SOLUTION: We travel to Grandmother's house with an average speed of 40 mph (snow storm) and return the same but our average speed is 60 mph, what was our average speed for the round trip?

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Question 1104927: We travel to Grandmother's house with an average speed of 40 mph (snow storm) and return the same but our average speed is 60 mph, what was our average speed for the round trip?
I keep getting 50 as the answer but my professor said its not 50
Found 3 solutions by jim_thompson5910, math_helper, Alan3354:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!

d = distance from your house to your grandmother's house
t1 = time it takes to go from your house to grandmother's house (going 40 mph)
t2 = time it takes to go from grandmother's house back to your house (going 60 mph)
The distance is in miles. The time values are in hours.

I'm going to use d = r*t
where
d = distance traveled
r = rate or speed
t = time

Going from your house to your grandmother's house, you travel 40 mph and you take t1 hours. So
d = r*t
d = 40*t1
d/40 = t1
t1 = d/40
The last equation is the time equation for going from your house to your grandmother's house.

Similarly,
d = r*t
d = 60*t2
d/60 = t2
t2 = d/60
represents how long it takes to go from your grandmother's house to your house

Add up the two time values to get the total time
t1+t2 = (d/40) + (d/60)
t1+t2 = (3d/120) + (2d/120)
t1+t2 = (3d+2d)/120
t1+t2 = (5d)/120
Let's call this total time y
So y = (5d)/120
where d is the unknown distance

You travel a total of 2*d miles when you make the round trip. We want to find the r value (rate) when you travel 2*d miles and take y hours to do so
distance = rate*time
2*d = r*y
2*d = r*(t1+t2)
2*d = r*(5d/120)
120*2*d = r*5d
240*d = r*5d
240*d/d = r*5d/d
240 = 5r
240/5 = 5r/5
48 = r
r = 48

So you were fairly close when you got 50. The answer is 48 miles per hour

An alternative method is to use this formula

which is effectively what happened earlier. In this case
and
It's possible to divide every term by d. Doing so leads us from this

to this

So plugging in r1 = 40 and r2 = 60 leads to

which simplifies (skipping a few steps) to

Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website!

On the way to Grandmother's: d = 40mi/hr * t1
On the way back: d = 60mi/hr * t2
Notice here . This is the key to tell you you can not just take (60+40)/2 as the average.
So 40*t1 = 60*t2 �>
Average speed =
d/t2 = 60 (from the return trip) �> Average speed = mi/hr.
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Basically, in the general case, you can not average speeds (rates). You must take apart the problem and figure out average from the individual parts. The only exception is if the TIMES are the same for each leg (e.g. drive 1hr at 60mi/hr then drive 1hr at 40mi/hr, then the average is indeed 50mi/hr).

Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website!
Avg speed of a round trip is similar to parallel resistors, parallel flows, parallel work, etc.
-------
Avg = 2*r1*r2/(r1+r2)
= 2*60*40/100
= 48 mi/hr
===================
It's not 50 because more time is spent at 40 mi/hr than at 60 mi/hr.
I'm not in favor of memorizing a lot of formulas, but this is an easy one.