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.	two airplanes fly the same exact route from atlanta to chicago . plane a flies 200 mph while plane b flies 300 mph (assume that simple model versus actual acceleration / deceleration near the airports) . if plane a took off exactly 40 minutes before plane b , after how many minutes will plane b overtake plane a ? a) 65 , b) 80 , c) 90 , d) 115 , e) 120	we can use some form of the equation d = rt [ distance = rate * time ] rate of plane a : 200 mph rate of plane b : 300 mph plane a will be in the sky 40 minutes longer than plane b (2 / 3 of an hour) time of plane a when b overtakes it : t + (2 / 3) use 2 / 3 of an hour since the rate is in hours time of plane b when b overtakes a : t at the time that b overtakes a , they will have travelled the same distance so rt is equal for each plane : 200 * (t + 2 / 3) = 300 * t 200 t + 400 / 3 = 300 t 400 / 3 = 100 t 4 / 3 = t this is 1 and a third hour , so 60 minutes + 20 minutes = 80 minutes b
Solve the problem systematically, showing each step clearly.	a , b and c have rs . 450 between them , a and c together have rs . 200 and b and c rs . 350 . how much does c have ? a) a) 100 , b) b) 110 , c) c) 120 , d) d) 130 , e) e) 140	"a + b + c = 450 a + c = 200 b + c = 350 - - - - - - - - - - - - - - a + b + 2 c = 550 a + b + c = 450 - - - - - - - - - - - - - - - - c = 100 answer : a"
Explain the solution process for the given equation.	there are 600 employees in a room . 99 % are managers . how many managers must leave the room to bring down the percentage of managers to 98 % ? a) 100 , b) 200 , c) 250 , d) 300 , e) 400	"there are 594 managers and 6 others . the 6 others would compose 2 % of the total number of people if there were 300 people in the room . thus 300 managers must leave . the answer is d ."
Provide a detailed solution to the given mathematical problem.	a group of 4 investment bankers and 5 clients recently frequented the bonbon ribs restaurant . the total bill for the meal , including 20 % gratuity , came to $ 756 . on average , how much did the meal of each individual cost before gratuity ? a) $ 160 , b) $ 70 , c) $ 90 , d) $ 80 , e) $ 55	4 ibs and 5 clients - so total 9 people the bill $ 756 includes 20 % gratuity . . . so the actual cost of dinner was $ 630 now , the cost per person will be $ 630 / 9 which is $ 70 option b
Work through the equation methodically, ensuring clarity at each step.	if a - b = 3 and a (power 2) + b (power 2) = 21 , find the value of ab . a) 5 , b) 8 , c) 4 , d) 6 , e) 3	"2 ab = (a (power 2) + b (power 2) - (a - b) (power 2) = 21 - 9 = 12 ab = 6 . answer is d ."
Work through the equation methodically, ensuring clarity at each step.	worker a takes 10 hours to do a job . worker b takes 15 hours to do the same job . how long it take both a & b , working together but independently , to do the same job ? a) 40 days , b) 40 / 9 days , c) 39 / 9 days , d) 6 days , e) 60 / 9 days	"a' s one hour work = 1 / 10 . b' s one hour work = 1 / 15 . (a + b)' s one hour work = 1 / 10 + 1 / 15 = 5 / 30 = 1 / 6 . both a & b can finish the work in 6 days d"
Break down and solve the equation step by step.	a person is traveling at 100 km / hr and reached his destiny in 5 then find the distance ? a) 250 km , b) 300 km , c) 500 km , d) 450 km , e) 350 km	"t = 5 hr s = 100 km / hr d = t * s = 100 * 5 = 500 km answer is c"
Break down and solve the equation step by step.	what is the difference between the largest and the smallest number written with 7 , 7 , 0 , 7 ? a) 6893 , b) 6993 , c) 6093 , d) 693 , e) none	"7770 7077 - - - - - - - - - - - - - 693 answer : d"
Explain the solution process for the given equation.	a chemical supply company has 60 liters of a 25 % hno 3 solution . how many liters of pure undiluted hno 3 must the chemists add so that the resultant solution is a 50 % solution ? a) 12 , b) 15 , c) 20 , d) 24 , e) 30	"60 liters of a 25 % hno 3 solution means hno 3 = 15 liters in 60 liters of the solution . now , let x be the pure hno 3 added . as per question , 15 + x = 50 % of (60 + x) or x = 30 . hence , e"
Work through the equation methodically, ensuring clarity at each step.	find the lowest common multiple of 25 , 35 and 50 . a) 350 , b) 420 , c) 510 , d) 320 , e) 280	lcm = 2 * 5 * 5 * 7 = 350 . answer is a
Solve the problem systematically, showing each step clearly.	find the average of all the numbers between 40 and 80 which are divisible by 3 . a) 15 , b) 70 , c) 20 , d) 60 , e) none	"sol . average = (42 + 45 + 48 + 51 + 54 + 57 + 60 + 63 + 66 + 69 + 72 + 75 + 78 / 13) = 780 / 5 = 60 . answer d"
Solve the problem systematically, showing each step clearly.	arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 11 from the product of their ages a year from today . what is their age today ? a) 2 . , b) 4 . , c) 5 . , d) 7 . , e) 9 .	"ad = (a + 1) (d + 1) - 11 0 = a + d - 10 a + d = 10 a = d (as they are twin brothers) a = d = 5 c is the answer"
Break down and solve the equation step by step.	a classroom has equal number of boys and girls . 8 girls left to play kho - kho , leaving twice as many boys as girls in the classroom . what was the total number of girls and boys present initially ? a) 16 , b) 24 , c) 32 , d) 48 , e) 54	after 8 girls left remaining 8 girls now boys 16 are twice as many as remaining girls . initially boys = 16 and girls = 16 . answer : c
Break down and solve the equation step by step.	nd the area of trapezium whose parallel sides are 20 cm and 15 cm long , and the distance between them is 14 cm ? a) 230 cm 2 , b) 245 cm 2 , c) 255 cm 2 , d) 260 cm 2 , e) 280 cm 2	"area of a trapezium = 1 / 2 (sum of parallel sides) * (perpendicular distance between them) = 1 / 2 (20 + 15) * (14) = 245 cm 2 answer : b"
Provide a detailed solution to the given mathematical problem.	if a certain toy store' s revenue in november was 2 / 5 of its revenue in december and its revenue in january was 1 / 5 of its revenue in november , then the store' s revenue in december was how many times the average (arithmetic mean) of its revenues in november and january ? a) 5 / 2 , b) 10 / 3 , c) 15 / 4 , d) 25 / 3 , e) 25 / 6	"n = 2 d / 5 j = n / 5 = 2 d / 25 the average of november and january is (n + j) / 2 = 12 d / 25 / 2 = 6 d / 25 d is 25 / 6 times the average of november and january . the answer is e ."
Break down and solve the equation step by step.	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 . 2640 in all from both of them as interest . the rate of interest per annum is ? a) 16 % , b) 12 % , c) 74 % , d) 10 % , e) 45 %	"let the rate be r % p . a . then , (5000 * r * 2) / 100 + (3000 * r * 4) / 100 = 2640 100 r + 120 r = 2640 r = 12 % answer : b"
Solve the problem systematically, showing each step clearly.	an empty wooden vessel weighs 16 % of its total weight when filled with paint . if the weight of a partially filled vessel is one half that of a completely filled vessel , what fraction of the vessel is filled . a) 3 / 5 , b) 5 / 9 , c) 1 / 24 , d) 4 / 9 , e) 7 / 15	"an empty wooden vessel weighs 16 % of its total weight when filled with paint : vessel = 0.16 (vessel + paint) ; 16 v = v + p (so the weight of completely filled vessel is 16 v) p = 15 v (so the weight of the paint when the vessels is completely filled is 15 v) . the weight of a partially filled vessel is one half that of a completely filled vessel : v + p' = 1 / 2 * 16 v ; p' = 7 v (so the weight of the paint when the vessels is partially filled is 7 v) . what fraction of the vessel is filled ? so , we need to find the ratio of the weight of the paint when the vessel iscompletely filledto the weight of the paint when the vessel ispartially filled : p' / p = 7 v / 15 v = 7 / 15 . answer : e ."
Solve the problem systematically, showing each step clearly.	a regular hexagon is there . the mid points of the sides were joined and formed another hexagon . then what is the percentage reduction in area . a) 25 % , b) 35 % , c) 45 % , d) 55 % , e) 65 %	let abcdef be the regular hexagon with sides ab and ed parallel . p and q are mid points of af and ef resp . ae = root (3) * s (s is side of hexagon) then pq = (root (3) * s) / 2 mid point theorem now u know side of inner hexagon use formula area of hexagon = (3 * root (3)) 2 * s ^ 2 and find diff in area of two hexagon multiply by 100 and divide the product by area of bigger hexagon ans 25 % answer : a
Provide a detailed solution to the given mathematical problem.	in what ratio should a variety of rice costing rs . 4.5 per kg be mixed with another variety of rice costing rs . 8.75 per kg to obtain a mixture costing rs . 7.50 per kg ? a) 5 / 6 , b) 5 / 9 , c) 5 / 1 , d) 5 / 3 , e) 5 / 12	"let us say the ratio of the quantities of cheaper and dearer varieties = x : y by the rule of allegation , x / y = (8.75 - 7.50) / (7.50 - 4.5) = 5 / 12 answer : e"
Provide a detailed solution to the given mathematical problem.	a sum of money at simple interest amounts to 815 in 3 years and to 854 in 4 years . the sum is : a) 698 , b) 387 , c) 373 , d) 593 , e) 372	"a 698 s . i . for 1 year = (854 - 815) = 39 . s . i . for 3 years = (39 x 3) = 117 . principal = (815 - 117) = 698 ."
Work through the equation methodically, ensuring clarity at each step.	claire has a total of 92 pets consisting of gerbils and hamsters only . one - quarter of the gerbils are male , and one - third of the hamsters are male . if there are 25 males altogether , how many gerbils does claire have ? a) 39 , b) 68 , c) 54 , d) 57 , e) 60	"g + h = 92 . . . 1 ; g / 4 + h / 3 = 25 . . . . 2 or 3 g + 4 h = 25 * 12 = 300 g = 92 - h or 3 (92 - h) + 4 h = 300 h = 300 - 276 = 24 then g = 92 - 24 = 68 b"
Explain the solution process for the given equation.	100 people are attending a newspaper conference . 40 of them are writers and more than 38 are editors . of the people at the conference , x are both writers and editors and 2 x are neither . what is the largest possible number of people who are both writers and editors ? a) 21 , b) 24 , c) 28 , d) 30 , e) 32	"{ total } = { writers } + { editors } - { both } + { neither } . { total } = 100 ; { writers } = 40 ; { editors } > 38 ; { both } = x ; { neither } = 2 x ; 100 = 40 + { editors } - x + 2 x - - > x = 60 - { editors } . we want to maximize x , thus we should minimize { editors } , minimum possible value of { editors } is 39 , thus x = { both } = 60 - 39 = 21 . answer : a ."
Provide a detailed solution to the given mathematical problem.	a sum of money at simple interest amounts to rs . 1072 in 2 years and to rs . 1144 in 4 years . the sum is : a) rs . 1000 , b) rs . 850 , c) rs . 500 , d) rs . 1200 , e) rs . 900	"s . i . for 2 years = rs . (1144 - 1072) = rs . 72 . s . i . for 1 year = rs . 72 / 2 = rs . 36 . principal = rs . (1072 - 72) = rs . 1000 . answer : option a"
Work through the equation methodically, ensuring clarity at each step.	the average of the marks of 12 students in a class is 36 . if the marks of each student are doubled , find the new average ? a) 72 , b) 23 , c) 29 , d) 26 , e) 13	"sum of the marks for the 12 students = 12 * 36 = 432 . the marks of each student are doubled , the sum also will be doubled . the new sum = 432 * 2 = 864 . so , the new average = 864 / 12 = 72 . answer : a"
Provide a detailed solution to the given mathematical problem.	walking at 5 / 6 of its usual speed , a train is 10 minutes too late . what is usual time to cover the journey ? a) 70 minutes , b) 2 hours , c) 2 / 3 hours , d) 5 / 6 hours , e) 85 minutes	"et speed be s and time to cover the journey be t s * t = d (distance of journey) - - - 1 now as per the question stem we have (5 / 6) s * (t + 10) = d - - - - 2 equating value of d from (1) in (2) we get t = 50 mins or 5 / 6 hrs . answer : d"
Provide a detailed solution to the given mathematical problem.	a sum of money is to be distributed among a , b , c , d in the proportion of 5 : 2 : 4 : 3 . if c gets rs . 3000 more than d , what is b' s share ? a) a) 8239 , b) b) 2900 , c) c) 6000 , d) d) 2393 , e) e) 2009	"let the shares of a , b , c and d be 5 x , 2 x , 4 x and 3 x rs . respectively . then , 4 x - 3 x = 3000 = > x = 3000 . b' s share = rs . 2 x = 2 * 3000 = rs . 6000 . answer : c"
