SOLUTION: A train travels from A to B at 10km per hour and returns from B to A at 15km per hour. What is the average speed of the train? I thought it was 12.5kmph using this: 10+15=25

Question 465894: A train travels from A to B at 10km per hour and returns from B to A at 15km per hour. What is the average speed of the train?
I thought it was 12.5kmph using this:
10+15=25
25/2 = 12.5
The answer is 12kmph, though, and I can't figure out why. Can anyone help? Please?
Found 2 solutions by bucky, robertb:
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
This type of problem is a common "trick" question. You can't average different rates to get the average rate. Here's what you have to do.
.
First recognize that the Distance, Rate, Time relationship is given by the equation Distance = Rate times Time, or D = R*T.
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Then recognize that the average rate will be the total distance traveled divided by the total time taken.
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We'll just say that L is the distance from point A to point B. (Obviously, L is also the distance from point B to point A.)
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For the first leg of the trip (going from A to B) the rate is 10 km per hour. Substituting L and 10 km/h into the formula D = R*T results in the equation:
.

.
We can solve this for the time it took to travel from A to B by dividing both sides of this equation by 10 to get:
.

.
So the time taken for the first leg from A to B is
.
The distance traveled for the second leg (B to A} is L and the rate is 15 km/h. Substituting these values into the equation D = R*T results in:
.

.
Divide both sides of this equation by the rate of 15 km/h to solve for the time it takes for the return trip gives:
.

.
Now we can sum the two times to find the total travel time taken to make the round trip. The sum is:
.

.
Add these by using the common denominator of 30 to give a total time of:
.

.
And the total distance covered is 2*L (comprised of the first leg L from A to B and the second leg L from B to A).
.
Substitute these total values into the Distance = Rate * Time equation to get:
.

.
L is a common factor on both sides, so dividing both sides by L gets rid of it and results in:
.

.
Get rid of the denominator of 30 by multiplying both sides of this equation by 30:
.

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Solve for R by dividing both sides of the equation by 5 to get:
.

.
and the "average" rate, which equals the total distance divided by the total time turns out to be:
.

.
or
.
km per hour
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Just as you said it should be.
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Sure is easy to fall into the trap of just adding the two rates and dividing by 2 though, isn't it?
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Hope this helps you to understand the process. You can use this new knowledge of the process to prove to your friends that if they think you should add the two rates and divide by 2, they're not as knowledgeable as you are. Have fun!!!

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Take the harmonic mean of the two rates:
kph.
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