SOLUTION: Barbara and Ramir raced each other. Barbara averaged 9 mph for the entire course while Ramir averaged 8 mph for the fist half of the course and 10 mph for the second half. Who won

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Question 175252: Barbara and Ramir raced each other. Barbara averaged 9 mph for the entire course while Ramir averaged 8 mph for the fist half of the course and 10 mph for the second half. Who won?
Found 3 solutions by solver91311, josmiceli, gonzo:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
We don't know the distance they raced, so let's just call that . Since we know that distance equals rate times time or , we can also say that time equals distance divided by rate or .

The question we are trying to answer is who had the shorter time. Because Barbara averaged 9 mph for the entire course, we can describe her time .

Ramir's time for the first half of the race must be: and the second half: . Then Ramir's total time must be the sum of these two:

It is convenient in this case to use 80 as a common denominator, .

Now go back and examine Barbara's time: . If we multiply by 1 in the form of , then her time can be expressed as .

If two fractions have the same numerator, the one with the larger denominator is the smaller number, hence Barbara wins. (But just by the hair on her chinny-chin-chin -- a little more than 6 tenths of a second over a 200 meter race)

Answer by josmiceli(19441)   (Show Source):
You can put this solution on YOUR website!
The length of the course isn't given,
so I can pick any length. I'll
pick miles
Using ,
Barbara's time for the race is:

hrs
------------
Ramir's time for the 1st half is:

hrs
Ramir's time for the 2nd half is:

hrs

hrs
Ramir took hrs longer, or
min, or
sec longer to finish, so
Barbara won

Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website!
it appears that barbara is the winner.
here's why:
---
let t = time it took barbara.
since rate * time = distance, then
9*t = d
which means that
t = d/9
barbara took d/9 hours to cover the distance.
---
let t1 = time it took ramir to cover half the distance at 8 mph.
since rate * time = distance, then
8*t1 = d/2
which means that
t1 = d/16
---
let t2 = time it took ramir to cover half the distance at 10 mph.
since rate * time = distance, then
10*t2 = d/2
which means that
t2 = d/20
---
the total time it took ramir is equal to t1 + t2.
since t1 = d/16 and t2 = d/20, then
t1 + t2 = d/16 + d/20
---
d/16 + d/20 = (20d + 16d) / (16*20)
which equals
36d / 320
which equals
d/8.888888888..... hours
---
t1 + t2 = d/8.8888888.... hours which is the time it took ramir.
---
since d/9 is a smaller number than d/8.8888888....., this means that barbara is the winner.
---
this answer should be good at any distance.
as an example, let d = 90
it took barbara 10 hours
it took ramir 10.125 hours.
this is 45/8 + 45/10 since he covered half the distance at 8 mph and half the distance at 10 mph.
---
let d = 1500 miles
it took barbara 166.7 hours (rounded)
it took ramir 168.75 hours.
this is 750/8 + 750/10 since he covered half the distance at 8 mph and half the distance at 10 mph.
---
answer checks out ok.
barbara is the winner at any distance.