Question

1. (page 11) the dimensions of a rectangle room are 18 feet by 12 feet. (18 ft by 12 ft) what is the side length in feet of the largest square tile that can tile the room completely? 2. (page 12) the dimensions of a rectangular tile measures 15 inches by 6 inches. what is the side length in inches of the smallest square that can be created with rectangular tiles of this size?

Explanation:

Step1: Recall GCD concept for largest - square tile

To find the side - length of the largest square tile that can tile a rectangle completely, we need to find the greatest common divisor (GCD) of the dimensions of the rectangle.
For a room of dimensions 18 feet by 12 feet, first convert to inches (1 foot = 12 inches), so 18 feet = 18×12 = 216 inches and 12 feet = 12×12 = 144 inches. Then find GCD(216, 144).
Using the Euclidean algorithm:
216=144×1 + 72
144=72×2+0
So GCD(216, 144) = 72 inches, and in feet, $\frac{72}{12}=6$ feet.

Step2: Recall GCD concept for smallest - square with rectangular tiles

To find the side - length of the smallest square that can be created with rectangular tiles of dimensions 15 inches by 6 inches, we find the least common multiple (LCM) of 15 and 6.
Prime - factorize: 15 = 3×5 and 6 = 2×3.
LCM(15, 6)=2×3×5 = 30 inches.

Answer:

1. 6 feet
2. 30 inches