Solve the problem systematically, showing each step clearly.	multiplication of 2 numbers is 9375 and the quotient , when the larger one is divided by the smalleris 15 . find the sum of the numbers ? a) 200 , b) 300 , c) 400 , d) 500 , e) 600	let the numbers be x and y . then , xy = 9375 and x = 15 . y xy = 9375 (x / y) 15 y 2 = 625 . y = 25 . x = 15 y = (15 x 25) = 375 . sum of the numbers = x + y = 375 + 25 = 400 . c
Explain the solution process for the given equation.	roja and pooja start moving in the opposite directions from a pole . they are moving at the speeds of 6 km / hr and 3 km / hr respectively . after 4 hours what will be the distance between them ? a) 22 km , b) 20 km , c) 65 km , d) 18 km , e) 36 km	distance = relative speed * time = (6 + 3) * 4 = 36 km [ they are travelling in the opposite direction , relative speed = sum of the speeds ] . answer : e
Provide a detailed solution to the given mathematical problem.	the length of a rectangular field is 7 / 5 its width . if the perimeter of the field is 384 meters , what is the width of the field ? a) 50 , b) 60 , c) 11 , d) 80 , e) 90	"let l be the length and w be the width . l = (7 / 5) w perimeter : 2 l + 2 w = 384 , 2 (7 / 5) w + 2 w = 384 solve the above equation to find : w = 80 m and l = 112 m . correct answer d) 80"
Explain the solution process for the given equation.	there are 6 more women than there are men on a local co - ed softball team . if there are a total of 24 players on the team , what is the ratio of men to women ? a) 10 / 16 , b) 9 / 15 , c) 4 / 16 , d) 6 / 10 , e) 4 / 10	"w = m + 6 w + m = 24 m + 6 + m = 24 2 m = 18 m = 9 w = 15 ratio : 9 : 15 ans : b"
Work through the equation methodically, ensuring clarity at each step.	if the wheel is 15 cm then the number of revolutions to cover a distance of 1056 cm is ? a) 18 , b) 11.2 , c) 14 , d) 12 , e) 91	"2 * 22 / 7 * 15 * x = 1056 = > x = 11.2 answer : b"
Work through the equation methodically, ensuring clarity at each step.	look carefully for the pattern , and then choose which pair of numbers comes next . 28 25 5 21 18 5 14 a) 11 5 , b) 10 7 , c) 11 8 , d) 5 10 , e) 10 5	explanation : this is an alternating subtraction series with the interpolation of a random number , 5 , as every third number . in the subtraction series , 3 is subtracted , then 4 , then 3 , and so on . answer : option a
Break down and solve the equation step by step.	the average of runs of a cricket player of 20 innings was 32 . how many runs must he make in his next innings so as to increase his average of runs by 6 ? a) 96 , b) 106 , c) 158 , d) 116 , e) 122	"average = total runs / no . of innings = 32 so , total = average x no . of innings = 32 * 20 = 640 now increase in avg = 4 runs . so , new avg = 32 + 6 = 38 runs total runs = new avg x new no . of innings = 38 * 21 = 798 runs made in the 11 th inning = 798 - 640 = 158 answer : c"
Solve the problem systematically, showing each step clearly.	there are 9 teams in a certain league and each team plays each of the other teams exactly once . what is the total number of games played ? a) 24 , b) 25 , c) 28 , d) 32 , e) 36	"9 c 2 = 36 the answer is e"
Break down and solve the equation step by step.	if 60 % of (x - y) = 30 % of (x + y) then what percent of x is y ? a) 33 % , b) 10 % , c) 5 % , d) 15 % , e) 25 %	"60 % of (x - y) = 30 % of (x + y) (60 / 100) (x - y) = (30 / 100) (x + y) 6 (x - y) = 3 (x + y) 3 x = 9 y x = 3 y therefore required percentage = ((y / x) x 100) % = ((y / 3 y) x 100) = 33 % answer is a ."
Solve the problem systematically, showing each step clearly.	during a sale of 20 % on everything in a store , a kid is successful in convincing the store manager to give him 20 candies for the discounted price of 14 candies . the store still makes a profit of 10 % on this sale . what is the mark up percentage on each candy ? a) 100 % , b) 80 % , c) 75 % , d) 66 + 2 / 3 % , e) 55 %	"hi akhil , i can share the way i solved it . . let' s say marked price = m so , there' s a discount of 20 % on m so , new s . p . = 80 % of m now , the child convinces the owner to sell 20 candies for the price of 14 candies . let' s say each candy after discount is 1 $ . so , s . p . of 20 candies = 20 $ . the child bought it for 14 $ so , he got a discount of 6 / 20 * 100 = 30 % so , the latest s . p . = 70 % of 80 % of m = 0.7 * 0.8 m now , we are given that the shopkeeper still makes a profit of 12 % . so we have , 0.7 * 0.8 * m = 1.12 c . p so , we get , m = 2 c . p . i . e . marked price was kept 100 % above c . p . c"
Solve the problem systematically, showing each step clearly.	the difference between c . i . and s . i . on an amount of $ 10,000 for 2 years is $ 49 . what is the rate of interest per annum ? a) 5 % , b) 6 % , c) 7 % , d) 8 % , e) 9 %	$ 49 is the interest on the first year of interest . let x be the interest rate . the interest after the first year is 10000 * x . the interest on the first year' s interest is 10000 * x * x 10000 * x ^ 2 = 49 x = 0.07 the answer is c .
Provide a detailed solution to the given mathematical problem.	from a container , 6 liters milk was drawn out and was replaced by water . again 6 liters of mixture was drawn out and was replaced by the water . thus the quantity of milk and water in the container after these two operations is 9 : 16 . the quantity of mixture is : a) 15 , b) 26 , c) 22 , d) 27 , e) 11	explanation : let quantity of mixture be x liters . suppose a container contains x units of liquid from which y units are taken out and replaced by water . after operations , the quantity of pure liquid = units , where n = no of operations . so , quantity of milk = given that , milk : water = 9 : 16 milk : (milk + water) = 9 : (9 + 16) milk : mixture = 9 : 25 answer : a) 15
Break down and solve the equation step by step.	how many 3 digit no' s are between 100 - 500 , where sum of two digit is 3 rd ? a) 24 , b) 54 , c) 60 , d) 84 , e) 94	total permutations possible = 18 + 16 + 14 + 12 = 60 answer : c
Work through the equation methodically, ensuring clarity at each step.	a certain music store stocks 800 cellos and 600 violas . of these instruments , there are 120 cello - viola pairs , such that a cello and a viola were both made with wood from the same tree (each tree can make at most one viola and one cello , so there are no pairs other than these 90) . if one viola and one cello are chosen at random , what is the probability that the two instruments are made with wood from the same tree ? a) 3 / 16,000 , b) 1 / 8,000 , c) 3 / 1,600 , d) 1 / 90 , e) 2 / 45	"solution provided by stanford 2012 is correct : 120 / 800 choosing one of the cellos which has a pair viola , 1 / 600 choosing the viola which is the pair of chosen cello - - > p = 120 / 800 * 1 / 600 = 1 / 8,000 . answer : b ."
Provide a detailed solution to the given mathematical problem.	a tank is filled by 3 pipes a , b , c in 5 hours . pipe c is twice as fast as b and b is twice as fast as a . how much will pipe a alone take to fill the tank ? a) 25 hr , b) 35 hr , c) 40 hr , d) 20 hr , e) 50 hr	"suppose pipe a alone take x hours to fill the tank then pipe b and c will take x / 2 and x / 4 hours respectively to fill the tank . 1 / x + 2 / x + 4 / x = 1 / 5 7 / x = 1 / 5 x = 35 hours answer is b"
Work through the equation methodically, ensuring clarity at each step.	a sum of money lent out at s . i . amounts to rs . 780 after 2 years and to rs . 1020 after a further period of 5 years . the sum is ? a) s . 684 , b) s . 500 , c) s . 540 , d) s . 740 , e) s . 840	"s . i for 5 years = (1020 - 780) = rs . 240 . s . i . for 2 years = 240 / 5 * 2 = rs . 96 . principal = (780 - 96) = rs . 684 answer : a"
Solve the problem systematically, showing each step clearly.	in expressing a length of 61.472 km as nearly as possible with the 3 significant digits , find the percentage error ? a) 0.35 % , b) 0.4 % , c) 0.045 % , d) 0.6 % , e) none of these	explanation : error = (61.5 - 61.472) = 0.028 required percentage = 0.028 / 61.472100 = 0.045 answer : c
Explain the solution process for the given equation.	how many numbers from 49 to 79 are exactly divisible by 11 ? a) 5 , b) 7 , c) 9 , d) 3 , e) 12	"49 / 11 = 4 and 79 / 11 = 7 = = > 7 - 4 = 3 numbers answer : d"
Provide a detailed solution to the given mathematical problem.	a candidate got 25 % of the votes polled and he lost to his rival by 4000 votes . how many votes were cast ? a) 7500 , b) 2028 , c) 2775 , d) 5496 , e) 8000	"25 % - - - - - - - - - - - l 75 % - - - - - - - - - - - w - - - - - - - - - - - - - - - - - - 50 % - - - - - - - - - - 4000 100 % - - - - - - - - - ? = > 8000 answer : e"
Explain the solution process for the given equation.	if 12 men or 20 women can do a piece of work in 27 days , then in how many days can 9 men and 12 women together do the work ? a) 10 days , b) 30 days , c) 20 days , d) 80 days , e) 40 days	"c 20 days given that 12 m = 20 w = > 3 m = 5 w 9 men + 12 women = 15 women + 12 women = 27 women 20 women can do the work in 27 days . so , 27 women can do it in (20 * 27) / 27 = 20 days ."
Provide a detailed solution to the given mathematical problem.	a big container is 50 % full with water . if 20 liters of water is added , the container becomes 3 / 4 full . what is the capacity of the big container in liters ? a) 80 , b) 90 , c) 100 , d) 110 , e) 120	"20 liters is 25 % of the capacity c . 20 = 0.25 c c = 80 liters . the answer is a ."
Break down and solve the equation step by step.	the sides of a triangle are in the ratio 3 : 5 : 7 . if the pereimeter is 60 cms , the longest side is a) 7 , b) 30 , c) 28 , d) 25 , e) 16	"3 x + 5 x + 7 x = 60 15 x = 60 x = 4 longest side = 7 * 4 = 28 answer : c"
Work through the equation methodically, ensuring clarity at each step.	a cylinder is inserted in a sphere d / h = 2 / 3 . find the surface area of the cylinder ? ['a) 23.12','b) 24.12','c) 25.12','d) 26.12','e) 27.12']	take r = 1 , h = 3 = 2 * 3.14 * r * r + 2 * 3.14 * r * h = 2 * 3.14 * r (r + h) = 2 * 3.14 * 1 (1 + 3) = 2 * 3.14 (4) = 25.12 answer : c
Break down and solve the equation step by step.	287 x 287 + 269 x 269 - 2 x 287 x 269 = ? a) 534 , b) 446 , c) 354 , d) 324 , e) none of them	= a ^ 2 + b ^ 2 - 2 ab where a = 287 and b = 269 = ( a - b ) ^ 2 = ( 287 - 269 ) ^ 2 = ( 18 ) ^ 2 = 324 answer is d
Solve the problem systematically, showing each step clearly.	a small water pump would take 4 hours to fill an empty tank . a larger pump would take 1 / 2 hour to fill the same tank . how many hours would it take both pumps , working at their respective constant rates , to fill the empty tank if they began pumping at the same time ? a ) 1 / 6 , b ) 4 / 9 , c ) 1 / 4 , d ) 5 / 9 , e ) 1 / 3	"rate of the small pump is 1 / 4 tank / hour rate of the larger pump is 1 / ( 1 / 2 ) or 2 tank / hour ; combined rate of the two pumps is 1 / 4 + 2 = 9 / 4 tank / hour , together they will fill the empty tank in 1 / ( 9 / 4 ) ) = 4 / 9 hours ( time = job / rate ) . answer : b"
Explain the solution process for the given equation.	what is the total number of prime numbers less than 70 ? a ) 17 , b ) 18 , c ) 19 , d ) 20 , e ) 21	solution prime numbers less than 70 are = 12 . 2,3 , 5,7 , 11,13 , 17,19 , 23,29 , 31,37 , 41,43 , 47,53 , 59,61 , and 67 their number is 19 . answer c
Break down and solve the equation step by step.	in a sample of 800 high school students in which all students are either freshmen , sophomores , juniors , or seniors , 22 percent are juniors and 75 percent are not sophomores . if there are 160 seniors , how many more freshmen than sophomores are there among the sample of students ? a ) 42 , b ) 48 , c ) 64 , d ) 76 , e ) 88	"200 are sophomores . the number of freshmen is 600 - 160 - 0.22 ( 800 ) = 264 the answer is c ."
Break down and solve the equation step by step.	the value of ( 34.31 * 0.473 * 1.567 ) / ( 0.0673 * 23.25 * 7.57 ) is close to a ) 2 , b ) 1.15 , c ) 2.05 , d ) 2.15 , e ) 2.35	"( 34.31 * 0.473 * 1.567 ) / ( 0.0673 * 23.25 * 7.57 ) = 25.4303 / 11.845 = 2.15 answer : d"
