Along a straight passage, 27 flags are to be fixed at intervals of 2 m. The flags are stored at the position of the middle-most flag. Ruchi places the flags, carrying only one at a time and returning to the store each time. How much total distance does she cover to complete the job and return? What is the maximum distance she travels while carrying a flag?
Total distance: 364 m; Maximum carrying distance: 26 m
Step 1: Total number of flags = 27.
Step 2: The middle-most flag is number 14 (because 13 flags on the left, 1 in the middle, 13 on the right). The store is exactly at this 14th flag’s position.
Step 3: The middle flag does not need any movement, because it is already at the store position.
Step 4: On each side (left and right), there are 13 flags. These flags are placed at distances of 2 m, 4 m, 6 m, …, up to 26 m away from the store.
Step 5: Ruchi carries one flag at a time. For a flag kept at distance \(d\) metres, she walks from the store to that place and then comes back. So the distance covered = \(2d\) metres (a round trip).
Step 6: Total distance for one side:
\[ 2(2 + 4 + 6 + \cdots + 26) \]
This is 2 times the sum of the first 13 even numbers.
Sum of first 13 even numbers = \(2 + 4 + 6 + \cdots + 26 = 2(1+2+3+\cdots+13)\).
\[ 1+2+3+\cdots+13 = \dfrac{13 \times 14}{2} = 91. \]
So, sum = \(2 \times 91 = 182\).
Therefore, distance for one side = \(2 \times 182 = 364\,\text{m}.\)
Step 7: The other side is exactly the same, so total distance both sides = \(364\,\text{m}\).
Step 8: The farthest flag is at 26 m. This means the maximum distance she carries a flag away from the store is \(26\,\text{m}\).
Final Answer: Total distance = 364 m, Maximum carrying distance = 26 m.