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Break down and solve the equation step by step. | in a can , there is a mixture of milk and water in the ratio 5 : 3 . if the can is filled with an additional 8 liters of milk , the can would be full and the ratio of milk and water would become 2 : 1 . find the capacity of the can ? a) 56 , b) 60 , c) 64 , d) 68 , e) 72 | "let c be the capacity of the can . (5 / 8) * (c - 8) + 8 = (2 / 3) * c 15 c - 120 + 192 = 16 c c = 72 the answer is e ." |
Provide a detailed solution to the given mathematical problem. | out of 470 students of a school , 325 play football , 175 play cricket and 50 neither play football nor cricket . how many students play both football and cricket ? a) 80 , b) 150 , c) 100 , d) 180 , e) 220 | "n (a) = 325 , n (b) = 175 , n (aub) = 470 - 50 = 420 . required number = n (anb) = n (a) + n (b) - n (aub) = 325 + 175 - 420 = 80 . answer is a" |
Work through the equation methodically, ensuring clarity at each step. | an ant walks an average of 500 meters in 12 minutes . a beetle walks 15 % less distance at the same time on the average . assuming the beetle walks at her regular rate , what is its speed in km / h ? a) 2.215 , b) 2.5 . , c) 2.775 , d) 3.2 , e) 3.5 | "the ant walks an average of 500 meters in 12 minutes or , 500 meters in 1 / 5 hours the beetle walks 15 % less distance = 500 - 75 = 425 meters in 12 minutes or , 0.425 km in 12 / 60 = 1 / 5 hours speed = 0.425 * 5 = 2.125 km / h ans should be 2.125 answer : a" |
Solve the problem systematically, showing each step clearly. | in triangle pqr , the angle q = 90 degree , pq = 3 cm , qr = 8 cm . x is a variable point on pq . the line through x parallel to qr , intersects pr at y and the line through y , parallel to pq , intersects qr at z . find the least possible length of xz a) 3.6 cm , b) 2.4 cm , c) 4.8 cm , d) 2.16 cm , e) 3.2 cm | "look at the diagram below : now , in case when qy is perpendicular to pr , two right triangles pqr and pqy are similar : qy : qp = qr : pr - - > qy : 3 = 8 : 10 - - > qy = 2.4 . answer : b ." |
Break down and solve the equation step by step. | barbata invests $ 2000 in the national bank at 5 % . how much additional money must she invest at 8 % so that the total annual income will be equal to 6 % of her entire investment ? a) 1200 , b) 3000 , c) 1000 , d) 3600 , e) 2400 | "let the additional invested amount for 8 % interest be x ; equation will be ; 2000 + 0.05 * 2000 + x + 0.08 x = 2000 + x + 0.06 (2000 + x) 0.05 * 2000 + 0.08 x = 0.06 x + 0.06 * 2000 0.02 x = 2000 (0.06 - 0.05) x = 2000 * 0.01 / 0.02 = 1000 ans : ` ` c''" |
Solve the problem systematically, showing each step clearly. | a , b and c can do a piece of work in 30 days , 30 days and 40 days respectively . they began the work together but c left 4 days before the completion of the work . in how many days was the work completed ? a) 12 days , b) 16 days , c) 18 days , d) 11 days , e) 38 days | "one day work of a , b and c = 1 / 30 + 1 / 30 + 1 / 40 = 1 / 11 work done by a and b together in the last 4 days = 4 * (1 / 30 + 1 / 30) = 4 / 15 remaining work = 11 / 15 the number of days required for this initial work = 8 days . the total number of days required = 4 + 8 = 12 days . answer : a" |
Work through the equation methodically, ensuring clarity at each step. | if the sum of two positive integers is 20 and the difference of their squares is 40 , what is the product of the two integers ? a) 108 , b) 99 , c) 128 , d) 135 , e) 143 | "let the 2 positive numbers x and y x + y = 20 - - 1 x ^ 2 - y ^ 2 = 40 = > (x + y) (x - y) = 40 - - 2 using equation 1 in 2 , we get = > x - y = 2 - - 3 solving equation 1 and 3 , we get x = 11 y = 9 product = 11 * 9 = 99 answer b" |
Break down and solve the equation step by step. | how many digits are used to number a book containg 300 pages . a) 694 , b) 792 , c) 698 , d) 784 , e) 789 | explanation : number of digits in 1 digit numbers = 1 to 9 = 9 number of digits in 2 digit numbers = 10 to 99 = 90 (2) = 180 number of digits in 3 digit numbers = 100 to 300 = 201 (3) = 603 total = 792 answer : option b |
Explain the solution process for the given equation. | the unit digit in the product (224 * 607 * 214 * 863) is : a) 2 , b) 5 , c) 6 , d) 8 , e) 10 | "explanation : unit digit in the given product = unit digit in (4 * 7 * 4 * 3) = 6 answer : c" |
Provide a detailed solution to the given mathematical problem. | what is the units digit of the expression 17 ^ 7 - 2 ? a) 1 , b) 2 , c) 3 , d) 7 , e) 9 | "7 ^ 1 = 7 7 ^ 2 = 49 7 ^ 3 = 343 7 ^ 4 = 1 (last digit) 7 ^ 5 = 7 (last digit) and the cycle repeats after every 4 powers therefore , last digit of 17 ^ 7 = 3 3 - 2 = 1 answer a" |
Explain the solution process for the given equation. | jim drove 923 miles of a 1200 miles journey . how many more miles does he need to drive to finish his journey ? a) 113 miles , b) 277 miles , c) 456 miles , d) 887 miles , e) 767 miles | the number of miles to drive to finish his journey is given by 1200 - 923 = 277 miles correct answer b |
Solve the problem systematically, showing each step clearly. | if the sides of a square are multiplied by sqrt (5) , the area of the original square is how many times as large as the area of the resultant square ? ['a) 5 %','b) 20 %','c) 50 %','d) 100 %','e) 120 %'] | let x be the original length of one side . then the original area is x ^ 2 . the new square has sides of length sqrt (5) * x , so the area is 5 x ^ 2 . the area of the original square is 1 / 5 = 20 % times the area of the new square . the answer is b . |
Break down and solve the equation step by step. | a train 125 m long passes a man , running at 8 km / hr in the same direction in which the train is going , in 10 seconds . the speed of the train is : a) 53 , b) 50 , c) 99 , d) 288 , e) 12 | speed of the train relative to man = (125 / 10) m / sec = (25 / 2) m / sec . [ (25 / 2) * (18 / 5) ] km / hr = 45 km / hr . let the speed of the train be x km / hr . then , relative speed = (x - 8) km / hr . x - 8 = 45 = = > x = 53 km / hr . answer : a |
