The width of a rectangular room is $\dfrac{3}{5}$ of its length $x$ metres. If its perimeter is $y$ metres, write an equation connecting $x$ and $y$. Find the floor area of the room if its perimeter is 32 m.

Given,

Length of rectangular room = x meters

Width of rectangular room = $\dfrac{3}{5}x$ meters.

Perimeter = y meters.

We know that,

Perimeter = 2(l + b)

Substituting the values we get,

$\Rightarrow y = 2\Big[x + \dfrac{3}{5}x\Big] \\[1em] \Rightarrow y = 2\Big[\dfrac{5x + 3x}{5}\Big] \\[1em] \Rightarrow 5y = 2 \times 8x \\[1em] \Rightarrow 5y = 16x \text{ …….. (1)}$

The above equation is the required relation between x and y.

Given, perimeter = y = 32 m.

Now substituting the value of y in equation (1)

⇒ 16x = 5 × 32

⇒ x = $\dfrac{160}{16}$ = 10 m,

⇒ Breadth = $\dfrac{3}{5} \times x = \dfrac{3}{5} \times 10$ = 6 m.

Floor area of the room = l × b

= 10 × 6

= 60 m².

Hence, 16x = 5y is the equation connecting x and y and the floor area of room = 60 m².