Provide a detailed solution to the given mathematical problem.	let p be a prime number . if n = 7 p , how many divisors does n have ? a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5	two . n = 7 * p , and both 7 and p are prime . so the answer is b .
Solve the problem systematically, showing each step clearly.	every student in a room is either a junior or a senior . there is at least one junior and at least one senior in the room . if 2 / 5 of the juniors is equal to 1 / 3 of the seniors , what fraction of the students in the room are juniors ? a ) 3 / 20 , b ) 1 / 3 , c ) 5 / 12 , d ) 12 / 17 , e ) 5 / 11	"let total number of juniors = j total number of seniors = s ( 2 / 5 ) j = ( 1 / 3 ) s = > s = 6 / 5 j total number of students = j + s = ( 11 / 5 ) j fraction of the students in the room are juniors = j / ( j + s ) = j / [ ( 11 / 5 ) j ] = 5 / 11 answer e"
Explain the solution process for the given equation.	in a division sum , the remainder is 8 and the divisor is 3 times the quotient and is obtained by adding 3 to the thrice of the remainder . the dividend is : a ) 200 , b ) 251 , c ) 250 , d ) 262 , e ) 254	"diver = ( 8 * 3 ) + 3 = 27 3 * quotient = 27 quotient = 9 dividend = ( divisor * quotient ) + remainder dividend = ( 27 * 9 ) + 8 = 251 b"
Explain the solution process for the given equation.	31 of the scientists that attended a certain workshop were wolf prize laureates , and 12 of these 31 were also nobel prize laureates . of the scientists that attended that workshop and had not received the wolf prize , the number of scientists that had received the nobel prize was 3 greater than the number of scientists that had not received the nobel prize . if 50 of the scientists attended that workshop , how many of them were nobel prize laureates ? a ) a ) 11 , b ) b ) 23 , c ) c ) 24 , d ) d ) 29 , e ) d ) 36	"lets solve by creating equation . . w = 31 . . total = 50 . . not w = 50 - 31 = 19 . . now let people who were neither be x , so out of 19 who won nobel = x + 3 . . so x + x + 3 = 19 or x = 8 . . so who won nobel but not wolf = x + 3 = 11 . . but people who won both w and n = 12 . . so total who won n = 11 + 12 = 23 . . b"
Provide a detailed solution to the given mathematical problem.	a square mirror has exactly half the area of the rectangular wall on which it is hung . if each side of the mirror is 24 inches and the width of the wall is 42 inches , what is the length of the wall , in inches ? a ) 21.4 , b ) 23.4 , c ) 25.4 , d ) 26.4 , e ) 27.4	"since the mirror is 42 inches in all sides , it must be a square . area of a square is a = a ^ 2 ; 24 ^ 2 = 576 . area of rectangle is double of that 2 * 576 = 1152 . now a = lw and we need find w so a / l = w ; 1152 / 42 = 27.4 answer ! answer : e"
Provide a detailed solution to the given mathematical problem.	the sum of money will be double itself in 20 years and simple interest find rate of interest ? a ) 10 % , b ) 5 % , c ) 15 % , d ) 20 % , e ) 25 %	"t = 20 years p = principle amount = x a = total amount = 2 x si = simple interest = a - p = 2 x - x = x r = 100 si / pt = 100 x / 20 x = 5 % answer is b"
Break down and solve the equation step by step.	bert and rebecca were looking at the price of a condominium . the price of the condominium was 110 % more than bert had in savings , and separately , the same price was also 20 % more than rebecca had in savings . what is the ratio of what bert has in savings to what rebecca has in savings . a ) 1 : 5 , b ) 1 : 4 , c ) 2 : 3 , d ) 4 : 7 , e ) 1 : 2	"suppose bert had 100 so price becomes 210 , this 210 = 1.2 times r' s saving . . so r' s saving becomes 175 so required ratio is 100 : 175 = 4 : 7 answer : d"
Work through the equation methodically, ensuring clarity at each step.	a bullet train 200 m long is running with a speed of 69 kmph . in what time will it pass a man who is running at 3 kmph in the direction opposite to that in which the bullet train is going ? a ) 10 sec , b ) 15 sec , c ) 12 sec , d ) 11 sec , e ) 16 sec	"a 10 sec speed of the bullet train relative to man = ( 69 + 3 ) kmph = 72 * 5 / 18 m / sec = 60 / 3 m / sec . time taken by the bullet train to cross the man = time taken by it to cover 200 m at ( 60 / 3 ) m / sec = ( 200 * 3 / 60 ) sec = 10 sec"
Provide a detailed solution to the given mathematical problem.	in a village there are 150 men and 90 women in present time . if in next year population will be p = ( a ^ 2 + b ^ 2 ) ^ 1 / 2 , and in every year men are reduces 6 % . what is population of after 2 year . a ) 140 , b ) 141 , c ) 142 , d ) 143 , e ) 145	"next year total population = [ 150 ^ 2 + 90 ^ 2 ] ^ . 5 = 174.92 = 175 man decreased by 6 % so total man = 150 * . 94 = 141 women will be = 175 - 141 = 34 so population after two years = [ 135 ^ 2 + 34 ^ 2 ] ^ . 5 = 145.04 = 145 so population after two year = 145 answer : e"
Explain the solution process for the given equation.	the value of a machine depreciates at 20 % per annum . if its present value is rs . 1 , 50,000 , at what price should it be sold after two years such that a profit of rs . 20,000 is made ? a ) 328897 , b ) 120000 , c ) 877888 , d ) 116000 , e ) 188871	"the value of the machine after two years = 0.8 * 0.8 * 1 , 50,000 = rs . 96,000 sp such that a profit of rs . 20,000 is made = 96,000 + 20,000 = rs . 1 , 16,000 answer : d"
Solve the problem systematically, showing each step clearly.	a distributor sells a product through an on - line store , which take a commission of 20 % of the price set by the distributor . the distributor obtains the product from a producer at the price of $ 15 per item . what is the price that the buyer observers on - line if the distributor wants to maintain a 35 % profit on the cost of the item ? a ) 18 , b ) 21.6 , c ) 22 , d ) 22.5 , e ) 27.7	"producer price = $ 15 ; the distributor wants to maintain a 20 % profit on the cost of the item , thus he must get $ 15 * 1.2 = $ 18 after the store takes a commission of 35 % of the final price - - > ( final price ) * 0.65 = $ 18 - - > ( final price ) = $ 27.7 . answer : e ."
Break down and solve the equation step by step.	a bag holds 4 red marbles and 5 green marbles . if you removed two randomly selected marbles from the bag , without replacement , what is the probability that both would be red ? a ) 1 / 10 , b ) 1 / 6 , c ) 3 / 10 , d ) 2 / 5 , e ) 1 / 4	"probability of selecting first red marble = 4 / 9 probability of selecting second red marble without replacement = 3 / 8 final probability = 4 / 9 * 3 / 8 = 1 / 6 the correct answer is b ."
Work through the equation methodically, ensuring clarity at each step.	if 9 - 4 / x = 7 + 8 / x , then x = a ) 15 , b ) 6 , c ) 1 , d ) 5 , e ) 12	we' re given the equation 9 - 4 / x = 7 + 8 / x . we' re asked for the value of x . the common - denominator of these 4 numbers is x , so we have to multiple 2 of the numbers to give them that denominator . . . 9 x - 4 x / x = 7 x + 8 x / x we can then eliminate that denominator , which gives us . . . . 9 x - 4 = 7 x + 8 12 = 2 x both 2 x and 12 have a common denominator of 2 , so we divide both sides by 2 for our final answer . . . 6 = x b
Solve the problem systematically, showing each step clearly.	if ( 5 ^ 13 ) ( 9 ^ 7 ) = 3 ( 15 ^ x ) , what is the value of x ? a ) 7 , b ) 9 , c ) 11 , d ) 13 , e ) 15	( 5 ^ 13 ) ( 9 ^ 7 ) = 3 ( 15 ^ x ) = > 5 ^ 13 * 3 ^ 14 = 3 * 3 ^ x * 5 ^ x = > 5 ^ 13 * 3 ^ 14 = 3 ^ ( x + 1 ) * 5 ^ x value of x = 13 answer d
Provide a detailed solution to the given mathematical problem.	a and b started a business in partnership investing rs . 20,000 and rs . 15,000 respectively . after 6 months , c joined them with rs . 20,000 . whatwill be b' s share in total profit of rs . 21,000 earned at the end of 2 years from the startingof the business ? a ) s . 5,000 , b ) s . 5,500 , c ) s . 5,700 , d ) s . 6,300 , e ) s . 7,500	"a : b : c = ( 20,000 x 24 ) : ( 15,000 x 24 ) : ( 20,000 x 18 ) = 4 : 3 : 3 . b' s share = rs . 21000 x 3 / 10 = rs . 6,300 . d"
Break down and solve the equation step by step.	a couple married in 1980 had two children , one in 1982 and the other in 1984 . their combined ages will equal the years of the marriage in ? a ) 1986 , b ) 1985 , c ) 1987 , d ) 1988 , e ) 1989	sum of year = 1 + 9 + 8 + 0 = ( 18 ) 1 + 8 = 9 1 + 9 + 8 + 2 = ( 20 ) 2 + 0 = 2 1 + 9 + 8 + 4 = ( 13 ) 1 + 3 = 4 now 9 + 2 + 4 = ( 15 ) 1 + 5 = 6 nd option a 1 + 9 + 8 + 6 = 24 ( 2 + 4 ) = 6 answer : a
Solve the problem systematically, showing each step clearly.	how many trailing zeroes does 49 ! + 50 ! have ? a ) 9 , b ) 10 , c ) 11 , d ) 12 , e ) 22	"49 ! + 50 ! = 51 * 49 ! no of trailing 0' s = no of 5 * 2 no of 2' s = 49 / 2 + 49 / 4 + 49 / 8 + 49 / 16 + 49 / 32 = d no of 5' s = 49 / 5 + 49 / 25 = 10 = p now since p < d no of trailing 0' s = 10 answer : b"
Explain the solution process for the given equation.	a new tower has just been built at the verbico military hospital ; the number of beds available for patients at the hospital is now 6 times the number available before the new tower was built . currently , 1 / 3 of the hospital' s original beds , as well as 1 / 5 of the beds in the new tower , are occupied . for the purposes of renovating the hospital' s original wing , all of the patients in the hospital' s original beds must be transferred to beds in the new tower . if patients are neither admitted nor discharged during the transfer , what fraction of the beds in the new tower will be unoccupied once the transfer is complete ? a ) 11 / 30 , b ) 29 / 60 , c ) 17 / 30 , d ) 19 / 30 , e ) 11 / 15	"i think e - 11 / 15 is the correct answer . here goes : lets assume originally the number of beds = x after the new tower , the total combined no of beds = 6 x so old = x , new = 5 x now 1 / 3 of x are occupied and 1 / 5 of 5 x are occupied which simplifies to ( 5 / 5 ) x we are shifting 1 / 3 of x to the new ward so there will now be : 1 / 3 of x plus 5 / 5 of x occupied in the new ward . add them up to get 4 / 3 of x there are 5 x beds in new tower so ratio is : ( 4 / 3 ) x / 5 x = 4 / 15 of x subtract that from 15 / 15 of x and you get the number of un - occupied beds to total capacity of new tower = 11 / 15 . e"
Explain the solution process for the given equation.	if x < y < z and y - x > 3 , where x is an even integer and y and z are odd integers , what is the least possible value of z - x ? a ) 7 , b ) 6 , c ) 8 , d ) 9 , e ) 10	"we have : 1 ) x < y < z 2 ) y - x > 3 3 ) x = 2 k ( x is an even number ) 4 ) y = 2 n + 1 ( y is an odd number ) 5 ) z = 2 p + 1 ( z is an odd number ) 6 ) z - x = ? least value z - x = 2 p + 1 - 2 k = 2 p - 2 k + 1 = 2 ( p - k ) + 1 - that means that z - x must be an odd number . we are asked to find the least value , so we have to pick the least numbers since y is odd and x is even , y - x must be odd . since y - x > 3 , the least value for y - x must be 5 , the least value for x must be 2 , and , thus , the least possible value for y must be 7 ( y - 2 = 5 , y = 7 ) 2 < 7 < z , since z is odd , the least possible value for z is 9 z - x = 9 - 2 = 7 answer a"
Solve the problem systematically, showing each step clearly.	a work which could be finished in 11 days was finished 3 days earlier after 10 more men joined . the number of men employed was ? a ) 22 , b ) 20 , c ) 88 , d ) 71 , e ) 12	"x - - - - - - - 11 ( x + 10 ) - - - - 6 x * 11 = ( x + 10 ) 6 x = 12 \ answer : e"
Work through the equation methodically, ensuring clarity at each step.	how many positive integers less than 243 are multiple of 4 but not multiples of 6 ? a ) 20 , b ) 31 , c ) 42 , d ) 40 , e ) 64	"240 / 4 = 60 multiples of 4 which are a multiple of 6 will be of the form 2 * 2 * 3 = 12 n where n > 0 240 / 12 = 20 60 - 20 = 40 answer : d"
Solve the problem systematically, showing each step clearly.	john traveled 80 % of the way from yellow - town to green - fields by train at an average speed of 80 miles per hour . the rest of the way john traveled by car at an average speed of v miles per hour . if the average speed for the entire trip was 45 miles per hour , what is v in miles per hour ? a ) 30 , b ) 40 , c ) 50 , d ) 55 , e ) 70	"hibunuel the question seems incorrect . it should not be 80 % at the speed of 80 . however if it' s 20 % at the speed of 80 , answer comes out 55 . the question is correct . here' s the explanation : let distance be d . we can find the total timeequate it , which comes as : 0.8 d / 80 + 0.2 d / v = d / 40 = > v = 40 ( option b ) ."
