . thus , we can write ( 1 + 0.003 ) ^ 4 is nearly equal to ( 1 + 4 * 0.003 ) = 1.012 . as the question asks for approximation to three decimal places , the further terms will anyways not come into picture . d .
Work through the equation methodically, ensuring clarity at each step.	the mean of 40 observations was 100 . it was found later that an observation 50 was wrongly taken as 75 the corrected new mean is a ) 125.8 , b ) 102.8 , c ) 99.375 , d ) 100.52 , e ) 122	"explanation : correct sum = ( 100 * 40 + 50 - 75 ) = 3975 . correct mean = = 3975 / 40 = 99.375 answer : c"
Break down and solve the equation step by step.	in a college , the ratio of the number of boys to girls is 8 : 5 . if there are 200 girls , the total number of students in the college is a ) 420 , b ) 520 , c ) 620 , d ) 620 , e ) none of these	explanation : let the boy are 8 x and girls are 5 x = > 5 x = 200 = > x = 40 total students = 8 x + 5 x = 13 x = 13 ( 40 ) = 520 answer : b
Break down and solve the equation step by step.	how many x ways can you group 3 people from 4 sets of twins if no two people from the same set of twins can be chosen ? a ) 3 , b ) 16 , c ) 28 , d ) 32 , e ) 56	ways to select 3 people from 8 people ( 4 twins x 2 ) = 8 c 3 = 56 ways to select 1 twin + 1 people = 4 c 1 * 6 c 1 = 24 ways to select a group 3 people from 4 sets of twins if no two people from the same set of twins can be chosen x = 56 - 24 = 32 ans : d
Work through the equation methodically, ensuring clarity at each step.	ram covers a part of the journey at 20 kmph and the balance at 70 kmph taking total of 8 hours to cover the distance of 400 km . how many hours has been driving at 20 kmph ? a ) 3.2 hr , b ) 3.4 hr , c ) 3.6 hr , d ) 3.8 hr , e ) 4.2 hr	let see d / t = v then x / 20 + ( 400 - x ) / 70 = 8 x = 64 kmh t = 64 / 20 = 3.2 hr answer : a
Break down and solve the equation step by step.	a can finish a work in 10 days and b can do same work in half the time taken by a . then working together , what part of same work they can finish in a day ? a ) 1 / 5 , b ) 1 / 6 , c ) 1 / 7 , d ) 3 / 10 , e ) none of these	"explanation : please note in this question , we need to answer part of work for a day rather than complete work . it was worth mentioning here because many do mistake at this point in hurry to solve the question so lets solve now , a' s 1 day work = 1 / 10 b' s 1 day work = 1 / 5 [ because b take half the time than a ] ( a + b )' s one day work = ( 1 / 10 + 1 / 5 ) = 3 / 10 so in one day 3 / 10 work will be done answer : d"
Provide a detailed solution to the given mathematical problem.	pascal has 96 miles remaining to complete his cycling trip . if he reduced his current speed by 4 miles per hour , the remainder of the trip would take him 16 hours longer than it would if he increased his speed by 50 % . what is his current speed z ? a ) 6 , b ) 8 , c ) 10 , d ) 12 , e ) 16	"let the current speed be x miles per hour . time taken if speed is 50 % faster ( i . e . 3 x / 2 = 1.5 x ) = 96 / 1.5 x time taken if speed is reduced by 4 miles / hr ( i . e . ( x - 4 ) ) = 96 / ( x - 4 ) as per question , 96 / ( x - 4 ) - 96 / 1.5 x = 16 solving this z we get x = 8 . b ."
Solve the problem systematically, showing each step clearly.	carol and jordan draw rectangles of equal area . if carol' s rectangle measures 5 inches by 24 inches and jordan' s rectangle is 3 inches long , how wide is jordan' s rectangle , in inches ? a ) 25 , b ) 43 , c ) 42 , d ) 40 , e ) 18	"area of carol' s rectangle = 24 * 5 = 120 let width of jordan' s rectangle = w since , the areas are equal 3 w = 120 = > w = 40 answer d"
Provide a detailed solution to the given mathematical problem.	one hour after adrienne started walking the 60 miles from x to y , william started walking from x to y as well . adrienne walks 3 miles per hour and william walks 1 mile per hour faster than adrienne . how far from x will william be when he catches up to adrienne ? a ) 8 miles , b ) 9 miles , c ) 10 miles , d ) 11 miles , e ) 12 miles	first , determine how far adrienne has walked in the one hour . she has walked three miles which means she is three miles ahead of william when he sets off . william walks at four miles / hour which means that every hour , william will get one mile closer to adrienne . if he gets one mile closer every hour , it will take him three hours to catch up to her which means he travels 3 hours * 4 miles / hour = 12 miles and she travels 4 hours * 3 miles / hour = 12 miles . he will be 12 miles from x when he catches up to her . a slightly different way to solve . . . we do n' t know how long they will walk before they catch up to one another but we do know that a walks for one hour more than j . j = t and a = t + 1 . we are looking for the distance at which they meet up which means the distance will be the same . d = r * t so , r * t ( william ) = r * t ( adrienne ) r * ( t ) = r * ( t + 1 ) 4 t = 3 t + 3 t = 3 d = r * t d = 4 * 3 d = 12 d ) 12 miles
Provide a detailed solution to the given mathematical problem.	for every even positive integer m , f ( m ) represents the product of all even integers from 2 to m , inclusive . for example , f ( 12 ) = 2 x 4 x 6 x 8 x 10 x 12 . what is the greatest prime factor of f ( 28 ) ? a ) 2 , b ) 7 , c ) 11 , d ) 13 , e ) 17	"f ( 28 ) = 2 * 4 * 6 * 8 * 10 * 12 * 14 * 16 * 18 * 20 * 22 * 24 * 26 * 28 the greatest prime factor in this list is 13 . the answer is d ."
Explain the solution process for the given equation.	how many numbers from 2 to 15 are exactly divisible by 2 ? a ) a ) 2 , b ) b ) 3 , c ) c ) 5 , d ) d ) 7 , e ) e ) 8	"2 / 2 = 1 and 15 / 2 = 7 7 - 1 = 6 6 + 1 = 7 numbers . answer : d"
Solve the problem systematically, showing each step clearly.	the side of a square is increased by 30 % then how much % does its area increases ? a ) 52.65 , b ) 69 , c ) 50.75 , d ) 42.75 , e ) 52.75	"a = 100 a 2 = 10000 a = 130 a 2 = 16900 - - - - - - - - - - - - - - - - 10000 - - - - - - - - - 6900 100 - - - - - - - ? = > 69 % answer : b"
Provide a detailed solution to the given mathematical problem.	average weight of 10 people increased by 3 kg when one person of 45 kg is replaced by a new man . then weight of the new man is a ) 50 , b ) 55 , c ) 60 , d ) 65 , e ) 75	explanation : total weight increased is 3 * 10 = 30 . so weight of new person is 45 + 30 = 75 answer : option e
Provide a detailed solution to the given mathematical problem.	the first half of the 20 th century , the population of a particular country increased by 200 percent . in the second half of the century , the population increased by 300 percent . what was the percent increase for the 20 th century as a whole ? a ) 500 % , b ) 600 % , c ) 800 % , d ) 1100 % , e ) 1200 %	say initially population was 100 . what is 200 % of 100 ? it is 200 / 100 * 100 = 200 . an increase of 200 % means the new population became 100 + 200 = 300 what is 300 % of 300 ? it is 300 / 100 * 300 = 900 an increase of 300 % means the new population now is 300 + 900 = 1200 so from 100 , the population increased to 1200 i . e . an increase of 1100 . 1100 is what percent of 100 ? 1100 = x / 100 * 100 i . e . it is 1100 % d
Break down and solve the equation step by step.	a man engaged a servant on the condition that he would pay him rs . 900 and a uniform after 1 year service . he served only for 9 months and receiveduniform and rs . 650 , find the priceof the uniform ? a ) rs . 90 , b ) rs . 100 , c ) rs . 120 , d ) rs . 150 , e ) rs . 160	"9 / 12 = 3 / 4 * 900 = 675 650 - - - - - - - - - - - - - 25 1 / 4 - - - - - - - - 25 1 - - - - - - - - - ? = > rs . 100 b"
Work through the equation methodically, ensuring clarity at each step.	a train which has 600 m long , is running 60 kmph . in what time will it cross a person moving at 6 kmph in same direction ? a ) 42 sec , b ) 44 sec , c ) 46 sec , d ) 40 sec , e ) 50 sec	"time taken to cross a moving person = length of train / relative speed time taken = 600 / ( ( 60 - 6 ) ( 5 / 18 ) = 600 / 54 * ( 5 / 18 ) = 600 / 15 = 40 sec answer : d"
Explain the solution process for the given equation.	if the cost price of 50 articles is equal to the selling price of 35 articles , then the gain or loss percent is ? a ) 16 , b ) 127 , c ) 12 , d ) 18 , e ) 43	"percentage of profit = 15 / 35 * 100 = 43 % answer : e"
Solve the problem systematically, showing each step clearly.	a money lender finds that due to a fall in the annual rate of interest from 8 % to 7 3 / 4 % his yearly income diminishes by rs . 64.50 , his capital is ? a ) 24,602 , b ) 28,888 , c ) 25,800 , d ) 24,628 , e ) 24,6012	"let the capital be rs . x . then , ( x * 8 * 1 ) / 100 - ( x * 31 / 4 * 1 / 100 ) = 64.50 32 x - 31 x = 6450 * 4 x = 25,800 . answer : c"
Explain the solution process for the given equation.	at what price must an article costing rs . 66.5 be marked in order that after deducting 5 % from the list price . it may be sold at a profit of 25 % on the cost price ? a ) 87.5 , b ) 89.5 , c ) 82.3 , d ) 82.1 , e ) 92.4	"cp = 66.5 sp = 66.5 * ( 125 / 100 ) = 83.125 mp * ( 95 / 100 ) = 59.375 mp = 87.5 answer : a"
Provide a detailed solution to the given mathematical problem.	jack , jill , and sandy each have one try to make a basket from half court . if their individual probabilities of making the basket are 1 / 6 , 1 / 7 , and 1 / 8 respectively , what is the probability that jack and jill will make a basket but sandy will miss ? a ) 1 / 8 , b ) 1 / 16 , c ) 1 / 48 , d ) 1 / 96 , e ) 1 / 168	"the probability that jack and jill will make a basket but sandy will miss is 1 / 6 * 1 / 7 * 7 / 8 = 1 / 48 . the answer is c ."
Work through the equation methodically, ensuring clarity at each step.	divide rs . 32000 in the ratio 3 : 7 ? a ) 12000 , 20000 , b ) 9600 , 22400 , c ) 12000 , 20007 , d ) 12000 , 20006 , e ) 12000 , 20001	3 / 10 * 32000 = 9600 7 / 8 * 32000 = 22400 answer : b
Work through the equation methodically, ensuring clarity at each step.	a and b complete a work in 3 days . a alone can do it in 9 days . if both together can do the work in how many days ? a ) 3.75 days , b ) 4.75 days , c ) 3.25 days , d ) 3.15 days , e ) 2.75 days	"1 / 3 + 1 / 9 = 4 / 9 9 / 4 = 4.75 days answer : b"
