You can write a quadratic equation, or you can estimate, or you can make an educated guess.

108 = 1*108 = 2*54 = 3*36 = 4*27 = 6*18 = 9*12

The dimensions 9 by 12 are each 3 ft less than the dimensions 12 by 15. The border will be half of 3 ft, that is 1.5 ft.

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The perimeter of the room is 2(12+15)=54 ft. The difference between the area of the room and the area of the rug is 180 ft^2 - 108 ft^2 = 72 ft^2, so the width of the border can be estimated as slightly more than 72ft^2/(54ft), about 1.5 ft.

108 = 1*108 = 2*54 = 3*36 = 4*27 = 6*18 = 9*12

The dimensions 9 by 12 are each 3 ft less than the dimensions 12 by 15. The border will be half of 3 ft, that is 1.5 ft.

_____

The perimeter of the room is 2(12+15)=54 ft. The difference between the area of the room and the area of the rug is 180 ft^2 - 108 ft^2 = 72 ft^2, so the width of the border can be estimated as slightly more than 72ft^2/(54ft), about 1.5 ft.