Break down and solve the equation step by step. | a man swims downstream 45 km and upstream 25 km taking 5 hours each time , what is the speed of the man in still water ? a) 17 , b) 24 , c) 7 , d) 24 , e) 20 | "45 - - - 5 ds = 9 ? - - - - 1 25 - - - 5 us = 5 ? - - - - 1 m = ? m = (9 + 5) / 2 = 14 / 2 = 7 answer : c" |
Work through the equation methodically, ensuring clarity at each step. | if a coin has an equal probability of landing heads up or tails up each time it is flipped , what is the probability that the coin will land heads up exectly twice in 3 consecutive flips ? a) 0.125 , b) 0.25 , c) 0.375 , d) 0.5 , e) 0.666 | "p = ncm * p ^ m * q ^ (n - m) n - the total number of trials . m - the number of trials with heads . n - m - the number of trials with tails . p - probability of heads . q - probability of tails . p = 3 c 2 * (1 / 2) ^ 2 * (1 / 2) ^ 1 = 3 / 8 = 0.375 answer : c" |
Provide a detailed solution to the given mathematical problem. | in a graduating class of 236 students , 144 took geometry and 119 took biology . what is the difference between the greatest possible number and the smallest possible number of students that could have taken both geometry and biology ? a) 144 , b) 119 , c) 113 , d) 117 , e) 31 | "greatest possible number taken both should be 144 (as it is maximum for one) smallest possible number taken both should be given by total - neither = a + b - both both = a + b + neither - total (neither must be 0 to minimize the both) so 144 + 119 + 0 - 236 = 27 greatest - smallest is 144 - 27 = 117 so answer must be ... |
Solve the problem systematically, showing each step clearly. | sum of two numbers prime to each other is 14 and their l . c . m . is 45 . what are the numbers ? a) 6 and 5 , b) 8 and 5 , c) 9 and 5 , d) 8 and 5 , e) 3 and 5 | "as two numbers are prime , satisfies all but option c will make the product of numbers i . e 45 answer : c" |
Break down and solve the equation step by step. | a fashion designer sold a pair of jeans to a retail store for 40 percent more than it cost to manufacture the pair of jeans . a customer bought the pair of jeans for 40 percent more than the retailer paid for them . the price the customer paid was what percent greater than the cost of manufacturing the jeans ? a) 65 % ... | "find the product of the two increases : (14 / 10) * (14 / 10) which is 1.96 and a 96 % increase . e" |
Provide a detailed solution to the given mathematical problem. | if the least common addition of two prime numbers x and y is 36 , where x < y , then the value of 4 x + y is a) 32 , b) 45 , c) 59 , d) 73 , e) 87 | (x + y) = 36 and both x an y are prime . the only values of x and y can be 17 and 19 (x = 17 and y = 19) 4 x + y = 4 * 17 + 19 = 87 correct option : e |
Solve the problem systematically, showing each step clearly. | if the area of a circle is 36 pi square feet , find its circumference . a) 65 pi feet , b) 43 pi feet , c) 12 pi feet , d) 18 pi feet , e) 64 pi feet | "the area is given by pi * r * r . hence pi * r * r = 81 pi r * r = 36 ; hence r = 36 feet the circumference is given by 2 * pi * r = 2 * pi * 6 = 128 pi feet correct answer c" |
Explain the solution process for the given equation. | a train 300 meters long completely crosses a 300 meters long bridge in 45 seconds . what is the speed of the train is ? a) 32 , b) 48 , c) 29 , d) 27 , e) 21 | "s = (300 + 300) / 45 = 600 / 45 * 18 / 5 = 48 answer : b" |
Provide a detailed solution to the given mathematical problem. | what is the remainder when 3 ^ 381 is divided by 5 ? a) 0 , b) 1 , c) 2 , d) 3 , e) 4 | i also agree that the remainder is' 3' (using the last digit of the powers of 7) . could we have the official answer please ? d |
Break down and solve the equation step by step. | from a vessel on the first day , 1 / 3 rd of the liquid evaporates . on the second day 3 / 4 th of the remaining liquid evaporates . what fraction of the volume is present at the end of the 2 day a) 1 / 3 , b) 1 / 4 , c) 1 / 5 , d) 1 / 6 , e) 1 / 7 | let x be the volume . . . after 1 st day volume remaining = (x - x / 3) = 2 x / 3 after 2 st day volume remaining = (2 x / 3) - ((2 x / 3) * (3 / 4)) = (2 x / 3) (1 - 3 / 4) = (2 x / 3) * (1 / 4) = x / 6 answer : d |
Solve the problem systematically, showing each step clearly. | there is a right angle triangle inside a circle of diameter 18 cm . inside this triangle , there is another circle of diameter 6 cm . find the perimeter of the triangle . ['a) 24 cm','b) 36 * 2 ^ 1 / 2 cm','c) 36 cm','d) 54 cm','e) 42 cm'] | 1) the radius of the incircle of a right triangle with legs a and b and hypotenuse c is r = (a + b - c) / 2 2) the radius of the circumcircle is half the length of the hypotenuse r = c / 2 3) thus the sum of the circumradius and the inradius is half the sum of the legs r + r = (a + b) / 2 from 2 nd rule c = 18 cm from ... |
Work through the equation methodically, ensuring clarity at each step. | a store owner estimates that the average price of type a products will increase by 25 % next year and that the price of type b products will increase by 10 % next year . this year , the total amount paid for type a products was $ 4500 and the total price paid for type b products was $ 8300 . according to the store owne... | "cost of type a products next year = 1.25 * 4500 = 5625 cost of type b products next year = 1.1 * 8300 = 9130 total 5625 + 9130 = 14755 answer : a" |
Work through the equation methodically, ensuring clarity at each step. | in the fifth grade at parkway elementary school there are 420 students . 312 students are boys and 250 students are playing soccer . 82 % of the students that play soccer are boys . how many girl student are in parkway that is not playing soccer ? a) 63 . , b) 73 . , c) 81 , d) 91 , e) 108 | "total students = 420 boys = 312 , girls = 108 total playing soccer = 250 82 % of 250 = 205 are boys who play soccer . girls who play soccer = 45 . total girls who do not play soccer = 108 - 45 = 63 . correct option : a" |