Provide a detailed solution to the given mathematical problem.	a watch was sold at a loss of 10 % . if it was sold for rs . 210 more , there would have been a gain of 4 % . what is the cost price ? a ) s . 1000 , b ) s . 1009 , c ) s . 1007 , d ) s . 1006 , e ) s . 1500	explanation : 90 % 104 % - - - - - - - - 14 % - - - - 210 100 % - - - - ? = > rs . 1500 answer : e
Explain the solution process for the given equation.	a leak in the bottom of a tank can empty the full tank in 6 hours . an inlet pipe fills water at the rate of 3.5 liters per minute . when the tank is full in inlet is opened and due to the leak the tank is empties in 8 hours . the capacity of the tank is ? a ) 5729 , b ) 5040 , c ) 2889 , d ) 2870 , e ) 2799	"1 / x - 1 / 6 = - 1 / 8 x = 24 hrs 24 * 60 * 3.5 = 5040 . answer : b"
Explain the solution process for the given equation.	sonika deposited rs . 8000 which amounted to rs . 9200 after 3 years at simple interest . had the interest been 2.5 % more . she would get how much ? a ) 9800 , b ) 96288 , c ) 26667 , d ) 1662 , e ) 2882	"( 8000 * 3 * 2.5 ) / 100 = 600 9200 - - - - - - - - 9800 answer : a"
Break down and solve the equation step by step.	in 10 years , a will be twice as old as b was 10 years ago . if a is now 4 years older than b the present age of b is a ) 37 , b ) 38 , c ) 39 , d ) 34 , e ) 41	let present age of a be a and b be b a + 10 = 2 * ( b - 10 ) = > 2 b - a = 30 . . . . . . ( i ) a = b + 4 = > 2 b - b - 4 = 30 b = 34 so the present age of b is 34 years answer : d
Explain the solution process for the given equation.	in a lake , there is a patch of lily pads . every day , the patch doubles in size . it takes 50 days for the patch to cover the entire lake , how many days would it take the patch to cover half of the lake ? a ) 36 , b ) 2 ^ 4 * 3 , c ) 24 , d ) 49 , e ) 47	"working backward from the day it' s covered : day 50 : fully covered day 49 : half covered so 49 days answer : d"
Explain the solution process for the given equation.	a jogger running at 9 kmph alongside a railway track in 240 meters ahead of the engine of a 120 metres long train running at 45 kmph in the same direction . in how much time will the train pass the jogger ? a ) 18 seconds , b ) 26 seconds , c ) 36 seconds , d ) 16 seconds , e ) 46 seconds	total distance ( d ) = 240 + 120 = 360 meter relative speed ( s ) = 45 - 9 = 36 kmph = 36 * 5 / 18 = 10 mps d = s * t 360 = 10 * t t = 36 seconds answer : c
Work through the equation methodically, ensuring clarity at each step.	find the total number of prime factors in the expression ( 4 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 a ) 26 , b ) 22 , c ) 25 , d ) 27 , e ) 33	( 4 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 = ( 2 x 2 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 = 2 ^ 13 x 2 ^ 13 x 7 ^ 5 x 11 ^ 2 = 2 ^ 26 x 7 ^ 5 x 11 ^ 2 total number of prime factors = ( 26 + 5 + 2 ) = 33 . answer is e .
Provide a detailed solution to the given mathematical problem.	find the missing figures : 0.3 % of ? = 0.15 a ) 50 , b ) 45 , c ) 150 , d ) 450 , e ) 500	let 0.3 % of x = 0.15 . then , 0.30 * x / 100 = 0.15 x = [ ( 0.15 * 100 ) / 0.3 ] = 50 . answer is a .
Break down and solve the equation step by step.	in a rectangular axis system , what is the area of a parallelogram with the coordinates : ( 5,3 ) , ( 8,3 ) , ( 4,5 ) , ( 7,5 ) ? a ) 21 . , b ) 28 . , c ) 35 . , d ) 49 . , e ) 6 .	"delta x will give us the dimension of one side of the parallelogram = 8 - 5 = 3 unit delta y will give us the dimension of the other side of parallelogram = 5 - 3 = 2 unit area of parallelogram = 3 * 2 = 6 answer is e"
Solve the problem systematically, showing each step clearly.	a certain company has records stored with a record storage firm in 15 - inch by 12 - inch by 10 - inch boxes . the boxes occupy 1.08 million cubic inches of space . if the company pays $ 0.4 per box per month for the record storage , what is the total amount that the company pays each month for record storage ? a ) a . 150 , b ) b . 300 , c ) c . 600 , d ) d . 240 , e ) e . 200	"volume per box : 15 x 12 x 10 = 1,800 total volume : 1 , 080,000 number of boxes : total volume / volume per box = 1 , 080,000 / 1,800 = 600 price per month : number of boxes * price per box = 600 * 0.4 = 240 answer : d"
Provide a detailed solution to the given mathematical problem.	two trains 110 m and 160 m long run at the speed of 60 km / hr and 40 km / hr respectively in opposite directions on parallel tracks . the time which they take to cross each other is ? a ) 10.9 , b ) 10.7 , c ) 10.3 , d ) 10.8 , e ) 9.72	"relative speed = 60 + 40 = 100 km / hr . = 100 * 5 / 18 = 250 / 9 m / sec . distance covered in crossing each other = 110 + 160 = 270 m . required time = 270 * 9 / 250 = 243 / 25 = 9.72 sec .' answer : e"
Solve the problem systematically, showing each step clearly.	fresh grapes contain 70 % by weight while dried grapes contain 10 % water by weight . what is the weight of dry grapes available from 100 kg of fresh grapes ? a ) 18.33 kg , b ) 21.50 kg , c ) 25.00 kg , d ) 33.33 kg , e ) 40.39 kg	"from the question we know : 100 kg * 70 % = 70 kg of water in the fresh grapes 100 kg - 70 kg of water = 30 kg of non - water mass we are looking for the weight of the dry grapes ( x ) . since the question tells us that 10 % of the weight of the dry graps is water and we know that 30 kg is non - water mass we can set up the following equation : x = 1 / 10 ( x ) + 30 kg 9 / 10 ( x ) = 30 kg x = 33.33 kg answer - d"
Break down and solve the equation step by step.	what is the sum of all 3 digit numbers that leave a remainder of' 2' when divided by 5 ? a ) 897 , b ) 98,910 , c ) 64,749 , d ) 49,700 , e ) 56,720	find the number , upon sum of 3 digits of a number gives a reminder 2 when it is divided by 5 seeing the options after dividing and finding the reminder of 2 my answer was b
Explain the solution process for the given equation.	how many 5 digit number contain number 3 ? a ) 31512 , b ) 32512 , c ) 33512 , d ) 34512 , e ) 37512	"total 5 digit no . = 9 * 10 * 10 * 10 * 10 = 90000 not containing 3 = 8 * 9 * 9 * 9 * 9 = 52488 total 5 digit number contain 3 = 90000 - 52488 = 37512 answer : e"
Work through the equation methodically, ensuring clarity at each step.	the manufacturing cost of a shoe is rs . 190 and the transportation lost is rs . 500 for 100 shoes . what will be the selling price if it is sold at 20 % gains a ) s 234 , b ) s 216 , c ) s 220 , d ) s 210 , e ) s 217	"explanation : total cost of a watch = 190 + ( 500 / 100 ) = 195 . gain = 20 % = > sp = 1.2 cp = 1.2 x 195 = 234 answer : a"
Work through the equation methodically, ensuring clarity at each step.	what is the maximum number t of 27 cubic centimetre cubes that can fit in a rectangular box measuring 8 centimetre x 9 centimetre x 12 centimetre ? ['a ) 36','b ) 32','c ) 24','d ) 21','e ) 15']	27 cubic centimetre cubes gives side = 3 cm so if : l * w * h is 9 * 12 * 8 , then max . cube we can have are 3 * 4 * 2 = 24 l * w * h is 9 * 8 * 12 , then max . cube we can have are 3 * 2 * 4 = 24 l * w * h is 12 * 8 * 9 , then max . cube we can have are 4 * 2 * 3 = 24 l * w * h is 12 * 9 * 8 , then max . cube we can have are 4 * 3 * 2 = 24 l * w * h is 8 * 12 * 9 , then max . cube we can have are 2 * 4 * 3 = 24 l * w * h is 8 * 9 * 12 , then max . cube we can have are 2 * 3 * 4 = 24 in all cases we get t = 24 cubes . ans . c
Explain the solution process for the given equation.	how many multiples of 8 are there between 100 and 200 ( both are inclusive ) ? a ) 27 , b ) 4 , c ) 6 , d ) 13.5 , e ) 8	"the answer is ( 200 - 100 ) / 8 + 1 = 13.5 answer is d"
Solve the problem systematically, showing each step clearly.	a train passes a station platform in 36 sec and a man standing on the platform in 22 sec . if the speed of the train is 54 km / hr . what is the length of the platform ? a ) 288 , b ) 210 , c ) 881 , d ) 1277 , e ) 121	"speed = 54 * 5 / 18 = 15 m / sec . length of the train = 15 * 22 = 330 m . let the length of the platform be x m . then , ( x + 330 ) / 36 = 15 = > x = 210 m . answer : b"
Work through the equation methodically, ensuring clarity at each step.	a train 140 m long is running with a speed of 50 km / hr . in what time will it pass a man who is running at 4 km / hr in the direction opposite to that in which the train is going ? a ) 8.1 , b ) 7.0 , c ) 9.33 , d ) 8 , e ) 9	"speed of train relative to man = 50 + 4 = 54 km / hr . = 54 * 5 / 18 = 15 m / sec . time taken to pass the men = 140 * 1 / 15 = 9.33 sec . answer : option c"
Work through the equation methodically, ensuring clarity at each step.	in the xy - plane , a triangle has vertices ( 0,0 ) , ( 4,0 ) and ( 4,7 ) . if a point ( a , b ) is selected at random from the triangular region , what is the probability that a - b > 0 ? a ) 4 / 7 , b ) 2 / 3 , c ) 3 / 5 , d ) 3 / 7 , e ) 1 / 2	"the area of the right triangle is ( 1 / 2 ) * 4 * 7 = 14 . only the points ( a , b ) below the line y = x satisfy a - b > 0 . the part of the triangle which is below the line y = x has an area of ( 1 / 2 ) ( 4 ) ( 4 ) = 8 . p ( a - b > 0 ) = 8 / 14 = 4 / 7 the answer is a ."
Explain the solution process for the given equation.	the cash difference between the selling prices of an article at a profit of 4 % and 6 % is rs 3 . the ratio of two selling prices is a ) 51 : 52 , b ) 52 : 53 , c ) 53 : 54 , d ) 54 : 55 , e ) none of these	"explanation : let the cost price of article is rs . x required ratio = 104 % of x / 106 % of x = 104 / 106 = 52 / 53 = 52 : 53 option b"
Work through the equation methodically, ensuring clarity at each step.	in what time will a train 300 meters long cross an electric pole , if its speed is 90 km / hr a ) 5 seconds , b ) 12 seconds , c ) 3 seconds , d ) 2.5 seconds , e ) none of these	"explanation : first convert speed into m / sec speed = 90 * ( 5 / 18 ) = 25 m / sec time = distance / speed = 300 / 25 = 12 seconds answer : b"
Provide a detailed solution to the given mathematical problem.	evaluate 2 + 22 + 222 + 2.22 a ) 269.72 , b ) 268.22 , c ) 248.21 , d ) 248.22 , e ) 268.21	"2 + 22 + 222 + 2.22 = 248.22 option d"
Solve the problem systematically, showing each step clearly.	the list price of an article is rs . 70 . a customer pays rs . 61.11 for it . he was given two successive discounts , one of them being 10 % . the other discount is ? a ) 8 % , b ) 7 % , c ) 10 % , d ) 3 % , e ) 4 %	"explanation : 70 * ( 90 / 100 ) * ( ( 100 - x ) / 100 ) = 61.11 x = 3 % d"

