## PROBLEM: Average Rate

Danielsville is 20 miles frm Jim's home in Athens. He Drives 60 mph going to a meeting (he is almost late) but coming home the weather is bad and drives 30 mph. What is his average speed for the time he is on the road?

A. Discuss why 45 mph is not correct.
Going to D'ville took 1/3 of an hour. Coming home took 2/3 of an hour. So the total 40 miles took one hour. The speed is 40 mph.

B. Use d = r t to verify and develop an understanding of how to find average rates. Athens to Danielsville = 20 miles Athens to Danielsville = 1/3 hour Danielsville to Athens = 2/3 hour Total travel time = 1/3 + 2/3 = 1 hour Total travel distance = 20 miles + 20 miles = 40 miles

A. When determining an average speed one traveled while going from one place to another one cannot just average the different speeds together. This is true because each different speed occurred over a given time. (It is worth noting here that if each different speed was traveled for the same length of time, one can average all the speeds together to obtain the overall average speed.) If the speeds occurred over different lengths of time, then in order to determine the overall average speed, the given speeds need to weighted according to the amountof time they took. For example, in this problem, it took 1/3 of an hour to go to Danielsville at 60mph and 2/3 of an hour to go from Danielsville to Athens at 30mph. Therefore, it took twice as long to return from Danielsville as it took to go to Danielsville. From this, one can see that there will be 3 speeds that need to be average to determine the overall average speed: 60mph from the trip to Danielsville and a pair of 30 mph speeds. There is a pair of 30mph speeds, one for each 1/3 of an hour from Danielsville back to Athens. When determining average rates, one needs to compare rates over equal amounts of time. Therefore, the average speed for the trip would be (60+30+30)/3 = 40mph.

B. Now let us look at this problem using the distance equation d=rt (distance=rate*time). To find the average speed, we need to compute the total distance traveled and the total travel time. From the table above, the total distance traveled was 40 miles and the total time to travel the 4 miles was 1 hour. Therefore, the average speed for the entire trip would be

40 miles = 1 hour * rate
This impies: rate = 40 miles/1 hour = 40mph

So in order to use the distance formula, one needs to have the total distance traveled and the total time it took to travel that distance.

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