Question

An area is paved with square tiles of a certain size and the number required is 128. If the tiles had been 2 cm smaller each way, 200 tiles would have been needed to pave the area. Find the size of the larger tiles.

Answer 1

Let the size of square of tile = x cm

x² cm² therefore, the area of one tile = no. of tiles requires to pave a certain area = 128.

x² cm²    given : area = 128 ............(i)

now, If the tiles had been 2 cm smaller each way, then

size of square tile = (x - 2) cm

(x - 2)² cm²              area of one tile = no. of tiles = 200

area paved by these tiles = 200 .............(ii)

(x - 2)² cm²

from (i) and (ii), we get,

128x² = 200(x - 2)²

⇒ 128x² = 200(x² + 4 - 4x)

⇒ 128x² = 200x² + 800 - 800x

or

72x² - 800x + 800 = 0

⇒ 9x² - 100x + 100 = 0

x = 100 +_ √10000 - 3600 / 18

x = 100 +_√6400 /18

x = 100 +_ 80 /18

x = 100 + 80 /18 , 100 - 80 /18

x = 180 /18 , 20 /18

x = 10, 10 /9

x = 10/9 is impossible

∴, size of larger square tile = 10 cm

Answer 2

Solution:-
Let the side of the square tile = 'x' cm
Therefore, the area = x*x = x² sq cm.
Number of tiles required to pave a certain area = 128
So, area of these tiles = 128x² sq cm.    ........(1)
Now, if the tiles had been 2 cm smaller each way, then
Side of square tile = (x - 2) cm
Area of one tile = (x - 2)² sq cm
Total number of tiles = 200
Area of these 200 tiles = 200(x - 2)² sq cm.    .....(2)
From (1) and (2), we get.
128x² = 200(x - 2)² 
128x² = 200(x² - 4x + 4)
128x² = 200x² - 800x +800
⇒ 200x² - 128x² - 800x + 800 = 0
⇒ 72x² - 800x + 800 = 0
Dividing it by 8, we get
⇒ 9x² - 100x + 100 = 0 
⇒ 9x² - 90x - 10x + 100 = 0
⇒ 9x(x - 10) - 10(x - 10) = 0
(x - 10) (9x - 10) = 0
x = 10 or x = 10/9
x = 10/9 is not possible.
So, x = 10 is correct.
The size of larger tile is 10 cm.
Answer.