The Greatest of All

Transcript

The Greatest of AllWe start with a question Sameeksha is building her new house. The main room of the house is 12 ft by 16 ft. She feels that the room would look nice if the floor is covered with square tiles of the same size. She also wants to use as few tiles as possible, and for the length of the tile to be a whole number of feet. What size tiles should she buy? Let’s try to draw an image here For tiles to Fit the length (16 ft) – Side of tile should be a factor of 16 Fit the breadth (12 ft) – Side of tile should be a factor of 12 Thus, side of square tile Should be a factor of both 16 and 12 Should be largest common factor – since we want fewest tiles possible Let’s find the factors of 16 & 12 separately Factors of 16 1 × 16 = 16 2 × 8 = 16 4 × 4 = 16 8 × 2 = 16 We stop here as 8 and 2 have occurred earlier Factors of 16 are 1, 2, 4, 8, 16 Factors of 12 1 × 12 = 12 2 × 6 = 12 3 × 4 = 12 4 × 3 = 12 We stop here as 4 and 3 have occurred earlier So factors of 12 are 1, 2, 3, 4, 6, 12 Thus, common factors are 1, 2, and 4 Since 4 is the largest common factor Thus, tile will be of side 4 ft Here, 4 is the Highest Common Factor of 16 & 12 The Highest Common Factor (HCF) of two or more numbers, is the highest (or greatest) of their common factors. It is also known as the Greatest Common Divisor (GCD). There are two questions in the bubble at the top of the second page. Let's solve them! Question A: How many tiles of this size should she purchase? We know the room is 12 ft by 16 ft , and the tiles are 4 ft by 4 ft . Along the width (𝟏𝟐" " 𝐟𝐭) : She fits 12÷4=3 tiles. Along the length (𝟏𝟔" " 𝐟𝐭) : She fits 16÷4=4 tiles. Total Tiles: We multiply the rows and columns. 𝟑 × 𝟒=𝟏𝟐" tiles " We can also do this by Number of tiles = (𝐴𝑟𝑒𝑎 𝑜𝑓 ℎ𝑜𝑢𝑠𝑒)/(𝐴𝑟𝑒𝑎 𝑜𝑓 1 𝑡𝑖𝑙𝑒) = (𝟏𝟔 × 𝟏𝟐)/(𝟒 × 𝟒) = (3 × 4)/(1 × 1) = 12 Question B: What if the tile length could be a fraction? Would the answer change? If we want the tiles to fit perfectly into 12 ft and 16 ft , the size of the tile still has to divide both numbers evenly. The largest number that divides 12 and 16 is 4 . Even if we allowed fractions (like 4.5 ft ), 4.5 doesn't fit into 12 or 16 evenly. Answer: No, the answer for the largest square tile would not change. 4 ft is the mathematical limit for these room dimensions!