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      Question

      The speed of the boat in still water is โ€˜yโ€™ % more than the speed of the boat in upstream. The time taken by the boat to cover (2d+50) km distance in downstream is 3.5 hours less than the time taken by the same boat to cover โ€˜dโ€™ km distance in upstream. The same boat can cover (d+75) km distance in still water in (y/6) hours. If the speed of stream is 30 km/h, then which of the following statements is/are true? (i) The value of โ€˜yโ€™ is the multiple of 5. (ii) The speed of boat in still water should not be more than 60 km/h. (iii) The value of โ€˜dโ€™ is a four digit number.

      A Only (i) Correct Answer Incorrect Answer
      B Only (ii) Correct Answer Incorrect Answer
      C Only (iii) Correct Answer Incorrect Answer
      D Only (ii) and (iii) Correct Answer Incorrect Answer
      E Only (i) and (iii) Correct Answer Incorrect Answer

      Solution

      Letโ€™s assume the speed of boat in still water and the speed of stream are โ€˜Bโ€™ and โ€˜Cโ€™ respectively. If the speed of stream is 30 km/h. C = 30 km/h The speed of the boat in still water is โ€˜yโ€™ % more than the speed of the boat in upstream. B = (100+y)% of (B-C) Put the value of โ€˜Cโ€™ in the above equation. B = (100+y)% of (B-30) 100B = (100+y)x(B-30) [100B/(B-30)] = (100+y) y = [100B/(B-30)] - 100ย  ย  Eq.(i) The time taken by the boat to cover (2d+50) km distance in downstream is 3.5 hours less than the time taken by the same boat to cover โ€˜dโ€™ km distance in upstream. [(2d+50)/(B+C)] = [d/(B-C)] - 3.5 Put the value of โ€˜Cโ€™ in the above equation. [(2d+50)/(B+30)] = [d/(B-30)] - 3.5 [d/(B-30)] - [(2d+50)/(B+30)] = 3.5ย  ย  Eq.(ii) The same boat can cover (d+75) km distance in still water in (y/6) hours. (d+75)/B = (y/6) Put the value of โ€˜yโ€™ from Eq.(i) in the above equation. (d+75)/B = [[100B/(B-30)] - 100]/6 (d+75) = B[[100B/(B-30)] - 100]/6 d = B[[100B/(B-30)] - 100]/6 - 75ย  ย  Eq.(iii) Put the value of โ€˜dโ€™ from Eq.(iii) to Eq.(ii). [[B[[100B/(B-30)] - 100]/6 - 75]/(B-30)] - [(2x[B[[100B/(B-30)] - 100]/6 - 75]+50)/(B+30)] = 3.5 After solving the above equation, there will be three values of โ€˜Bโ€™. But two of them will be negative. So these can be eliminated. Hence B = 70 which is the only possible value. Put the value of โ€˜Bโ€™ in Eq.(i). y = [100x70/(70-30)] - 100 y = [7000/40] - 100 y = 175 - 100 y = 75 Put the value of โ€˜Bโ€™ in Eq.(ii). [d/(70-30)] - [(2d+50)/(70+30)] = 3.5 [d/40] - [(2d+50)/100] = 3.5 [d/40] - [(d+25)/50] = 3.5 [5d/200] - [4(d+25)/200] = 3.5 5d - 4d - 100 = 3.5x200 d - 100 = 700 d = 700+100 d = 800 (i) The value of โ€˜yโ€™ is the multiple of 5. The above given statement is true. (ii) The speed of boat in still water should not be more than 60 km/h. The above given statement is not true. (iii) The value of โ€˜dโ€™ is a four digit number. The above given statement is not true.

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