Provide a detailed solution to the given mathematical problem.

if a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster , he would have covered 70 more miles than he actually did . how many more miles would he have covered than he actually did if he had driven 1 hours longer and at an average rate of 10 miles per hour faster on that day ? a ) 100 , b ) 120 , c ) 130 , d ) 150 , e ) 160

"case 1 : let rate = r , time = t , and distance = d so d = rt case 2 : ( d + 70 ) = ( r + 5 ) ( t + 1 ) case 3 : ( d + x ) = ( r + 10 ) ( t + 1 ) x = 130 ans c"

Explain the solution process for the given equation.

what is the ratio g of the surface area of a cube to the surface area of a rectangular solid identical to the cube in all ways except that its length has been doubled ? ['a ) 1 / 4','b ) 3 / 8','c ) 1 / 2','d ) 3 / 5','e ) 2']

one side surface area of a cube = x * x = x ^ 2 total 6 sides = 6 x ^ 2 as for the rectangular , height ( h ) and width ( w ) are same as cube , x . only length = 2 x . l x h = 2 x * x = 2 x ^ 2 - - - - > 4 sides = 2 x ^ 2 * 4 = 8 x ^ 2 w * h = x * x = x ^ 2 - - - - - - > 2 sides = x ^ 2 * 2 = 2 x ^ 2 total 6 sides = 8 x ^ 2 + 2 x ^ 2 = 10 x ^ 2 ratio of cube area to rectangular area g = 6 x ^ 2 / 10 x ^ 2 - - - - > 6 / 10 - - - - > 3 / 5 ( d )

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

the minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time . how much does the clock gain or lose in 12 hours ? a ) 27.273 seconds , b ) 29.273 seconds , c ) 37.273 seconds , d ) 47.273 seconds , e ) 17.273 seconds

after 60 minute rotation the minute hand will make an angle of 30 degree with hour hand . after that let the time required to make an overtake is t minute in this time angle covered by minute hand = ( 360 / 60 ) t = 6 t similarly angle covered by hour hand = ( 30 / 60 ) t = t / 2 so for overtake 6 t = 30 + t / 2 or t = 60 / 11 min = 5 min and 5 / 11 min = 5 min and 300 / 11 sec = 5 min and 27.273 sec so clock loses 27.273 seconds answer : a

Provide a detailed solution to the given mathematical problem.

in a hostel , the number of students decreased by 8 % and the price of food increased by 20 % over the previous year . if each student consumes the same amount of food then by how much should the consumption of food be cut short by every student , so that the total cost of the food remains the same as that of the previous year ? a ) 19 % , b ) 15 % , c ) 25 % , d ) 40 % , e ) 9.4 %

"cost of food ( c ) = food consumed per student ( f ) * number of students ( n ) * price of food ( p ) originally , c = fnp when number of students decrease by 8 % , and the price of food increases by 20 % , c = f ( new ) * ( 0.92 n ) * ( 1.2 p ) = > f ( new ) = f / ( 0.92 * 1.2 ) = > f ( new ) = 0.906 f therefore the new cost of food must be 90.6 % of the old cost , or the cost of food must decrease by 9.4 % answer : e"

Explain the solution process for the given equation.

the value of a machine depreciates at 25 % per annum . if its present value is $ 1 , 50,000 , at what price should it be sold after two years such that a profit of $ 24,000 is made ? a ) $ 258375 , b ) $ 438375 , c ) $ 128375 , d ) $ 108375 , e ) $ 158375

"the value of the machine after two years = 0.75 * 0.75 * 1 , 50,000 = $ 84,375 sp such that a profit of $ 24,000 is made = 84,375 + 24,000 = $ 1 , 08,375 d"

Break down and solve the equation step by step.

a milk man has 15 liters of milk . if he mixes 5 liters of water , which is freely available , in 20 liters of pure milk . if the cost of pure milk is rs . 18 per liter , then the profit of the milkman , when he sells all the mixture at cost price is : a ) 20 % , b ) 25 % , c ) 33.33 % , d ) 18 % , e ) none of these

"explanation : when the water is freely available and all the water is sold at the price of the milk , then the water gives the profit on the cost of 20 liters of milk . therefore , profit percentage = 20 % . answer : a"

Solve the problem systematically, showing each step clearly.

how many cubes of 2 cm edge can be cut out of a cube of 24 cm edge a ) 1136 , b ) 1728 , c ) 1216 , d ) 1484 , e ) none of these

"explanation : number of cubes = ( 24 x 24 x 24 ) / ( 2 x 2 x 2 ) = 1728 answer : b"

Break down and solve the equation step by step.

how many integers from 10 to 180 , inclusive , are divisible by 3 but not divisible by 7 ? a ) 45 , b ) 49 , c ) 50 , d ) 52 , e ) 56

"we should find # of integers divisible by 3 but not by 3 * 7 = 21 . # of multiples of 21 in the range from 10 to 180 , inclusive is ( 168 - 21 ) / 21 + 1 = 8 ; 57 - 8 = 49 . answer : b ."

