Puzzle 70 - Solution

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 The puzzle is

A car goes uphill a certain distance at a speed of A km per hour. Then it comes back the same distance at a speed of B km per hour. A and B are not equal. 

Which of the averages are greater - Average of A and B or average speed of the total travel? 

The answer is average of A and B.

Solution :
  
Let us say the distance the car travels in one direction is D. Let the time taken for uphill travel is t1 and time taken for downhill travel is t2.

Total distance traveled = 2D

Total time = t1+t2 = D/A + D/B = DB+DA / AB = D (A+B)/AB

Average speed of the travel = distance / time = 2D /(D (A+B) / AB) 
                   = 2AB/(A+B)   

Average of A and B  = (A+B)/2

Which of these two is greater?

2AB/(A+B)                     A+B / 2

2AB                                 (A+B)*(A+B)/2      --multiplying both the sides by A+B

4AB                                  (A+B)*(A+B)        ---Multiplying both sides by 2

4AB                                   A2+B2+2AB         --Expanding A+B * A+B
0                                           A2+B2-2AB        --Adding -4AB to both sides
0                                          (A-B)2

Since A is not equal to B, A-B is not zero. Which means (A-B)2 is positive. Second column is greater than 0.

So (A+B)/2 is greater than 2AB/(A+B)

Average of A and B is greater than the average speed of the journey. 

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