Work through the equation methodically, ensuring clarity at each step. | what is the remainder when 46 * 49 is divided by 8 ? a) 1 , b) 3 , c) 4 , d) 6 , e) 7 | "we can make use of the rule : remainder of { (a * b) / n } } = remainder of (a / n) * remainder of (b / n) here remainder of { 46 * 49) / 8 } } = remainder of (46 / 8) * remainder of (49 / 8) = 6 * 1 = 6 answer : d" |
Explain the solution process for the given equation. | a and b started a business investing rs . 10,000 and rs 20,000 respectively . in what ratio the profit earned after 2 years be divided between a and b respectively ? a) 3 : 2 , b) 9 : 2 , c) 18 : 20 , d) 1 : 2 , e) 18 : 4 | "a : b = 10000 : 20000 = 1 : 2 answer : d" |
Work through the equation methodically, ensuring clarity at each step. | water consists of hydrogen and oxygen , and the approximate ratio , by mass , of hydrogen to oxygen is 2 : 16 . approximately how many grams of oxygen are there in 153 grams of water ? a) 16 , b) 72 , c) 112 , d) 136 , e) 142 | "since the ratio , by mass , of hydrogen to oxygen is 2 : 16 , then oxygen is 16 / (2 + 16) = 8 / 9 of mass of water . therefore there are 153 * 8 / 9 = 136 grams of oxygen in 153 grams of water . answer : d ." |
Provide a detailed solution to the given mathematical problem. | the sum of three consecutive even numbers is 24 . find the middle number of the three ? a) 14 , b) 16 , c) 18 , d) 24 , e) 34 | "three consecutive even numbers (2 p - 2) , 2 p , (2 p + 2) . (2 p - 2) + 2 p + (2 p + 2) = 24 6 p = 24 = > p = 4 . the middle number is : 4 p = 16 . answer : b" |
Solve the problem systematically, showing each step clearly. | machine t takes 2 more hours than machine b to make 20 widgets . if working together , the machines can make 25 widgets in 3 hours , how long will it take machine t to make 40 widgets ? a) 5 , b) 6 , c) 8 , d) 10 , e) 12 | i approached this one by plugging in numbers . . . started with c . if 40 are made in 8 hours , then 20 are made in 4 hours . so time of t is 4 , and time of b is 2 . rate together : 20 / 4 + 20 / 2 = 5 + 10 = 15 . so in 1 hour , together make 15 widgets . in 3 hours = 45 . way too much . we can eliminate right away c ... |
Solve the problem systematically, showing each step clearly. | a person x working alone can complete a work in 5 days . a person y completes the same amount of work in 20 days , and a person z when working alone can complete the same amount of work in 30 days . all 3 people work together for 2 days but then x and y leave . how many more days does z need to complete the work after ... | the fraction of work completed in two days is 2 / 5 + 2 / 20 + 2 / 30 = 17 / 30 the fraction of work remaining is 1 - 17 / 30 = 13 / 30 the number of days required for z is 13 days . the answer is a . |
Work through the equation methodically, ensuring clarity at each step. | walking 3 / 2 of his usual rate , a boy reaches his school 4 min early . find his usual time to reach the school ? a) 12 , b) 99 , c) 27 , d) 28 , e) 20 | "speed ratio = 1 : 3 / 2 = 2 : 3 time ratio = 3 : 2 1 - - - - - - - - 3 4 - - - - - - - - - ? 12 m . answer : a" |
Solve the problem systematically, showing each step clearly. | 1907 x 1907 = ? a) a) 3623216 , b) b) 3624216 , c) c) 3624316 , d) d) 3636649 , e) e) 3625216 | "1907 x 1907 = (1907) 2 = (1900 + 7) 2 = (1900) 2 + (7) 2 + (2 x 1900 x 7) = 3610000 + 49 + 26600 . = 3636649 . d)" |
Break down and solve the equation step by step. | a pizzeria in castroville creates its own special blend of shredded cheese at a cost of $ 696.05 per kilogram . the blend includes mozzarella , at $ 504.35 per kilogram , and romano cheese , at $ 887.75 per kilogram . if the employees have already shredded 19 kilograms of mozzarella cheese , how many kilograms of roman... | 504.35 (19) + 887.75 x = 696.05 (19 + x) solve the equation . 504.35 (19) + 887.75 x = 696.05 (19 + x) 9 , 582.65 + 887.75 x = 13 , 224.95 + 696.05 x 9 , 582.65 + 191.7 x = 13 , 224.95 191.7 x = 3 , 642.3 x = 19 . answer is a . |
Solve the problem systematically, showing each step clearly. | how many of the positive factors of 24 , 16 and how many common factors are there in numbers ? a) 1 , b) 2 , c) 3 , d) 4 , e) 5 | "factors of 24 - 1 , 2 , 3 , 4 , 8 , 12 and 24 factors of 16 - 1 , 2 , 4 , 8 and 16 comparing both , we have three common factors of 45,16 - 4 answer d" |
Work through the equation methodically, ensuring clarity at each step. | according to a recent student poll , 3 / 4 out of 20 members of the finance club are interested in a career in investment banking . if two students are chosen at random , what is the probability that at least one of them is interested in investment banking ? a) 1 / 14 , b) 4 / 49 , c) 18 / 19 , d) 45 / 49 , e) 13 / 14 | "15 students are interested , 5 are not interested prob = 1 - 5 c 2 / 20 c 2 = 1 - (5 * 4 / (20 * 19)) = 1 - 1 / 19 = 18 / 19 answer : c" |
Explain the solution process for the given equation. | a 250 m long train is running at a speed of 55 km / hr . it crossed a platform of length 520 m in ? a) 41.1 sec , b) 20.2 sec , c) 31.8 sec , d) 50.4 sec , e) none of the above | "speed = 55 km / hr (to convert km / hr in to m / s) = 55 x 5 / 18 m / s distance = 250 m + 520 m (if questions is about train crossing a post you need to consider only the length of train ,) = 770 m time = distance / speed = 770 x 18 / (5 x 55) = 50.4 sec ans is : d" |
Provide a detailed solution to the given mathematical problem. | a train with 120 wagons crosses john who is going in the same direction , in 36 seconds . it travels for half an hour from the time it starts overtaking the john (he is riding on the horse) before it starts overtaking the mike (who is also riding on his horse) coming from the opposite direction in 24 seconds . in how m... | "let the length of the train be l metres and speeds of the train arun and sriram be r , a and s respectively , then - - - - - - - - - - (i) and - - - - - - - - - (ii) from eq . (i) and (ii) 3 (r - a) = 2 (r + k) r = 3 a + 2 k in 30 minutes (i . e 1800 seconds) , the train covers 1800 r (distance) but the arun also cove... |