Provide a detailed solution to the given mathematical problem.	three numbers are in the ratio of 2 : 3 : 4 and their l . c . m . is 264 . what is their h . c . f . ? a ) 18 , b ) 22 , c ) 24 , d ) 28 , e ) 32	"let the numbers be 2 x , 3 x , and 4 x . lcm of 2 x , 3 x and 4 x is 12 x . 12 x = 264 x = 22 hcf of 2 x , 3 x and 4 x = x = 22 the answer is b ."
Provide a detailed solution to the given mathematical problem.	one - tenth of the students at a nursery school are 4 years old or older . if 20 students have not yet reached their third birthday , and a total of 50 students are not between 3 years old and 4 years old , how many children are in the nursery school ? a ) 150 , b ) 300 , c ) 450 , d ) 490 , e ) 620	x / 10 students are > 4 yrs 20 students are < 3 yrs x / 10 + 20 = 50 x / 10 = 30 x = 300 answer : b
Provide a detailed solution to the given mathematical problem.	the speed at which a man can row a boat in still water is 18 kmph . if he rows downstream , where the speed of current is 6 kmph , what time will he take to cover 100 metres ? a ) 6.67 seconds , b ) 18 seconds , c ) 26 seconds , d ) 14.99 seconds , e ) 6 seconds	"speed of the boat downstream = 18 + 6 = 24 kmph = 24 * 5 / 18 = 6.67 m / s hence time taken to cover 100 m = 100 / 6.67 = 14.99 seconds . answer : d"
Provide a detailed solution to the given mathematical problem.	calculate how many seconds it eill take for 4 bells to toll together again , given that they begin to toll together respectively at the intervals of 2 , 5 , 8 and 11 seconds . ? a ) 440 , b ) 420 , c ) 410 , d ) 442 , e ) 422	"lcm of 2 , 5 , 8 and 11 is 440 lcm = 440 answer : a"
Work through the equation methodically, ensuring clarity at each step.	on a partly cloudy day , derek decides to walk back from work . when it is sunny , he walks at a speed of s miles / hr ( s is an integer ) and when it gets cloudy , he increases his speed to ( s + 1 ) miles / hr . if his average speed for the entire distance is 2.8 miles / hr , what fraction r of the total distance did he cover while the sun was shining on him ? a ) 1 / 4 , b ) 4 / 5 , c ) 1 / 5 , d ) 1 / 6 , e ) 1 / 7	if s is an integer and we know that the average speed is 2.8 , s must be = 2 . that meanss + 1 = 3 . this implies that the ratio of time for s = 2 is 1 / 4 of the total time . the formula for distance / rate is d = rt . . . so the distance travelled when s = 2 is 2 t . the distance travelled for s + 1 = 3 is 3 * 4 t or 12 t . therefore , total distance covered while the sun was shining over him is r = 2 / 14 = 1 / 7 . answer : e
Break down and solve the equation step by step.	a woman complete a journey in 30 hours . she travels first half of the journey at the rate of 21 km / hr and second half at the rate of 24 km / hr . find the total journey in km . a ) 334 km . , b ) 216 km . , c ) 314 km . , d ) 224 km . , e ) 672 km .	"0.5 x / 21 + 0.5 x / 24 = 30 - - > x / 21 + x / 24 = 60 - - > x = 672 km . e"
Solve the problem systematically, showing each step clearly.	for the positive integers x , x + 2 , x + 4 , x + 7 , and x + 27 , the mean is how much greater than the median ? a ) 0 , b ) 1 , c ) 2 , d ) 4 , e ) 7	"mean = ( x + x + 2 + x + 4 + x + 7 + x + 27 ) / 5 = ( 5 x + 40 ) / 5 = x + 8 median = x + 4 thus mean - median = x + 8 - ( x + 4 ) = 4 answer = d"
Break down and solve the equation step by step.	a meeting has to be conducted with 4 managers . find the number of ways in which the managers be selected from among 7 managers , if 2 managers will not attend the meeting together ? a ) 25 , b ) 20 , c ) 35 , d ) 36 , e ) 38	"we can either choose all 4 people from 5 manager who have no problems or choose 3 from the 5 and 1 from the 2 managers who have a problem sitting together so 5 c 4 + ( 5 c 3 * 2 c 1 ) this is 5 + 20 = 25 answer : a"
Solve the problem systematically, showing each step clearly.	a certain number when divided by 110 leaves a remainder 25 , what is the remainder if the same no . be divided by 15 ? a ) 4 , b ) 5 , c ) 6 , d ) 8 , e ) 9	"explanation : 110 + 25 = 135 / 15 = 9 ( remainder ) e"
Work through the equation methodically, ensuring clarity at each step.	a , b and c are partners . a receives 2 / 3 of profits , b and c dividing the remainder equally . a' s income is increased by rs . 200 when the rate to profit rises from 5 to 7 percent . find the capital of a ? a ) 3377 , b ) 2899 , c ) 2500 , d ) 2778 , e ) 10000	"a : b : c = 2 / 3 : 1 / 6 : 1 / 6 = 4 : 1 : 1 x * 2 / 100 * 2 / 3 = 200 a capital = 15000 * 2 / 3 = 10000 answer : e"
Break down and solve the equation step by step.	how many seconds will a 400 metre long train take to cross a man running with a speed of 6 km / hr in the direction of the moving train if the speed of the train is 46 km / hr ? a ) 25 , b ) 30 , c ) 40 , d ) 45 , e ) 36	"explanation : speed of train relatively to man = ( 46 - 6 ) km / hr = 40 km / hr = ( 40 x 5 / 18 ) m / sec = 11.11 m / sec time taken to pass the man = ( 400 / 11.11 ) sec = 36 sec . answer : e"
Break down and solve the equation step by step.	one night 15 percent of the female officers on a police force were on duty . if 300 officers were on duty that night and half of these were female officers , how many female officers were on the police force ? a ) 90 , b ) 180 , c ) 270 , d ) 500 , e ) 1,000	"let total number of female officers in the police force = f total number of officers on duty on that night = 300 number of female officers on duty on that night = 300 / 2 = 150 ( 15 / 100 ) * f = 150 = > f = 1000 answer e"
Solve the problem systematically, showing each step clearly.	the distance from steve' s house to work is 40 km . on the way back steve drives twice as fast as he did on the way to work . altogether , steve is spending 6 hours a day on the roads . what is steve' s speed on the way back from work ? a ) 5 . , b ) 10 . , c ) 14 . , d ) 15 . , e ) 20 .	"time is in the ratio 2 : 1 : : to : fro office therefore , 2 x + 1 x = 6 hrs time take to come back - 2 hrs , distance travelled - 40 km = > speed = 20 kmph e"
Break down and solve the equation step by step.	if x is real , find the maximum value of the expression - 2 x ^ 2 + 9 x + 11 . a ) 12.125 , b ) 15.125 , c ) 18.125 , d ) 21.125 , e ) 24.125	"this is an equation of a downward facing parabola . the maximum value is the top point of the parabola . - 2 x ^ 2 + 9 x + 11 = ( - 2 x + 11 ) ( x + 1 ) the roots are 11 / 2 and - 1 . the maximum value must be when x is halfway between these two points . x = 2.25 the maximum value is - 2 ( 2.25 ) ^ 2 + 9 ( 2.25 ) + 11 = 21.125 the answer is d ."
Explain the solution process for the given equation.	milk contains 5 % water . what quantity of pure milk should be added to 10 litres of milk to reduce this 2 % ? a ) 15 litres , b ) 16 litres , c ) 17 litres , d ) 25 litres , e ) 35 litres	present ration of milk to water is 95 : 5 amount of milk is 10 litres hence , pure milk content is 9.5 litre and water content is 0.5 litres now , we should not disturb the water content so , 0.5 litres of water is equivalent to 2 % so 100 % mixture is : ( 0.5 / 2 ) * 100 = 25 litre thus , milk should be added = 25 - 10 = 15 litres of pure milk answer : a
Provide a detailed solution to the given mathematical problem.	the speed of a boat in still water is 15 km / hr and the rate of current is 3 km / hr . the distance travelled downstream in 12 minutes is a ) 1.2 km , b ) 1.8 km , c ) 2.4 km , d ) 3.6 km , e ) 1.6 km	"b = 15 km / hr , s = 3 km / hr , downstream b + s = 15 + 3 = 18 km / hr down stream time t = 12 min , we have to change it to hours so t = 12 / 60 = 1 / 5 hrs . distance = speed * time = 18 * 1 / 5 = 3.6 km . answer : d"
Solve the problem systematically, showing each step clearly.	what is the angle between the hands of a clock when time is 8 : 40 ? a ) 65 deg , b ) 75 deg , c ) 45 deg , d ) 15 deg , e ) 100 deg	"angle between two hands = 40 h - 11 / 2 m = 40 * 8 - 40 * 11 / 2 = 320 - 220 = 100 deg answer : e"
Break down and solve the equation step by step.	farm tax is levied on the 75 % of the cultivated land . the tax department collected total $ 3840 through the farm tax from the village of mr . william . mr . william paid only $ 480 as farm tax . the percentage of total land of mr . willam over the total taxable land of the village is : a ) 15 % , b ) 25 % , c ) 0.125 % , d ) 0.2083 % , e ) 16.667 %	"this will be equal to the percentage of total cultivated land he holds over the total cultivated land in the village . that leads to ( 480 / 3840 ) x 100 = 12.5 % in percentage terms . but the question asks ratio between his total land to total cultivated land . hence the answer is 12.5 % x ( 100 / 75 ) = 16.667 % the correct answer is ( e ) ."
Explain the solution process for the given equation.	a certain taxi company charges $ 3.00 for the first 1 / 5 of a mile plus $ 0.40 for each additional 1 / 5 of a mile . what would this company charge for a taxi ride that was 8 miles long ? a ) 15.6 , b ) 16.0 , c ) 17.5 , d ) 18.6 , e ) 19.1	"a certain taxi company charges $ 3.00 for the first 1 / 5 of a mile plus $ 0.40 for each additional 1 / 5 of a mile . what would this company charge for a taxi ride that was 8 miles long ? a . 15.60 b . 16.00 c . 17.50 d . 18.70 e . 19.10 1 / 5 miles = 0.2 miles . the cost of 8 miles long ride would be $ 3.00 for the first 0.2 miles plus ( 8 - 0.2 ) / 0.2 * 0.4 = $ 3.0 + $ 15.6 = $ 18.6 . answer : d ."
Work through the equation methodically, ensuring clarity at each step.	a hall is 15 m long and 12 m broad . if the sum of the areas of the floor and the ceiling is equal to the sum of the areas of four walls , the volume of the hall is : a ) 3348 , b ) 3898 , c ) 1200 , d ) 2881 , e ) 7881	answer : c ) 1200
Solve the problem systematically, showing each step clearly.	a vessel of capacity 90 litres is fully filled with pure milk . nine litres of milk is removed from the vessel and replaced with water . nine litres of the solution thus formed is removed and replaced with water . find the quantity of pure milk in the final milk solution ? a ) 73.9 , b ) 72.9 , c ) 73.0 , d ) 73.5 , e ) 92.2	"let the initial quantity of milk in vessel be t litres . let us say y litres of the mixture is taken out and replaced by water for n times , alternatively . quantity of milk finally in the vessel is then given by [ ( t - y ) / t ] n * t for the given problem , t = 90 , y = 9 and n = 2 . hence , quantity of milk finally in the vessel = [ ( 90 - 9 ) / 90 ] 2 ( 90 ) = 72.9 litres . answer : option b"
Break down and solve the equation step by step.	the tax on a commodity is diminished by 30 % but its consumption is increased by 15 % . find the decrease percent in the revenue derived from it ? a ) 20 % , b ) 18 % , c ) 15 % , d ) 12 % , e ) 14 %	"100 * 100 = 10000 70 * 115 = 8050 10000 - - - - - - - 1950 100 - - - - - - - ? = 20 % answer : a"
Explain the solution process for the given equation.	if both 7 ^ 2 and 3 ^ 4 are factors of the number a * 4 ^ 3 * 6 ^ 2 * 13 ^ 11 , then what is the smallest possible value of a ? a ) 120 , b ) 256 , c ) 343 , d ) 441 , e ) 512	"the number a must include at least 3 ^ 2 * 7 ^ 2 = 441 the answer is d ."
Work through the equation methodically, ensuring clarity at each step.	sakshi invests a part of rs . 12000 in 12 % stock at rs . 120 and the remainder in 15 % stock at rs . 125 . if his total dividend per annum is rs . 1360 , how much does he invest in 12 % stock at rs . 120 ? a ) rs . 4000 , b ) rs . 4500 , c ) rs . 5500 , d ) rs . 6000 , e ) rs . 6500	let investment in 12 % stock be rs . x . then , investment in 15 % stock = rs . ( 12000 - x ) . 12 / 120 * x + 15 / 125 * ( 12000 - x ) = 1360 x / 10 + 3 / 25 ( 12000 - x ) = 1360 5 x + 72000 - 6 x = 1360 x 50 x = 4000 . answer = a