Break down and solve the equation step by step.

what is the compound interest paid on a sum of rs . 1500 for the period of 2 years at 10 % per annum . a ) 230 , b ) 175 , c ) 225 , d ) 315 , e ) none of these

"solution = interest % for 1 st year = 10 interest % for 2 nd year = 10 + 10 % of 10 = 10 + 10 * 10 / 100 = 11 total % of interest = 10 + 11 = 21 total interest = 21 % 1500 = 1500 * ( 21 / 100 ) = 315 answer d"

Provide a detailed solution to the given mathematical problem.

pipe a fills a tank of capacity 950 liters at the rate of 40 liters a minute . another pipe b fills the same tank at the rate of 30 liters a minute . a pipe at the bottom of the tank drains the tank at the rate of 20 liters a minute . if pipe a is kept open for a minute and then closed and pipe b is open for a minute and then closed and then pipe c is open for a minute and then closed and the cycle is repeated , when will the tank be full ? a ) 57 minutes , b ) 14 minutes , c ) 39 minutes , d ) 40 minutes 20 seconds , e ) none of these

in one cycle they fill 40 + 30 - 20 = 50 liters 950 = 50 * n = > n = 19 here n = number of cycles . total time = 19 * 3 = 57 as in one cycle there are 3 minutes . thus 57 minutes answer : a

Break down and solve the equation step by step.

the cost of registration at a professional association meeting was $ 50 per person ; a lunch for registrants only was available for an additional $ 22 per person . if the number of registrants who paid for lunch was 20 more than the number who did not , and if receipts for registration and lunch totaled $ 50,240 , how many people paid just for registration at the meeting ? a ) 700 , b ) 800 , c ) 1,300 , d ) 1,500 , e ) 400

hope this might be useful to you . let the number of people who have opted only to register = x now since the registration cost is 50 $ per person , the total amount sums to = 50 x $ as per the information given in the question , the number of registrants who paid for lunch was 20 more than the number who did not . that means , total number of people who registered and paid for lunch = 20 + x . for the people who registered for lunch the cost is 50 $ ( for the event registration ) + 22 $ ( for lunch ) = 72 $ . total amount in this case sums to = 72 ( 20 + x ) = 1440 + 72 x now , total amount received was 50240 . thus , from the above data , 50 x + 1440 + 72 x = 50240 122 x = 50240 - 1440 122 x = 48800 x = 400 . hence the correct ans is e

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

at what rate percent of simple interest will a sum of money double itself in 66 years ? a ) 6.54 % , b ) 4.54 % , c ) 1.51 % , d ) 1.94 % , e ) 5.54 %

"let sum = x . then , s . i . = x . rate = ( 100 * s . i . ) / ( p * t ) = ( 100 * x ) / ( x * 66 ) = 100 / 66 = 1.51 % answer : c"

Provide a detailed solution to the given mathematical problem.

in a garden , 26 trees are planted at equal distances along a yard 300 metres long , one tree being at each end of the yard . what is the distance between two consecutive trees ? a ) 10 metres , b ) 20 metres , c ) 14 metres , d ) 12 metres , e ) 13 metres

"between 26 trees , there are 25 gaps length of each gap = 300 / 25 = 12 i . e . , distance between two consecutive trees = 12 answer : d"

Solve the problem systematically, showing each step clearly.

if two integers x , y ( x > y ) are selected from - 5 to 5 ( inclusive ) , how many cases are there ? a ) 55 , b ) 60 , c ) 65 , d ) 70 , e ) 75

"there are 11 integers from - 5 to 5 inclusive . 11 c 2 = 55 . the answer is a ."

Provide a detailed solution to the given mathematical problem.

john and david can finish a job together in 1 hours . if john can do the job by himself in 2 hours , what percent of the job does david do ? a ) 40 % , b ) 45 % , c ) 50 % , d ) 55 % , e ) 60 %

you can also plug in numbers . for example , bob and alice work at a donut factory and make 10 donuts which is the job ( i picked this as a smart number ) . john on his own works 10 / 2 = 5 donuts per hour . john and david work 10 / 1 = 10 donuts per hour so david works 5 donuts / hour to find out the percentage , david works 5 donuts / hr x 1 hours = 5 donuts per hour . therefore 5 donuts / 10 donuts = 1 / 2 = 50 % answer : c

Explain the solution process for the given equation.

a number consists of 3 digits whose sum is 10 . the middle digit is equal to the sum of the other two and the number will be increased by 99 if its digits are reversed . the number is a ) 145 , b ) 185 , c ) 253 , d ) 370 , e ) none

solution let the number be x . then 2 x = 10 or x = 5 . so , the number is either 253 or 352 . since the number increases on reversing the digits , so the hundred' s digit is smaller than the units digit . hence , required number = 253 . answer c

Explain the solution process for the given equation.

excluding stoppages , the average speed of a bus is 60 km / hr and including stoppages , the average speed of the bus is 30 km / hr . for how many minutes does the bus stop per hour ? a ) 22 , b ) 30 , c ) 77 , d ) 20 , e ) 99

"in 1 hr , the bus covers 60 km without stoppages and 30 km with stoppages . stoppage time = time take to travel ( 60 - 30 ) km i . e 30 km at 60 km / hr . stoppage time = 30 / 60 hrs = 30 min . answer : b"

Break down and solve the equation step by step.

( 26.3 * 12 * 20 ) / 3 + 125 = ? a ) 2339 , b ) 2229 , c ) 4429 , d ) 5529 , e ) 2669

b 2229 ? = [ ( 26.3 * 12 * 20 ) / 3 ] + 125 ? = 2104 + 125 = 2229

Explain the solution process for the given equation.

a test has 180 questions . each question has 5 options , but only 1 option is correct . if test - takers mark the correct option , they are awarded 1 point . however , if an answer is incorrectly marked , the test - taker loses 0.25 points . no points are awarded or deducted if a question is not attempted . a certain group of test - takers attempted different numbers of questions , but each test - taker still received the same net score of 40 . what is the maximum possible number of such test - takers ? a ) 29 , b ) 33 , c ) 35 , d ) 40 , e ) 42

"a correct answers get you 1 point , an incorrect answer gets you minus 1 / 4 point and a skipped question gets you 0 points . since there are 200 total questions , there are a variety of ways to get a total of 40 points . let c be the number of correct answers and let i be the number of incorrect answers . to get 40 points , a test taker must have at least 40 correct answers . then c = > 40 . for every correct question above 40 , the test taker has 4 incorrect answers . then , the i = 4 * ( c - 40 ) . also , i + c < = 180 . thus 5 c < = 340 and so c < = 68 . then 40 < = c < = 68 and c can have 29 possible values . the answer is a ."

Explain the solution process for the given equation.

a and b invests rs . 6000 and rs . 9000 in a business . after 4 months , a withdraws half of his capital and 2 months later , b withdraws one - third of his capital . in what ratio should they share the profits at the end of the year ? a ) 32 : 99 , b ) 82 : 31 , c ) 24 : 45 , d ) 34 : 89 , e ) 35 : 21

"a : b ( 6000 * 4 ) + ( 3000 * 8 ) : ( 9000 * 6 ) + ( 6000 * 6 ) 48000 : 90000 24 : 45 answer : c"

Explain the solution process for the given equation.

a candidate appearing for an examination has to secure 42 % marks to pass paper i . but he secured only 42 marks and failed by 22 marks . what is the maximum mark for paper i ? a ) 110 , b ) 120 , c ) 130 , d ) 152 , e ) 150

"he secured 42 marks nd fail by 22 marks so total marks for pass the examinatn = 64 let toal marks x x * 42 / 100 = 64 x = 152 answer : d"

Solve the problem systematically, showing each step clearly.

b completes a work in 8 days . a alone can do it in 10 days . if both work together , the work can be completed in how many days ? a ) 3.75 days , b ) 4.44 days , c ) 5 days , d ) 6.44 days , e ) 7 days

"1 / 8 + 1 / 10 = 9 / 40 40 / 9 = 4.44 days answer : b"

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

if 21 lines are drawn in a plane such that no two of them are parallel and no three are concurrent , then in how many points do they intersect ? a ) 150 , b ) 170 , c ) 190 , d ) 210 , e ) 230

"if two lines are not parallel , then they will intersect in exactly one point . lines can be extended infinitely on both ends so somewhere they will intersect with each other if they are not parallel . we are also given that no three lines are concurrent . this means that no three lines intersect at the same point . so every pair of two lines we select will have a unique point of intersection which they will not share with any third line . the number of ways to select 2 lines from 21 lines is 21 c 2 = 210 the answer is d ."

Solve the problem systematically, showing each step clearly.

what is the remainder when 43 ^ 91 is divided by 5 ? a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4

"the units digit of the exponents of 3 cycle in a group of 4 : { 3 , 9 , 7 , 1 } 91 has the form 4 k + 3 so the units digit of 43 ^ 91 is 7 . the remainder when dividing by 5 is 2 . the answer is c ."

Explain the solution process for the given equation.

how many integerskgreater than 100 and less than 1200 are there such that if the hundreds and the units digits ofkare reversed , the resulting integer is k + 99 ? a ) 50 , b ) 60 , c ) 70 , d ) 80 , e ) 100

"numbers will be like 102 = > 201 = 102 + 99 203 = > 302 = 103 + 99 so the hundereth digit and units digit are consecutive where unit digit is bigger than hundred digit . there will be ten pairs of such numbers for every pair there will 10 numbers like for 12 = > 102 , 112,132 , 142,152 , 162,172 , 182,192 . total = 10 * 10 = 100 hence e"

Solve the problem systematically, showing each step clearly.

the total price of a basic computer and printer are $ 2,500 . if the same printer had been purchased with an enhanced computer whose price was $ 500 more than the price of the basic computer , then the price of the printer would have been 1 / 3 of that total . what was the price of the basic computer ? a ) 1500 , b ) 1600 , c ) 1750 , d ) 1900 , e ) 2000

"let the price of basic computer be c and the price of the printer be p : c + p = $ 2,500 . the price of the enhanced computer will be c + 500 and total price for that computer and the printer will be 2,500 + 500 = $ 3,000 . now , we are told that the price of the printer is 1 / 3 of that new total price : p = 1 / 3 * $ 3,000 = $ 1000 plug this value in the first equation : c + 1000 = $ 2,500 - - > c = $ 1,500 . answer : a ."

