Let’s suppose you are working on problems involving distance, speed and time. You might consider posing these problems one at a time.

**Q1:** Suppose I travel 2 hours at 40 mph and then 2 hours at 20 mph. What is my average speed for the trip?

**Q2:** Suppose I travel 2 hours at 40 mph and then 4 hours at 20 mph. What is my average speed for the trip?

**Q3:** Suppose the cities Abat and Boto are 80 miles apart and I travel from Abat to Boto at 40 mph and from Boto to Abat at 20 mph. What is my average speed for the trip?

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I should point out that I am now a units freak since my engineering husband always insisted that I include units when talking to him. In becoming such, I have learn to avoid many traps such as answering 30 mph to Q3.

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Average speed = total distance/total time. Notice that the unit is correct!

**A1: ** The total distance is: 2 hrs*40 mph+2 hrs*20 mph= 120 miles. The total time is: 2 hrs+2 hrs=4 hrs.

average speed = 120 miles/4hrs = **30 mph**

**A2: ** The total distance is: 2 hrs*40 mph+4 hrs*20 mph= 160 miles. The total time is: 2 hrs+4 hrs = 6 hrs.

average speed = 160 miles/6 hrs **= 26.7 mph**

**A3: ** The total distance is: 80 miles+80 miles=160 miles.

The total time must be calculated and is: 80 miles/40 mph + 80 miles/20 mph =2 hrs+4 hrs=6 hrs.

So this is the same problem as Q2. I drive twice as long a time coming back at 20 mph as going at 40 mph .

average speed = **26.7 mph**

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**Resource: **These questions arose when I was thinking about the 2nd comment to my last blog about **harmonic averages** (thank-you Math Maker!) and so I looked around for more information at wikipedia (more about this to come in future blogs).

## 12 May 2009

### Thinking questions for rates

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I like the last two examples which show the relationship between distance, time and speed in different ways. A rather strange thing is that I found that students struggle more with the third form than the second form and yet neither is really more complicated than the other. Perhaps it’s the way students are taught the formula, s=ut (s=distance,u=speed), and then have difficulty recognizing its other forms (apart from algebraic difficulty of changing the subject of the equation). Thanks again for the interesting post!

Comment by The Math Maker — 18 May 2009 @ 11:43 am