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P _ _ N T Clue
THE TRAIN IS ABOUT TO LEAVE AND YOU WILL NOT BE ABLE TO CATCH THE TRAIN. THE AMOUNT OF TIME TAKEN TO TRAVEL TO THE HALFWAY POINT AT 15 MILES AN HOUR IS THE SAME AMOUNT OF TIME TAKEN TO REACH THE TRAIN STATION WHEN TRAVELLING TO THE RAILROAD STATION AT 30 MILES AN HOUR. FOR EXAMPLE, LET'S ASSUME THAT THE DISTANCE TO TRAVEL IS 30 MILES. WHEN TRAVELLING AT 30 MILES AN HOUR, IT TAKES 1 HOUR. WHEN TRAVELLING AT 15 MILES AN HOUR, IT WILL ALREADY TAKE 1 HOUR TO REACH THE HALFWAY POINT THE AVERAGE SPEED IS 24 MPH, NOT 25MPH. LET D BE THE DISTANCE TO THE STORE, T BE THE TIME IT GETS TO GET THERE, T BE THE TIME IT TAKES TO GET BACK, AND A BE THE AVERAGE SPEED (WHICH IS WHAT WE WANT TO FIND OUT). AS WE KNOW FROM ELEMENTARY MATHEMATICS, DISTANCE EQUALS RATE TIMES TIME: D = 20T T = D/20 D = 30T T = D/30 NOW THAT WE HAVE EXPRESSIONS FOR T AND T, WE CAN COME UP WITH AN EQUATION THAT DESCRIBES THE ROUND TRIP: 2D = A(T + T) 2D = A(D/20 + D/30) 2D = A(3D/60 + 2D/60) 2D = A(5D/60) A = 120D/5D A = 24 SO THE AVERAGE SPEED IS 24 MPH. IF THAT IS STRANGE, ASSUME THAT THE DISTANCE TO THE STORE IS 60 MILES. WHEN YOU DRIVE AT 20 MPH, YOU WILL TAKE 3 HOURS. WHEN YOU DRIVE AT 30 MPH, YOU WILL TAKE 2 HOURS. THE TOTAL TIME TAKEN IS 5 HOURS AND THE TOTAL DISTANCE TRAVELLED IS 120 MILES. SO THE AVERAGE SPEED IS: 120 / 5 = 24 MPH THERE IS NO WAY TO GET AN AVERAGE SPEED OF 30 MPH (MILES PER HOUR). IF YOUR AVERAGE SPEED FOR THE 2 MILES STRETCH MUST BE 30 MPH, YOU SHOULD DO IT IN 4 MINUTES. DURING THE ASCENT, YOU WOULD HAVE ALREADY TAKEN 4 MINUTES TO TRAVEL 1 MILE AT 15 MPH. SINCE YOU HAVE TAKEN 4 MINUTES TO GO UP THE HILL, YOU WILL NEED TO TAKE 0 SECONDS TO TRAVEL THE 1 MILE DESCENT IN ORDER TO GET AN AVERAGE SPEED OF 30 MPH. THIS IS NOT POSSIBLE THE STATION CLOCK IS 3 MINUTES FAST. THE MORNING JOURNEY TOOK 65 MINUTES, AND THE EVENING JOURNEY THEREFORE TOOK 52 MINUTES, AND THE TRAIN ARRIVED 57 MINUTES AFTER IT SHOULD HAVE LEFT, THAT IS, 3 MINUTES EARLY