Hello, foreverbrokenpromises

The size of the tiles in irrelevant.

We are concerned with the number of tiles (whole tiles).

Janet and Tom are making a mosaic to hang in their living room.

The mosaic is going to be 40 tiles high by 60 tiles long

and they want to have a border that leaves 84% of the original area in the centre.

How wide should the border be and will it work nicely with tiles of this size? Code:

: - - - 60 - - - - :
- *-------------------* -
: | | x
: | *---------- * | -
: | |///////////| | :
40 | |///////////| | 40-2x
: | |///////////| | :
: | *-----------* | -
: | | x
- *-------------------* -
: x : - 60-2x - : x :

The height of the mosaic is 40 tiles, its length is 60 tiles.

The border will be $\displaystyle x$ tiles wide.

The height of the shaded region is: .$\displaystyle 40-2x$ tiles.

The length of the shaded region is: .$\displaystyle 60-2x$ tiles.

The area of the shaded region is: .$\displaystyle (40-2x)(60-2x) \:=\:2400 - 200x + 4x^2$ tiles

. . which will be 84% of (40 × 60) = 2016 tiles.

There is our equation! . . . . $\displaystyle 2400 - 200x + 4x^2\:=\:2016 \quad\Rightarrow\quad x^2 - 50x + 96 \:=\:0$

. . which factors: .$\displaystyle (x - 2)(x - 48) \:=\:0$

. . and has roots: .$\displaystyle x \:=\:2,\;\;{\color{red}\rlap{///////}}x\:=\:48 $

Therefore, the border will be 2 tiles wide.

[Since the tiles are 10 cm × 10 cm, the border will be 20 cm wide.]