Provide a detailed solution to the given mathematical problem.

two airplanes fly the same exact route from atlanta to chicago . plane a flies 200 mph while plane b flies 300 mph (assume that simple model versus actual acceleration / deceleration near the airports) . if plane a took off exactly 40 minutes before plane b , after how many minutes will plane b overtake plane a ? a) 65 , b) 80 , c) 90 , d) 115 , e) 120

we can use some form of the equation d = rt [ distance = rate * time ] rate of plane a : 200 mph rate of plane b : 300 mph plane a will be in the sky 40 minutes longer than plane b (2 / 3 of an hour) time of plane a when b overtakes it : t + (2 / 3) use 2 / 3 of an hour since the rate is in hours time of plane b when b overtakes a : t at the time that b overtakes a , they will have travelled the same distance so rt is equal for each plane : 200 * (t + 2 / 3) = 300 * t 200 t + 400 / 3 = 300 t 400 / 3 = 100 t 4 / 3 = t this is 1 and a third hour , so 60 minutes + 20 minutes = 80 minutes b

Solve the problem systematically, showing each step clearly.

a , b and c have rs . 450 between them , a and c together have rs . 200 and b and c rs . 350 . how much does c have ? a) a) 100 , b) b) 110 , c) c) 120 , d) d) 130 , e) e) 140

"a + b + c = 450 a + c = 200 b + c = 350 - - - - - - - - - - - - - - a + b + 2 c = 550 a + b + c = 450 - - - - - - - - - - - - - - - - c = 100 answer : a"

Explain the solution process for the given equation.

there are 600 employees in a room . 99 % are managers . how many managers must leave the room to bring down the percentage of managers to 98 % ? a) 100 , b) 200 , c) 250 , d) 300 , e) 400

"there are 594 managers and 6 others . the 6 others would compose 2 % of the total number of people if there were 300 people in the room . thus 300 managers must leave . the answer is d ."

Provide a detailed solution to the given mathematical problem.

a group of 4 investment bankers and 5 clients recently frequented the bonbon ribs restaurant . the total bill for the meal , including 20 % gratuity , came to $ 756 . on average , how much did the meal of each individual cost before gratuity ? a) $ 160 , b) $ 70 , c) $ 90 , d) $ 80 , e) $ 55

4 ibs and 5 clients - so total 9 people the bill $ 756 includes 20 % gratuity . . . so the actual cost of dinner was $ 630 now , the cost per person will be $ 630 / 9 which is $ 70 option b

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

if a - b = 3 and a (power 2) + b (power 2) = 21 , find the value of ab . a) 5 , b) 8 , c) 4 , d) 6 , e) 3

"2 ab = (a (power 2) + b (power 2) - (a - b) (power 2) = 21 - 9 = 12 ab = 6 . answer is d ."

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

worker a takes 10 hours to do a job . worker b takes 15 hours to do the same job . how long it take both a & b , working together but independently , to do the same job ? a) 40 days , b) 40 / 9 days , c) 39 / 9 days , d) 6 days , e) 60 / 9 days

"a' s one hour work = 1 / 10 . b' s one hour work = 1 / 15 . (a + b)' s one hour work = 1 / 10 + 1 / 15 = 5 / 30 = 1 / 6 . both a & b can finish the work in 6 days d"

Break down and solve the equation step by step.

a person is traveling at 100 km / hr and reached his destiny in 5 then find the distance ? a) 250 km , b) 300 km , c) 500 km , d) 450 km , e) 350 km

"t = 5 hr s = 100 km / hr d = t * s = 100 * 5 = 500 km answer is c"

Break down and solve the equation step by step.

what is the difference between the largest and the smallest number written with 7 , 7 , 0 , 7 ? a) 6893 , b) 6993 , c) 6093 , d) 693 , e) none

"7770 7077 - - - - - - - - - - - - - 693 answer : d"

Explain the solution process for the given equation.

a chemical supply company has 60 liters of a 25 % hno 3 solution . how many liters of pure undiluted hno 3 must the chemists add so that the resultant solution is a 50 % solution ? a) 12 , b) 15 , c) 20 , d) 24 , e) 30

"60 liters of a 25 % hno 3 solution means hno 3 = 15 liters in 60 liters of the solution . now , let x be the pure hno 3 added . as per question , 15 + x = 50 % of (60 + x) or x = 30 . hence , e"

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

find the lowest common multiple of 25 , 35 and 50 . a) 350 , b) 420 , c) 510 , d) 320 , e) 280

lcm = 2 * 5 * 5 * 7 = 350 . answer is a

Solve the problem systematically, showing each step clearly.

find the average of all the numbers between 40 and 80 which are divisible by 3 . a) 15 , b) 70 , c) 20 , d) 60 , e) none

"sol . average = (42 + 45 + 48 + 51 + 54 + 57 + 60 + 63 + 66 + 69 + 72 + 75 + 78 / 13) = 780 / 5 = 60 . answer d"

Solve the problem systematically, showing each step clearly.

arnold and danny are two twin brothers that are celebrating their birthday . the product of their ages today is smaller by 11 from the product of their ages a year from today . what is their age today ? a) 2 . , b) 4 . , c) 5 . , d) 7 . , e) 9 .

"ad = (a + 1) (d + 1) - 11 0 = a + d - 10 a + d = 10 a = d (as they are twin brothers) a = d = 5 c is the answer"

Break down and solve the equation step by step.

a classroom has equal number of boys and girls . 8 girls left to play kho - kho , leaving twice as many boys as girls in the classroom . what was the total number of girls and boys present initially ? a) 16 , b) 24 , c) 32 , d) 48 , e) 54

after 8 girls left remaining 8 girls now boys 16 are twice as many as remaining girls . initially boys = 16 and girls = 16 . answer : c

Break down and solve the equation step by step.

nd the area of trapezium whose parallel sides are 20 cm and 15 cm long , and the distance between them is 14 cm ? a) 230 cm 2 , b) 245 cm 2 , c) 255 cm 2 , d) 260 cm 2 , e) 280 cm 2

"area of a trapezium = 1 / 2 (sum of parallel sides) * (perpendicular distance between them) = 1 / 2 (20 + 15) * (14) = 245 cm 2 answer : b"

Provide a detailed solution to the given mathematical problem.

if a certain toy store' s revenue in november was 2 / 5 of its revenue in december and its revenue in january was 1 / 5 of its revenue in november , then the store' s revenue in december was how many times the average (arithmetic mean) of its revenues in november and january ? a) 5 / 2 , b) 10 / 3 , c) 15 / 4 , d) 25 / 3 , e) 25 / 6

"n = 2 d / 5 j = n / 5 = 2 d / 25 the average of november and january is (n + j) / 2 = 12 d / 25 / 2 = 6 d / 25 d is 25 / 6 times the average of november and january . the answer is e ."

Break down and solve the equation step by step.

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 . 2640 in all from both of them as interest . the rate of interest per annum is ? a) 16 % , b) 12 % , c) 74 % , d) 10 % , e) 45 %

"let the rate be r % p . a . then , (5000 * r * 2) / 100 + (3000 * r * 4) / 100 = 2640 100 r + 120 r = 2640 r = 12 % answer : b"

Solve the problem systematically, showing each step clearly.

an empty wooden vessel weighs 16 % of its total weight when filled with paint . if the weight of a partially filled vessel is one half that of a completely filled vessel , what fraction of the vessel is filled . a) 3 / 5 , b) 5 / 9 , c) 1 / 24 , d) 4 / 9 , e) 7 / 15

"an empty wooden vessel weighs 16 % of its total weight when filled with paint : vessel = 0.16 (vessel + paint) ; 16 v = v + p (so the weight of completely filled vessel is 16 v) p = 15 v (so the weight of the paint when the vessels is completely filled is 15 v) . the weight of a partially filled vessel is one half that of a completely filled vessel : v + p' = 1 / 2 * 16 v ; p' = 7 v (so the weight of the paint when the vessels is partially filled is 7 v) . what fraction of the vessel is filled ? so , we need to find the ratio of the weight of the paint when the vessel iscompletely filledto the weight of the paint when the vessel ispartially filled : p' / p = 7 v / 15 v = 7 / 15 . answer : e ."

Solve the problem systematically, showing each step clearly.

a regular hexagon is there . the mid points of the sides were joined and formed another hexagon . then what is the percentage reduction in area . a) 25 % , b) 35 % , c) 45 % , d) 55 % , e) 65 %

let abcdef be the regular hexagon with sides ab and ed parallel . p and q are mid points of af and ef resp . ae = root (3) * s (s is side of hexagon) then pq = (root (3) * s) / 2 mid point theorem now u know side of inner hexagon use formula area of hexagon = (3 * root (3)) 2 * s ^ 2 and find diff in area of two hexagon multiply by 100 and divide the product by area of bigger hexagon ans 25 % answer : a

Provide a detailed solution to the given mathematical problem.

in what ratio should a variety of rice costing rs . 4.5 per kg be mixed with another variety of rice costing rs . 8.75 per kg to obtain a mixture costing rs . 7.50 per kg ? a) 5 / 6 , b) 5 / 9 , c) 5 / 1 , d) 5 / 3 , e) 5 / 12

"let us say the ratio of the quantities of cheaper and dearer varieties = x : y by the rule of allegation , x / y = (8.75 - 7.50) / (7.50 - 4.5) = 5 / 12 answer : e"

Provide a detailed solution to the given mathematical problem.

a sum of money at simple interest amounts to 815 in 3 years and to 854 in 4 years . the sum is : a) 698 , b) 387 , c) 373 , d) 593 , e) 372

"a 698 s . i . for 1 year = (854 - 815) = 39 . s . i . for 3 years = (39 x 3) = 117 . principal = (815 - 117) = 698 ."

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

claire has a total of 92 pets consisting of gerbils and hamsters only . one - quarter of the gerbils are male , and one - third of the hamsters are male . if there are 25 males altogether , how many gerbils does claire have ? a) 39 , b) 68 , c) 54 , d) 57 , e) 60

"g + h = 92 . . . 1 ; g / 4 + h / 3 = 25 . . . . 2 or 3 g + 4 h = 25 * 12 = 300 g = 92 - h or 3 (92 - h) + 4 h = 300 h = 300 - 276 = 24 then g = 92 - 24 = 68 b"

Explain the solution process for the given equation.

100 people are attending a newspaper conference . 40 of them are writers and more than 38 are editors . of the people at the conference , x are both writers and editors and 2 x are neither . what is the largest possible number of people who are both writers and editors ? a) 21 , b) 24 , c) 28 , d) 30 , e) 32

"{ total } = { writers } + { editors } - { both } + { neither } . { total } = 100 ; { writers } = 40 ; { editors } > 38 ; { both } = x ; { neither } = 2 x ; 100 = 40 + { editors } - x + 2 x - - > x = 60 - { editors } . we want to maximize x , thus we should minimize { editors } , minimum possible value of { editors } is 39 , thus x = { both } = 60 - 39 = 21 . answer : a ."

