From January 1, 2021, the price of petrol (in Rupees per litre) on m ^ {th} day of the year is 80 + 0.1 m, where m = 1, 2, 3, … , 100 and thereafter remains constant. On the other hand, the price of diesel (in Rupees per litre) on n ^ {th} day of 2021 is 69 + 0.15 n for any n. On which date in the year 2021 are the prices of these two fuels equal?

The problem requires us to find the day when the price of petrol and diesel become equal.

1. Petrol price formula: P_petrol = 80 + 0.1m for m = 1, 2, ..., 100 P_petrol = 80 + 0.1 * 100 = 90 for m > 100 (remains constant after 100 days).

2. Diesel price formula: P_diesel = 69 + 0.15n for any day n.

We need to find the day 'd' (which is 'm' or 'n') when P_petrol = P_diesel.

Let's first assume the prices become equal on or before the 100th day (d 100). In this case, the petrol price is constant at 90 Rupees per litre. So, we set the constant petrol price equal to the diesel price formula: 90 = 69 + 0.15d 90 - 69 = 0.15d 21 = 0.15d d = 21 / 0.15 d = 21 / (15/100) d = 21 * 100 / 15 d = 7 * 100 / 5 (dividing numerator and denominator by 3) d = 7 * 20 d = 140

So, the prices of petrol and diesel become equal on the 140th day of the year 2021.

Now, we need to find which date corresponds to the 140th day of 2021. 2021 is not a leap year. January: 31 days February: 28 days March: 31 days April: 30 days Total days till the end of April = 31 + 28 + 31 + 30 = 120 days.

To reach the 140th day, we need 140 - 120 = 20 more days. These 20 days fall in May. Therefore, the date is May 20th.

Comparing with the options: A) 21st May B) 20th May C) 19th May D) 18th May

The calculated date is 20th May, which matches option B.

The final answer is B.