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Funda 1: Average Speed

We know that the average speed during a journey is given by (Total Distance Covered) / (TotalTime Taken); but there are a few special cases which might help in solving questions,

- If the distance covered is constant(d1 = d2 = d3 = dn) in each part of the journey, thenthe average speed is the Harmonic Mean of the values.

SpeedAvg = n / (1/s1 + 1/s2 + 1/s3 + 1/sn)

- If the time taken is constant(t1 = t2 = t3 = tn) in each part of the journey then the averagespeed is Arithmetic Mean of the values.

SpeedAvg = (s1 + s2 + s3 +sn)/n

Funda 2: Using Progressions (Arithmetic & Harmonic)

In many questions, you will come across a situation when a person is going from point A topoint B at various speeds and taking various times. We know that ifdistance is constant,speed and time are inversely proportional to each other. But this information can also be usedto deduce the following two facts,

- If the various speeds which are mentioned are in AP, then the corresponding times taken willbe in HP.

- If the various speeds which are mentioned are in HP, then the corresponding times taken willbe in AP.

Let us use these ideas to solve couple of quant questions.

Example 1

Arun, Barun and Kiranmala start from the same place and travel in the same direction atspeeds of 30, 40 and 60 km per hour respectively. Barun starts two hours after Arun. If Barunand Kiranmala overtake Arun at the same instant, how many hours after Arun did Kiranmalastart? [from CAT 2006]

(Some useless information: Arun Barun Kiranmala is a 1968 Bangladeshi film. Now you canguess what inspires the CAT question setters. Here is a song from the film.)

Solution As you can see that the speeds are in HP, so we can say that the times taken will bein AP. Time difference between Arun and Barun is 2 hours, so the time difference betweenBarun and Kiranbala will also be 2 hours.

Hence, Kiranbala started 4 hours after Arun.

Example 2

Rishi Kapoor can swim a certain course against the river flow in 84 minutes; he can swim thesame course with the river flow in 9 minutes less than he can swim in still water. How longwould he take to swim the course with the river flow?

http://www.youtube.com/watch?v=2t2un2Mj_d0http://www.youtube.com/watch?v=2t2un2Mj_d0http://www.youtube.com/watch?v=2t2un2Mj_d0

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Solution

Let us say Speed of Rishi Kapoor in still water is RK and Speed of the river is R. Hence,Rishi Kapoors speeds against the river flow, in still water and with the river flow are,

RK R, RK and RK + R.

As you can see, they are in AP.

Hence, the corresponding times taken will be in HP.

Let us say that the time taken to row down with the stream is t, then 84, t+9 and t are in HP.So,

t + 9 = (2 * 84 * t) / (84 + t)

t2 + 93t + 756 = 168t

t2 75t + 756 = 0

t = 63 or 12

Funda 3: Special Case

Let us say that two bodies a & b start at the same time from two points P & Q towards eachother and meet at a point R in between. After meeting at R, a takes ta time to reach itsdestination (Q) and b takes tb time to reach its destination (P). Then,

Sa / Sb = (tb / ta)

Also, the time taken by a & b to meet (i.e. to reach point R from P & Q respectively) is givenby,

t = (ta * t b)

Note: The same formulae will be valid if two bodies a & b start at different times from twopoints P & Q towards each other. They meet at a point R in between after travelling for ta andtb time respectively. After meeting, they take the same amount of time (t) to reach theirrespective destinations (Q & P).

I hope that these ideas will help you reduce the number of equations that you form whilesolving TSD problems if not completely eliminate them.