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

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.
RELATED QUESTIONS
Ben and Ron raced each other. Ben averaged 9 miles per hour for the entire course while... (answered by Mathtut)
Question: I had a flat tire after biking 3/4 of the way home. I walked the rest of the (answered by Mathtut,solver91311)
S.F. and Santa Barbara are 690 miles apart. Jaclyn leaves Santa Barbara at the same time... (answered by harpazo,lwsshak3)
Frank drove 100 mi. For the first 55 mi. he averaged 50 mph and for the last 45 mi he... (answered by stanbon,Deina)
Frank drove 100 mi. For the first 55 mi he averaged 50 mph, and for the last 45 mi he... (answered by ewatrrr)
The Daytona 500 started in 1959 with an average speed of 135 mph but before they raced on (answered by richwmiller)
An automobile averaged 45 mph for the first part of a trip and 50 mph for the second... (answered by ankor@dixie-net.com)
A truck driver began driving at 8 am,drove 570 miles,and arrived at his destination at 6... (answered by ankor@dixie-net.com)
You are traveling to grandmother's house and get stuck in traffic so that you averaged... (answered by Alan3354)