Question

The cost of white washing the four walls of a room is Rs. 100. Calculate the cost of white-washing a room twice in length and breadth and one fourth in height.

Answer 1

Hint: Apply the formula, $2h\left( {l + b} \right)$ to find the area of the four walls of the original room and equate it to the cost of white washing, which is Rs. 100. After this, translate the given phrase or condition into the mathematical equation and apply the formula, $2h\left( {l + b} \right)$ to the new room. At this stage you will develop a link or a common factor between the two equations you have formed, from where you can get your required value.

Complete step by step Answer :

Let us begin by denoting the dimensions of the original rectangle. Let the length of the rectangle be l, its breadth be b and its height be h.

It is given that the cost of white washing the four walls of a room is Rs. 100. Apply the formula to find the area of all the four walls and equate it to the given cost which is 100.
\[
   \Rightarrow {\text{Area of four walls}} = 2h\left( {l + b} \right) \\
   \Rightarrow {\text{Area of four walls: Cost}} \\
   \Rightarrow 2h\left( {l + b} \right):100{\text{ ......(1)}} \\
\]

Now, we have that the dimensions of the new room are twice in length and breadth and one fourth in height of the original room.
\[
   \Rightarrow l' = 2l \\
   \Rightarrow b' = 2b \\
   \Rightarrow h' = \dfrac{1}{4}h \\
 \]


Again apply the formula to find the area of all the four walls of the new room.
\[
   \Rightarrow {\text{Area of four walls}} = 2h'\left( {l' + b'} \right) \\
   \Rightarrow {\text{Area of four walls}} = 2\dfrac{h}{4}\left( {2l + 2b} \right) \\
   \Rightarrow {\text{Area of four walls: }}\dfrac{{2h}}{4} \times 2\left( {l + b} \right) \\
   \Rightarrow {\text{Area of four walls}} = \dfrac{1}{2} \times 2h\left( {l + b} \right) \\
\]

We can see from equation (1), $2h\left( {l + b} \right) = 100$ .
\[
   \Rightarrow \dfrac{1}{2} \times 2h\left( {l + b} \right) = \dfrac{1}{2} \times 100 \\
   = 50 \\
\]

Thus, the cost of white-washing a room twice in length and breadth and one fourth in height is Rs. 50.

Note: You can also find the area of walls by multiplying the two dimensions one by one and then add all of them to get the total area of all the four walls. But it is better to remember the formula used in the solution above in order to save your time and error percentage in these types of questions. Also, while forming the dimensions of the new room, carefully read the question and be sure of one-fourth and twice means and what needs to be doubled, either the new dimensions or the old ones. A major care to be taken while doing this, because interchanging even one condition will lead to a wrong answer. Also, try to not make any calculation errors.