Can any one demostrate it?...............A room can be paved with 1210 square tiles.?

Question

A room can be paved with 1210 square tiles. If the side of each tile were to increase by 1 cm, it would only take 1000 tiles to pave the room. Find the dimension of each tile.



(Answer is 10 cm) Can any one demostrate it?

Answer 1

x=number of tiles for width
y=number of tiles for length
Area=x*y*w^2=1210w^2=1000(w+1cm)^2
=> 1210w^2=1000(w^2 +2*wcm+1cm)
=> 2*wcm+1cm=210xm Where x is unknown multiplier of meters
setting w=10 we get
20cm+1cm=210xm
setting xm=mm we have a working formula.

Answer 2

Suppose that the width of the tile is w.

You can tile the room with x tiles across and y tiles down and use 1210 tiles, so
xy=1210.

If the width of the tile is w+1 then you can tile the room with a tiles across and b tiles down and use 1000 tiles, so
ab=1000

Either way the area of the room is the same so
xy w^2 = ab (w+1)^2

Since xy=1210 and ab=1000 then
1210 w^2 = 1000(w+1)^2
1210 w^2 = 1000 w^2 + 2000 w + 1000
or
210 w^2 - 2000w - 1000 = 0

Solving using the quadratic equation we get
w = (2000 +- sqrt(2000^2 + 4(1000)(210))/420
w = (2000 +- 2200)/420
w = 10 or -200/420

Since the negative answer does not make sense we can take the positive answer, w=10.

Answer 3

Let x = the length in cm of each side of a tile.
Let a and b be the length and width of the room in tiles.

Therefore the length of the room in cm = ax
and the width of the room in cm = bx.

Thus the area of the room = ax * bx = abx^2 cm^2.
But we are given ab = 1210.
So the area of the room = 1210x^2 cm^2.

If each tile is increased by 1 cm,
then each tile is now (x + 1) cm by (x + 1) cm.

If c and d is now the length and width of the room in tiles,
then area of the room = c(x + 1) * d(x + 1) = cd(x + 1)^2 cm^2.
But, given that cd = 1000,
the area of the room = 1000(x + 1)^2 cm^2

Of course, the area of the room in cm^2 is the same for both, so
1210x^2 = 1000(x + 1)^2

Divide through by the common factor 10.
121x^2 = 100(x + 1)^2

Multiply out the RHS.
121x^2 = 100x^2 + 200x + 100

Subtract the RHS from both sides.
21x^2 - 200x - 100 = 0

Factorise.
(21x + 10)(x - 10) = 0

Thus, x = -10/21 or 10
The negative value doesn't mean anything here,
so x = 10 cm is the dimension of each tile.

So we see that 1210 * 10 * 10 cm^2 = 1000 * 11 * 11 cm^2