Every morning, Dhaniram lights an agarbatti and puts it out once it has burned…

Concept: When a quantity repeatedly shrinks to a fixed fraction r of its current value every period, the amount remaining after n periods is rn times the original. If the process only happens once per period (so partial periods aren't physically meaningful), the number of periods needed to reach a target percentage is the smallest whole number n for which rn is at or below that target — found by first solving rn = target using logarithms, then checking the surrounding whole numbers.

Application: Apply this to the agarbatti:

1. Every morning the agarbatti burns down to one-third of the previous day's length, so after n days the remaining height is (1/3)n of the original.

2. Set (1/3)n = 0.10 (the 10% target) and solve using logarithms: n = ln(0.10) / ln(1/3) ≈ 2.096.

3. Since n must be a whole number of days (the agarbatti is only checked once each morning), round up to the next whole day, n = 3, and verify it against the neighbouring days.

Cross-check: Verify against the neighbouring whole days:

1. After 2 days: (1/3)2 = 1/9 ≈ 11.11% — still above 10%, so 2 days are not enough.

2. After 3 days: (1/3)3 = 1/27 ≈ 3.70% — at or below 10%, confirming 3 is the first full day the target is reached.

Result: The agarbatti first drops to at most 10% of its original height after 3 days.