Question

How many bricks of size \[22{\rm{cm}} \times 10{\rm{cm}} \times {\rm{7cm}}\]are required to construct a wall \[11{\rm{m}} \times 3.5{\rm{m}}\] m long and high, and 40 cm thick, if cement and sand used on the construction occupy \[{\left( {\dfrac{1}{{10}}} \right)^{{\rm{th}}}}\] part of the wall?

Answer 1

Hint: Here, we need to find the number of bricks required to construct the wall. We will assume the number of bricks required to be \[x\]. First, we will find out the volume of the wall and 1 brick. Then, we will find out the volume of the wall taken up by the cement and sand. Using that, we will find the volume of the wall taken up by the bricks. Finally, using the volume of 1 brick and the volume of the wall taken up by bricks, we will form an equation in terms of \[x\]. We will solve the equation to find the value of \[x\], and hence, the number of bricks required to construct the wall.

Formula used: We will use the formula of the volume of a cuboid is given by \[l \times b \times h\], where \[l\] is the length, \[b\] is the breadth, and \[h\] is the height.

Complete step-by-step answer:
Let the number of bricks required be \[x\].
Let us convert all the given dimensions to centimetres.
We know that 1 metre is equal to 100 centimetres.
Therefore, we get
Length of the wall \[ = 11{\rm{ m}} = 11 \times 100{\rm{ cm}} = 1100{\rm{ cm}}\]
Height of the wall \[ = 3.5{\rm{ m}} = 3.5 \times 100{\rm{ cm}} = 350{\rm{ cm}}\]
Next, we need to find out the volume of the wall and the volume of each brick.
We know that the brick and the wall are in the shape of a cuboid, so we will use the formula of the volume of a cuboid.
Substituting \[l = 1100{\rm{ cm}}\], \[b = 40{\rm{ cm}}\], and \[h = 350{\rm{ cm}}\] in the formula \[l \times b \times h\], we get
Volume of the wall \[ = 1100 \times 40 \times 350{\rm{ c}}{{\rm{m}}^3}\]
Substituting \[l = 22{\rm{ cm}}\], \[b = 10{\rm{ cm}}\], and \[h = 7{\rm{ cm}}\] in the formula \[l \times b \times h\], we get
Volume of 1 brick \[ = 22 \times 10 \times 7{\rm{ c}}{{\rm{m}}^3}\]
Now, we will find the volume of the wall taken up by the cement and sand.
It is given that cement and sand used on the construction occupy \[{\left( {\dfrac{1}{{10}}} \right)^{{\rm{th}}}}\] part of the wall.
Therefore, we get
Volume occupied by cement and sand \[ = \dfrac{1}{{10}}\]of Volume of the wall \[ = \dfrac{1}{{10}}\left( {1100 \times 40 \times 350} \right){\rm{ c}}{{\rm{m}}^3}\]
The volume occupied by the bricks is the difference in the volume of the wall and the volume of the wall occupied by cement and sand.
Therefore, we get
Volume of wall occupied by bricks \[ = {\rm{Volume\, of\, wall}} - {\rm{Volume\, occupied\, by \,cement\, and\, sand}}\]
Thus, we have
Volume of wall occupied by bricks \[ = 1100 \times 40 \times 350{\rm{ c}}{{\rm{m}}^3} - \dfrac{1}{{10}}\left( {1100 \times 40 \times 350} \right){\rm{ c}}{{\rm{m}}^3}\]
Simplifying the expression, we get
Volume of wall occupied by bricks \[ = \dfrac{{10}}{{10}}\left( {1100 \times 40 \times 350} \right){\rm{ c}}{{\rm{m}}^3} - \dfrac{1}{{10}}\left( {1100 \times 40 \times 350} \right){\rm{ c}}{{\rm{m}}^3}\]
Therefore, we get
Volume of wall occupied by bricks \[ = \dfrac{9}{{10}}\left( {1100 \times 40 \times 350} \right){\rm{ c}}{{\rm{m}}^3}\]
Now, we will form an equation in terms of \[x\].
The total volume occupied by the bricks on the wall is equal to the product of the volume of 1 brick multiplied by the number of bricks used to construct the wall.
Therefore, we get
Total volume occupied by the \[x\] bricks on the wall \[ = 22 \times 10 \times 7 \times x{\rm{ c}}{{\rm{m}}^3}\]
Substituting the total volume occupied by the bricks on the wall as \[\dfrac{9}{{10}}\left( {1100 \times 40 \times 350} \right){\rm{ c}}{{\rm{m}}^3}\], we get
\[ \Rightarrow \dfrac{9}{{10}}\left( {1100 \times 40 \times 350} \right) = 22 \times 10 \times 7 \times x\]
We will simplify this equation to get the value of \[x\].
Rewriting the equation, we get
\[ \Rightarrow \dfrac{{9 \times 1100 \times 40 \times 350}}{{10 \times 22 \times 10 \times 7}} = x\]
Simplifying the expression on the left hand side, we get
\[ \Rightarrow 9 \times 20 \times 50 = x\]
Multiplying the terms, we get
\[ \Rightarrow x = 9000\]
\[\therefore\] The total number of bricks required to construct the wall is 9000.

Note: Here in the question, we have been provided with measurements in two units i.e. centimetre and metre. We need to convert either meter into centimetre or vice versa. In order to perform a mathematical calculation it is important that all the measurements should have the same unit or else we will get the wrong answer.