Work through the equation methodically, ensuring clarity at each step. | the sum of 111 consecutive integers is 11211 . what is the greatest integer in the set ? a) 148 , b) 150 , c) 152 , d) 154 , e) 156 | "let x be the first integer in the set , then x + 110 is the largest integer . the sum is : x + (x + 1) + (x + 2) + . . . + (x + 110) = 111 x + 110 * 111 / 2 = 111 (x + 55) then x + 55 = 101 x = 46 the largest integer in the set is 46 + 110 = 156 the answer is e ." |
Break down and solve the equation step by step. | in a certain large company , the ratio of college graduates with a graduate degree to non - college graduates is 1 : 8 , and ratio of college graduates without a graduate degree to non - college graduates is 2 : 3 . if one picks a random college graduate at this large company , what is the probability e this college gr... | in believe the answer is d . please see below for explanation . 0) we are told the following ratios cgd - college graduate with degree ncg - non college graduate cgn - college graduate no degree cgd ncg cgn 1 8 3 2 in order to make cgd and cgn comparable we need to find the least common multiple of 8 and 3 and that is ... |
Work through the equation methodically, ensuring clarity at each step. | a small college reduced its faculty by approximately 13 percent to 195 professors . what was the original number of faculty members ? a) 182 , b) 208 , c) 224 , d) 254 , e) 302 | "f x is the original number of faculty members , then after 13 % reduction in faculty members number is . 87 x but we are given . 87 x = 195 x = 224 so the original number of faculty members is 224 correct answer - c" |
Break down and solve the equation step by step. | when positive integer n is divided by 3 , the remainder is 2 . when n is divided by 7 , the remainder is 6 . how many values less than 100 can n take ? a) 0 , b) 2 , c) 3 , d) 6 , e) 5 | "a quick approac to this q is . . the equation we can form is . . 3 x + 2 = 7 y + 5 . . 3 x - 3 = 7 y . . . 3 (x - 1) = 7 y . . . so (x - 1) has to be a multiple of 7 as y then will take values of multiple of 3 . . here we can see x can be 1 , 8,15 , 22,29 so 5 values till 100 is reached as (29 - 1) * 3 = 84 and next m... |
Work through the equation methodically, ensuring clarity at each step. | a car gets 32 miles to the gallon . if it is modified to use a solar panel , it will use only 80 percent as much fuel as it does now . if the fuel tank holds 12 gallons , how many more miles will the car be able to travel , per full tank of fuel , after it has been modified ? a) 84 , b) 88 , c) 92 , d) 96 , e) 100 | originally , the distance the car could go on a full tank was 12 * 32 = 384 miles . after it has been modified , the car can go 32 / 0.8 = 40 miles per gallon . on a full tank , the car can go 12 * 40 = 480 miles , thus 96 miles more . the answer is d . |
Explain the solution process for the given equation. | in a fuel station the service costs $ 2.30 per vehicle and every liter of fuel costs $ 0.70 . assuming that you fill up 4 mini - vans and 2 trucks , what will be the total cost , if a mini - van' s tank is 65 liters and a truck' s tank is 120 % bigger and they are all empty ? a) $ 396 , b) $ 400 , c) $ 404 , d) $ 408 ,... | "the service cost of 4 vans and 2 trucks is 6 * 2.30 $ 13.80 the fuel in 4 vans is 4 * 65 = 260 liters the fuel in 2 trucks is 2 * 65 * 2.2 = 286 liters the total fuel (vans + trucks) = 546 liters the total fuel cost is 546 * 0.7 = $ 382.20 the total cost is $ 382.20 + $ 13.80 = $ 396 the answer is a ." |
Explain the solution process for the given equation. | in a group of 65 people , 40 like cricket , 10 like both cricket and tennis . how many like tennis ? a) 30 , b) 35 , c) 40 , d) 45 , e) 50 | "make a venn diagram , and enter your data . let the number of people who like only tennis be x x + 10 + 30 = 65 x = 25 so number who like tennis = 10 + 25 = 35 answer b" |
Provide a detailed solution to the given mathematical problem. | rajat , vikas and abhishek are submitting questions in the ratio 7 : 3 : 2 . if total number of questions submitted by them is 24 . find the number of questions submitted by vikas . a) 3 , b) 4 , c) 5 , d) 6 , e) 7 | explanation : number of questions submitted by vikas = (24 * 3) / (7 + 3 + 2) = 6 answer : d |
Explain the solution process for the given equation. | how many times in a day , the hands of a clock are straight ? a) 22 , b) 24 , c) 44 , d) 48 , e) 52 | "in 12 hours , the hands coincide or are in opposite direction 22 times . in 24 hours , the hands coincide or are in opposite direction 44 times a day . answer : option c" |
Work through the equation methodically, ensuring clarity at each step. | 12,21 , 48,87 , __ a) 176 , b) 186 , c) 184 , d) 174 , e) 164 | "21 = 12 * 2 - 3 48 = 21 * 2 + 6 87 = 48 * 2 - 9 so next number is 87 * 2 + 12 = 186 answer : b" |
Provide a detailed solution to the given mathematical problem. | x varies inversely as square of y . given that y = 2 for x = 1 . the value of x for y = 6 will be equal to : a) 1 / 2 , b) 1 / 3 , c) 1 / 5 , d) 1 / 9 , e) 1 / 3 | "given x = k / y 2 , where k is a constant . now , y = 2 and x = 1 gives k = 4 . x = 4 / y 2 = > x = 4 / 62 , when y = 6 = > x = 4 / 36 = 1 / 9 . answer : d" |
Work through the equation methodically, ensuring clarity at each step. | if the sum of the 4 th term and the 12 th term of an arithmetic progression is 8 , what is the sum of the first 15 terms of the progression ? a) 60 , b) 120 , c) 160 , d) 240 , e) 840 | "let a = first term , d = common difference . 4 th term = a + 3 d 12 th term = a + 11 d a + 3 d + a + 11 d = 8 2 a + 14 d = 8 a + 7 d = 4 . sum of the first 15 terms = 15 / 2 (2 a + 14 d) = 15 (a + 7 d) = 15 (4) = 60 . ans (a)" |
