Here's another example using this formula you might find helpful:
Tracey ran to the top of a steep hill at an average pace of 6 miles per hour. She took the exact same trail back down. To her relief, the descent was much faster; her average speed rose to 14 miles per hour. If the entire run took Tracey exactly one hour to complete and she did not make any stops, how many miles is the trail one way?
ANSWER: For the way up the hill, we know that D = 6mph x T.
For the way down the hill, we know that D = 14mph x T. Since we went know that the distance up the hill was the same as the distance down the hill, we can pick a number for D. Let’s choose “84″ since it is a multiple of both 6 and 14. If 84 = 6mph x T, then we know that T = 14 hours. If 84 = 14mph x T, then we know that T = 6 hours.
Now we can use another formula, the Average Rate formula, to find the average speed for the WHOLE trip. Average Rate = Total Distance / Total Time
Using our Picked Number of 84, we know that the Total Distance traveled would be 168 miles. The Total Time is 14 hours + 6 hours = 20 hours. So the Average Rate = 168 miles / 20 hours = 8.4 mph.
It doesn’t matter that Tracey didn’t “really” go 168 miles, or that we know she didn’t “really” go 20 hours. We Picked a Number just so that we could find the ratio of the Total Distance to the Total Time in order to calculate the Average Rate of the ENTIRE journey.
Now that we have found the Average Rate for the whole trip, we can plug it in to the “DIRT” formula to find the ACTUAL distance for the entire journey.
D = R x T
D = 8.4mph x 1 hour
We know that T = 1 hour because the problem told us so. Therefore, the actual distance for the entire trip was 8.4 miles. The problem asks how many miles the trail was one way. 8.4 / 2 = 4.2. The answer to the question is 4.2 miles.
