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Hallway Borders

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

Weekly Problem 23 - 2013

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Let the number of ivy, nightshade and triffid plants be $i$, $n$ and $t$ respectively.

Then:

$2i + 9n + 12t = 120$ and $i + n+ t = 20$, where $i> 0$; $n> 0$; $t> 0$.

Multiplying the second equation by $2$ and subtracting the new equation from the first:
$$7n + 10t=80$$
Thus $$7n = 10(8-t)$$
Therefore $n$ is a multiple of $10$ and since 1 \le n < 20 , $n=10$

Hence $8-t=7$ and therefore $t=1$.

This problem is taken from the UKMT Mathematical Challenges.

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