Story problem explanation?

Question

Here is the scenario: There is a 6 mile race. Here are the two contestants and their speeds. Who won?

Mary: 3 miles at 6mph and 3 miles at 3mph
Steve: all 6 miles at 4 mph

The answer is that they tie. But the question I am having a hard time explaining is, why can't you figure Mary's overall speed by just taking her average speed (4 1/2 mph)? I know how to work the problem, I am not sure how to explain why this does not work without just showing an example. Any better explanations?

d=rt so t=d/r
Mary: 3/6+3/3=1 1/2 hr
Steve: 6/4=1 1/2 hr

Why can't you figure Mary is faster by averaging her speed and saying it is r=(6+3)/2=4.5mph and therfore she is faster than Steve?

Answer 1

Time determines who will win the race

time = distance/rate

Time of Mary:
t = 3/6 + 3/3
t = 1.5hr

Time of Steve:
t = 6/4
t = 1.5hr

They both finished 6 miles after 1.5 hrs. So it's a tie.


You can't find the average speed by adding the two speed together, then divide by 2

Average speed = total distance / total time
average speed = (6) / (1.5)
average speed = 4 mi/hr

average speed of Mary is 4 mi/hr. Not 4.5 mi/hr

Answer 2

Because that is not Mary's average speed, over time - it is the average over speed.

Mary runs at 6mph for half an hour then at 3mph for an hour, so her average over time is (6 * 1/2 + 3 * 1) / 1.5 = 6 / 1.5 = 4 mph

Answer 3

there is actual speed and distance, then you should use the actual speed. when you use to tied result you will note that the average spped of Mary is equal to Steve average speed.

from the formular
Mary: 3/6+3/3=>(3+6)/6 =>9/6 =>3/2 hrs

the average speed for the whole distance is

6/(3/2) => 4mph

Answer 4

4.5mph isnt marys average speed. you cant get it just by adding both and dividing by 2 cause marys running 3mph twice the time she running 6mph. so the real average speed is
=(2*3+1*6)/(2+1)
=12/3
=4mph, same as steves.

Answer 5

Because you need to look at the math behind it.
The math is: TIME = sum (distance/mph). Note that speed is in the DENOMINATOR; you can't take an average of denominators without changing the numerators in some way.