Work through the equation methodically, ensuring clarity at each step. | pipe a fills a tank of capacity 900 liters at the rate of 40 liters a minute . another pipe b fills the same tank at the rate of 30 liters a minute . a pipe at the bottom of the tank drains the tank at the rate of 20 liters a minute . if pipe a is kept open for a minute and then closed and pipe b is open for a minute a... | "in one cycle they fill 40 + 30 - 20 = 50 liters 900 = 50 * n = > n = 18 here n = number of cycles . total time = 18 * 3 = 54 as in one cycle there are 3 minutes . thus 54 minutes answer : c" |
Provide a detailed solution to the given mathematical problem. | in a certain state , the ratio of registered republicans to registered democrats is 3 to 2 , and every registered voter is either a republican or a democrat . if 70 percent of the republicans and 25 percent of the democrats are expected to vote for candidate x , and everyone else is expected to vote for candidate y , b... | since we were expected to find a percentage figure - it thought that it might be easier to pick a' smart number' to represent the total number of voters (republicans and democrats) . therefore , i picked 100 (as the total number of voters) and thus 60 : 40 represents the number ratio of republicans : democrats . if 70 ... |
Work through the equation methodically, ensuring clarity at each step. | by weight , liquid x makes up 0.7 percent of solution p and 1.2 percent of solution q . if 200 grams of solution p are mixed with 800 grams of solution q , then liquid x accounts for what percent of the weight of the resulting solution ? a) 1.2 % , b) 1.1 % , c) 1.0 % , d) 0.9 % , e) 0.8 % | "the number of grams of liquid x is 0.7 (200) / 100 + 1.2 (800) / 100 = 1.4 + 9.6 = 11 grams . 11 / 1000 = 1.1 % the answer is b ." |
Solve the problem systematically, showing each step clearly. | a cab driver 5 days income was $ 400 , $ 250 , $ 650 , $ 400 , $ 500 . then his average income is ? a) $ 400 , b) $ 420 , c) $ 440 , d) $ 460 , e) $ 480 | "avg = sum of observations / number of observations avg income = (400 + 250 + 650 + 400 + 500) / 5 = 440 answer is c" |
Break down and solve the equation step by step. | the ratio of investments of two partners p and q is 7 : 5 and the ratio of their profits is 7 : 10 . if p invested the money for 8 months , find for how much time did q invest the money ? a) 11 , b) 10 , c) 18 , d) 16 , e) 12 | "7 * 5 : 8 * x = 7 : 10 x = 16 answer : d" |
Solve the problem systematically, showing each step clearly. | one pipe can fill a tank four times as fast as another pipe . if together the two pipes can fill the tank in 40 minutes , then the slower pipe alone will be able to fill the tank in ? a) 100 min , b) 150 min , c) 200 min , d) 250 min , e) 300 min | "let the slower pipe alone fill the tank in x minutes then , faster pipe will fill it in x / 4 minutes 1 / x + 4 / x = 1 / 40 5 / x = 1 / 40 x = 200 min answer is c" |
Work through the equation methodically, ensuring clarity at each step. | if m and n are positive integers and m ^ 2 + n ^ 2 = 58 , what is the value of m ^ 3 + n ^ 3 ? a) 72 , b) 224 , c) 370 , d) 512 , e) 1,600 | "you need to integers which squared are equal 58 . which could it be ? let' s start with the first integer : 1 ^ 2 = 1 2 ^ 2 = 4 3 ^ 2 = 9 4 ^ 2 = 16 5 ^ 2 = 25 6 ^ 2 = 36 7 ^ 2 = 49 stop . the integers ca n' t be greater than 7 or we will score above 58 . the second integer need to be picked up the same way . 1 ^ 2 = ... |
Break down and solve the equation step by step. | an aeroplane covers a certain distance at a speed of 240 kmph in 5 hours . to cover the same distance in 1 hours , it must travel at a speed of : a) 300 kmph , b) 360 kmph , c) 600 kmph , d) 720 kmph , e) 750 kmph | "explanation : distance = (240 x 5) = 1200 km . speed = distance / time speed = 1200 / (5 / 3) km / hr . required speed = (1200 x 3 / 5) km / hr = 720 km / hr . answer is d" |
Explain the solution process for the given equation. | n people were sitting on a circular manner , if each of two present in the party except the pairs were adjacent sang a song , if a song lasted for 2 mins and 28 minutes was taken for singing song , find n ['a) 11','b) 12','c) 13','d) 14','e) 15'] | each of 2 present in the party = > no . of pairs possible = nc 2 possible adjacent pairs = n total required combinations = nc 2 - n time taken by each combination to sing a song = 2 min total time taken = 2 (nc 2 - n) = 28 on solving the above equation , we get n = 14 answer : d |
Work through the equation methodically, ensuring clarity at each step. | what is the greatest positive integer n such that 3 ^ n is a factor of 54 ^ 100 ? a) 100 , b) 200 , c) 300 , d) 600 , e) 900 | "54 = 3 ^ 3 * 2 . 54 ^ 100 = 3 ^ 300 * 2 ^ 100 the answer is c ." |
Solve the problem systematically, showing each step clearly. | half a number plus 5 is 13 . what is the number ? a) 8 , b) 16 , c) 10 , d) 11 , e) 12 | "let x be the number . always replace ` ` is'' with an equal sign (1 / 2) x + 5 = 13 (1 / 2) x = 13 - 5 (1 / 2) x = 8 x = 16 correct answer is b" |
Solve the problem systematically, showing each step clearly. | a man is walking at a speed of 6 km per hour . after every km he takes rest for 6 minutes . how much time will he take to cover a distance of 12 km . a) 3 hr , b) 5 hr 42 min , c) 4 hrs . 10 min , d) 3 hr . 6 min , e) none of these | "explanation : for 12 km he takes 11 times rest . 11 x 6 = 66 minutes rest = 1 hr 6 min for 12 km , 2 hr is required + rest = 2 + 1 hr 6 m = 3 hr . 6 min answer : option d" |