Solve the problem systematically, showing each step clearly.

if a 2 + b 2 + c 2 = 941 and ab + bc + ca = 10 , then a + b + c is a ) 44 , b ) 56 , c ) 16 , d ) 32 , e ) 31

"by formula , ( a + b + c ) ^ 2 = a ^ 2 + b ^ 2 + c ^ 2 + 2 ( ab + bc + ca ) , since , a ^ 2 + b ^ 2 + c ^ 2 = 941 and ab + bc + ca = 10 , ( a + b + c ) ^ 2 = 941 + 2 ( 10 ) = 961 = 31 ^ 2 therefore : a + b + c = 31 answer : e"

Provide a detailed solution to the given mathematical problem.

a merchant has 1000 kg of sugar part of which he sells at 8 % profit and the rest at 18 % profit . he gains 14 % on the whole . the quantity sold at 18 % profit is ? a ) 700 kg , b ) 500 kg , c ) 600 kg , d ) 800 kg , e ) 900 kg

"by the rule of alligation : profit of first part profit of second part so , ratio of 1 st and 2 nd parts = 4 : 6 = 2 : 3 . quantity of 2 nd kind = ( 3 / 5 x 1000 ) kg = 600 kg answer : c"

Provide a detailed solution to the given mathematical problem.

the total marks obtained by a student in mathematics and physics is 20 and his score in chemistry is 20 marks more than that in physics . find the average marks scored in mathamatics and chemistry together . a ) 40 , b ) 26 , c ) 20 , d ) 28 , e ) 22

"let the marks obtained by the student in mathematics , physics and chemistry be m , p and c respectively . given , m + c = 20 and c - p = 20 m + c / 2 = [ ( m + p ) + ( c - p ) ] / 2 = ( 20 + 20 ) / 2 = 20 . answer : c"

Provide a detailed solution to the given mathematical problem.

the price of a t . v . set worth rs . 8000 is to be paid in 20 installments of rs . 750 each . if the rate of interest be 6 % per annum , and the first installment be paid at the time of purchase , then the value of the last installment covering the interest as well will be ? a ) 7250 , b ) 7200 , c ) 7820 , d ) 6000 , e ) 2799

"money paid in cash = rs . 750 balance payment = ( 8000 - 1000 ) = rs . 7250 answer : a"

Provide a detailed solution to the given mathematical problem.

a shopkeeper sold an article offering a discount of 10 % and earned a profit of 25 % . what would have been the percentage of profit earned if no discount was offered ? a ) 39 % , b ) 52 % , c ) 15 % , d ) 21 % , e ) 28 %

"let c . p . be $ 100 . then , s . p . = $ 125 let marked price be $ x . then , 90 / 100 * x = 125 x = 12500 / 90 = $ 139 now , s . p . = $ 139 , c . p . = rs . 100 profit % = 39 % . answer : a"

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

one side of the bottom layer of a triangular pyramid has 12 balls . how many are there in the whole pyramid ? note that the pyramid is equilateral and solid . ['a ) 286','b ) 364','c ) 504','d ) 164','e ) 165']

as there are 12 balls along one side , it means that there are 12 layers of balls . the top most layer has 1 ball . the second layer has 3 ( 1 + 2 ) balls . the third layer has 6 ( 1 + 2 + 3 ) balls . the fourth layer has 10 ( 1 + 2 + 3 + 4 ) balls . the fifth layer has 15 ( 1 + 2 + 3 + 4 + 5 ) balls . similarly , there are 21 , 28 , 36 , 45 , 55 , 66 and 78 balls in the remaining layers . hence , the total number of balls are = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 balls answer : b

Break down and solve the equation step by step.

if n = 2 ^ 0.25 and n ^ b = 8 , b must equal a ) 3 / 80 , b ) 3 / 5 , c ) 12 , d ) 5 / 3 , e ) 80 / 3

"25 / 100 = 1 / 4 n = 2 ^ 1 / 4 n ^ b = 2 ^ 3 ( 2 ^ 1 / 4 ) ^ b = 2 ^ 3 b = 12 answer : c"

Solve the problem systematically, showing each step clearly.

susan finds that she spends her free time during summer vacation between swimming , reading , and hanging out with friends in a ratio of 1 : 4 : 10 . how many hours did she spend reading if she hung out with her friends for 20 hours ? a ) 2 , b ) 4 , c ) 5 , d ) 8 , e ) 10

the ratio is 1 hour swimming : 4 hours reading : 10 hours hanging out with friends . divide 20 by 10 to find 1 ` ` part'' of the ratio . 20 / 10 = 2 multiply this by 4 to find the hours spent reading . 2 * 4 = 8 the answer is d .

Explain the solution process for the given equation.

20 ^ 10 / 120 ^ 5 = ? a ) 6 ^ 5 , b ) 5 ^ 6 , c ) ( 10 / 3 ) ^ 5 , d ) 6 ^ 3 , e ) 15 ^ 3

"20 ^ 10 / 120 ^ 5 = ? a . 6 ^ 5 b . 5 ^ 6 c . ( 10 / 3 ) ^ 5 d . 6 ^ 3 e . 15 ^ 3 - > 20 ^ 10 / 120 ^ 5 = ( 20 ^ 10 ) / ( 6 ^ 5 ) ( 20 ^ 5 ) = ( 20 ^ 5 ) / ( 6 ^ 5 ) = ( 2 ^ 5 ) ( 10 ^ 5 ) / ( 2 ^ 5 ) ( 3 ^ 5 ) = ( 10 / 3 ) ^ 5 . thus , c is the answer ."

Provide a detailed solution to the given mathematical problem.

the amount of principal rs . 8000 at compound interest at the ratio of 5 % p . a . for 6 years is a ) s . 9621 , b ) s . 6921 , c ) s . 10721 , d ) s . 6261 , e ) s . 6361

"c . i = p ( 1 + r / 100 ) ^ n = 8000 ( 1 + 5 / 100 ) ^ 6 = rs 10721 answer : c"

Explain the solution process for the given equation.

an alloy weighing 48 ounces is 25 % gold . how many ounces of pure gold must be added to create an alloy that is 40 % gold ? a ) 15 , b ) 20 , c ) 12 , d ) 14 , e ) 7

an alloy of 48 oz which is 25 % gold means there is 12 oz of gold . to get to an alloy that is 40 % gold , let' s use this expression : ( 12 + x ) / ( 48 + x ) = 0.40 with x representing the amount of pure gold that must be added to get to 40 % . the expression we are using represents the new total weight of pure gold over the new total weight of the alloy and this fraction should represent 40 % or 0.4 . you will see that 12 is the correct answer , as 24 / 60 = 0.4 choose c

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

a and b can do a piece of work in 4 days . b and c can do it in 12 days and a and c in 16 days . working together they will complete the work in how many days ? a ) 4.05 days , b ) 5.05 days , c ) 14.76 days , d ) 8.05 days , e ) 15.05 days

"a + b = 1 / 4 b + c = 1 / 12 c + a = 1 / 16 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2 ( a + b + c ) = 1 / 4 + 1 / 12 + 1 / 16 = 19 / 48 a + b + c = 19 / 96 96 / 19 = 5.05 days answer : b"

Break down and solve the equation step by step.

how many pieces of 75 cm can be cut from a rope 57 meters long ? a ) 30 , b ) 40 , c ) 76 , d ) none , e ) can not be determined

"explanation : total pieces of 75 cm that can be cut from a rope of 57 meters long is = ( 57 meters ) / ( 75 cm ) = ( 57 meters ) / ( 0.75 meters ) = 76 answer : c"

Provide a detailed solution to the given mathematical problem.

the sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 480 m , its area is ? a ) 150 , b ) 240 , c ) 277 , d ) 261 , e ) 281

"5 x + 12 x + 13 x = 480 = > x = 16 a = 80 , b = 192 , c = 208 s = ( 80 + 192 + 208 ) / 2 = 240 answer : b"

Provide a detailed solution to the given mathematical problem.

the area of an isosceles trapezoid with sides of length 5 and bases of length 6 and 12 is ? a ) 44 , b ) 42 , c ) 40 , d ) 38 , e ) 36

"isosceles trapezoid indicates that if we draw perpendicular from two vertex of the smaller side , the longer side with side 12 will be divided into 3 parts = 6 , 3 and 3 ( making one square and two right triangles ) for the right triangle , height will be = ( 5 ^ 2 - 3 ^ 2 ) ^ 1 / 2 = 4 and the area of trapezoid = 1 / 2 ( b 1 + b 2 ) h = 1 / 2 * 18 * 4 = 36 e"

Provide a detailed solution to the given mathematical problem.

a cistern which could be filled in 9 hours takes 1 hour more to be filled owing to a leak in its bottom . if the cistern is full , in what time will the leak empty it ? a ) 50 hours , b ) 62 hours , c ) 90 hours , d ) 75 hours , e ) 100 hours

"let the leak empty the full cistern in x hours 9 x / x - 9 = 9 + 1 x = 90 hours answer is c"

Explain the solution process for the given equation.

a call center has two teams . each member of team a was able to process 2 / 5 calls as compared to each member of team b . if team a has 5 / 8 as many number of call center agents as team b , what fraction of the total calls was processed by team b ? a ) 3 / 2 , b ) 3 / 4 , c ) 4 / 5 , d ) 1 / 2 , e ) 1 / 5

let team b has 8 agents , so team a has 5 agents let each agent of team b picked up 5 calls , so total calls by team b = 40 so , each agent in team a picked up 2 calls , so total calls for team a = 10 fraction for team b = 40 / ( 40 + 10 ) = 4 / 5 = answer = c

Explain the solution process for the given equation.

of the 720 students at a certain university , 25 percent are seniors . if x seniors were to graduate early and leave the university and no additional students entered or left the university , what value of x would reduce the number of seniors at the university to 20 percent ? a ) 30 , b ) 36 , c ) 42 , d ) 45 , e ) 48

sol : 180 - x = 20 % of ( 720 - x ) 4 x = 180 x = 45 answer : d

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

if 2 is the remainder when m is divided by 5 , what is the remainder when 3 m is divided by 5 ? a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 4

let' s find a number which when divided by 5 leaves remainder 2 = 5 + 2 = 7 i . e . one possible value of m = 7 value of 3 m = 3 * 7 = 21 when em is divided by 5 then remainder = rem ( 21 / 5 ) = 1 answer : option b

Provide a detailed solution to the given mathematical problem.

two sides of a triangle have lengths x and w and meet at a right angle . if the perimeter of the triangle is 4 x , what is the ratio of x to w ? ['a ) a ) 2 : 3','b ) b ) 3 : 4','c ) c ) 4 : 3','d ) d ) 3 : 2','e ) e ) 2 : 1']

ahhhh . . . . my bad . thank you , you' re correct . 8 x = 6 w - - > x / w = 3 / 4 . actually b is the right answer .

