Question

A floor which measures 15m x 8m is to be laid with tiles measuring 50cm x 25cm. Find the number of tiles required.
Further, if a carpet is laid on the floor so that a space of 1m exists between its edges and the edges on the floor, what fraction of the floor is uncovered.
A. $960;\dfrac{7}{20}$
B. $860;\dfrac{20}{7}$
C. $760;\dfrac{7}{13}$
D. $660;\dfrac{7}{15}$

Answer 1

Hint: To answer this question, we first calculate the area of the floor in ${{m}^{2}}.$ We then calculate the area of 1 tile again in ${{m}^{2}}.$ To obtain the number of tiles, we divide the total area of the floor by the area of one tile. Now given that there exists a gap between the edges of the carpet and floor, we calculate the area of the edges and divide it by the total area of the floor to obtain the fraction of floor left uncovered.

Complete step by step solution:
Given that the floor measures 15m x 8m. These represent the length and breadth of the floor. To find the area of the floor, we just need to simply multiply the length and breadth.
$\Rightarrow \text{Area of the floor=15}\times \text{8=120}{{m}^{2}}$
Now we calculate the area of one tile. For this we convert the length and breadth of the tile into metres. We know that 1 metre is equal to 100 centimetres. Therefore, 1 centimetre will be $\dfrac{1}{100}$ metre. Therefore, 50cm in metres is 0.5m. Similarly, 25cm in metres is 0.25m. Multiplying the two will give us the area of one tile in ${{m}^{2}}.$
$\Rightarrow \text{Area of one tile=0}\text{.5}\times 0.25\text{=0}\text{.125}{{m}^{2}}$
To find the number of tiles, we just divide the area of the floor by the area of one tile.
$\Rightarrow \text{Number of tiles = }\dfrac{\text{Area of the floor}}{\text{Area of one tile}}$
Substituting the values,
$\Rightarrow \text{Number of tiles = }\dfrac{\text{120}}{\text{0}\text{.125}}$
Dividing the two numbers,
$\Rightarrow \text{Number of tiles = 960}$
Hence, we require 960 tiles to cover the floor measuring 15m x 8m where each tile side is 50cm x 25cm.
Now it says that the carpet is laid such that there exits a 1m gap between it and the wall. This means that there is a 1m gap on top and 1m gap below. This reduces the length by 2m. Similarly, there is a 1m gap on the left and 1m gap on the right. This reduces the breadth by 2m.
Therefore, the length of the carpet is 2m less than the length of the room.
$\Rightarrow \text{Length of carpet=}\left( 15-2 \right)m=13m$
Similarly, the breadth of the carpet is 2m less than the length of the room.
$\Rightarrow \text{Breadth of carpet=}\left( 8-2 \right)m=6m$
Now we can calculate the area of this carpet by taking the product of the length and breadth.
$\Rightarrow \text{Area of carpet=13}\times \text{6}m=78m$
We can now find the fraction of the floor left uncovered by subtracting this with the total area of the floor and dividing it by the total area of the floor.
$\Rightarrow \text{Area not covered by carpet=}120-78=42m$
Now we divide this by the total area of the floor.
$\Rightarrow \text{Portion of floor left uncovered=}\dfrac{\text{Area not covered by carpet}}{\text{Area of the floor}}$
Substituting the values,
$\Rightarrow \text{Portion of floor left uncovered=}\dfrac{42}{120}$
Dividing both the numerator and denominator by 6,
$\Rightarrow \text{Portion of floor left uncovered=}\dfrac{7}{20}$
Hence, the fraction of the floor left uncovered is $\dfrac{7}{20}.$ Hence, the correct option is option A.

Note: Students must know the conversion from one unit to another in order to simplify and solve this question easily. Care must be taken when subtracting the length to find the carpet length. It is to be noted that we subtract it as 2m since it is 1 m from the top edge and 1 m from the bottom edge and same for the breadth.