A person travel first half of the total distance with speed `v_1`. In the second half distance, constant speed is `1/3 rd` time is `v_2` and in remaining `2/3 rd` time the constant speed is `v_3`. The average speed during the complete journey is :

Explanation

Let `d` is the total distance travelled by the person.

Let `t_1` and `t_2` is the time taken by in the first and second half of the distance.

Then,

`d = v_1t_1+v_2t_2/3+v_3(2t_2)/3`

`=>3d = 3v_1t_1 +v_2t_2 + 2v_3t_2->(1)`

Now, ` t_1 = (d/2)/(v_1) = d/(2v_1)`

`t_2 = (d/2)/(1/3v_2)+ (d/2)/(2/3v_3) =(3d)/(2(v_2+2v_3))`

Now, Average speed `= (d/2+d/2)/(t_1+t_2) = d/( d/(2v_1)+(3d)/(2(v_2+2v_3)))`

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