Solve the problem systematically, showing each step clearly. | s (n) is a n - digit number formed by attaching the first n perfect squares , in order , into one integer . for example , s (1) = 1 , s (2) = 14 , s (3) = 149 , s (4) = 14916 , s (5) = 1491625 , etc . how many digits w are in s (99) ? a) w = 350 , b) w = 353 , c) w = 354 , d) 356 , e) 357 | "focus on the points where the number of digits in squares change : 1 , 2 , 3 - single digit squares . first 2 digit number is 10 . 4 , 5 , . . . 9 - two digit squares . to get 9 , the last number with two digit square , think that first 3 digit number is 100 which is 10 ^ 2 . so 9 ^ 2 must be the last 2 digit square .... |
Break down and solve the equation step by step. | if a large pizza has a radius that is 40 % larger that that of a medium pizza , what is the percent increase in area between a medium and a large pizza ? a) 40 % , b) 80 % , c) 96 % , d) 112 % , e) 124 % | let the radius of medium pizza be r . then the radius of large pizza is 1.4 r . the area of the medium pizza is pi * r ^ 2 the area of the large pizza is pi * (1.4 * r) ^ 2 = 1.96 * pi * r ^ 2 , an increase of 96 % . the answer is c . |
Explain the solution process for the given equation. | a school has received 40 % of the amount it needs for a new building by receiving a donation of $ 400 each from people already solicited . people already solicited represent 40 % of the people from whom the school will solicit donations . how much average contribution is requited from the remaining targeted people to c... | "let the amount school needs = x let total people school plans to solicit = t school has received 60 % of x = > (3 / 5) x people already solicited = 40 % of t = > (2 / 5) t now , as per the information given in the question : (3 / 5) x = $ 400 . (2 / 5) . t - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 remani... |
Explain the solution process for the given equation. | in feb mrs bil' s earnings were 70 percent of the bil family' s total income . in june mrs bil earned 10 percent more than in may . if the rest of the family' s income was the same both months , then , in june , mrs bil' s earnings were approximately what percent of the bil family' s total income ? a) 72 , b) 34 , c) 7... | lets say the family income is 100 in may , bil earned 70 family income is 30 in june , bil earned 10 % more than may , so it is (70 + 10 * 70 / 100 = 77) family income is same 30 in june bil' s income percent is 77 * 100 / 107 ~ 72 ans is a |
Break down and solve the equation step by step. | the average salary of the employees in a office is rs . 120 / month . the avg salary of officers is rs . 440 and of non officers is rs 110 . if the no . of officers is 15 , then find the no of nonofficers in the office . a) 400 , b) 420 , c) 480 , d) 450 , e) 510 | "let no . of non - officers be x 15 * 440 + x * 110 = (x + 15) 120 x = 480 c" |
Explain the solution process for the given equation. | a 30 % alcohol mixture is added to a 50 % alcohol mixture to form a 10 litre mixture of 45 % alcohol . how much of the 30 % mixture was used ? a) 1.5 liters , b) 2.5 liters , c) 3.5 liters , d) 4.5 liters , e) 5.5 liters | solution a = 50 % solution solution b = 30 % solution 30 % - 45 % = 15 % solution b 50 % - 45 % = 5 % solution a so the ratio is 3 : 1 for 30 % : 50 % solutions 3 / 4 * 10 liter = 7.5 for 30 % solution and 2.5 for 50 % solution . . answer : b |
Solve the problem systematically, showing each step clearly. | two numbers are in the ratio of 5 : 7 . if 20 is subtracted from each number , they are in the ratio of 5 : 9 . what are the two numbers ? a) 35 , 49 , b) 40 , 56 , c) 45 , 63 , d) 50 , 70 , e) 55 , 77 | "(5 x - 20) : (7 x - 20) = 5 y : 9 y 5 x - 20 = 5 y 7 x - 20 = 9 y 140 - 100 = 45 y - 35 y 40 = 10 y y = 4 5 x - 20 = 20 x = 8 the two numbers are 5 x and 7 x which are 40 and 56 . the answer is b ." |
Break down and solve the equation step by step. | mother , her daughter and her grand child weighs 160 kg . daughter and her daughter (child) weighs 60 kg . child is 1 / 5 th of her grand mother . what is the age of the daughter ? a) 46 , b) 47 , c) 48 , d) 49 , e) 40 | "mother + daughter + child = 160 kg daughter + child = 60 kg mother = 160 - 60 = 100 kg child = 1 / 5 th of mother = (1 / 5) * 100 = 20 kg so now daughter = 120 - (mother + child) = 160 - (100 + 20) = 40 kg answer : e" |
Provide a detailed solution to the given mathematical problem. | running at the same rate , 8 identical machines can produce 560 paperclips a minute . at this rate , how many paperclips could 10 machines produce in 6 minutes ? a) 1344 , b) 4200 , c) 8400 , d) 50400 , e) 67200 | 8 machines produce 560 in 1 min 8 machines produce 560 * 6 in 6 min 10 machine produce 560 * 6 * (10 / 8) in 6 minutes 560 * 6 * 10 / 8 = 4200 answer is b . |
Solve the problem systematically, showing each step clearly. | a train length 100 m going with speed 10 m / sec another train 150 m long going with speed 15 m / sec . the distance between two train is 50 m . then how much time second train will take to cross the first train ? a) 60 sec , b) 40 sec , c) 30 sec , d) 45 sec , e) 1 min 10 sec | net speed = 15 - 10 = 5 m / sec net distance = 100 + 150 + 50 = 300 total time = 300 / 5 = 60 sec answer a |
Break down and solve the equation step by step. | a quiz has one mcq question with a , b and c as options . and two questions with true / false answers . what is the probability of giving all 3 answers correct ? a) 1 / 12 , b) 1 / 3 , c) 1 / 4 , d) 1 / 2 , e) 1 / 6 | probability of getting mcq answer correct = 1 / 3 probability of getting one true / false answer correct = 1 / 2 probability of getting all 3 answers correct = 1 / 3 * 1 / 2 * 1 / 2 = 1 / 12 answer : a |
Break down and solve the equation step by step. | a merchant has selected two items to be placed on sale , one of which currently sells for 25 percent less than the other . if he wishes to raise the price of the cheaper item so that the two items are equally priced , by what percentage must he raise the price of the less expensive item ? a) 52 % , b) 40 % , c) 30 % , ... | "expensive item = $ 100 ; cheap item = $ 75 ; we must increase $ 75 to $ 100 , so by $ 25 , which is approximately 30 % increase : (100 - 75) / 75 = 1 / 3 = ~ 0.33 answer : c ." |
