Question

Find the least number of square tiles required to pave the ceiling of a room 15m 17cm long and 9m 2cm broad.

Answer 1

Hint: We convert the length and breadth in centimeters using conversion. Calculate the area of the rectangular ceiling. Using the measurements of length and breadth, calculate the highest common factor of length and breadth. Use the HCF as the length of a square and calculate the area of one square. Divide the area of the rectangular ceiling by the area of a square to find the number of square tiles.
* A rectangle is a quadrilateral having four sides, where opposite sides are equal and parallel. If length of rectangle is ‘l’ and breadth of rectangle is ‘b’ then area of rectangle is \[l \times b\]
* A square is a quadrilateral having four equal sides. If length of a side of square is ‘a’ then area of square is \[{a^2}\]
* HCF of two or more numbers is the highest common factor which is given by the multiplication of all common prime numbers that are common in their prime factorization.
* 1 meter has 100 centimeters

Complete step by step answer:
We have a rectangular ceiling of a room having length 15m 17cm and breadth 9m 2cm
Length of ceiling \[ = 15\]m\[ + 17\]cm
Since we know 1 meter has 100 cm
\[ \Rightarrow \]Length of ceiling \[ = 15 \times 100\]cm\[ + 17\]cm
\[ \Rightarrow \]Length of ceiling \[ = (1500 + 17)\]cm
\[ \Rightarrow \]Length of ceiling \[ = 1517\]cm
\[\Rightarrow l = 1517\]
Similarly, Breadth of ceiling \[ = 9\]m\[ + 2\]cm
Since we know 1 meter has 100 cm
\[ \Rightarrow \]Breadth of ceiling \[ = 9 \times 100\]cm\[ + 2\]cm
\[ \Rightarrow \]Breadth of ceiling \[ = (900 + 2)\]cm
\[ \Rightarrow \]Breadth of ceiling \[ = 902\]cm
\[\Rightarrow b = 902\]
Area of rectangular ceiling of the room \[ = l \times b\]
Substitute value of \[l = 1517,b = 902\]
\[ \Rightarrow \]Area of rectangular ceiling of the room \[ = l \times b\]
\[ \Rightarrow \]Area of rectangular ceiling of the room \[ = 1517 \times 902\]\[c{m^2}\]
\[ \Rightarrow \]Area of rectangular ceiling of the room \[ = 1368334\]\[c{m^2}\] …………….… (1)
Now let us assume the length of the square tile be ‘$a$’.
We find the length of the side of a square tile by taking HCF of the measurements of the ceiling. As the number obtained from HCF will give us the highest common factor of both length and breadth, so if we take a square of that length then we will be able to cover the ceiling entirely with several square tiles.
To calculate HCF of 1517 and 902, write the prime factorization
Prime factorization of 1517 and 902 is:
\[\Rightarrow 1517 = 37 \times 41\]
\[\Rightarrow 902 = 2 \times 11 \times 41\]
We look at the highest common factor from the two prime factorizations.
Highest common factor is 41
\[ \Rightarrow \]HCF \[ = 41\]
We take length of side of square, \[a = 41\]cm
Then area of square is \[{a^2}\]
\[ \Rightarrow \]Area of one square tile\[ = {41^2}\]
\[ \Rightarrow \]Area of one square tile\[ = 1681\] \[c{m^2}\] …………..… (2)
Now we find the number of square tiles that pave the ceiling of a rectangular room by dividing the area of the rectangular ceiling by the area of a square tile. Divide equation (1) by (2)
\[ \Rightarrow \]Number of square tiles \[ = \dfrac{{1368334}}{{1681}}\]
Cancel same terms from numerator and denominator
\[ \Rightarrow \]Number of square tiles \[ = 814\]

\[\therefore \] 814 square tiles are required to pave the ceiling of the room.

Note:
Students might get confused as to why we take HCF of two numbers here and not LCM, as HCF will give us a number that divides both the length and breadth, so length of a square tile will be less than any of measurements of rectangular ceiling. Whereas, LCM of two numbers will give us a length of side of square greater than both length and breadth of ceiling, which is wrong.