SOLUTION: You travel the first 48 miles of a journey at an average of 36 mph and the second 48 miles at an average of v mph. What must be v if you want to average 45 mph over the 96 mile jou

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Question 1140236: You travel the first 48 miles of a journey at an average of 36 mph and the second 48 miles at an average of v mph. What must be v if you want to average 45 mph over the 96 mile journey?
Found 4 solutions by Alan3354, Theo, josgarithmetic, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
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You travel the first 48 miles of a journey at an average of 36 mph and the second 48 miles at an average of v mph. What must be v if you want to average 45 mph over the 96 mile journey?
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Avg speed of a round trip or of equal parts is:
Avg = 2*r1*r2/(r1 + r2)
----
45 = 2*36*r2/(36+r2)
45*(36+r2) = 72r2
1620 + 45r2 = 72r2
27r2 = 1620
r2 = 60 mi/hr
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IMO, this is a formula you should remember. It's similar to parallel work, parallel flow, parallel resistors, etc. Only the factor of 2 is different.
You can remember it, or not.
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It's also the basis for the famous Michelson-Morley experiment that showed no effects of the "ether" and dispelled belief in the existence of the ether. Ether is now stored in the same closet as phlogiston.

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
r * t = d

r = rate
t = time
d = distance.

for the first 48 miles, the equation becomes r1 * t1 = 48
for the second 48 miles, the equation becomes r2 * t2 = 48

since r1 = 36, the first equation becomes 36 * t1 = 48
solve for t1 to get t1 = 48/36

since r2 = v, the second equation becomes v * t2 = 48

for the overall trip, r * t = 96
r is the overall rate
t is the overall time

since r = 45, then r * t = 96 becomes 45 * t = 96.
solve for t to get t = 96/45

since t1 + t2 = t, you get 48/36 + t2 = 96/45
that makes t2 = 96/45 - 48/36 = .8

you now have:

r1 = 36
t1 = 48/36
r2 = v
t2 = .8

you know that r2 * t2 = 48 and you know that r2 = v and t2 = .8, so you get v * .8 = 48.
solve for v to get v = 48/.8 = 60

that's your solution.

you now have:

r1 = 36
t1 = 48/36
r2 = 60
t2 = .8

since r1 * t1 + r2 * t2 = r * t, the formula becomes:

36 * 48/36 + 60 * .8 = 45 * 96/45

simplify to get 48 + 48 = 96 which is true, confirming the solution is correct.





Answer by josgarithmetic(39631) About Me  (Show Source):
You can put this solution on YOUR website!
total time:
48%2F36%2B48%2Fv

total distance:
96 miles

EQUATION
96%2F%2848%2F36%2B48%2Fv%29=45-----solve for v.
-
2%2F%281%2F36%2B1%2Fv%29=45
2%2F%28%28v%2B36%29%2F36v%29=45
%282%2A36v%29%2F%28v%2B36%29=45
%282%2A12v%29%2F%28v%2B36%29=15
%282%2A4v%29%2F%28v%2B36%29=5
8v=5%28v%2B36%29
8v=5v%2B5%2A36
3v=5%2A3%2A12
v=60

Answer by ikleyn(52925) About Me  (Show Source):
You can put this solution on YOUR website!
.
1.  "You travel the first 48 miles of a journey at an average of 36 mph".


    It means that you spent  48%2F36 = 112%2F36 hours = 11%2F3 hours = 1 hour and 20 minutes for this part of the journey.



2.  "you want to average 45 mph over the 96 mile journey".


    It means that you spend  96%2F45 = 26%2F45 hours = 22%2F15 hours = 28%2F60 hours = 2 hours and 8 minutes for the entire journey of 96 miles.



3.  So, you have  2 hours 8 minutes - 1 hour 20 minutes = 128 minutes - 80 minutes = 48 minutes for the second part of 48 miles long.



4.  Thus your speed on the seconf leg of 48 miles should be  48%2F48 = 1 miles per minute or 60 miles per hour.      ANSWER

Solved.

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You do not need use complicated formula.

You do not need use equations, at all.

To solve the problem, all you need is to apply your common sense.