Madhu’s flower garden is now a square. If she enlarges it by increasing the width 1 metre and the length 3 metres, the area will be 19 square metres more than the present area. What is the length of a side now?

Hint: We first assume the length of a side at present and find its area. Then we enlarge the sides and find the area of the rectangle garden. We take the difference as 19. We solve the algebraic equation to get the solution of the variable.

Complete step by step answer:
Madhu’s flower garden is now a square. If she enlarges it by increasing the width 1 metre and the length 3 metres, the area will be 19 square metres more than the present area.Madhu’s garden changes from the shape of a square to a rectangle.Let us assume the side length for the square garden was $a$ meters.So, the area was $a\times a={{a}^{2}}$ square meters.

The dimensions change to length being $a+3$ meters and width being $a+1$ meters. So, the area was $\left( a+3 \right)\times \left( a+1 \right)={{a}^{2}}+4a+3$ square meters.The difference is 19 square metres. Therefore, the equation becomes
$\left( {{a}^{2}}+4a+3 \right)-{{a}^{2}}=19$.
Simplifying the equation, we get
$\left( {{a}^{2}}+4a+3 \right)-{{a}^{2}}=19 \\
\Rightarrow {{a}^{2}}+4a+3-{{a}^{2}}=4a+3=19 \\
\Rightarrow 4a=19-3=16 \\
\therefore a=\dfrac{16}{4}=4 \\ $
Therefore, the length of a side of the present square garden is 4 metres.

Note: We can see the enlargement of the sides increases the area of the square to the area of the rectangle. The formula for the area in both cases is the multiplication of the consecutive sides. In the case of squares, the sides are equal and that’s why the area is the square of the side length.