Provide a detailed solution to the given mathematical problem.

a sum of money at simple interest amounts to rs . 1072 in 2 years and to rs . 1144 in 4 years . the sum is : a) rs . 1000 , b) rs . 850 , c) rs . 500 , d) rs . 1200 , e) rs . 900

"s . i . for 2 years = rs . (1144 - 1072) = rs . 72 . s . i . for 1 year = rs . 72 / 2 = rs . 36 . principal = rs . (1072 - 72) = rs . 1000 . answer : option a"

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

the average of the marks of 12 students in a class is 36 . if the marks of each student are doubled , find the new average ? a) 72 , b) 23 , c) 29 , d) 26 , e) 13

"sum of the marks for the 12 students = 12 * 36 = 432 . the marks of each student are doubled , the sum also will be doubled . the new sum = 432 * 2 = 864 . so , the new average = 864 / 12 = 72 . answer : a"

Provide a detailed solution to the given mathematical problem.

walking at 5 / 6 of its usual speed , a train is 10 minutes too late . what is usual time to cover the journey ? a) 70 minutes , b) 2 hours , c) 2 / 3 hours , d) 5 / 6 hours , e) 85 minutes

"et speed be s and time to cover the journey be t s * t = d (distance of journey) - - - 1 now as per the question stem we have (5 / 6) s * (t + 10) = d - - - - 2 equating value of d from (1) in (2) we get t = 50 mins or 5 / 6 hrs . answer : d"

Provide a detailed solution to the given mathematical problem.

a sum of money is to be distributed among a , b , c , d in the proportion of 5 : 2 : 4 : 3 . if c gets rs . 3000 more than d , what is b' s share ? a) a) 8239 , b) b) 2900 , c) c) 6000 , d) d) 2393 , e) e) 2009

"let the shares of a , b , c and d be 5 x , 2 x , 4 x and 3 x rs . respectively . then , 4 x - 3 x = 3000 = > x = 3000 . b' s share = rs . 2 x = 2 * 3000 = rs . 6000 . answer : c"

Solve the problem systematically, showing each step clearly.

multiplication of 2 numbers is 9375 and the quotient , when the larger one is divided by the smalleris 15 . find the sum of the numbers ? a) 200 , b) 300 , c) 400 , d) 500 , e) 600

let the numbers be x and y . then , xy = 9375 and x = 15 . y xy = 9375 (x / y) 15 y 2 = 625 . y = 25 . x = 15 y = (15 x 25) = 375 . sum of the numbers = x + y = 375 + 25 = 400 . c

Explain the solution process for the given equation.

roja and pooja start moving in the opposite directions from a pole . they are moving at the speeds of 6 km / hr and 3 km / hr respectively . after 4 hours what will be the distance between them ? a) 22 km , b) 20 km , c) 65 km , d) 18 km , e) 36 km

distance = relative speed * time = (6 + 3) * 4 = 36 km [ they are travelling in the opposite direction , relative speed = sum of the speeds ] . answer : e

Provide a detailed solution to the given mathematical problem.

the length of a rectangular field is 7 / 5 its width . if the perimeter of the field is 384 meters , what is the width of the field ? a) 50 , b) 60 , c) 11 , d) 80 , e) 90

"let l be the length and w be the width . l = (7 / 5) w perimeter : 2 l + 2 w = 384 , 2 (7 / 5) w + 2 w = 384 solve the above equation to find : w = 80 m and l = 112 m . correct answer d) 80"

Explain the solution process for the given equation.

there are 6 more women than there are men on a local co - ed softball team . if there are a total of 24 players on the team , what is the ratio of men to women ? a) 10 / 16 , b) 9 / 15 , c) 4 / 16 , d) 6 / 10 , e) 4 / 10

"w = m + 6 w + m = 24 m + 6 + m = 24 2 m = 18 m = 9 w = 15 ratio : 9 : 15 ans : b"

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

if the wheel is 15 cm then the number of revolutions to cover a distance of 1056 cm is ? a) 18 , b) 11.2 , c) 14 , d) 12 , e) 91

"2 * 22 / 7 * 15 * x = 1056 = > x = 11.2 answer : b"

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

look carefully for the pattern , and then choose which pair of numbers comes next . 28 25 5 21 18 5 14 a) 11 5 , b) 10 7 , c) 11 8 , d) 5 10 , e) 10 5

explanation : this is an alternating subtraction series with the interpolation of a random number , 5 , as every third number . in the subtraction series , 3 is subtracted , then 4 , then 3 , and so on . answer : option a

Break down and solve the equation step by step.

the average of runs of a cricket player of 20 innings was 32 . how many runs must he make in his next innings so as to increase his average of runs by 6 ? a) 96 , b) 106 , c) 158 , d) 116 , e) 122

"average = total runs / no . of innings = 32 so , total = average x no . of innings = 32 * 20 = 640 now increase in avg = 4 runs . so , new avg = 32 + 6 = 38 runs total runs = new avg x new no . of innings = 38 * 21 = 798 runs made in the 11 th inning = 798 - 640 = 158 answer : c"

Solve the problem systematically, showing each step clearly.

there are 9 teams in a certain league and each team plays each of the other teams exactly once . what is the total number of games played ? a) 24 , b) 25 , c) 28 , d) 32 , e) 36

"9 c 2 = 36 the answer is e"

Break down and solve the equation step by step.

if 60 % of (x - y) = 30 % of (x + y) then what percent of x is y ? a) 33 % , b) 10 % , c) 5 % , d) 15 % , e) 25 %

"60 % of (x - y) = 30 % of (x + y) (60 / 100) (x - y) = (30 / 100) (x + y) 6 (x - y) = 3 (x + y) 3 x = 9 y x = 3 y therefore required percentage = ((y / x) x 100) % = ((y / 3 y) x 100) = 33 % answer is a ."

Solve the problem systematically, showing each step clearly.

during a sale of 20 % on everything in a store , a kid is successful in convincing the store manager to give him 20 candies for the discounted price of 14 candies . the store still makes a profit of 10 % on this sale . what is the mark up percentage on each candy ? a) 100 % , b) 80 % , c) 75 % , d) 66 + 2 / 3 % , e) 55 %

"hi akhil , i can share the way i solved it . . let' s say marked price = m so , there' s a discount of 20 % on m so , new s . p . = 80 % of m now , the child convinces the owner to sell 20 candies for the price of 14 candies . let' s say each candy after discount is 1 $ . so , s . p . of 20 candies = 20 $ . the child bought it for 14 $ so , he got a discount of 6 / 20 * 100 = 30 % so , the latest s . p . = 70 % of 80 % of m = 0.7 * 0.8 m now , we are given that the shopkeeper still makes a profit of 12 % . so we have , 0.7 * 0.8 * m = 1.12 c . p so , we get , m = 2 c . p . i . e . marked price was kept 100 % above c . p . c"

Solve the problem systematically, showing each step clearly.

the difference between c . i . and s . i . on an amount of $ 10,000 for 2 years is $ 49 . what is the rate of interest per annum ? a) 5 % , b) 6 % , c) 7 % , d) 8 % , e) 9 %

$ 49 is the interest on the first year of interest . let x be the interest rate . the interest after the first year is 10000 * x . the interest on the first year' s interest is 10000 * x * x 10000 * x ^ 2 = 49 x = 0.07 the answer is c .

Provide a detailed solution to the given mathematical problem.

from a container , 6 liters milk was drawn out and was replaced by water . again 6 liters of mixture was drawn out and was replaced by the water . thus the quantity of milk and water in the container after these two operations is 9 : 16 . the quantity of mixture is : a) 15 , b) 26 , c) 22 , d) 27 , e) 11

explanation : let quantity of mixture be x liters . suppose a container contains x units of liquid from which y units are taken out and replaced by water . after operations , the quantity of pure liquid = units , where n = no of operations . so , quantity of milk = given that , milk : water = 9 : 16 milk : (milk + water) = 9 : (9 + 16) milk : mixture = 9 : 25 answer : a) 15

Break down and solve the equation step by step.

how many 3 digit no' s are between 100 - 500 , where sum of two digit is 3 rd ? a) 24 , b) 54 , c) 60 , d) 84 , e) 94

total permutations possible = 18 + 16 + 14 + 12 = 60 answer : c

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

a certain music store stocks 800 cellos and 600 violas . of these instruments , there are 120 cello - viola pairs , such that a cello and a viola were both made with wood from the same tree (each tree can make at most one viola and one cello , so there are no pairs other than these 90) . if one viola and one cello are chosen at random , what is the probability that the two instruments are made with wood from the same tree ? a) 3 / 16,000 , b) 1 / 8,000 , c) 3 / 1,600 , d) 1 / 90 , e) 2 / 45

"solution provided by stanford 2012 is correct : 120 / 800 choosing one of the cellos which has a pair viola , 1 / 600 choosing the viola which is the pair of chosen cello - - > p = 120 / 800 * 1 / 600 = 1 / 8,000 . answer : b ."

Provide a detailed solution to the given mathematical problem.

a tank is filled by 3 pipes a , b , c in 5 hours . pipe c is twice as fast as b and b is twice as fast as a . how much will pipe a alone take to fill the tank ? a) 25 hr , b) 35 hr , c) 40 hr , d) 20 hr , e) 50 hr

"suppose pipe a alone take x hours to fill the tank then pipe b and c will take x / 2 and x / 4 hours respectively to fill the tank . 1 / x + 2 / x + 4 / x = 1 / 5 7 / x = 1 / 5 x = 35 hours answer is b"

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

a sum of money lent out at s . i . amounts to rs . 780 after 2 years and to rs . 1020 after a further period of 5 years . the sum is ? a) s . 684 , b) s . 500 , c) s . 540 , d) s . 740 , e) s . 840

"s . i for 5 years = (1020 - 780) = rs . 240 . s . i . for 2 years = 240 / 5 * 2 = rs . 96 . principal = (780 - 96) = rs . 684 answer : a"

Solve the problem systematically, showing each step clearly.

in expressing a length of 61.472 km as nearly as possible with the 3 significant digits , find the percentage error ? a) 0.35 % , b) 0.4 % , c) 0.045 % , d) 0.6 % , e) none of these

explanation : error = (61.5 - 61.472) = 0.028 required percentage = 0.028 / 61.472100 = 0.045 answer : c

Explain the solution process for the given equation.

how many numbers from 49 to 79 are exactly divisible by 11 ? a) 5 , b) 7 , c) 9 , d) 3 , e) 12

"49 / 11 = 4 and 79 / 11 = 7 = = > 7 - 4 = 3 numbers answer : d"

Provide a detailed solution to the given mathematical problem.

a candidate got 25 % of the votes polled and he lost to his rival by 4000 votes . how many votes were cast ? a) 7500 , b) 2028 , c) 2775 , d) 5496 , e) 8000

"25 % - - - - - - - - - - - l 75 % - - - - - - - - - - - w - - - - - - - - - - - - - - - - - - 50 % - - - - - - - - - - 4000 100 % - - - - - - - - - ? = > 8000 answer : e"

Explain the solution process for the given equation.

if 12 men or 20 women can do a piece of work in 27 days , then in how many days can 9 men and 12 women together do the work ? a) 10 days , b) 30 days , c) 20 days , d) 80 days , e) 40 days

"c 20 days given that 12 m = 20 w = > 3 m = 5 w 9 men + 12 women = 15 women + 12 women = 27 women 20 women can do the work in 27 days . so , 27 women can do it in (20 * 27) / 27 = 20 days ."

Provide a detailed solution to the given mathematical problem.

a big container is 50 % full with water . if 20 liters of water is added , the container becomes 3 / 4 full . what is the capacity of the big container in liters ? a) 80 , b) 90 , c) 100 , d) 110 , e) 120

"20 liters is 25 % of the capacity c . 20 = 0.25 c c = 80 liters . the answer is a ."

Break down and solve the equation step by step.

the sides of a triangle are in the ratio 3 : 5 : 7 . if the pereimeter is 60 cms , the longest side is a) 7 , b) 30 , c) 28 , d) 25 , e) 16

"3 x + 5 x + 7 x = 60 15 x = 60 x = 4 longest side = 7 * 4 = 28 answer : c"

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

a cylinder is inserted in a sphere d / h = 2 / 3 . find the surface area of the cylinder ? ['a) 23.12','b) 24.12','c) 25.12','d) 26.12','e) 27.12']

take r = 1 , h = 3 = 2 * 3.14 * r * r + 2 * 3.14 * r * h = 2 * 3.14 * r (r + h) = 2 * 3.14 * 1 (1 + 3) = 2 * 3.14 (4) = 25.12 answer : c

Break down and solve the equation step by step.

287 x 287 + 269 x 269 - 2 x 287 x 269 = ? a) 534 , b) 446 , c) 354 , d) 324 , e) none of them

= a ^ 2 + b ^ 2 - 2 ab where a = 287 and b = 269 = ( a - b ) ^ 2 = ( 287 - 269 ) ^ 2 = ( 18 ) ^ 2 = 324 answer is d

Solve the problem systematically, showing each step clearly.

a small water pump would take 4 hours to fill an empty tank . a larger pump would take 1 / 2 hour to fill the same tank . how many hours would it take both pumps , working at their respective constant rates , to fill the empty tank if they began pumping at the same time ? a ) 1 / 6 , b ) 4 / 9 , c ) 1 / 4 , d ) 5 / 9 , e ) 1 / 3

"rate of the small pump is 1 / 4 tank / hour rate of the larger pump is 1 / ( 1 / 2 ) or 2 tank / hour ; combined rate of the two pumps is 1 / 4 + 2 = 9 / 4 tank / hour , together they will fill the empty tank in 1 / ( 9 / 4 ) ) = 4 / 9 hours ( time = job / rate ) . answer : b"

Explain the solution process for the given equation.

what is the total number of prime numbers less than 70 ? a ) 17 , b ) 18 , c ) 19 , d ) 20 , e ) 21

solution prime numbers less than 70 are = 12 . 2,3 , 5,7 , 11,13 , 17,19 , 23,29 , 31,37 , 41,43 , 47,53 , 59,61 , and 67 their number is 19 . answer c

Break down and solve the equation step by step.

in a sample of 800 high school students in which all students are either freshmen , sophomores , juniors , or seniors , 22 percent are juniors and 75 percent are not sophomores . if there are 160 seniors , how many more freshmen than sophomores are there among the sample of students ? a ) 42 , b ) 48 , c ) 64 , d ) 76 , e ) 88

"200 are sophomores . the number of freshmen is 600 - 160 - 0.22 ( 800 ) = 264 the answer is c ."

Break down and solve the equation step by step.

the value of ( 34.31 * 0.473 * 1.567 ) / ( 0.0673 * 23.25 * 7.57 ) is close to a ) 2 , b ) 1.15 , c ) 2.05 , d ) 2.15 , e ) 2.35

"( 34.31 * 0.473 * 1.567 ) / ( 0.0673 * 23.25 * 7.57 ) = 25.4303 / 11.845 = 2.15 answer : d"

Provide a detailed solution to the given mathematical problem.

let p be a prime number . if n = 7 p , how many divisors does n have ? a ) 1 , b ) 2 , c ) 3 , d ) 4 , e ) 5

two . n = 7 * p , and both 7 and p are prime . so the answer is b .

