Question 521406: Tahir travels from A to B at a speed of 40 mph, and returns along the same route a speed of 80 mph. what is his average speed for the round trip (in mph)?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You have to be careful with this type of problem. When you think of finding an average you usually think of adding some numbers together and dividing by the number of terms you added. For example, in this case you may think that you find the average speed by adding 40 mph to 80 mph to get 120 and then dividing by 2 to get an average speed of 60 mph. That is NOT the correct answer.
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It perhaps is easier to understand if we use some actual numbers. Because 40 and 80 both can be divided into 160, let's assume that the distance between A and B is 160 miles. At 40 miles per hour it will take Tahir 4 hours to go from A to B because Time equals the Distance divided by the rate:
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Time = D/R = 160/40 = 4 hours
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However, for the return trip from B to A, Tahir drives the 160 miles at a rate of 80 miles per hour. Therefore, the Time required to make this trip is:
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Time = D/R = 160/80 = 2 hours
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Now we can determine the average speed. The average speed will be the total distance of the trips divided by the total time of the trips. The total distance is 320 miles consisting of the 160 miles to go from A to B and the 160 miles to drive back from B to A. As we just determined above the total time to complete these trips will be the sum of the 4 hours to go from A to B plus the 2 hours to go from B to A. Therefore, the total travel time is 6 hours.
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The average speed will be the total distance divided by the total time. So in this case the speed or rate is:
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R = D/T = 320/6 = 53.333 ... miles per hour
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In more general terms (rather than use specific numbers) you can do the problem this way:
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Call D the distance from A to B. Therefore, the total distance to be driven is 2D. Recognize that from the equation Distance = Rate * Time that the times involved for each of the two trips involved can be found by dividing the distance D by the speed or rate. So the two times are:
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T = D/40 and T = D/80
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The sum of these two times is D/40 + D/80 and using a common denominator of 80 so that we can add these two terms together we get that the total time is:
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T = D/40 + D/80 = 2D/80 + D/80 = 3D/80
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Now we can compute the Average Rate by dividing the total time into the total distance as follows:
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R = (total distance)/(total time) = 2D/(3D/80)
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Since we are dividing by a fraction [3D/80] we can do the division by inverting it and using this inversion to multiply the numerator. The inverted denominator is [80/3D]. Multiply the numerator of 2D by this inverted fraction. Notice that there is a D in the numerator and a D in the denominator of this product. They cancel.
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R = (2D*80)/3D = (2*80)/3 = 160/3 = 53.333 ...
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You can see that the cancelling of the D terms makes this independent of how far the trip from A to B actually is. As long as the distance for the two legs remains constant, if you travel one leg at 40 mph and the return leg at 80 mph, the average speed will be 53.333 ... mph.
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Hope this helps you to understand the problem a little better. Good luck to you!
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