Question

l
33. A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. It is to be paved with
square tiles of the same size. Find the least possible number of such tiles.​

Answer 1

Question :

A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles.

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Answer :

The least possible number of tiles required to pave the rectangular courtyard is $\large{\mathfrak{4290}}$.

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$\bf\:‎ Step~wise~Calculation~:$

Given :

• Length of rectangular courtyard = 18m 72cm

• Breadth of rectangular courtyard = 13m 20cm

• Tiles used for paving the courtyard is squared shape

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To find :

• The least possible number of tiles required to pave the rectangular courtyard.

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Concept

In order to solve this question we will proceed by calculating the area of the rectangular courtyard. Then by using the concept of HCF in the given length and breadth of the courtyard we will get the size of the side of a square tile. Then after calculating the area of the square tile we will acquire the least number of tiles required to pave the rectangular courtyard.

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Formulas to be used :

➷ Area of rectangular courtyard = $\small{\underline{\boxed{\mathrm{l \times b}}}}$

where : l stands for length and b for breadth.

➷ Area of square tiles = $\small{\underline{\boxed{\mathrm{(s)^{2}}}}}$

where : s stands for side.

➷ Number of tiles = $\small{\underline{\boxed{\mathrm{\frac{Area~of~courtyard}{Area~of~tiles}}}}}$

Hope am clear let's solve :D~

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Solution :

Calculating the area of the rectangular courtyard -

Length of the courtyard = 18m 72cm = 1872cm

Breadth of the courtyard = 13m 20cm = 1320cm

⠀⠀⠀ ⠀⠀ $\tt[\because\:1m~=~100cm]$

Therefore ,

Area of the courtyard = $\tt\:(1872 \times 1320)sq.cm$

Area of the courtyard = $\tt\color{purple}{2471040~sq.cm}$

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Now we know the size of the side of the square tile is equal to the HCF of the length and breadth of the rectangular courtyard.

Henceforth ,

Prime factorisation of 1872 = $\tt\:2^{4} \times 3^{2} \times 13$

Prime factorisation of 1320 = $\tt\:2^{3} \times 3 \times 5 \times 11$

HCF of 1872 and 1320 = $\tt\:2^{3} \times 3$ = $\tt\color{purple}{24}$

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Hence we got the length of the square tile to be as 24cm . Now calculating the area of the square tile :

Area of the square tile = $\tt\:(24 \times 24)sq.cm$

Area of the square tile = $\tt\color{purple}{576~sq.cm}$

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Finally,

$\sf\implies\:Number~of~tiles$ = $\tt\:\frac{Area~of~courtyard}{Area~of~tiles}$

Substituting the values we got

$\sf\implies\:Number~of~tiles$ = $\tt\:\frac{2471040}{576}$

Reducing the fraction to the lower terms

$\sf\implies\:Number~of~tiles$ = $\tt\:\cancel{\frac{2471040}{576}}$

$\sf\implies\:Number~of~tiles$ = $\tt\:\cancel{\frac{1235520}{288}}$

$\sf\implies\:Number~of~tiles$ = $\tt\:\cancel{\frac{308880}{72}}$

$\sf\implies\:Number~of~tiles$= $\large{\mathfrak\red{4290}}$

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Note :- HCF also known as Greatest Common Divisor ( GCD ) is the largest number which could divide the given number .

• In this particular question I have calculated HCF by prime factorisation method.

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