Consider the following problem:
If Lucia walks to work at a rate of 4 miles per hour, but she walks home by the same route at a rate of 6 miles per hour, what is Lucia’s average walking rate for the round trip?
It is very tempting to find an average rate as you would find any other average: add and divide. Thus, you might say that Lucia’s average rate is 5 miles per hour (4 + 6 = 10 and 10 ÷ 2 = 5). However, this is INCORRECT!
If an object moves the same distance twice, but at different rates, then the average rate will NEVER be the average of the two rates given for the two legs of the journey. In fact, because the object spends more time traveling at the slower rate, the average rate will be closer to the slower of the two rates than to the faster.
To find the average rate, you must first find the total combined time for the trips and the total combined distance for the trips.
First, you need a value for the distance. All you need to know to determine the average rate is the total time and total distance, so you can actually pick any number for the distance. The portion of the total distance represented by each part of the trip (“Going” and “Return”) will dictate the time.
Pick a Smart Number for the distance. Because you would like to choose a multiple of the two rates in the problem, 4 and 6, 12 is an ideal choice.
Set up a Multiple RTD chart:
| Rate (mi/hr) |
× | Time (hr) |
− | Distance (mi) |
|
| Going | 4 | × |
t1 | = |
12 |
| Return | 6 | × |
t2 | = |
12 |
| Total | r | × |
t3 | = |
24 |
The times can be found using the RTD equation. For the Going trip, 4t1 = 12, so t1 is 3 hours. For the Return trip, 6t2 = 12, so t2 is 2 hours. Thus, the total time is 5 hours. Now plug in these numbers:
| Rate (mi/hr) |
× | Time (hr) |
= | Distance (mi) |
|
| Going | 4 | × |
3 | = |
12 |
| Return | 6 | × |
2 | = |
12 |
| Total | r | × |
5 | = |
24 |
Now that you have the total time and the total distance, you can find the average rate using the RTD equation:
Again, 4.8 miles per hour is not the simple average of 4 miles per hour and 6 miles per hour. In fact, it is the weighted average of the two rates, with the times as the weights. Because of that, the average rate is closer to the slower of the two rates.
You can test different numbers for the distance (try 24 or 36) to prove that you will get the same answer, regardless of the number you choose for the distance.
Juan bikes halfway to school at 9 miles per hour, and walks the rest of the distance at 3 miles per hour. What is Juan’s average speed for the whole trip?