What is the largest square tile that can pave a floor 15 m 17 cm long and 9 m 2 cm wide, and how many such tiles are needed?
SHORT ANSWER
Convert to centimetres: 1517 and 902. HCF(1517, 902) = 41, so the largest tile is 41 cm by 41 cm. Tiles needed = (1517/41) x (902/41) = 37 x 22 = 814.
Explain it simply
The tile side must fit exactly along the length and exactly along the width, otherwise you would have to cut tiles. So it must be a common divisor of both measurements. The largest common divisor is the HCF. Always convert both sides to the same unit first, because 15 m 17 cm and 9 m 2 cm cannot be compared directly.
Example
Question: Floor = 15 m 17 cm by 9 m 2 cm. Largest square tile and number of tiles?
- In cm: 1517 and 902.
- HCF by division:
- 1517 = 902 x 1 + 615
- 902 = 615 x 1 + 287
- 615 = 287 x 2 + 41
- 287 = 41 x 7 + 0 So HCF = 41 cm.
- Tiles along length: 1517 / 41 = 37. Along width: 902 / 41 = 22.
- Total tiles = 37 x 22 = 814.
js
const g = (a, b) => (b === 0 ? a : g(b, a % b));
const side = g(1517, 902);
console.log(side); // Output: 41
console.log((1517 / side) * (902 / side)); // Output: 814What to say in the interview
"The tile side has to divide both dimensions exactly, and I want the largest, so it is the HCF of 1517 cm and 902 cm, which is 41 cm by Euclid's method. That needs 37 by 22, or 814 tiles."
Common mistakes
- Forgetting to convert metres to centimetres, which gives a meaningless HCF.
- Using area divided by tile area without checking that the side divides both lengths; here it does, but it is the HCF step that guarantees no cut tiles.