1. ## Mosaics

Janet and Tom are making a mosaic out of 10cm x 10cm tiles 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 for a more detailed design in the centre. How wide should the border be and will it work nicely with tiles of this size?

I have the answer, i just need to know how to do it

thankkks : )

but whyy??

2. Originally Posted by foreverbrokenpromises
Janet and Tom are making a mosaic out of 10cm x 10cm tiles 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 for a more detailed design in the centre. How wide should the border be and will it work nicely with tiles of this size?

I have the answer, i just need to know how to do it

thankkks : )

but whyy??
Here's my method, there's probably a neater way than this but oh well

You mosaic is 40 tiles by 60, this means a total of 2400 tiles.

If you wish to have 84% left in the middle, this means that the border will 16% of the total mosaic, or 384 tiles.

Now consider the perimeter of the border. You have two sides of 60, and two sides of 38 (sides of 40, but you will have already counted the two corners with the 60 if that makes sense), this gives you 196 tiles in your first border.

Now your second border, there are two sides of 58, and two sides of 36 (for the same reason above it's not sides of 38), giving you 188 tiles in your second border.

Adding these together you get exactly 384, leaving the 84% free in the middle.

Your border is 2 tiles wide, hence the 20cm.

Hope this helps

3. thankks : )
i understand it more now.
but how would you solve this problem using a quadratic equation?

4. 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.]