This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Convert the dimensions of the room from meters to centimeters to work with integers. The length of the room is 5.4 m. The width of the room is 4.2 m. Since 1 m = 100 cm: Length (L) = 5.4 × 100 cm = 540 cm Width (W) = 4.2 × 100 cm = 420 cm Step 2: Determine the side length of the largest possible square tile. To cover the rectangular floor with square tiles without cutting any tiles, the side length of the square tile must be a common divisor of both the length and the width of the room. To find the minimum number of tiles, we need to use the largest possible square tiles. This means the side length of the tile must be the Greatest Common Divisor (GCD) of the room's length and width. We need to find GCD(540, 420). We can use prime factorization: 540 = 2 × 270 = 2^2 × 135 = 2^2 × 3 × 45 = 2^2 × 3^2 × 15 = 2^2 × 3^3 × 5 420 = 2 × 210 = 2^2 × 105 = 2^2 × 3 × 35 = 2^2 × 3 × 5 × 7 The common prime factors with their lowest powers are 2^2, 3^1, and 5^1. GCD(540, 420) = 2^2 × 3^1 × 5^1 = 4 × 3 × 5 = 60 So, the side length of the largest square tile is 60 cm. Step 3: Calculate the number of tiles needed along the length and width. Number of tiles along the length = Length of roomSide length of tile = 540 cm60 cm = 9 tiles Number of tiles along the width = Width of roomSide length of tile = 420 cm60 cm = 7 tiles Step 4: Calculate the total minimum number of square tiles. The total number of tiles is the product of the number of tiles along the length and the number of tiles along the width. Total number of tiles = 9 × 7 = 63 The minimum number of square tiles that can be used to cover the floor is 63. Send me the next one 📸