From January 1, 2021, the price of petrol (in Rupees per litre) on m^th day of the year is 80 + 0.1m, 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 + 0.15n for any n. On which date in the year 2021 are the prices of these two fuels equal?

Correct Answer: Option 20th May

Let us step-by-step calculate the date on which the price of petrol and diesel become equal in the year 2021.

Step 1: Understand the price functions of petrol and diesel

The price of petrol on the $m$^th day of the year is given by:

$\text{Petrol Price}=80+0.1m$

for $1\le m\le 100$, and after the 100^th day, it remains constant at the price of the 100^th day.

Let's find the maximum petrol price on the 100^th day ($m=100$):

$\text{Petrol Price on 100th day}=80+0.1\times 100=80+10=90\text{Rupees}$

The price of diesel on the $n$^th day of the year is given by:

$\text{Diesel Price}=69+0.15n$

Step 2: Equate the two prices

Assuming the prices become equal on day $d$ where $d\le 100$, we set the two equations equal:

$80+0.1d=69+0.15d$

Now, simplify the equation to solve for $d$:

$80-69=0.15d-0.1d$

$11=0.05d$

$d=\frac{11}{0.05}=220$

Since $220>100$, our initial assumption that $d\le 100$ was incorrect. Thus, the petrol price has already reached its maximum constant value of 90 Rupees after the 100^th day.

Step 3: Solve for the day when Diesel Price reaches 90 Rupees

Since petrol price stays constant at 90 Rupees for any day $n>100$, we set the diesel price equal to 90 Rupees:

$69+0.15n=90$

$0.15n=90-69$

$0.15n=21$

$n=\frac{21}{0.15}=140$

So, the prices of both fuels become equal on the 140^th day of the year 2021.

Step 4: Find the date corresponding to the 140^th day of 2021

Note that 2021 is a non-leap year (365 days). Let us count the number of days in each month starting from January 1, 2021:

• January: 31 days
• February: 28 days
• March: 31 days
• April: 30 days

Total days up to April 30 = $31+28+31+30=120\text{days}$

To reach the 140^th day, we need:

$140-120=20\text{more days in May}$

Therefore, the 140^th day of the year 2021 is 20th May.