Example 1

कश्चिन्नरः प्रयाति त्रिभिरादा उत्तरैस्तथाष्टाभिः।

नियतगतिरेकविंशतिरनयोः कः प्राप्तकालः स्यात्॥६.३२०॥

A person starts journey with a speed of 3 and then increasing regularly by 8 as the successive common difference. Another person travels at a constant speed of 21. What is the time of their meeting again, if they start from the same place, at the same time, and move in the same direction?

Example 2

षड् योजनानि कश्चित्पुरुषस्त्वपरः प्रयाति च त्रिणि।

उभयोरभिमुखगत्योरष्टोत्तरशतकयोजनं गम्यम्।

प्रत्येकं च तयोः स्यात्कालः किं गणक कथय मे शीघ्रम्॥६.३२१॥

One man travels at the speed of 6 yojanas and another at the speed of 3 yojanas. The average distance to be covered by either of them moving in opposite directions is 108 yojanas. Tell me quickly, O Mathematician, the time of their meeting?

The above interesting problems are from the book Ganitasarasangraha written around 850 CE by a Digambara Jain monk Mahavira (815 – 877 CE).

Let us consider the first problem:

Let the time of the second meeting of both the persons be t.

The varying speeds of the first person follow the arithmetic progression.

Let the starting speed of the first person be u and let the increase in speed at regular intervals be a.

Then the successive speeds will be .

Then the speed of the first person at the time of second meeting will be 

Then the average speed of the first person during the entire journey will be

Let the speed of the second person be v.

At the time of the second meeting both the persons had travelled equal distances, hence we have:

Now let us consider example 2.

Refer Figure 1.

Figure 1

Let one person travel from A to B and the other travel from B to A. Let the distance between A and B be d. Let them meet at point M at time t. Let the distance between A and M be x. Then the distance between B and M will be d – x.

Let the speed of the person starting his travel from point A be v_A and the speed of the one who starts his travel from point B be v_B.

These 2 formulas were given by Mahavira in the following sutra:

ध्रुवगतिरादिविहीनश्चयदलभक्तस्सरुपकः कालः।

द्विगुणो मार्गस्तद्गतियोगहृतो योगकालस्स्यात्॥६.३१९॥

The constant speed is subtracted by the first term of the speed in Arithmetic progression, and is then divided by the half of the common difference. On adding one to the resulting quantity you get the time of the meeting.

When two persons are travelling in opposite directions, each with a constant speed, twice the average distance to be covered by each of them is the distance between their starting points. This when divided by the sum of their speeds gives us the time of their meeting.

Let us now use this wisdom to solve the given problems.

Solution 1

Solution 2

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