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

in a two digit number it is known that its unit' s digit exceeds its 10' s digit by 4 and that the product of the given number and the sum of its digits is equal to 208 , what is the number ? a ) 24 , b ) 25 , c ) 26 , d ) 27 , e ) 28

xy is 2 digit no . y - x = 4 ; ( 10 x + y ) * ( 2 x + 4 ) = 208 ; sub y = x + 4 ; we get 26 answer : c

Provide a detailed solution to the given mathematical problem.

if n is a prime number greater than 5 , what is the remainder when n ^ 2 is divided by 23 ? a ) 0 , b ) 1 , c ) 2 , d ) 3 , e ) 5

"there are several algebraic ways to solve this question including the one under the spoiler . but the easiest way is as follows : since we can not have two correct answersjust pick a prime greater than 5 , square it and see what would be the remainder upon division of it by 23 . n = 5 - - > n ^ 2 = 49 - - > remainder upon division 49 by 23 is 3 . answer : d ."

Break down and solve the equation step by step.

a retailer buys a radio for rs 225 . his overhead expenses are rs 28 . he sellis the radio for rs 300 . the profit percent of the retailer is a ) 10 % , b ) 50 % , c ) 25 % , d ) 52 % , e ) 18.5 %

"explanation : cost price = ( 225 + 28 ) = 253 sell price = 300 gain = ( 47 / 253 ) * 100 = 18.5 % . answer : e"

Explain the solution process for the given equation.

a lent rs . 5000 to b for 2 years and rs . 3000 to c for 4 years on simple interest at the same rate of interest and received rs . 3300 in all from both of them as interest . the rate of interest per annum is ? a ) 16 % , b ) 12 % , c ) 74 % , d ) 10 % , e ) 15 %

"let the rate be r % p . a . then , ( 5000 * r * 2 ) / 100 + ( 3000 * r * 4 ) / 100 = 3300 100 r + 120 r = 3300 r = 15 % answer : e"

Solve the problem systematically, showing each step clearly.

an equilateral triangle t 2 is formed by joining the mid points of the sides of another equilateral triangle t 1 . a third equilateral triangle t 3 is formed by joining the mid - points of t 2 and this process is continued indefinitely . if each side of t 1 is 50 cm , find the sum of the perimeters of all the triangles . a ) 180 cm , b ) 220 cm , c ) 240 cm , d ) 270 cm , e ) 300 cm

"we have 50 for first triangle , when we join mid - points of first triangle we get the second equilateral triangle then the length of second one is 25 and continues . so we have 50,25 , 12.5 , . . . we have ratio = 1 / 2 , and it is gp type . sum of infinite triangle is a / 1 - r = 50 / 1 - ( 1 / 2 ) = 100 equilateral triangle perimeter is 3 a = 3 * 100 = 300 . so option e ."

Break down and solve the equation step by step.

when a natural number n is successively divided by 12 , 34 the remainders are 56 , 78 . what will be the sum of the remainders if the order of the division is reversed ? a ) 11 , b ) 12 , c ) 10 , d ) 2 , e ) 7

1 2 3 4 5 6 7 8 leave the top right - most number 8 start with bottom right - most number 5 8 * 3 + 7 = 31 31 * 2 + 6 = 68 68 * 1 + 5 = 73 this is the number required now , do the successive division in the reverse order the sum of the remainders is 10 hence , the correct option is c

Explain the solution process for the given equation.

johnny travels a total of one hour to and from school . on the way there he jogs at 5 miles per hour and on the return trip he gets picked up by the bus and returns home at 25 miles per hour . how far is it to the school ? a ) 2 miles , b ) 4 miles , c ) 6.6 miles , d ) 8 miles , e ) 10 miles

answer : c ) 6.6 miles . average speed for round trip = 2 * a * b / ( a + b ) , where a , b are speeds so , average speed was = 2 * 5 * 25 / ( 5 + 25 ) = 6.6 m / hr the distance between schoolhome should be half of that . ie . 6.6 miles answer c

Break down and solve the equation step by step.

( x + 7 ) is a factor in x ^ 2 - mx - 35 . what is the value of m ? a ) - 3 , b ) - 2 , c ) - 4 , d ) - 5 , e ) - 6

"i solved the second degree equation and found it like this : x ^ 2 - mx - 35 ( x - 5 ) ( x + 7 ) x = 5 or x = - 7 substituting both values for x in the equation we find : x ^ 2 - mx - 35 = > ( - 7 ) ^ 2 - m ( - 7 ) = 35 = > 49 + 7 m = 35 = > 7 m = 35 - 49 = - 14 = > m = - 2 and with 5 , using a similar process we end up with : ( 5 ) ^ 2 - m ( 5 ) = 35 - 10 = 5 m m = - 2 ao , ans b"

Break down and solve the equation step by step.

rounded to 3 decimal places , 1.003 ^ 4 = a ) 1.004 , b ) 1.006 , c ) 1.008 , d ) 1.012 , e ) 1.016

as compared to 1 , 0.003 is a very small quantity . thus , we can write ( 1 + 0.003 ) ^ 4 is nearly equal to ( 1 + 4 * 0.003 ) = 1.012 . as the question asks for approximation to three decimal places , the further terms will anyways not come into picture . d .

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

the mean of 40 observations was 100 . it was found later that an observation 50 was wrongly taken as 75 the corrected new mean is a ) 125.8 , b ) 102.8 , c ) 99.375 , d ) 100.52 , e ) 122

"explanation : correct sum = ( 100 * 40 + 50 - 75 ) = 3975 . correct mean = = 3975 / 40 = 99.375 answer : c"

Break down and solve the equation step by step.

in a college , the ratio of the number of boys to girls is 8 : 5 . if there are 200 girls , the total number of students in the college is a ) 420 , b ) 520 , c ) 620 , d ) 620 , e ) none of these

explanation : let the boy are 8 x and girls are 5 x = > 5 x = 200 = > x = 40 total students = 8 x + 5 x = 13 x = 13 ( 40 ) = 520 answer : b

Break down and solve the equation step by step.

how many x ways can you group 3 people from 4 sets of twins if no two people from the same set of twins can be chosen ? a ) 3 , b ) 16 , c ) 28 , d ) 32 , e ) 56

ways to select 3 people from 8 people ( 4 twins x 2 ) = 8 c 3 = 56 ways to select 1 twin + 1 people = 4 c 1 * 6 c 1 = 24 ways to select a group 3 people from 4 sets of twins if no two people from the same set of twins can be chosen x = 56 - 24 = 32 ans : d

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

ram covers a part of the journey at 20 kmph and the balance at 70 kmph taking total of 8 hours to cover the distance of 400 km . how many hours has been driving at 20 kmph ? a ) 3.2 hr , b ) 3.4 hr , c ) 3.6 hr , d ) 3.8 hr , e ) 4.2 hr

let see d / t = v then x / 20 + ( 400 - x ) / 70 = 8 x = 64 kmh t = 64 / 20 = 3.2 hr answer : a

Break down and solve the equation step by step.

a can finish a work in 10 days and b can do same work in half the time taken by a . then working together , what part of same work they can finish in a day ? a ) 1 / 5 , b ) 1 / 6 , c ) 1 / 7 , d ) 3 / 10 , e ) none of these

"explanation : please note in this question , we need to answer part of work for a day rather than complete work . it was worth mentioning here because many do mistake at this point in hurry to solve the question so lets solve now , a' s 1 day work = 1 / 10 b' s 1 day work = 1 / 5 [ because b take half the time than a ] ( a + b )' s one day work = ( 1 / 10 + 1 / 5 ) = 3 / 10 so in one day 3 / 10 work will be done answer : d"

Provide a detailed solution to the given mathematical problem.

pascal has 96 miles remaining to complete his cycling trip . if he reduced his current speed by 4 miles per hour , the remainder of the trip would take him 16 hours longer than it would if he increased his speed by 50 % . what is his current speed z ? a ) 6 , b ) 8 , c ) 10 , d ) 12 , e ) 16

"let the current speed be x miles per hour . time taken if speed is 50 % faster ( i . e . 3 x / 2 = 1.5 x ) = 96 / 1.5 x time taken if speed is reduced by 4 miles / hr ( i . e . ( x - 4 ) ) = 96 / ( x - 4 ) as per question , 96 / ( x - 4 ) - 96 / 1.5 x = 16 solving this z we get x = 8 . b ."

