Question
The speed of the boat in still water is βyβ % 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 βdβ 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 βyβ 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 βdβ is a four digit number.Solution
Letβs assume the speed of boat in still water and the speed of stream are βBβ and βCβ respectively. If the speed of stream is 30 km/h. C = 30 km/h The speed of the boat in still water is βyβ % more than the speed of the boat in upstream. B = (100+y)% of (B-C) Put the value of βCβ 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)] - 100Β Β 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 βdβ km distance in upstream. [(2d+50)/(B+C)] = [d/(B-C)] - 3.5 Put the value of βCβ in the above equation. [(2d+50)/(B+30)] = [d/(B-30)] - 3.5 [d/(B-30)] - [(2d+50)/(B+30)] = 3.5Β Β 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 βyβ 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 - 75Β Β Eq.(iii) Put the value of βdβ 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 βBβ. 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 βBβ in Eq.(i). y = [100x70/(70-30)] - 100 y = [7000/40] - 100 y = 175 - 100 y = 75 Put the value of βBβ 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 βyβ 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 βdβ is a four digit number. The above given statement is not true.
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