Tuesday, November 18, 2014

Distance and Average Speed

(From February 2011)

Last night, I worked with a Geometry student in Highland Park. After the session, we discussed a few math problems. Here was one:   A car travels at a speed of 30 mph over a certain distance. Then, returns over the same distance at a speed of 20 mph. What is the average speed for the entire trip?  (Hint: It is not 25 mph) 

Continue to see the solution...

Many people are quick to answer 25mph, but that is incorrect..
For example: suppose the distance traveled is 60 miles each way. The trip there takes 60/30 = 2 hours... And, the return trip takes 60/20 = 3 hours.. the total time for 120 miles is 5 hours, which equals 24mph!!
Let's confirm this with a general solution:
Let d = distance traveled one way.
      t = time it takes to get there.
       T =  time it takes to return.
       A = average time.
We know distance = rate * time
So, d = 30t   (going there)   and  d =20T (coming back)...
Then,  t = d/30   and   T = d/20
Now that we have expressions for T and t, let's set up the equation to find the average speed A..
d + d = A (t + T)
2d = A (d/30 + d/20)
2d = A (2d/60 + 3d/60)
2d = A (5d/60)
A = 120d/5d
A = 24
So, the average speed over the round trip is 24 mph. If this seems strange to you, consider that more time is spent going 20mph therefore the "20mph" figure should weigh more toward the average.

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