Solve the problem systematically, showing each step clearly.

every student in a room is either a junior or a senior . there is at least one junior and at least one senior in the room . if 2 / 5 of the juniors is equal to 1 / 3 of the seniors , what fraction of the students in the room are juniors ? a ) 3 / 20 , b ) 1 / 3 , c ) 5 / 12 , d ) 12 / 17 , e ) 5 / 11

"let total number of juniors = j total number of seniors = s ( 2 / 5 ) j = ( 1 / 3 ) s = > s = 6 / 5 j total number of students = j + s = ( 11 / 5 ) j fraction of the students in the room are juniors = j / ( j + s ) = j / [ ( 11 / 5 ) j ] = 5 / 11 answer e"

Explain the solution process for the given equation.

in a division sum , the remainder is 8 and the divisor is 3 times the quotient and is obtained by adding 3 to the thrice of the remainder . the dividend is : a ) 200 , b ) 251 , c ) 250 , d ) 262 , e ) 254

"diver = ( 8 * 3 ) + 3 = 27 3 * quotient = 27 quotient = 9 dividend = ( divisor * quotient ) + remainder dividend = ( 27 * 9 ) + 8 = 251 b"

Explain the solution process for the given equation.

31 of the scientists that attended a certain workshop were wolf prize laureates , and 12 of these 31 were also nobel prize laureates . of the scientists that attended that workshop and had not received the wolf prize , the number of scientists that had received the nobel prize was 3 greater than the number of scientists that had not received the nobel prize . if 50 of the scientists attended that workshop , how many of them were nobel prize laureates ? a ) a ) 11 , b ) b ) 23 , c ) c ) 24 , d ) d ) 29 , e ) d ) 36

"lets solve by creating equation . . w = 31 . . total = 50 . . not w = 50 - 31 = 19 . . now let people who were neither be x , so out of 19 who won nobel = x + 3 . . so x + x + 3 = 19 or x = 8 . . so who won nobel but not wolf = x + 3 = 11 . . but people who won both w and n = 12 . . so total who won n = 11 + 12 = 23 . . b"

Provide a detailed solution to the given mathematical problem.

a square mirror has exactly half the area of the rectangular wall on which it is hung . if each side of the mirror is 24 inches and the width of the wall is 42 inches , what is the length of the wall , in inches ? a ) 21.4 , b ) 23.4 , c ) 25.4 , d ) 26.4 , e ) 27.4

"since the mirror is 42 inches in all sides , it must be a square . area of a square is a = a ^ 2 ; 24 ^ 2 = 576 . area of rectangle is double of that 2 * 576 = 1152 . now a = lw and we need find w so a / l = w ; 1152 / 42 = 27.4 answer ! answer : e"

Provide a detailed solution to the given mathematical problem.

the sum of money will be double itself in 20 years and simple interest find rate of interest ? a ) 10 % , b ) 5 % , c ) 15 % , d ) 20 % , e ) 25 %

"t = 20 years p = principle amount = x a = total amount = 2 x si = simple interest = a - p = 2 x - x = x r = 100 si / pt = 100 x / 20 x = 5 % answer is b"

Break down and solve the equation step by step.

bert and rebecca were looking at the price of a condominium . the price of the condominium was 110 % more than bert had in savings , and separately , the same price was also 20 % more than rebecca had in savings . what is the ratio of what bert has in savings to what rebecca has in savings . a ) 1 : 5 , b ) 1 : 4 , c ) 2 : 3 , d ) 4 : 7 , e ) 1 : 2

"suppose bert had 100 so price becomes 210 , this 210 = 1.2 times r' s saving . . so r' s saving becomes 175 so required ratio is 100 : 175 = 4 : 7 answer : d"

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

a bullet train 200 m long is running with a speed of 69 kmph . in what time will it pass a man who is running at 3 kmph in the direction opposite to that in which the bullet train is going ? a ) 10 sec , b ) 15 sec , c ) 12 sec , d ) 11 sec , e ) 16 sec

"a 10 sec speed of the bullet train relative to man = ( 69 + 3 ) kmph = 72 * 5 / 18 m / sec = 60 / 3 m / sec . time taken by the bullet train to cross the man = time taken by it to cover 200 m at ( 60 / 3 ) m / sec = ( 200 * 3 / 60 ) sec = 10 sec"

Provide a detailed solution to the given mathematical problem.

in a village there are 150 men and 90 women in present time . if in next year population will be p = ( a ^ 2 + b ^ 2 ) ^ 1 / 2 , and in every year men are reduces 6 % . what is population of after 2 year . a ) 140 , b ) 141 , c ) 142 , d ) 143 , e ) 145

"next year total population = [ 150 ^ 2 + 90 ^ 2 ] ^ . 5 = 174.92 = 175 man decreased by 6 % so total man = 150 * . 94 = 141 women will be = 175 - 141 = 34 so population after two years = [ 135 ^ 2 + 34 ^ 2 ] ^ . 5 = 145.04 = 145 so population after two year = 145 answer : e"

Explain the solution process for the given equation.

the value of a machine depreciates at 20 % per annum . if its present value is rs . 1 , 50,000 , at what price should it be sold after two years such that a profit of rs . 20,000 is made ? a ) 328897 , b ) 120000 , c ) 877888 , d ) 116000 , e ) 188871

"the value of the machine after two years = 0.8 * 0.8 * 1 , 50,000 = rs . 96,000 sp such that a profit of rs . 20,000 is made = 96,000 + 20,000 = rs . 1 , 16,000 answer : d"

Solve the problem systematically, showing each step clearly.

a distributor sells a product through an on - line store , which take a commission of 20 % of the price set by the distributor . the distributor obtains the product from a producer at the price of $ 15 per item . what is the price that the buyer observers on - line if the distributor wants to maintain a 35 % profit on the cost of the item ? a ) 18 , b ) 21.6 , c ) 22 , d ) 22.5 , e ) 27.7

"producer price = $ 15 ; the distributor wants to maintain a 20 % profit on the cost of the item , thus he must get $ 15 * 1.2 = $ 18 after the store takes a commission of 35 % of the final price - - > ( final price ) * 0.65 = $ 18 - - > ( final price ) = $ 27.7 . answer : e ."

Break down and solve the equation step by step.

a bag holds 4 red marbles and 5 green marbles . if you removed two randomly selected marbles from the bag , without replacement , what is the probability that both would be red ? a ) 1 / 10 , b ) 1 / 6 , c ) 3 / 10 , d ) 2 / 5 , e ) 1 / 4

"probability of selecting first red marble = 4 / 9 probability of selecting second red marble without replacement = 3 / 8 final probability = 4 / 9 * 3 / 8 = 1 / 6 the correct answer is b ."

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

if 9 - 4 / x = 7 + 8 / x , then x = a ) 15 , b ) 6 , c ) 1 , d ) 5 , e ) 12

we' re given the equation 9 - 4 / x = 7 + 8 / x . we' re asked for the value of x . the common - denominator of these 4 numbers is x , so we have to multiple 2 of the numbers to give them that denominator . . . 9 x - 4 x / x = 7 x + 8 x / x we can then eliminate that denominator , which gives us . . . . 9 x - 4 = 7 x + 8 12 = 2 x both 2 x and 12 have a common denominator of 2 , so we divide both sides by 2 for our final answer . . . 6 = x b

Solve the problem systematically, showing each step clearly.

if ( 5 ^ 13 ) ( 9 ^ 7 ) = 3 ( 15 ^ x ) , what is the value of x ? a ) 7 , b ) 9 , c ) 11 , d ) 13 , e ) 15

( 5 ^ 13 ) ( 9 ^ 7 ) = 3 ( 15 ^ x ) = > 5 ^ 13 * 3 ^ 14 = 3 * 3 ^ x * 5 ^ x = > 5 ^ 13 * 3 ^ 14 = 3 ^ ( x + 1 ) * 5 ^ x value of x = 13 answer d

Provide a detailed solution to the given mathematical problem.

a and b started a business in partnership investing rs . 20,000 and rs . 15,000 respectively . after 6 months , c joined them with rs . 20,000 . whatwill be b' s share in total profit of rs . 21,000 earned at the end of 2 years from the startingof the business ? a ) s . 5,000 , b ) s . 5,500 , c ) s . 5,700 , d ) s . 6,300 , e ) s . 7,500

"a : b : c = ( 20,000 x 24 ) : ( 15,000 x 24 ) : ( 20,000 x 18 ) = 4 : 3 : 3 . b' s share = rs . 21000 x 3 / 10 = rs . 6,300 . d"

Break down and solve the equation step by step.

a couple married in 1980 had two children , one in 1982 and the other in 1984 . their combined ages will equal the years of the marriage in ? a ) 1986 , b ) 1985 , c ) 1987 , d ) 1988 , e ) 1989

sum of year = 1 + 9 + 8 + 0 = ( 18 ) 1 + 8 = 9 1 + 9 + 8 + 2 = ( 20 ) 2 + 0 = 2 1 + 9 + 8 + 4 = ( 13 ) 1 + 3 = 4 now 9 + 2 + 4 = ( 15 ) 1 + 5 = 6 nd option a 1 + 9 + 8 + 6 = 24 ( 2 + 4 ) = 6 answer : a

Solve the problem systematically, showing each step clearly.

how many trailing zeroes does 49 ! + 50 ! have ? a ) 9 , b ) 10 , c ) 11 , d ) 12 , e ) 22

"49 ! + 50 ! = 51 * 49 ! no of trailing 0' s = no of 5 * 2 no of 2' s = 49 / 2 + 49 / 4 + 49 / 8 + 49 / 16 + 49 / 32 = d no of 5' s = 49 / 5 + 49 / 25 = 10 = p now since p < d no of trailing 0' s = 10 answer : b"

Explain the solution process for the given equation.

a new tower has just been built at the verbico military hospital ; the number of beds available for patients at the hospital is now 6 times the number available before the new tower was built . currently , 1 / 3 of the hospital' s original beds , as well as 1 / 5 of the beds in the new tower , are occupied . for the purposes of renovating the hospital' s original wing , all of the patients in the hospital' s original beds must be transferred to beds in the new tower . if patients are neither admitted nor discharged during the transfer , what fraction of the beds in the new tower will be unoccupied once the transfer is complete ? a ) 11 / 30 , b ) 29 / 60 , c ) 17 / 30 , d ) 19 / 30 , e ) 11 / 15

"i think e - 11 / 15 is the correct answer . here goes : lets assume originally the number of beds = x after the new tower , the total combined no of beds = 6 x so old = x , new = 5 x now 1 / 3 of x are occupied and 1 / 5 of 5 x are occupied which simplifies to ( 5 / 5 ) x we are shifting 1 / 3 of x to the new ward so there will now be : 1 / 3 of x plus 5 / 5 of x occupied in the new ward . add them up to get 4 / 3 of x there are 5 x beds in new tower so ratio is : ( 4 / 3 ) x / 5 x = 4 / 15 of x subtract that from 15 / 15 of x and you get the number of un - occupied beds to total capacity of new tower = 11 / 15 . e"

Explain the solution process for the given equation.

if x < y < z and y - x > 3 , where x is an even integer and y and z are odd integers , what is the least possible value of z - x ? a ) 7 , b ) 6 , c ) 8 , d ) 9 , e ) 10

"we have : 1 ) x < y < z 2 ) y - x > 3 3 ) x = 2 k ( x is an even number ) 4 ) y = 2 n + 1 ( y is an odd number ) 5 ) z = 2 p + 1 ( z is an odd number ) 6 ) z - x = ? least value z - x = 2 p + 1 - 2 k = 2 p - 2 k + 1 = 2 ( p - k ) + 1 - that means that z - x must be an odd number . we are asked to find the least value , so we have to pick the least numbers since y is odd and x is even , y - x must be odd . since y - x > 3 , the least value for y - x must be 5 , the least value for x must be 2 , and , thus , the least possible value for y must be 7 ( y - 2 = 5 , y = 7 ) 2 < 7 < z , since z is odd , the least possible value for z is 9 z - x = 9 - 2 = 7 answer a"

Solve the problem systematically, showing each step clearly.

a work which could be finished in 11 days was finished 3 days earlier after 10 more men joined . the number of men employed was ? a ) 22 , b ) 20 , c ) 88 , d ) 71 , e ) 12

"x - - - - - - - 11 ( x + 10 ) - - - - 6 x * 11 = ( x + 10 ) 6 x = 12 \ answer : e"

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

how many positive integers less than 243 are multiple of 4 but not multiples of 6 ? a ) 20 , b ) 31 , c ) 42 , d ) 40 , e ) 64

"240 / 4 = 60 multiples of 4 which are a multiple of 6 will be of the form 2 * 2 * 3 = 12 n where n > 0 240 / 12 = 20 60 - 20 = 40 answer : d"

Solve the problem systematically, showing each step clearly.

john traveled 80 % of the way from yellow - town to green - fields by train at an average speed of 80 miles per hour . the rest of the way john traveled by car at an average speed of v miles per hour . if the average speed for the entire trip was 45 miles per hour , what is v in miles per hour ? a ) 30 , b ) 40 , c ) 50 , d ) 55 , e ) 70

"hibunuel the question seems incorrect . it should not be 80 % at the speed of 80 . however if it' s 20 % at the speed of 80 , answer comes out 55 . the question is correct . here' s the explanation : let distance be d . we can find the total timeequate it , which comes as : 0.8 d / 80 + 0.2 d / v = d / 40 = > v = 40 ( option b ) ."

