I'm confused with the rate and average questions -

Question: G can copy 50 pages in 8 hrs, and S copy 50 pages in 6 hours. to copy 100 page together, they will use how many hours?

===>> (50/8+50/6)*time=100

Question: G drives 50 miles in 8 hrs, and S drives 50 miles in 6 hrs. To drive 100 miles together, they will use how many hours?

It seems like we can combine rates, but we cant do the same thing for average speed? Can anyone explain to me why?
Can we take average as rate?
how do we know when to calcuate rate and when to calculate average?


Thanks a lot.

I will answer your general question but I can't comment on the specific problems you've provided unless you cite the author of the problems - please don't forget to do this. (Or if you just made them up, say so.)

You can combine rates but you cannot combine average speeds. Average speeds are a function of distance / time. In order to calculate an overall average speed from two "sub-averages" (that is, averages that cover only parts of the total distance and time), you have to know the individual distances and times, add each up, and divide. Try it.

I drive 1 mile from my house to work. In the morning, it takes me 10 minutes. In the evening, it takes me 15 minutes.

Rate = Distance / Time

My average speed in the morning is therefore 1/(1/6) = 6mph. (Don't forget to convert to hours!) My average speed in the afternoon is 1/(1/4) = 4mph. (Slow commute!). Is my overall average (6+4)/2 = 5mph?

I go 2 miles total (1 there, 1 back). I spend 25 minutes total (10 there, 15 back). My speed is 2/(25/60) = 120/25 = a little bit under 5 miles per hour. (5 exactly would be 125/25.)

How come? Because although my distance is the same for both legs of the trip, my time is not. In the evening, I'm slower, so I take more time. That extra time skews the average speed a bit towards the slower speed.

If, on the other hand, someone just said: you spend 10 minutes driving 6 miles an hour and 10 minutes driving 4 miles an hour, what's your average speed? Then it really would be 5, because your spending the same amount of time driving at each speed. But in the above case, you're spending more time driving at the slower speed (in this case 4 mph), so your overall average can't be exactly in the middle.

And, in fact, when you're driving the same distance (in this case, 1 mile) at two different speeds, you'll see that the skew is always towards the slower of the two speeds, because you will always spend more time covering that distance at the slower speed than you will spend at the higher speed.