Solve the problem systematically, showing each step clearly.

carol and jordan draw rectangles of equal area . if carol' s rectangle measures 5 inches by 24 inches and jordan' s rectangle is 3 inches long , how wide is jordan' s rectangle , in inches ? a ) 25 , b ) 43 , c ) 42 , d ) 40 , e ) 18

"area of carol' s rectangle = 24 * 5 = 120 let width of jordan' s rectangle = w since , the areas are equal 3 w = 120 = > w = 40 answer d"

Provide a detailed solution to the given mathematical problem.

one hour after adrienne started walking the 60 miles from x to y , william started walking from x to y as well . adrienne walks 3 miles per hour and william walks 1 mile per hour faster than adrienne . how far from x will william be when he catches up to adrienne ? a ) 8 miles , b ) 9 miles , c ) 10 miles , d ) 11 miles , e ) 12 miles

first , determine how far adrienne has walked in the one hour . she has walked three miles which means she is three miles ahead of william when he sets off . william walks at four miles / hour which means that every hour , william will get one mile closer to adrienne . if he gets one mile closer every hour , it will take him three hours to catch up to her which means he travels 3 hours * 4 miles / hour = 12 miles and she travels 4 hours * 3 miles / hour = 12 miles . he will be 12 miles from x when he catches up to her . a slightly different way to solve . . . we do n' t know how long they will walk before they catch up to one another but we do know that a walks for one hour more than j . j = t and a = t + 1 . we are looking for the distance at which they meet up which means the distance will be the same . d = r * t so , r * t ( william ) = r * t ( adrienne ) r * ( t ) = r * ( t + 1 ) 4 t = 3 t + 3 t = 3 d = r * t d = 4 * 3 d = 12 d ) 12 miles

Provide a detailed solution to the given mathematical problem.

for every even positive integer m , f ( m ) represents the product of all even integers from 2 to m , inclusive . for example , f ( 12 ) = 2 x 4 x 6 x 8 x 10 x 12 . what is the greatest prime factor of f ( 28 ) ? a ) 2 , b ) 7 , c ) 11 , d ) 13 , e ) 17

"f ( 28 ) = 2 * 4 * 6 * 8 * 10 * 12 * 14 * 16 * 18 * 20 * 22 * 24 * 26 * 28 the greatest prime factor in this list is 13 . the answer is d ."

Explain the solution process for the given equation.

how many numbers from 2 to 15 are exactly divisible by 2 ? a ) a ) 2 , b ) b ) 3 , c ) c ) 5 , d ) d ) 7 , e ) e ) 8

"2 / 2 = 1 and 15 / 2 = 7 7 - 1 = 6 6 + 1 = 7 numbers . answer : d"

Solve the problem systematically, showing each step clearly.

the side of a square is increased by 30 % then how much % does its area increases ? a ) 52.65 , b ) 69 , c ) 50.75 , d ) 42.75 , e ) 52.75

"a = 100 a 2 = 10000 a = 130 a 2 = 16900 - - - - - - - - - - - - - - - - 10000 - - - - - - - - - 6900 100 - - - - - - - ? = > 69 % answer : b"

Provide a detailed solution to the given mathematical problem.

average weight of 10 people increased by 3 kg when one person of 45 kg is replaced by a new man . then weight of the new man is a ) 50 , b ) 55 , c ) 60 , d ) 65 , e ) 75

explanation : total weight increased is 3 * 10 = 30 . so weight of new person is 45 + 30 = 75 answer : option e

Provide a detailed solution to the given mathematical problem.

the first half of the 20 th century , the population of a particular country increased by 200 percent . in the second half of the century , the population increased by 300 percent . what was the percent increase for the 20 th century as a whole ? a ) 500 % , b ) 600 % , c ) 800 % , d ) 1100 % , e ) 1200 %

say initially population was 100 . what is 200 % of 100 ? it is 200 / 100 * 100 = 200 . an increase of 200 % means the new population became 100 + 200 = 300 what is 300 % of 300 ? it is 300 / 100 * 300 = 900 an increase of 300 % means the new population now is 300 + 900 = 1200 so from 100 , the population increased to 1200 i . e . an increase of 1100 . 1100 is what percent of 100 ? 1100 = x / 100 * 100 i . e . it is 1100 % d

Break down and solve the equation step by step.

a man engaged a servant on the condition that he would pay him rs . 900 and a uniform after 1 year service . he served only for 9 months and receiveduniform and rs . 650 , find the priceof the uniform ? a ) rs . 90 , b ) rs . 100 , c ) rs . 120 , d ) rs . 150 , e ) rs . 160

"9 / 12 = 3 / 4 * 900 = 675 650 - - - - - - - - - - - - - 25 1 / 4 - - - - - - - - 25 1 - - - - - - - - - ? = > rs . 100 b"

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

a train which has 600 m long , is running 60 kmph . in what time will it cross a person moving at 6 kmph in same direction ? a ) 42 sec , b ) 44 sec , c ) 46 sec , d ) 40 sec , e ) 50 sec

"time taken to cross a moving person = length of train / relative speed time taken = 600 / ( ( 60 - 6 ) ( 5 / 18 ) = 600 / 54 * ( 5 / 18 ) = 600 / 15 = 40 sec answer : d"

Explain the solution process for the given equation.

if the cost price of 50 articles is equal to the selling price of 35 articles , then the gain or loss percent is ? a ) 16 , b ) 127 , c ) 12 , d ) 18 , e ) 43

"percentage of profit = 15 / 35 * 100 = 43 % answer : e"

Solve the problem systematically, showing each step clearly.

a money lender finds that due to a fall in the annual rate of interest from 8 % to 7 3 / 4 % his yearly income diminishes by rs . 64.50 , his capital is ? a ) 24,602 , b ) 28,888 , c ) 25,800 , d ) 24,628 , e ) 24,6012

"let the capital be rs . x . then , ( x * 8 * 1 ) / 100 - ( x * 31 / 4 * 1 / 100 ) = 64.50 32 x - 31 x = 6450 * 4 x = 25,800 . answer : c"

Explain the solution process for the given equation.

at what price must an article costing rs . 66.5 be marked in order that after deducting 5 % from the list price . it may be sold at a profit of 25 % on the cost price ? a ) 87.5 , b ) 89.5 , c ) 82.3 , d ) 82.1 , e ) 92.4

"cp = 66.5 sp = 66.5 * ( 125 / 100 ) = 83.125 mp * ( 95 / 100 ) = 59.375 mp = 87.5 answer : a"

Provide a detailed solution to the given mathematical problem.

jack , jill , and sandy each have one try to make a basket from half court . if their individual probabilities of making the basket are 1 / 6 , 1 / 7 , and 1 / 8 respectively , what is the probability that jack and jill will make a basket but sandy will miss ? a ) 1 / 8 , b ) 1 / 16 , c ) 1 / 48 , d ) 1 / 96 , e ) 1 / 168

"the probability that jack and jill will make a basket but sandy will miss is 1 / 6 * 1 / 7 * 7 / 8 = 1 / 48 . the answer is c ."

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

divide rs . 32000 in the ratio 3 : 7 ? a ) 12000 , 20000 , b ) 9600 , 22400 , c ) 12000 , 20007 , d ) 12000 , 20006 , e ) 12000 , 20001

3 / 10 * 32000 = 9600 7 / 8 * 32000 = 22400 answer : b

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

a and b complete a work in 3 days . a alone can do it in 9 days . if both together can do the work in how many days ? a ) 3.75 days , b ) 4.75 days , c ) 3.25 days , d ) 3.15 days , e ) 2.75 days

"1 / 3 + 1 / 9 = 4 / 9 9 / 4 = 4.75 days answer : b"

Provide a detailed solution to the given mathematical problem.

a watch was sold at a loss of 10 % . if it was sold for rs . 210 more , there would have been a gain of 4 % . what is the cost price ? a ) s . 1000 , b ) s . 1009 , c ) s . 1007 , d ) s . 1006 , e ) s . 1500

explanation : 90 % 104 % - - - - - - - - 14 % - - - - 210 100 % - - - - ? = > rs . 1500 answer : e

Explain the solution process for the given equation.

a leak in the bottom of a tank can empty the full tank in 6 hours . an inlet pipe fills water at the rate of 3.5 liters per minute . when the tank is full in inlet is opened and due to the leak the tank is empties in 8 hours . the capacity of the tank is ? a ) 5729 , b ) 5040 , c ) 2889 , d ) 2870 , e ) 2799

"1 / x - 1 / 6 = - 1 / 8 x = 24 hrs 24 * 60 * 3.5 = 5040 . answer : b"

Explain the solution process for the given equation.

sonika deposited rs . 8000 which amounted to rs . 9200 after 3 years at simple interest . had the interest been 2.5 % more . she would get how much ? a ) 9800 , b ) 96288 , c ) 26667 , d ) 1662 , e ) 2882

"( 8000 * 3 * 2.5 ) / 100 = 600 9200 - - - - - - - - 9800 answer : a"

Break down and solve the equation step by step.

in 10 years , a will be twice as old as b was 10 years ago . if a is now 4 years older than b the present age of b is a ) 37 , b ) 38 , c ) 39 , d ) 34 , e ) 41

let present age of a be a and b be b a + 10 = 2 * ( b - 10 ) = > 2 b - a = 30 . . . . . . ( i ) a = b + 4 = > 2 b - b - 4 = 30 b = 34 so the present age of b is 34 years answer : d

Explain the solution process for the given equation.

in a lake , there is a patch of lily pads . every day , the patch doubles in size . it takes 50 days for the patch to cover the entire lake , how many days would it take the patch to cover half of the lake ? a ) 36 , b ) 2 ^ 4 * 3 , c ) 24 , d ) 49 , e ) 47

"working backward from the day it' s covered : day 50 : fully covered day 49 : half covered so 49 days answer : d"

Explain the solution process for the given equation.

a jogger running at 9 kmph alongside a railway track in 240 meters ahead of the engine of a 120 metres long train running at 45 kmph in the same direction . in how much time will the train pass the jogger ? a ) 18 seconds , b ) 26 seconds , c ) 36 seconds , d ) 16 seconds , e ) 46 seconds

total distance ( d ) = 240 + 120 = 360 meter relative speed ( s ) = 45 - 9 = 36 kmph = 36 * 5 / 18 = 10 mps d = s * t 360 = 10 * t t = 36 seconds answer : c

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

find the total number of prime factors in the expression ( 4 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 a ) 26 , b ) 22 , c ) 25 , d ) 27 , e ) 33

( 4 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 = ( 2 x 2 ) ^ 13 x ( 7 ) ^ 5 x ( 11 ) ^ 2 = 2 ^ 13 x 2 ^ 13 x 7 ^ 5 x 11 ^ 2 = 2 ^ 26 x 7 ^ 5 x 11 ^ 2 total number of prime factors = ( 26 + 5 + 2 ) = 33 . answer is e .