Work through the equation methodically, ensuring clarity at each step. | how much time will a train of length 300 m moving at a speed of 72 kmph take to cross another train of length 500 m , moving at 36 kmph in the same direction ? a) 50 , b) 80 , c) 88 , d) 76 , e) 12 | "the distance to be covered = sum of their lengths = 300 + 500 = 800 m . relative speed = 72 - 36 = 36 kmph = 36 * 5 / 18 = 10 mps . time required = d / s = 800 / 10 = 80 sec . answer : b" |
Break down and solve the equation step by step. | in a certain village , 150 litres of water are required per household per month . at this rate , if there are 10 households in the village , how long (in months) will 6000 litres of water last ? a) 8 , b) 10 , c) 6 , d) 2 , e) 4 | "i find it much easier to understand with real numbers , so choose (almost) any numbers to replace m , n and p : in a certain village , m 150 litres of water are required per household per month . at this rate , if there aren 10 households in the village , how long (in months) willp 2000 litres of water last ? water re... |
Explain the solution process for the given equation. | x is a positive integer less than 100 . when x is divided by 7 , the remainder is 1 ; when x is divided by 3 , the remainder is 2 . how many x are there ? a) 14 , b) 22 , c) 23 , d) 24 , e) 25 | "the nubmer which when divided by 7 leaves remainder 1 should be of the form 7 k + 1 this number when divided by 3 leaves remainder 2 . so , (7 k + 1) - 2 should be divisible by 3 or 7 k - 1 should be divisible by 3 . we now put the values of k starting from 0 to find first number divisible by 3 we find 1 st number at ... |
Solve the problem systematically, showing each step clearly. | there were two candidates in an election . winner candidate received 75 % of votes and won the election by 500 votes . find the number of votes casted to the winning candidate ? a) 228 , b) 744 , c) 750 , d) 199 , e) 231 | "w = 75 % l = 25 % 75 % - 25 % = 50 % 50 % - - - - - - - - 500 75 % - - - - - - - - ? = > 750 answer : c" |
Work through the equation methodically, ensuring clarity at each step. | divide rs . 1800 among a , b and c so that a receives 2 / 5 as much as b and c together and b receives 1 / 5 as a and c together . a' s share is ? a) 400 , b) 500 , c) 375 , d) 450 , e) 600 | a + b + c = 1800 a = 2 / 5 (b + c) ; b = 1 / 5 (a + c) a / (b + c) = 2 / 5 a = 1 / 6 * 3600 = > 600 answer : e |
Solve the problem systematically, showing each step clearly. | a train passes a man standing on a platform in 8 seconds and also crosses the platform which is 270 metres long in 20 seconds . the length of the train (in metres) is : a) 180 , b) 176 , c) 175 , d) 96 , e) none of these | "explanation : let the length of train be l m . acc . to question (270 + l) / 20 = l / 8 2160 + 8 l = 20 l l = 2160 / 12 = 180 m answer a" |
Explain the solution process for the given equation. | a specialized type of sand consists of 40 % mineral x by volume and 60 % mineral y by volume . if mineral x weighs 1.5 grams per cubic centimeter and mineral y weighs 3 grams per cubic centimeter , how many grams does a cubic meter of specialized sand combination weigh ? (1 meter = 100 centimeters) a) 2 , 400,000 , b) ... | "let the volume be 1 m ^ 3 = 1 m * 1 m * 1 m = 100 cm * 100 cm * 100 cm = 1 , 000,000 cm ^ 3 by volume 40 % is x = 400,000 cm ^ 3 60 % is y = 600,000 cm ^ 3 by weight , in 1 cm ^ 3 , x is 1.5 gms in 400,000 cm ^ 3 , x = 1.5 * 400,000 = 600,000 grams in 1 cm ^ 3 , y is 3 gms in 600,000 cm ^ 3 , y = 3 * 600,000 = 1 , 800... |
Work through the equation methodically, ensuring clarity at each step. | one woman and one man can build a wall together in four hours , but the woman would need the help of two girls in order to complete the same job in the same amount of time . if one man and one girl worked together , it would take them eight hours to build the wall . assuming that rates for men , women and girls remain ... | "solution : let work done by man , women and girl per hour be m , w , g respectively . then , m + w = 1 / 4 - - > (1) , w + 2 g = 1 / 4 - - > (2) and m + g = 1 / 8 - - > (3) . no . of hours it would take forone woman , one man , and one girl , working together , to build the wall , n = 1 / m + w + g from (1) and (2) , ... |
Work through the equation methodically, ensuring clarity at each step. | in a college the ratio of the numbers of boys to the girls is 8 : 5 . if there are 160 girls , the total number of students in the college is ? a) 562 , b) 356 , c) 452 , d) 416 , e) 512 | "let the number of boys and girls be 8 x and 5 x then , 5 x = 160 x = 32 total number of students = 13 x = 13 * 32 = 416 answer is d" |
Solve the problem systematically, showing each step clearly. | in traveling from a dormitory to a certain city , a student went 1 / 2 of the way by foot , 1 / 4 of the way by bus , and the remaining 6 kilometers by car . what is the distance , in kilometers , from the dormitory to the city ? a) 35 , b) 24 , c) 14 , d) 24 , e) 12 | "whole trip = distance by foot + distance by bus + distance by car x = 1 / 2 x + 1 / 4 x + 6 x - 1 / 2 x - 1 / 4 x = 6 x = 24 km option : d" |
Break down and solve the equation step by step. | in a factory there are 3 types of machine m 1 , m 2 and m which produces 25 % , 35 % and 40 % of the total products respectively . m 1 , m 2 and m 3 produces 2 % , 4 % and 5 % defective products , respectively . what is the percentage of non - defective products ? a) 89 % , b) 97.1 % , c) 96.1 % , d) 86.1 % , e) none o... | solution : non - defective products , { [ (25 * 0.98) + (35 * 0.96) + (40 * 0.95) ] / 100 } * 100 = 96.1 % . answer : option c |
