Question

Rs 250 is divided equally among a certain number of children. If there were 25 more children, each would have received 50 paise less. Find the number of children.
Or, the hypotenuse of a right-angled triangle is 6 cm more than twice the shortest side. If the third side is 2 cm less than the hypotenuse, find the sides of the triangle.

Answer 1

Let the required number of children be x.
And total amount = Rs 250 = 25000 paise
Then, share of each child = $\frac{25000}{x}$ paise
If there were (x + 25) children, share of each child = $\frac{25000}{\left(x+25\right)}$paise
∴ $\frac{25000}{x}-\frac{25000}{\left(x+25\right)}=\frac{50}{1}$
$\Rightarrow 25000\left(\frac{1}{x}-\frac{1}{x+25}\right)=\frac{50}{1}$
$\Rightarrow 25000\left[\frac{x+25-x}{x\left(x+25\right)}\right]=\frac{50}{1}$
$\Rightarrow \frac{25000\times 25}{50}=x^2+25x$
$\Rightarrow 12500=x^2+25x$
$\Rightarrow x^2+25x-12500=0$
$\Rightarrow x^2+125x-100x-12500=0$
$\Rightarrow x\left(x+125\right)-100\left(x+125\right)=0$
$\Rightarrow \left(x-100\right)\left(x+125\right)=0$
$\Rightarrow \left(x-100\right)=0\mathrm{or}\left(\mathrm{x}+125\right)=0$
$\Rightarrow \mathrm{x}=100\mathrm{or}\mathrm{x}=-125$
Since the number of children cannot be negative, x = 100

OR

Let the length of the shortest side be x cm.
Then, hypotenuse = (2x + 6) cm
And the third side = (2x + 6 - 2) cm = (2x + 4) cm
By Pythagoras' theorem, we have:
(2x+ 6)² = x² + (2x + 4)²
$$\begin{gathered} \Rightarrow {\left(2x\right)}^2+2\times 2x\times 6+6^2=x^2+{\left(2x\right)}^2+2\times 2x\times 4+4^2 \\ \Rightarrow 4x^2+24x+36=x^2+4x^2+16x+16 \end{gathered}$$
⇒ x² - 8x - 20 = 0
⇒ x² - 10x + 2x - 20 = 0
⇒ x(x - 10) + 2(x - 10) = 0
⇒ (x + 2)(x - 10) = 0
⇒ x = - 2 or x = 10
Since, the side of a triangle is never negative, the length of the shortest side = 10 cm
Length of the hypotenuse = (2x + 6) cm = 26 cm
Length of the third side = ( 26 - 2) = 24 cm
Hence, the sides of the triangle are 10 cm, 26 cm and 24 cm.