Question

Each tile has a perimeter of 12 units. If Paul connect us 2 tiles, he makes a rectangle with a perimeter of 18 units. If he connects 4 tiles to form a square, which is the perimeter of Paul’s new shape???

Answer 1

The perimeter would be 24 units.

Let L be the length and w be the width of each tile.

The perimeter is given by 2L+2w; we know this is 12, so we have the equaiton
2L+2w=12.

Placing two of thee tiles together changes the perimeter to 18.  Assuming we place one tile above the other, so that the length of two of them is in the middle, we have 4 w's and 2 L's on the outside of the tile; this gives the equation
4w+2L=18

(Side note: this will work the same way, with the same answers, if you place the tiles beside each other instead of one atop the other.)

This makes our system of equations
2w+2L=12
4w+2L=18

Since the coefficients of L are the same, we can eliminate the L's by subtracting:
(2w+2L=12)-(4w+2L=18)
2w-4w+2L-2L=12-18
-2w=-6

Divide both sides by -2:
-2w/-2=-6/-2
w=3

Substituting this into the first equation,
2L+2(3)=12
2L+6=12

Subtract both sides by 6:
2L+6-6=12-6
2L=6

Divide both sides by 2:
2L/2=6/2
L=3

This means that each tile is a 3 by 3 square.

If we arrange 4 of these to form a square, this would be 2 rows and 2 columns.
This means we would have 3+3=6 for each side of the square, and the perimeter would be 4(6)=24.