### Find the percentage increase in the area of a triangle if each side is increased to $5$ times.

$2400 \%$

Step by Step Explanation:
1. Consider a triangle $QRS$ with sides $a, b$ and $c.$
Let $S = \dfrac { a+b+c } { 2 }$
Area of triangle $QRS, A_1 = \sqrt{ S(S-a)(S-b)(S-c) }$
2. Increasing the side of each side by $5$ times, we get a new triangle $XYZ.$
$XYZ$ has sides $5a, 5b$ and $5c.$
By Heron's formula,
Area of new triangle $= \sqrt{ S_{1}(S_{1}-5a)(S_{1}-5b)(S_{1}-5c) }$
Where $S_1 = \dfrac { 5a + 5b + 5c } { 2 } = 5 \times \dfrac { a+b+c } { 2 }$
Area of $XYZ = \sqrt{ 5S(5S-5a)(5S-5b)(5S-5c) }$
$$= \sqrt{ 5^{4}S(S-a)(S-b)(S-c) } \\ = 5^2 \times A_1 \\ = 25 A_1$$
3. Percentage increase in the area of the triangle, \begin{align} &= \dfrac{ \text{ Area of Triangle XYZ } - \text{ Area of Triangle QRS } } { \text{Area of Triangle QRS} } \times 100 \\ &= \dfrac{ 25 A_1 - A_1 } { A_1 } \times 100 \\ &= \dfrac{ 24 A_1 }{ A_1 } \times 100 \\ &= 2400 \end{align}
4. This means the area of the triangle, $A_1$ is increased by $2400 \%.$ 