Ancient mathematics texts are a treasure trove of interesting problems. The way in which the problems are solved with reference to shlokas is very interesting and insightful. This article is written to provoke the inquisitiveness about our ancient literature and knowledge of maths. Let us take a look at the various examples given below.

 योजनत्रिशती पन्थाः पुरयोरन्तरं तयोः ।

एकादशगतिस्त्वेको नवयोजनगः परः ॥

युगपन्निर्गतौ स्वस्वपुरतो लिपिवाहकौ ।

समागमद्वयं ब्रूहि गच्छतोश्च निवृत्तयोः ॥ ४४ ॥

Meaning:

The distance between two towns is 300 yojanas. Two letter carriers start their journey from their respective towns (simultaneously) to the other town, one with a speed of 11 yojanas per day and the other with 9 yojanas per day. Tell us quickly the times of their two meetings, the first during their onward journey and second while returning back.

1 yojana = approximately 12kms as per Vishnu Purana.

The above problem is from the book Ganita Kaumudi (गणित कौमुदी) composed by Narayan Pandit in the year 1356 CE as mentioned in the final verses of the book.

How does one solve such a problem?

Narayan Pandit has given a sutra to solve just such kind of problems. The sutra is as given below:

अध्वनि गतियोगहृते प्रजायते प्रथमसङ्गमे कालः ।

तस्मिन् योगे द्विगुणे योगात् तस्मात् पुनर्योगः ॥३९॥

Meaning:

The time of the first meeting is given by the distance between the two towns divided by the sum of the speeds. The time interval between the first meeting and the second meeting during the return journey is twice the time taken for the first meeting.

Let us understand how we get these results.

Let A and B be the two towns and d be the distance between them. Let v_A and v_B be the speeds of the letter carriers starting from towns A and B respectively. Let the M₁ be the point of their first meeting and M₂ the point of their second meeting. Let the time of their first meeting from the start of their journey be t₁ days and the time of their second meeting be t₂ days.

Let distance AM₁ = x

and distance BM₂ = y

Therefore,

BM₁ = d - x

AM₂ = d - y

We have, time = distance / speed

Hence, we have

 

Also

Therefore, we have

Solving for x

Substituting this in (1), we get

This proves the first part of the sutra.

Similarly for the time of the second meeting:

Let the total distance covered by letter carriers from the town A for the second meeting be d_A and that from town B be d_B.

We have

 

Now, we have

 

Also

 

Therefore, we have

 

 

Solving for d_A, we get

Substituting this in (2), we have

The time interval between the first and second meeting will be:

 

This proves the second part of the sutra.

 

Now, let us apply this sutra to solve the given problem. Thus, we have

Time of the first meeting = 15 days

And time of the second meeting = 45 days

Thus, the time interval between the two meetings = 45 - 15 = 30 days.

Given below is another interesting example that provides sequential explanation on finding the time of meeting of 2 people who are walking on a circular path.

Two friends go for a morning walk on a circular road. They start their walk at the same time with different speeds. The speed of one friend is 5 kms/hr and of the other is 4 kms/hr. If the circumference of the road is 500 metres then what is the time of their meeting?

The method to solve such a problem is given in the book Ganita Kaumudi (गणित कौमुदी) composed by Narayan Pandit in the year 1356 CE. Narayan Pandit gives a sutra to solve the above kinds of problems.

 सङ्गमकालः परिधौ गत्यन्तरभाजिते भवति ॥४०॥

Meaning:

The time of the meeting is given by the circumference divided by the difference in the speeds.

Let us see how we get this result:

Let c be the circumference of the circular road. Let v₁ and v₂ be the speeds of the two friends and let v₁ > v₂. Also, let t be the time of their meeting.

Let x be the distance walked by the second friend till the time of their meeting.

The first friend who is faster will complete one more round of the road than the slower one. Therefore, the distance covered by the first friend at the time of their meeting is = c + x.

We have,

time = distance / speed

Therefore, for the first friend, we have

And for the second friend, we have

 

Hence, we have

Substituting this in (1), we have

 

This proves the sutra.

Let us apply the sutra to solve the given problem.

Hence, the two friends will meet after half hour of starting their walk.

 

Image Credits: Pixabay

 

 

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