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Mr. Lim wants to install new square-shaped tiles on his living room floor. If the floor measures 9.2 m by 6.4 m, find the longest possible side, in centimeters, for each tile.

Answer :

To solve this problem, we need to determine the longest possible side length, in cm, for each square tile that can completely cover the floor without cutting any tiles. To achieve this, we'll find the greatest common divisor (GCD) of the floor's dimensions, which measure 9.2 m by 6.4 m.

First, convert the measurements from meters to centimeters because the answer requires the tile size in centimeters:

  1. Convert 9.2 meters to centimeters:
    [tex]9.2 \text{ m} = 920 \text{ cm}[/tex]

  2. Convert 6.4 meters to centimeters:
    [tex]6.4 \text{ m} = 640 \text{ cm}[/tex]

Next, we find the GCD of 920 cm and 640 cm to determine the largest square tile size that fits perfectly into both dimensions. The Euclidean algorithm is a reliable method for this:

  1. Divide 920 by 640 and find the remainder:
    [tex]920 \div 640 = 1 \, \text{remainder} \, 280[/tex]

  2. Divide 640 by the remainder from the previous step, 280, to find the next remainder:
    [tex]640 \div 280 = 2 \, \text{remainder} \, 80[/tex]

  3. Divide 280 by 80:
    [tex]280 \div 80 = 3 \, \text{remainder} \, 40[/tex]

  4. Divide 80 by 40:
    [tex]80 \div 40 = 2 \, \text{remainder} \, 0[/tex]

Since the remainder is now 0, the divisor from the last non-zero remainder, which is 40, is the GCD.

Thus, the longest possible side length for each square tile is 40 cm. This size allows the tiles to fit perfectly without cutting any pieces, covering the entire floor area seamlessly.

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