From January 1, 2021, the price of petrol (in Rupees per letre) on mth day of the year is 80 + 0.1m, where m = 1,2,3.....,100and there after remains…

Correct Option (2)

The price of petrol on the $m^{th}$mth day of the year is given by $80 + 0.1m$80+0.1m Rupees per litre for $m = 1, 2, ..., 100$m=1,2,...,100. After the $100^{th}$100th day, the price remains constant.

• On the $100^{th}$100th day, the petrol price is calculated as $80 + 0.1 \times 100 = 80 + 10 = 90$80+0.1×100=80+10=90 Rupees per litre.
• Therefore, for any day $n > 100$n>100, the petrol price is 90 Rupees per litre.

The price of diesel on the $n^{th}$nth day of the year is given by $69 + 0.15n$69+0.15n Rupees per litre for any $n$n.

To determine the day when the prices of these two fuels are equal, we consider the period after the $100^{th}$100th day, as the petrol price stabilizes then. If we were to equate the initial formulas $80 + 0.1n = 69 + 0.15n$80+0.1n=69+0.15n, it would yield $11 = 0.05n$11=0.05n, resulting in $n = 220$n=220. Since $220 > 100$220>100, this indicates that the equality does not occur during the period when the petrol price is still increasing according to its initial formula. Thus, the equality must occur when the petrol price is constant at 90 Rupees per litre.

Setting the constant petrol price equal to the diesel price formula:

$90 = 69 + 0.15n$90=69+0.15n

$21 = 0.15n$21=0.15n

$n = \frac{21}{0.15} = \frac{2100}{15} = 140$n=0.1521​=152100​=140

The prices of the two fuels are equal on the $140^{th}$140th day of the year 2021.

To convert the $140^{th}$140th day into a specific date in 2021 (which is not a leap year):

• January: 31 days
• February: 28 days
• March: 31 days
• April: 30 days
• Total days up to April 30th = $31 + 28 + 31 + 30 = 120$31+28+31+30=120 days.
• The remaining days to reach the $140^{th}$140th day are $140 - 120 = 20$140−120=20 days.

Therefore, the $140^{th}$140th day of 2021 is May 20th.

Incorrect Options:

The prices of the two fuels are equal specifically on the $140^{th}$140th day of the year, which corresponds to May 20th. Other dates would result in different diesel prices compared to the constant petrol price of 90 Rupees per litre.

• For 18th May (the $138^{th}$138th day): Diesel price = $69 + 0.15 \times 138 = 69 + 20.7 = 89.7$69+0.15×138=69+20.7=89.7 Rupees.
• For 19th May (the $139^{th}$139th day): Diesel price = $69 + 0.15 \times 139 = 69 + 20.85 = 89.85$69+0.15×139=69+20.85=89.85 Rupees.
• For 21st May (the $141^{st}$141st day): Diesel price = $69 + 0.15 \times 141 = 69 + 21.15 = 90.15$69+0.15×141=69+21.15=90.15 Rupees.

None of these dates result in the prices of both fuels being equal to 90 Rupees per litre.