Provide a detailed solution to the given mathematical problem.

three numbers are in the ratio of 2 : 3 : 4 and their l . c . m . is 264 . what is their h . c . f . ? a ) 18 , b ) 22 , c ) 24 , d ) 28 , e ) 32

"let the numbers be 2 x , 3 x , and 4 x . lcm of 2 x , 3 x and 4 x is 12 x . 12 x = 264 x = 22 hcf of 2 x , 3 x and 4 x = x = 22 the answer is b ."

Provide a detailed solution to the given mathematical problem.

one - tenth of the students at a nursery school are 4 years old or older . if 20 students have not yet reached their third birthday , and a total of 50 students are not between 3 years old and 4 years old , how many children are in the nursery school ? a ) 150 , b ) 300 , c ) 450 , d ) 490 , e ) 620

x / 10 students are > 4 yrs 20 students are < 3 yrs x / 10 + 20 = 50 x / 10 = 30 x = 300 answer : b

Provide a detailed solution to the given mathematical problem.

the speed at which a man can row a boat in still water is 18 kmph . if he rows downstream , where the speed of current is 6 kmph , what time will he take to cover 100 metres ? a ) 6.67 seconds , b ) 18 seconds , c ) 26 seconds , d ) 14.99 seconds , e ) 6 seconds

"speed of the boat downstream = 18 + 6 = 24 kmph = 24 * 5 / 18 = 6.67 m / s hence time taken to cover 100 m = 100 / 6.67 = 14.99 seconds . answer : d"

Provide a detailed solution to the given mathematical problem.

calculate how many seconds it eill take for 4 bells to toll together again , given that they begin to toll together respectively at the intervals of 2 , 5 , 8 and 11 seconds . ? a ) 440 , b ) 420 , c ) 410 , d ) 442 , e ) 422

"lcm of 2 , 5 , 8 and 11 is 440 lcm = 440 answer : a"

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

on a partly cloudy day , derek decides to walk back from work . when it is sunny , he walks at a speed of s miles / hr ( s is an integer ) and when it gets cloudy , he increases his speed to ( s + 1 ) miles / hr . if his average speed for the entire distance is 2.8 miles / hr , what fraction r of the total distance did he cover while the sun was shining on him ? a ) 1 / 4 , b ) 4 / 5 , c ) 1 / 5 , d ) 1 / 6 , e ) 1 / 7

if s is an integer and we know that the average speed is 2.8 , s must be = 2 . that meanss + 1 = 3 . this implies that the ratio of time for s = 2 is 1 / 4 of the total time . the formula for distance / rate is d = rt . . . so the distance travelled when s = 2 is 2 t . the distance travelled for s + 1 = 3 is 3 * 4 t or 12 t . therefore , total distance covered while the sun was shining over him is r = 2 / 14 = 1 / 7 . answer : e

Break down and solve the equation step by step.

a woman complete a journey in 30 hours . she travels first half of the journey at the rate of 21 km / hr and second half at the rate of 24 km / hr . find the total journey in km . a ) 334 km . , b ) 216 km . , c ) 314 km . , d ) 224 km . , e ) 672 km .

"0.5 x / 21 + 0.5 x / 24 = 30 - - > x / 21 + x / 24 = 60 - - > x = 672 km . e"

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for the positive integers x , x + 2 , x + 4 , x + 7 , and x + 27 , the mean is how much greater than the median ? a ) 0 , b ) 1 , c ) 2 , d ) 4 , e ) 7

"mean = ( x + x + 2 + x + 4 + x + 7 + x + 27 ) / 5 = ( 5 x + 40 ) / 5 = x + 8 median = x + 4 thus mean - median = x + 8 - ( x + 4 ) = 4 answer = d"

Break down and solve the equation step by step.

a meeting has to be conducted with 4 managers . find the number of ways in which the managers be selected from among 7 managers , if 2 managers will not attend the meeting together ? a ) 25 , b ) 20 , c ) 35 , d ) 36 , e ) 38

"we can either choose all 4 people from 5 manager who have no problems or choose 3 from the 5 and 1 from the 2 managers who have a problem sitting together so 5 c 4 + ( 5 c 3 * 2 c 1 ) this is 5 + 20 = 25 answer : a"

Solve the problem systematically, showing each step clearly.

a certain number when divided by 110 leaves a remainder 25 , what is the remainder if the same no . be divided by 15 ? a ) 4 , b ) 5 , c ) 6 , d ) 8 , e ) 9

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

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

a , b and c are partners . a receives 2 / 3 of profits , b and c dividing the remainder equally . a' s income is increased by rs . 200 when the rate to profit rises from 5 to 7 percent . find the capital of a ? a ) 3377 , b ) 2899 , c ) 2500 , d ) 2778 , e ) 10000

"a : b : c = 2 / 3 : 1 / 6 : 1 / 6 = 4 : 1 : 1 x * 2 / 100 * 2 / 3 = 200 a capital = 15000 * 2 / 3 = 10000 answer : e"

Break down and solve the equation step by step.

how many seconds will a 400 metre long train take to cross a man running with a speed of 6 km / hr in the direction of the moving train if the speed of the train is 46 km / hr ? a ) 25 , b ) 30 , c ) 40 , d ) 45 , e ) 36

"explanation : speed of train relatively to man = ( 46 - 6 ) km / hr = 40 km / hr = ( 40 x 5 / 18 ) m / sec = 11.11 m / sec time taken to pass the man = ( 400 / 11.11 ) sec = 36 sec . answer : e"

Break down and solve the equation step by step.

one night 15 percent of the female officers on a police force were on duty . if 300 officers were on duty that night and half of these were female officers , how many female officers were on the police force ? a ) 90 , b ) 180 , c ) 270 , d ) 500 , e ) 1,000

"let total number of female officers in the police force = f total number of officers on duty on that night = 300 number of female officers on duty on that night = 300 / 2 = 150 ( 15 / 100 ) * f = 150 = > f = 1000 answer e"

Solve the problem systematically, showing each step clearly.

the distance from steve' s house to work is 40 km . on the way back steve drives twice as fast as he did on the way to work . altogether , steve is spending 6 hours a day on the roads . what is steve' s speed on the way back from work ? a ) 5 . , b ) 10 . , c ) 14 . , d ) 15 . , e ) 20 .

"time is in the ratio 2 : 1 : : to : fro office therefore , 2 x + 1 x = 6 hrs time take to come back - 2 hrs , distance travelled - 40 km = > speed = 20 kmph e"

Break down and solve the equation step by step.

if x is real , find the maximum value of the expression - 2 x ^ 2 + 9 x + 11 . a ) 12.125 , b ) 15.125 , c ) 18.125 , d ) 21.125 , e ) 24.125

"this is an equation of a downward facing parabola . the maximum value is the top point of the parabola . - 2 x ^ 2 + 9 x + 11 = ( - 2 x + 11 ) ( x + 1 ) the roots are 11 / 2 and - 1 . the maximum value must be when x is halfway between these two points . x = 2.25 the maximum value is - 2 ( 2.25 ) ^ 2 + 9 ( 2.25 ) + 11 = 21.125 the answer is d ."

Explain the solution process for the given equation.

milk contains 5 % water . what quantity of pure milk should be added to 10 litres of milk to reduce this 2 % ? a ) 15 litres , b ) 16 litres , c ) 17 litres , d ) 25 litres , e ) 35 litres

present ration of milk to water is 95 : 5 amount of milk is 10 litres hence , pure milk content is 9.5 litre and water content is 0.5 litres now , we should not disturb the water content so , 0.5 litres of water is equivalent to 2 % so 100 % mixture is : ( 0.5 / 2 ) * 100 = 25 litre thus , milk should be added = 25 - 10 = 15 litres of pure milk answer : a

Provide a detailed solution to the given mathematical problem.

the speed of a boat in still water is 15 km / hr and the rate of current is 3 km / hr . the distance travelled downstream in 12 minutes is a ) 1.2 km , b ) 1.8 km , c ) 2.4 km , d ) 3.6 km , e ) 1.6 km

"b = 15 km / hr , s = 3 km / hr , downstream b + s = 15 + 3 = 18 km / hr down stream time t = 12 min , we have to change it to hours so t = 12 / 60 = 1 / 5 hrs . distance = speed * time = 18 * 1 / 5 = 3.6 km . answer : d"

Solve the problem systematically, showing each step clearly.

what is the angle between the hands of a clock when time is 8 : 40 ? a ) 65 deg , b ) 75 deg , c ) 45 deg , d ) 15 deg , e ) 100 deg

"angle between two hands = 40 h - 11 / 2 m = 40 * 8 - 40 * 11 / 2 = 320 - 220 = 100 deg answer : e"

Break down and solve the equation step by step.

farm tax is levied on the 75 % of the cultivated land . the tax department collected total $ 3840 through the farm tax from the village of mr . william . mr . william paid only $ 480 as farm tax . the percentage of total land of mr . willam over the total taxable land of the village is : a ) 15 % , b ) 25 % , c ) 0.125 % , d ) 0.2083 % , e ) 16.667 %

"this will be equal to the percentage of total cultivated land he holds over the total cultivated land in the village . that leads to ( 480 / 3840 ) x 100 = 12.5 % in percentage terms . but the question asks ratio between his total land to total cultivated land . hence the answer is 12.5 % x ( 100 / 75 ) = 16.667 % the correct answer is ( e ) ."

Explain the solution process for the given equation.

a certain taxi company charges $ 3.00 for the first 1 / 5 of a mile plus $ 0.40 for each additional 1 / 5 of a mile . what would this company charge for a taxi ride that was 8 miles long ? a ) 15.6 , b ) 16.0 , c ) 17.5 , d ) 18.6 , e ) 19.1

"a certain taxi company charges $ 3.00 for the first 1 / 5 of a mile plus $ 0.40 for each additional 1 / 5 of a mile . what would this company charge for a taxi ride that was 8 miles long ? a . 15.60 b . 16.00 c . 17.50 d . 18.70 e . 19.10 1 / 5 miles = 0.2 miles . the cost of 8 miles long ride would be $ 3.00 for the first 0.2 miles plus ( 8 - 0.2 ) / 0.2 * 0.4 = $ 3.0 + $ 15.6 = $ 18.6 . answer : d ."

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

a hall is 15 m long and 12 m broad . if the sum of the areas of the floor and the ceiling is equal to the sum of the areas of four walls , the volume of the hall is : a ) 3348 , b ) 3898 , c ) 1200 , d ) 2881 , e ) 7881

answer : c ) 1200

Solve the problem systematically, showing each step clearly.

a vessel of capacity 90 litres is fully filled with pure milk . nine litres of milk is removed from the vessel and replaced with water . nine litres of the solution thus formed is removed and replaced with water . find the quantity of pure milk in the final milk solution ? a ) 73.9 , b ) 72.9 , c ) 73.0 , d ) 73.5 , e ) 92.2

"let the initial quantity of milk in vessel be t litres . let us say y litres of the mixture is taken out and replaced by water for n times , alternatively . quantity of milk finally in the vessel is then given by [ ( t - y ) / t ] n * t for the given problem , t = 90 , y = 9 and n = 2 . hence , quantity of milk finally in the vessel = [ ( 90 - 9 ) / 90 ] 2 ( 90 ) = 72.9 litres . answer : option b"

Break down and solve the equation step by step.

the tax on a commodity is diminished by 30 % but its consumption is increased by 15 % . find the decrease percent in the revenue derived from it ? a ) 20 % , b ) 18 % , c ) 15 % , d ) 12 % , e ) 14 %

"100 * 100 = 10000 70 * 115 = 8050 10000 - - - - - - - 1950 100 - - - - - - - ? = 20 % answer : a"

Explain the solution process for the given equation.

if both 7 ^ 2 and 3 ^ 4 are factors of the number a * 4 ^ 3 * 6 ^ 2 * 13 ^ 11 , then what is the smallest possible value of a ? a ) 120 , b ) 256 , c ) 343 , d ) 441 , e ) 512

"the number a must include at least 3 ^ 2 * 7 ^ 2 = 441 the answer is d ."

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

sakshi invests a part of rs . 12000 in 12 % stock at rs . 120 and the remainder in 15 % stock at rs . 125 . if his total dividend per annum is rs . 1360 , how much does he invest in 12 % stock at rs . 120 ? a ) rs . 4000 , b ) rs . 4500 , c ) rs . 5500 , d ) rs . 6000 , e ) rs . 6500

let investment in 12 % stock be rs . x . then , investment in 15 % stock = rs . ( 12000 - x ) . 12 / 120 * x + 15 / 125 * ( 12000 - x ) = 1360 x / 10 + 3 / 25 ( 12000 - x ) = 1360 5 x + 72000 - 6 x = 1360 x 50 x = 4000 . answer = a