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a freight train makes a regular run of 1300 km according to its present schedule. the train disparches want to schedule this run for less than one hour, but the average speed will have to be increased by 2 km per hour. what average speed will be required to make the run on the new schedule?

Time should be less than 1. Therefore, the total time is: (1300/R) - (1300/(R + 2)) < 1. We multiply both sides by (R)(R + 2) to remove the denominators. We have: (1300)(R + 2) - (1300)(R) < (R)(R + 2) --> 1300R + 2600 - 1300R < R^2 + 2R --> 0 < R^2 + 2R - 2600 --> 0 < (R - 50)(R + 52) --> Since there are no negative rates, we use: 0 < (R - 50) --> 50 < R (Original speed). Therefore, the train need to have a speed greater than 50 + 2 km/hr --> 52 km/hr.