Say, a car travels at S1 mph on a trip and at S2 mph on return trip. What is its average speed for the entire trip?

Solution:

*** Don't fall in the trap of just averaging the 2 speeds. Overall average speed is not (S1+S2)/2. ***

Total average speed is simply = Total distance/Total time

Lets say,

D = distance travelled by the car in EACH direction

t1 = time spent on onward trip

t2 = time spent on return trip

Thus, the total distance travelled by the car = D+D= 2D

And, by the formula, Speed = Distance/Time

S1 = D/t1 => t1 = D/S1

S2 = D/t2 => t2 = D/S2

Total average speed = Total Distance/Total time = 2D/(t1+t2) = 2D/(D/S1+D/S2) = 2S1*S2/(S1+S2)

Remember this general formula for a total average speed problems:

Total average speed = 2S1*S2/(S1+S2)

Example:

A car travels at 60 mph on a trip and at 100 mph on return trip. What was its average speed for the entire trip?

Solution:

*** Total average speed is not (60+100)/2 = 80 ***

Total average speed = 2*60*100/(100+60) = 2*60*100/160 = 2*60*5/8 = 60*5/4 = 15*5 = 75

Alternatively, you may want to check if the following trick saves you some time.

Calculate the ratio of the speeds r1:r2. In our example it is 60:100 = 3:5

Then divide the difference between the speeds (s2-s1) by r1+r2 to get one part. In our example (100-60)/(3+5) = 5 is one part

The required answer is r1 parts away from the lower speed. That is, 60+r1*5 = 60+3*5 = 75 mph

Lets check how it works for S1=20 mph and S2=40 mph

Method 1:

Using the formula Total avg speed = 2S1*S2/(s1+s2)

= 2*20*40/(20+40)

= 2*20*40/60

= 80/3 = 26.67 mph

Method 2:

Ratio r1:r2 = 20:40 = 1:2

r1+r2 = 3

1 part = (S2-S1)/(r1+r2) = (40-20)/3 = 20/3 = 6.67

Total Avg speed is r1 parts away from smaller speed

Therefore avg speed = 20+ r1*6.67 = 20+1*6.67 = 26.67 mph

Hope you find this useful.