#### 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 same area. Find the size of the larger tiles.

#### Solution

Let the size of the larger tiles be x cm.

Area of larger tiles = x^{2} cm^{2}

Number of larger tiles required to pave an area is 128.

So, the area needed to be paved = 128 x^{2} cm^{2} …. (1)

Size of smaller tiles = (x – 2)cm

Area of smaller tiles = (x – 2)^{2} cm^{2}

Number of larger tiles required to pave an area is 200.

So, the area needed to be paved = 200 (x – 2)^{2} cm^{2} …. (2)

Therefore, from (1) and (2), we have:

128 x^{2} = 200 (x – 2)^{2}

128 x^{2} = 200x^{2} + 800 – 800x

72x^{2} – 800x + 800 = 0

9x^{2} – 100x + 100 = 0

9x^{2} – 90x – 10x + 100 = 0

9x(x – 10) – 10(x – 10) = 0

(x – 10)(9x – 10) = 0

x = 10,10/9

if x = 10/9 then x - 2 = 10/9 - 2 = (10 - 18)/9 = (-8)/9 which is not possible.

Hence, the size of the larger tiles is 10 cm.