Question

An area is paved with square tiles of a certain size and the required is 600. If the tiles had been 1 cm smaller each way, 864 tiles would have been needed to pave the same area. Find the size of the larger tiles.

Answer 1

Hint: Here we will assume the length of the tile to be x. Then we will calculate the total area and form the equation using the given information and solve for the value of x to get the desired answer.

Complete step-by-step answer:
Let the length of the tile be x.
Now since it is given that the tile is of square shape and the area of a square with edge length a is given by:- \[area = {a^2}\]
Therefore, the area of each tile would be \[area = {x^2}\]
Therefore, the total area paved will be \[600{x^2}\]………………………… (1)
Now it is given that,
If the tiles had been 1 cm smaller each way, 864 tiles would have been needed to pave the same area.
The area of a square with edge length a is given by:- \[area = {a^2}\]
Therefore, area of each tile with length 1 cm smaller each way is given by:- \[area = {\left( {x - 1} \right)^2}\]
Therefore, the total area paved according to the given situation will be \[864{\left( {x - 1} \right)^2}\]……………………… (2)
Equating equations 1 and 2 we get:-
\[600{x^2} = 864{\left( {x - 1} \right)^2}\]
Now we know that,
\[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
Applying this identity we get:-
\[600{x^2} = 864\left( {{x^2} + 1 - 2x} \right)\]
Solving it further we get:-
\[600{x^2} = 864{x^2} + 864 - 1728x\]
Simplifying the above equation:-
\[864{x^2} - 600{x^2} - 1728x + 864 = 0\]
Solving further we get:-
\[264{x^2} - 1728x + 864 = 0\]
Dividing whole equation by 24 we get:-
\[11{x^2} - 72x + 36 = 0\]
Solving the equation by middle term split we get:-
\[11{x^2} - 66x - 6x + 36 = 0\]
Taking the terms common we get:-
\[11x\left( {x - 6} \right) - 6\left( {x - 6} \right) = 0\]
\[ \Rightarrow \left( {11x - 6} \right)\left( {x - 6} \right) = 0\]
Now solving for x we get:-
\[11x - 6 = 0;x - 6 = 0\]
\[ \Rightarrow x = \dfrac{6}{{11}};x = 6\]
Now since we have to take the larger size.

Hence, the length of the tile is 6cm.

Note: Students can also use the quadratic formula to solve the quadratic equation.
The quadratic formula for standard equation \[a{x^2} + bx + c = 0\] is given by:-
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
In such questions first we need to form the correct equation according to the given situation and then solve it to get the answer.