200 bottles are stacked in such a way that there are 20 bottles in the bottom…
2025
200 bottles are stacked in such a way that there are 20 bottles in the bottom , 19 in the next row , 18 in the row next and so on . In how many rows will 200 bottles be placed ?
- A.
20 rows
- B.
16 rows
- C.
12 rows
- D.
25 rows
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Correct answer: B
Answer: 16 rows
The number of bottles per row forms an arithmetic progression with first term 20 and common difference -1.
Use the sum formula S_n = n/2 [2a + (n - 1)d]. Setting S_n = 200 gives n/2 [2*20 + (n - 1)(-1)] = 200, which simplifies to n(41 - n) = 400.
Solve n(41 - n) = 400 to get n^2 - 41n + 400 = 0, so (n - 16)(n - 25) = 0 and n = 16 or n = 25.
Reject n = 25 because rows cannot exceed 20 (the bottom row has 20 bottles and each row above has one fewer).
Therefore n = 16. Check: the last (16th) row has 20 - 15 = 5 bottles, and the sum is (16/2) * (20 + 5) = 8 * 25 = 200.