Question

The rate for a 1.2m wide carpet is Rs.40 per meter, find the cost of covering a hall 45m long and 32m wide with this carpet. Also, find the cost of carpeting the same hall if the carpet, 80cm wide is at Rs.25 per meter.
\[\begin{align}
  & A.Rs.48000,Rs.45000 \\
 & B.Rs.52000,Rs.75000 \\
 & C.Rs.66000,Rs.35000 \\
 & D.Rs.17000,Rs.85000 \\
\end{align}\]

Answer 1

Hint: The floor is in the form of rectangle, so we will first calculate area of base of floor by using area of rectangle obtained by multiplying length of floor to breadth of floor, because the area of floor is equal to area of carpet. Length of carpet can be calculated by using formula:
\[\text{Length of carpet}=\dfrac{\text{Area of carpet}}{\text{Width of carpet}}\]
Finally, we can compute cost by multiplying length with the rate of carpet.

Complete step-by-step answer:
We are given a carpet of 1.20m width.
Also the dimensions of hall are given as:
Length of floor = 45m let it be 'l' and breadth of floor = 32m let it be 'b'.
Then, l = 45 and b = 32 of the hall.

Because the floor is in the form of a rectangle, we will find the area of the floor which can be computed by using the formula of area of rectangle.
Area A of the rectangle whose length l and breadth b is given by \[A=l\times b\]
Substituting l = 45 and b = 32 to obtain the area of floor using above formula, we get:
\[\begin{align}
  & \text{Area of floor}=45\times 32 \\
 & \Rightarrow 1440{{m}^{2}} \\
\end{align}\]
Therefore, the area of floor is \[1440{{m}^{2}}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (i)}\]
Now, the width of the carpet is given to be 1.20m. Because the carpet covers the whole of the floor so the length of carpet can be calculated by dividing the area of floor by width of carpet.
This is so as carpet is again a rectangle and area of floor = area of carpet.
\[\text{Length of carpet}~\times \text{breadth of carpet}=\text{area of floor}\]
\[\text{Length of carpet}=\dfrac{\text{Area of floor}}{\text{Breadth of carpet}}\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (ii)}\]
Substituting values we get:
\[\text{Length of carpet}=\dfrac{1440}{1.20}=\dfrac{14400}{120}=1200m\]
So the length of carpet is 1200 m.
Given that, the rate of 1.20m wide carpet is 40/m.
For a carpet of length 1200m the cost can be obtained by multiplying rate = 40 to length = 1200m.
Cost of covering given hall by 1.20m wide carpet \[\Rightarrow 40\times 1200=48000\]
Therefore, cost of covering given hall by 1.20m wide carpet is \[\text{Rs}.\text{ 48}000\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }\left( \text{iii} \right)\]
Again, we are given that the new carpet is 80cm wide.
\[\begin{align}
  & 80cm=0.80m\text{ as 100cm=1m} \\
 & \text{1cm=}\dfrac{1}{100}m \\
 & \Rightarrow 80cm=\dfrac{80}{100}=0.8m \\
\end{align}\]
So the width of another carpet is 0.8m.
Using equation (ii) to calculate length of another carpet, we have:
\[\begin{align}
  & \text{Length of new carpet}=\dfrac{\text{Area of floor}}{\text{Width of new carpet}} \\
 & \text{Length of new carpet}=\dfrac{1440}{0.80}=1800 \\
\end{align}\]
So, the length of the new carpet is 1800m.
Given that, the rate of 0.80m wide carpet is 24/m
Cost of covering given hall by 0.80m wide carpet \[\Rightarrow \text{rate of new carpet}\times \text{length of new carpet}\]
Cost of covering the given hall by 0.80m wide carpet.
\[\text{Carpet}=25\times 1800=45000\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (iv)}\]
So, from equation (iii) and (iv) we have obtained that, cost of covering the hall with 1.20m carpet is Rs.48000 and that with 0.80m width carpet is Rs.45000.
So, the correct answer is “Option A”.

Note: The key point to note in this question is that, the width of carpet does not matter while calculating cost of carpeting, it is length that matters, and this is so because carpeting cost is measured by the base of floor covered and not the height of floor.
If the width was required to measure cost then for two different widths having the same rate per cost of carpeting would be the same.