Question

Find the least number of square tiles that can be paved on the floor whose length is 120m and breadth is 84m: -
(a) 40
(b) 50
(c) 60
(d) 70

Answer 1

Hint: First of all find the length of the side of each square by finding the H.C.F. of sides of the floor, i.e. 120m and 84m. To find H.C.F., first write both the numbers as the product of their primes. Multiply the common factors present in both of them to get the length of the side of each square. Now, assume that the number of such square tiles are ‘n’. Apply the formula for area relation: - \[n\times {{a}^{2}}=120\times 84\], where ‘n’ is the number of square tiles, ‘a’ is the length of side of one square tile and \[\left( 120\times 84 \right){{m}^{2}}\] is the area of the floor, to find the value of ‘n’.

Complete step-by-step answer:
Here, we have been given a floor of length 120m and breadth 84m and we have to determine the least number of square tiles that can be paved on the floor. To do this, first we need to find the length of the side of each square.
Now, the length of the side of each square will be the H.C.F. of 120m and 84m. So, let us find the H.C.F. of 120 and 84.
We know that, H.C.F. of two numbers is the product of their common prime factors. So, write 120 and 84 as a product of their primes, we get,
\[\begin{align}
  & \Rightarrow 120=2\times 2\times 2\times 3\times 5 \\
 & \Rightarrow 84=2\times 2\times 3\times 7 \\
\end{align}\]
Clearly, we can see that the common factors are 2, 2 and 3. Therefore, taking their product, we get,
H.C.F = \[2\times 2\times 3\]
\[\Rightarrow \] H.C.F. = 12m
\[\Rightarrow \] side length of each square = 12m.
Let us assume this side length as ‘a’.
\[\Rightarrow \] a = 12m – (1)
Now, we are assuming the least number of square tiles that can be paved is ‘n’. Therefore, the area of such ‘n’ square tiles must be equal to the area of the floor. So, equating their area, we get,
\[\begin{align}
  & \Rightarrow n\times {{a}^{2}}=120\times 84 \\
 & \Rightarrow n=\dfrac{120\times 84}{{{a}^{2}}} \\
\end{align}\]
Substituting a = 12 from equation (i), we get,
\[\Rightarrow n=\dfrac{120\times 84}{12\times 12}\]
Cancelling the common factors, we get,
\[\Rightarrow n=70\]

So, the correct answer is “Option (d)”.

Note: One must note that we should not find the L.C.M. of dimensions of the floor to determine the side length of each square. This is because if we will take the L.C.M. of 120 and 84 then we will get a number greater than both 120 and 84 which will not be a possible length of side of square.