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

Step 1: Write down when the Darjeeling tea price formula applies.
The Darjeeling tea price is 100 + 0.1n rupees per kg only for n = 1, 2, ..., 100; after day 100 it stops changing and stays fixed at whatever it was on day 100. Its value on day 100 is 100 + 0.1(100) = 110 rupees per kg, so for every day after the 100th day of the year, Darjeeling tea costs a constant Rs. 110 per kg.
Step 2: Check whether the two prices can be equal while Darjeeling is still rising.
While n <= 100, setting the two formulas equal gives 100 + 0.1n = 85 + 0.15n, so 15 = 0.05n, which gives n = 300. This value of n is far outside the range n <= 100 where the formula 100+0.1n is even valid, so it is not a real solution; by day 100 Darjeeling tea is already fixed at Rs. 110 and no longer follows that formula.
Step 3: Solve for equality using the constant Darjeeling price.
For n > 100, Darjeeling tea is fixed at Rs. 110, so the two prices are equal when 85 + 0.15n = 110, which gives 0.15n = 25, so n = 25/0.15 = 166.67. Since n must be a whole day, check the nearby integer days directly: on day 166, Ooty tea costs 85 + 0.15(166) = 109.9 rupees, still below Rs. 110; on day 167, Ooty tea costs 85 + 0.15(167) = 110.05 rupees, which has caught up to Rs. 110. So the two prices become equal at day 167 of the year.
Step 4: Convert day 167 into a calendar date.
Counting days in a non-leap year: January has 31 days, February 28, March 31, April 30 and May 31, together totalling 31+28+31+30+31 = 151 days through the end of May. Day 167 is therefore 167 - 151 = 16 days into June, that is, 16th June.
Step 5: Match with the options.
This is option (2), 16th June. Option (3), 15th June, is day 166, one day too early, since Ooty is still slightly below Rs. 110 on that day. Options (1) and (4), both in late October, come from the invalid n = 300 solution obtained by ignoring the day-100 cutoff on the Darjeeling price.