Question

The price of Darjeeling tea (in rupees per kilogram) is \( 100 + 0.10n \) on the \( n^{\text{th}} \) day of 2007 (\( n = 1, 2, \ldots, 100 \)), and then remains constant. The price of Ooty tea (in rupees per kilogram) is \( 89 + 0.15n \) on the \( n^{\text{th}} \) day of 2007 (\( n = 1, 2, \ldots, 365 \)). On which date in 2007 will the prices of these two varieties of tea be equal?

• A. May 21
• B. April 11
• C. May 20
• D. April 10
• E. June 30

Hint:

When mapping \( n^{\text{th}} \) day to a calendar date, sum month days carefully considering leap years if applicable.

Answer 1

The Correct Option is B

Darjeeling tea price: For \( n \leq 100 \), \( P_D = 100 + 0.10n \), For \( n>100 \), \( P_D = 100 + 0.10 \times 100 = 110 \).
Ooty tea price: \( P_O = 89 + 0.15n \) for all \( n \).
Set \( P_D = P_O \). For \( n \leq 100 \): \( 100 + 0.10n = 89 + 0.15n \) → \( 11 = 0.05n \) → \( n = 220 \) (but \(>100 \)), so invalid in this range.
For \( n>100 \): \( 110 = 89 + 0.15n \) → \( 21 = 0.15n \) → \( n = 140 \).
Day 140 of 2007 = Jan(31) + Feb(28) + Mar(31) + Apr(30) = 120 days till April end; day 140 is 20 days into May? Wait — recalc: 120 days till April 30 means day 140 is May 20 — conflict with options.
Check counting: Jan(31) + Feb(28) + Mar(31) + Apr(11) = 101 days → not matching.
Actually: Jan(31) + Feb(28) + Mar(31) = 90 days till end of March. April has 30 days. Day 140 means 50 days into April → April has only 30, so 20 days into May (May 20). But May 20 is not equal price? Actually option (3) matches this. Final: n=140 → May 20.