Provide a detailed solution to the given mathematical problem.

find the missing figures : 0.3 % of ? = 0.15 a ) 50 , b ) 45 , c ) 150 , d ) 450 , e ) 500

let 0.3 % of x = 0.15 . then , 0.30 * x / 100 = 0.15 x = [ ( 0.15 * 100 ) / 0.3 ] = 50 . answer is a .

Break down and solve the equation step by step.

in a rectangular axis system , what is the area of a parallelogram with the coordinates : ( 5,3 ) , ( 8,3 ) , ( 4,5 ) , ( 7,5 ) ? a ) 21 . , b ) 28 . , c ) 35 . , d ) 49 . , e ) 6 .

"delta x will give us the dimension of one side of the parallelogram = 8 - 5 = 3 unit delta y will give us the dimension of the other side of parallelogram = 5 - 3 = 2 unit area of parallelogram = 3 * 2 = 6 answer is e"

Solve the problem systematically, showing each step clearly.

a certain company has records stored with a record storage firm in 15 - inch by 12 - inch by 10 - inch boxes . the boxes occupy 1.08 million cubic inches of space . if the company pays $ 0.4 per box per month for the record storage , what is the total amount that the company pays each month for record storage ? a ) a . 150 , b ) b . 300 , c ) c . 600 , d ) d . 240 , e ) e . 200

"volume per box : 15 x 12 x 10 = 1,800 total volume : 1 , 080,000 number of boxes : total volume / volume per box = 1 , 080,000 / 1,800 = 600 price per month : number of boxes * price per box = 600 * 0.4 = 240 answer : d"

Provide a detailed solution to the given mathematical problem.

two trains 110 m and 160 m long run at the speed of 60 km / hr and 40 km / hr respectively in opposite directions on parallel tracks . the time which they take to cross each other is ? a ) 10.9 , b ) 10.7 , c ) 10.3 , d ) 10.8 , e ) 9.72

"relative speed = 60 + 40 = 100 km / hr . = 100 * 5 / 18 = 250 / 9 m / sec . distance covered in crossing each other = 110 + 160 = 270 m . required time = 270 * 9 / 250 = 243 / 25 = 9.72 sec .' answer : e"

Solve the problem systematically, showing each step clearly.

fresh grapes contain 70 % by weight while dried grapes contain 10 % water by weight . what is the weight of dry grapes available from 100 kg of fresh grapes ? a ) 18.33 kg , b ) 21.50 kg , c ) 25.00 kg , d ) 33.33 kg , e ) 40.39 kg

"from the question we know : 100 kg * 70 % = 70 kg of water in the fresh grapes 100 kg - 70 kg of water = 30 kg of non - water mass we are looking for the weight of the dry grapes ( x ) . since the question tells us that 10 % of the weight of the dry graps is water and we know that 30 kg is non - water mass we can set up the following equation : x = 1 / 10 ( x ) + 30 kg 9 / 10 ( x ) = 30 kg x = 33.33 kg answer - d"

Break down and solve the equation step by step.

what is the sum of all 3 digit numbers that leave a remainder of' 2' when divided by 5 ? a ) 897 , b ) 98,910 , c ) 64,749 , d ) 49,700 , e ) 56,720

find the number , upon sum of 3 digits of a number gives a reminder 2 when it is divided by 5 seeing the options after dividing and finding the reminder of 2 my answer was b

Explain the solution process for the given equation.

how many 5 digit number contain number 3 ? a ) 31512 , b ) 32512 , c ) 33512 , d ) 34512 , e ) 37512

"total 5 digit no . = 9 * 10 * 10 * 10 * 10 = 90000 not containing 3 = 8 * 9 * 9 * 9 * 9 = 52488 total 5 digit number contain 3 = 90000 - 52488 = 37512 answer : e"

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

the manufacturing cost of a shoe is rs . 190 and the transportation lost is rs . 500 for 100 shoes . what will be the selling price if it is sold at 20 % gains a ) s 234 , b ) s 216 , c ) s 220 , d ) s 210 , e ) s 217

"explanation : total cost of a watch = 190 + ( 500 / 100 ) = 195 . gain = 20 % = > sp = 1.2 cp = 1.2 x 195 = 234 answer : a"

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

what is the maximum number t of 27 cubic centimetre cubes that can fit in a rectangular box measuring 8 centimetre x 9 centimetre x 12 centimetre ? ['a ) 36','b ) 32','c ) 24','d ) 21','e ) 15']

27 cubic centimetre cubes gives side = 3 cm so if : l * w * h is 9 * 12 * 8 , then max . cube we can have are 3 * 4 * 2 = 24 l * w * h is 9 * 8 * 12 , then max . cube we can have are 3 * 2 * 4 = 24 l * w * h is 12 * 8 * 9 , then max . cube we can have are 4 * 2 * 3 = 24 l * w * h is 12 * 9 * 8 , then max . cube we can have are 4 * 3 * 2 = 24 l * w * h is 8 * 12 * 9 , then max . cube we can have are 2 * 4 * 3 = 24 l * w * h is 8 * 9 * 12 , then max . cube we can have are 2 * 3 * 4 = 24 in all cases we get t = 24 cubes . ans . c

Explain the solution process for the given equation.

how many multiples of 8 are there between 100 and 200 ( both are inclusive ) ? a ) 27 , b ) 4 , c ) 6 , d ) 13.5 , e ) 8

"the answer is ( 200 - 100 ) / 8 + 1 = 13.5 answer is d"

Solve the problem systematically, showing each step clearly.

a train passes a station platform in 36 sec and a man standing on the platform in 22 sec . if the speed of the train is 54 km / hr . what is the length of the platform ? a ) 288 , b ) 210 , c ) 881 , d ) 1277 , e ) 121

"speed = 54 * 5 / 18 = 15 m / sec . length of the train = 15 * 22 = 330 m . let the length of the platform be x m . then , ( x + 330 ) / 36 = 15 = > x = 210 m . answer : b"

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

a train 140 m long is running with a speed of 50 km / hr . in what time will it pass a man who is running at 4 km / hr in the direction opposite to that in which the train is going ? a ) 8.1 , b ) 7.0 , c ) 9.33 , d ) 8 , e ) 9

"speed of train relative to man = 50 + 4 = 54 km / hr . = 54 * 5 / 18 = 15 m / sec . time taken to pass the men = 140 * 1 / 15 = 9.33 sec . answer : option c"

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

in the xy - plane , a triangle has vertices ( 0,0 ) , ( 4,0 ) and ( 4,7 ) . if a point ( a , b ) is selected at random from the triangular region , what is the probability that a - b > 0 ? a ) 4 / 7 , b ) 2 / 3 , c ) 3 / 5 , d ) 3 / 7 , e ) 1 / 2

"the area of the right triangle is ( 1 / 2 ) * 4 * 7 = 14 . only the points ( a , b ) below the line y = x satisfy a - b > 0 . the part of the triangle which is below the line y = x has an area of ( 1 / 2 ) ( 4 ) ( 4 ) = 8 . p ( a - b > 0 ) = 8 / 14 = 4 / 7 the answer is a ."

Explain the solution process for the given equation.

the cash difference between the selling prices of an article at a profit of 4 % and 6 % is rs 3 . the ratio of two selling prices is a ) 51 : 52 , b ) 52 : 53 , c ) 53 : 54 , d ) 54 : 55 , e ) none of these

"explanation : let the cost price of article is rs . x required ratio = 104 % of x / 106 % of x = 104 / 106 = 52 / 53 = 52 : 53 option b"

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

in what time will a train 300 meters long cross an electric pole , if its speed is 90 km / hr a ) 5 seconds , b ) 12 seconds , c ) 3 seconds , d ) 2.5 seconds , e ) none of these

"explanation : first convert speed into m / sec speed = 90 * ( 5 / 18 ) = 25 m / sec time = distance / speed = 300 / 25 = 12 seconds answer : b"

Provide a detailed solution to the given mathematical problem.

evaluate 2 + 22 + 222 + 2.22 a ) 269.72 , b ) 268.22 , c ) 248.21 , d ) 248.22 , e ) 268.21

"2 + 22 + 222 + 2.22 = 248.22 option d"

Solve the problem systematically, showing each step clearly.

the list price of an article is rs . 70 . a customer pays rs . 61.11 for it . he was given two successive discounts , one of them being 10 % . the other discount is ? a ) 8 % , b ) 7 % , c ) 10 % , d ) 3 % , e ) 4 %

"explanation : 70 * ( 90 / 100 ) * ( ( 100 - x ) / 100 ) = 61.11 x = 3 % d"