Question

A room is 6 meters 24 centimeters in length and 4 meters 32 centimeters in width. Find the least number of square tiles of equal size required to cover the entire floor of the room?

Answer 1

Hint: In this question, we convert all the dimensions into centimeters by multiplying the meters part by 100 and then add them to the remaining dimensions in centimeters. We will then calculate the HCF of these converted dimensions and solve for the required answer.

Complete step-by-step solution -
The length and breadth of the room are 6 meters 24 centimeters and 4 meters 32 centimeters respectively. Here, we multiply the meters part by 100 and convert the whole dimension into centimeters for ease of operation.
Therefore, \[6\text{ meters}=6\times 100cm=600cm.\]
And, \[\text{4 meters}=4\times 100cm=400cm.\]
Therefore, 6 meters 24 centimeters = 600 + 24 centimeters = 624 centimeters.
And, 4 meters 32 centimeters = 400 + 32 centimeters = 432 centimeters.
As we want to find the least number of square tiles of equal sizes, the length of each square tile should be as large as possible.
Also, the length of each square tile should be a factor of the length and breadth of the room.
Therefore, we calculate the HCF of these two numbers to get the largest numbers which are a factor of both.
Therefore, HCF (624, 432):
\[624=2\times 2\times 2\times 2\times 3\times 13={{2}^{4}}\times {{3}^{1}}\times {{13}^{1}}\]
\[432=2\times 2\times 2\times 2\times 3\times 3\times 3={{2}^{4}}\times {{3}^{3}}\]
\[HCF\left( 624,432 \right)={{2}^{4}}\times {{3}^{1}}=48\]
Thus, the number of square tiles required is,
\[\dfrac{\text{area of the room}}{\text{area per tile}}=\dfrac{624\times 432}{48\times 48}=13\times 9=117\]
Therefore, the total number of tiles required for the room is 117.

Note: Remember that here, we have to calculate the greatest number which is divisible by both the length and breadth. Therefore, we calculate the HCF here to find the dimensions of the tile and then divide the area of the room to the area of the tile to calculate the total number of tiles required. Here, we convert all the dimensions into any one unit for our convenience. To convert meters to centimeters, we multiply by 100 and to convert from centimeters to meters, we divide by 100.