Question

Find the least number of square tiles required to pave the ceiling of a room 15m, 17cm long and 9m, 2cm broad.
(a) $656$
(b) $814$
(c) $902$
(d) $738$

Answer 1

Hint: In this question, we will use the concept of highest common factor of two numbers and then apply the unitary method to solve the equation.

Complete step-by-step answer:
In the given question we are to find the number of such squares which will be required to pave the ceiling of a room whose length is 15m, 17cm and breadth is 9m, 2cm. So, this ceiling has the shape of a rectangle. Therefore, area of ceiling is given as,
Area of ceiling=length $\times $ breadth
                          $=15m,17cm\times 9m,2cm$
We know, 1m=100cm.
Using this, we get,
$\begin{align}
  & 15m,17cm=1500cm+17cm \\
 & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,1517cm \\
\end{align}$
And $9m,2cm=900cm+2cm$
                          $=902\,cm$
Therefore,
$\begin{align}
  & \text{Area of ceiling}=1517cm\times 902cm \\
 & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=1368334c{{m}^{2}} \\
\end{align}$
We are to find such square tiles which can completely cover this ceiling, without leaving any extra space and without tiles needing to be cut. Also, the total area of all the tiles which will cover the ceiling will be equal to the area of ceiling.
Now, squares have all their sides equal. So, when tiles are paved on ceiling, one of the tiles will be lying equal with length and one will be lying equal with breadth. So, the side of this tile will be a factor of both length and breadth. Also, we need tiles such that numbers of tiles required will be minimum. So, the area of tiles needs to be maximum.
For the area to be maximum, the side of the square will be maximum. Also, this side is a common factor of length and breadth. So, the side of the required square will be the highest common factor of length and breadth.
Therefore, the side of tile $=HCF\left( 1517,902 \right)cm$.
$1517=\,41\times 37$
$902=\,41\times 11\times 2$
So, the HCF of 1517 and 902 is 41.
Hence, side of tiles = 41 cm.
Therefore, area of one square tile will be,
\[\begin{align}
  & \text{Area of title}=\left( 41\times 41 \right)c{{m}^{2}} \\
 & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=1681c{{m}^{2}} \\
\end{align}\]
Now, $1681c{{m}^{2}}$ area is to be covered by = 1 tile.
So, $1368334c{{m}^{2}}$ area will be covered by.
$=\dfrac{1}{1681}\times 1368334$ tiles
$=814$ tiles
Hence, the correct answer is option (b).

Note: In this question, do not confuse HCF with LCM as the word ‘least’ is used in question. Number will be least only when the area will be maximum.