# The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every $2 \mathrm{~m}$. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?

Given:

The number of flags $= 27$

Distance between each flag $= 2\ m$. The flags are stored at the position of the middlemost flag.

Ruchi kept her books, where the flags were stored and she could carry only one flag at a time.

To do:

We have to find the distance she covered in completing the job and returning back to collect her books and the maximum distance she travelled carrying a flag.

Solution:

The flags are stored at the position of the middle most flag i.e., 14th flag.
Let us consider the distances covered in placing the flags to the right of the middlemost flag.

The distance travelled by Ruchi to place 2nd flag (considering the middle most flag as first flag) $=(2+2)\ m=4\ m$

The distance travelled by Ruchi to place 3rd flag $=(4+4)\ m=8\ m$ Similarly,

The distance travelled by Ruchi to place 13th flag $=(13\times2+13\times2)\ m=(26+26)\ m=52\ m$ The distances travelled by Ruchi in placing 13 flags to the right of middlemost flag (in m) is,

$4,8,12,.....,52$

This is in A.P. where

First term $a=4$

Common difference $d=8-4=4$

We know that,

Sum of $n$ terms in an A.P. $S_n=\frac{n}{2}[2a+(n-1)d$

$S_{13}=\frac{13}{2}[2(4)+(13-1)4]$

$=\frac{13}{2}[8+48]$

$=13(28)$

$=364\ m$

Similarly, the distance travelled by Ruchi in placing 13 flags to the left of middle most flag $=364\ m$

Therefore, the total distance travelled by Ruchi in completing the job $=(364+364)\ m= 728\ m$

The maximum distance Ruchi travelled carrying a flag $=$ Distance travelled by Ruchi during placing the 14th flag to her right or the last flag to her left

The maximum distance Ruchi travelled carrying a flag $=2\times13\ m=26\ m$.

Therefore, the distance she covered in completing the job and returning back to collect her books is 728 m and the maximum distance she travelled carrying a flag is 26 m.

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