

Intercalary Month and Mathematics 
Intercalary Month (अधिक महिना) and Mathematics Dear students, teachers and parents, Mathematics is related to a number of fields in our life. Calendars have been prepared since ancient times using astronomical observations and mathematical calculations. The insertion of a leap day or month into some calendar is known as intercalation. You are familiar with the idea of leap year in Gregorian or solar calendar. The earth makes one revolution around the sun in about 365.24 days. But the solar year consists of 365 whole days for convenience and an intercalary day is added every four years and we have 29 days in February in the leap year. To improve accuracy, years divisible by 100 (but not by 400) are not considered leap years. Another calendar is a lunar calendar. The moon is the only satellite of our earth and it rotates around the earth. A lunar month has approximately 29.531days according to the movement of the moon so a lunar year has 29.531 × 12 = 354.37 days. So there is the difference of approximately 10.87 days in solar and lunar year. This gap of about 11 days, every year, is approximately 1 month in 3 years. In India the Hindu calendar is a lunisolar calendar in which days of the month (तिथि) depend on waxing and waning of the moon but important festivals and religious days are according to seasons that are determined by the movement of the Sun. Hence in order to make correction, Hindu calendar (पंचांग), includes intercalary month after a period of approximately 28 to 35 months. This is done in order to make the calendar follow the seasons. This year there is intercalary month and it has started on 17^{th}^{ }June and it will come to end on 16^{th} July. It is named as अधिक आषाढ.From 17^{th} July there will be regular आषाढ month which is named as निज आषाढ. Such intercalary month has cultural significance too. Since it occurs once after about 3 years, it is considered special for some religious activities. One such activity is donation – (दान). In Sanskrit, दान means giving or the act of being generous. It is a pious deed or virtuous act. The practice of दान on special occasions has been well established in Indian culture since ancient times and it is very much in vogue even at present in almost all the parts of the country. Ancient mathematical texts in Sanskrit have utilized this common practice in framing mathematical problems. Please note that these problems are from texts written during 7^{th} to 14^{th} centuries, i.e. hundreds of years back. The coins द्रम्म, निष्क were being used at that time. Here are some numerical problems based on the concept of दान. Try to solve them. 1. One,who donates 4 द्रम्म on the first day and increases his donation successively by 5 द्रम्म every day, thus donates for fifteen days. O friend, tell his total donation quickly. 2. A rich man gives 3 निष्क on the first day and the same amount is increased by 2 निष्क on each subsequent day for half a month. O friend, if you know mathematics well, tell the last day’s amount, the average amount and the total amount. (1 month = 30 days) 3. A wealthy man donates 2½ rupees on the first day and increases the amount by ½ rupee every day. Find the total amount donated by him in the month (30 days). 4. In the month of कार्तिक a certain king daily gives away some money in charity, starting with 2 on the first day (of the month) and increasing that by 3 per day. Fifteen days having passed, there arrived a wellversed man. The amount for the next ten days was given to him; that for the remaining five days of the month to someone else. Calculate the amount received by the last two persons. 5. A (donor) gave 3 द्रम्म on the first day. He continued increasing his donation each day by 2 द्रम्म and the total amount donated by him was 360 द्रम्म. Then tell quickly the number of days of this donation. 6. Oyou,who possess enough strength of arms to cross the ocean of arithmetic, give out the total value of the offerings made in relation to 1000 cities, commencing with 4 and increasing it successively by 8. 7. One person bestows for 3 days and the increase is fivefold successively. If the initial amount is three and a half, tell the amount donated by him. 8. A rich man gave the amount twice of that given in the previous year, in each subsequent year for five years. The total amount became 186. O learned, tell the initial amount donated. 9. A king donated two pieces of gold on the first day and doubled the gift on every successive day. Then tell quickly the gold pieces donated by the king in ten days. 10. One man donates 6 rupees initially and donates successively for 10 days with threefold increase daily, how much total amount is donated by him?
