Question

The Price of Darjeeling Tea (in rupees per kilogram) is 100 + 0.1n, on the nth day of a non-leap year (n = 1, 2, 3, ... 100) and then remains constant. On the other hand the price of Ooty tea (in rupees per kilogram) is 85 + 0.15n, on the nth day (n = 1, 2, ..., 365). On which date of that year will the prices of these two varieties of the tea be equal?

• A. 27th October
• B. 16th June
• C. 15th June
• D. 28th October

Hint:

Darjeeling tea's price stops rising after day 100 at Rs 110; find when Ooty tea's rising price catches up to that fixed value.

Answer 1

The Correct Option is B

Instead of solving for the day number first, work directly from the fact that Darjeeling tea's price is capped. On day 100 of the year, Darjeeling tea reaches $100 + 0.1(100) = 110$ rupees per kg and then never changes again. Day 100 itself falls on $31 + 28 + 31 + 10 = 100$, that is, 10th April (January 31, February 28 and March 31 give 90 days, then 10 more days into April). So from 10th April onward, Darjeeling tea is a flat Rs. 110 per kg for the rest of the year.

Ooty tea's price keeps climbing linearly all year at $85 + 0.15n$. The two teas can only match once Ooty's rising price reaches this flat ceiling of Rs. 110, since after 10th April Darjeeling can no longer meet Ooty halfway. Set:

\[85 + 0.15n = 110 \quad\Rightarrow\quad 0.15n = 25 \quad\Rightarrow\quad n = \frac{25}{0.15} = \frac{500}{3} \approx 166.67\]

Because $n$ must be a whole calendar day, test the two integers bracketing this value. On $n=166$: $85 + 0.15(166) = 109.90$, still short of 110. On $n=167$: $85+0.15(167) = 110.05$, now at or past 110. So the crossing day is $n = 167$.

Convert day 167 to a date: the first five months of a non-leap year (January through May) span $31+28+31+30+31 = 151$ days, leaving $167 - 151 = 16$ days into June. That date is 16th June.

The alternative root $n = 300$, obtained by naively equating $100+0.1n$ and $85+0.15n$ without respecting the day-100 cutoff, lands in late October and produces the two distractor options; it is not valid because Darjeeling tea no longer follows that rising formula by then. The correct, valid crossing point is \(\boxed{\text{16th June}}\).