Break down and solve the equation step by step. | the ages of patrick and michael are in the ratio of 3 : 5 and that of michael and monica are in the ratio of 3 : 4 . if the sum of their ages is 132 , what is the difference between the ages of patrick and monica ? a) 27 , b) 48 , c) 45 , d) 72 , e) 33 | ages of p and mi = 3 x : 5 x ages of mi and mo = 3 x : 4 x rationalizing their ages . ratio of their ages will be 9 x : 15 x : 20 x sum = 44 x = 132 x = 3 difference if ages of pa and mo = 20 x - 9 x = 11 x = 11 * 3 = 33 answer e |
Solve the problem systematically, showing each step clearly. | the sum of four consecutive even numbers is 140 . what would be the smallest number ? a) 32 , b) 88 , c) 70 , d) 123 , e) 12 | "let the four consecutive even numbers be 2 (x - 2) , 2 (x - 1) , 2 x , 2 (x + 1) their sum = 8 x - 4 = 140 = > x = 18 smallest number is : 2 (x - 2) = 32 answer : a" |
Solve the problem systematically, showing each step clearly. | if m is an integer such that (- 2) ^ 2 m = 2 ^ (3 - m) then m = ? a) 1 , b) 2 , c) 3 , d) 5 , e) 6 | "(- 2) ^ (2 m) = 4 ^ m and 2 ^ (3 - m) = 4 ^ ((3 - m) / 2) therefore , m = (3 - m) / 2 2 m = 3 - m m = 1 answer a" |
Work through the equation methodically, ensuring clarity at each step. | the difference between two parallel sides of a trapezium is 4 cm . perpendicular distance between them is 10 cm . if the area of the trapezium is 250 cm 2 find the lengths of the parallel side ['a) 27 , 23','b) 25 , 25','c) 24 , 26','d) 23 , 29','e) 24 , 45'] | explanation : let the two parallel sides of the trapezium be a cm and b cm . then , a - b = 4 - - - - - - (1) and , (1 / 2) x (a + b) x 10 = 475 = > (a + b) = ((250 x 2) / 10) = > a + b = 50 - - - - - - - (2) solving 1 and 2 , we get : a = 27 , b = 23 so , the two parallel sides are 27 cm and 23 cm . answer : option a |
Explain the solution process for the given equation. | x ^ y + y ^ x = 2408 find the values of x ? a) 2407 , b) 2408 , c) 2409 , d) 2405 , e) 32 | 2407 ^ 1 + 1 ^ 2407 = 2408 answer : a |
Explain the solution process for the given equation. | a child want to buy 5 new toys . if there are 15 toys that satisfy the child' s criteria , in how many different ways can the child select the 5 new toys if the order of selection does n' t matter . a) 210 , b) 1365 , c) 3003 , d) 5005 , e) 3030 | the number of ways of choosing 5 toys out of 15 are counted by = 15 c 5 ways = 3003 answer : option c |
Work through the equation methodically, ensuring clarity at each step. | a shopkeeper buys two articles for rs . 1000 each and then sells them , making 20 % profit on the first article and 20 % loss on second article . find the net profit or loss percent ? a) 200 , b) 177 , c) 188 , d) 166 , e) 129 | "profit on first article = 20 % of 1000 = 200 . answer : a" |
Solve the problem systematically, showing each step clearly. | a man two flats for $ 675958 each . on one he gains 11 % while on the other he loses 11 % . how much does he gain or lose in the whole transaction ? a) 1.21 % , b) 2.56 % , c) 3.12 % , d) 4.65 % , e) 5.12 % | "in such a case there is always a loss loss % = (11 / 10) ^ 2 = 23 / 19 = 1.21 % answer is a" |
Explain the solution process for the given equation. | the number 219 can be written as sum of the squares of 3 different positive integers . what is the difference of these 2 different larger integers ? a) 6 , b) 2 , c) 3 , d) 5 , e) 4 | "sum of the squares of 3 different positive integers = 219 13 ^ 2 + 7 ^ 2 + 1 ^ 2 = 219 now , difference of these 2 different larger integers = 13 - 7 = 6 ans - a" |
Explain the solution process for the given equation. | a milk man has 10 liters of milk . if he mixes 5 liters of water , which is freely available , in 20 liters of pure milk . if the cost of pure milk is rs . 18 per liter , then the profit of the milkman , when he sells all the mixture at cost price is : a) 20 % , b) 15 % , c) 33.33 % , d) 18 % , e) none of these | "explanation : when the water is freely available and all the water is sold at the price of the milk , then the water gives the profit on the cost of 10 liters of milk . therefore , profit percentage = 15 % . answer : b" |
Explain the solution process for the given equation. | if the price of gasoline increases by 20 % and a driver intends to spend only 8 % more on gasoline , by how much percent should the driver reduce the quantity of gasoline that he buys ? a) 4 % , b) 5 % , c) 6 % , d) 8 % , e) 10 % | "let x be the amount of gasoline the driver buys originally . let y be the new amount of gasoline the driver should buy . let p be the original price per liter . (1.2 * p) y = 1.08 (p * x) y = (1.08 / 1.2) x = 0.9 x which is a reduction of 10 % . the answer is e ." |
Explain the solution process for the given equation. | at a certain organisation , the number of male members went up by 12 % in the year 2001 from year 2000 , and the number of females members went down by 5 % in the same time period . if the total membership at the organisation went up by 1.2 % from the year 2000 to 2001 , what was the ratio of male members to female mem... | "men increase by 12 % = = > 1.12 m = males in 2001 women decrease by 5 % = = > 0.95 f = women in 2001 total employees increase by 1.2 % = = > 1.012 * (m + f) = total number of employees in 2001 obviously (males in 2001) + (females in 2001) = total number of employees in 2001 1.12 m + 0.95 f = 1.012 * (m + f) 1.12 m + 0... |
Break down and solve the equation step by step. | in what time will a train 100 meters long cross an electric pole , if its speed is 360 km / hr a) 1 second , b) 4.5 seconds , c) 3 seconds , d) 2.5 seconds , e) none of these | "explanation : first convert speed into m / sec speed = 360 * (5 / 18) = 100 m / sec time = distance / speed = 100 / 100 = 1 second answer : a" |
Break down and solve the equation step by step. | 30 ^ 10 / 90 ^ 5 = ? a) 6 ^ 5 , b) 5 ^ 6 , c) 3 ^ 6 , d) 10 ^ 5 , e) 15 ^ 3 | "30 ^ 10 / 90 ^ 5 = ? a . 6 ^ 5 b . 5 ^ 6 c . 3 ^ 6 d . 10 ^ 5 e . 15 ^ 3 - > 30 ^ 10 / 90 ^ 5 = (30 ^ 10) / (3 ^ 5) (30 ^ 5) = (30 ^ 5) / (3 ^ 5) = (3 ^ 5) (10 ^ 5) / (3 ^ 5) = 10 ^ 5 . thus , d is the answer ." |
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