Question

Q10⟩⟩ John has 'x' children by his first wife. Mary has (x + 1) children by her first husband. They marry and have children of their own. The whole family has 24 children. Assuming that the children of the same parents do not fight, find the maximum number of fights that can take place.​

Answer 1

$\large\underline{\sf{Solution-}}$

★ Given that,

• John has 'x' children by his first wife.

• Mary has (x + 1) children by her first husband.

• They marry and have children of their own.

★ Let assume that

• John and Mary have 'y' children.

★ According to statement,

★ Total number of children in whole family = 24

$\rm :\longmapsto\:x + x + 1 + y = 24$

$\rm :\longmapsto\:2x + 1 + y = 24$

$\rm :\longmapsto\:2x + y = 24 - 1$

$\rm :\longmapsto\:2x + y = 23$

$\bf\implies \:y = 23 - 2x - - - (1)$

From equation (1), we concluded that x should not be more than 11.

Now,

★ It is given that children of same family donot fight with each other.

So 3 cases arises.

• When child of 'x' family fight with child of (x + 1)

• When child of 'x' family fight with child of 'y'

• When child of 'y' family fight with (x + 1)

★ Let number of fights be 'f'.

So,

$\rm :\longmapsto\:f = x(x + 1) + xy + (x + 1)y$

$\rm :\longmapsto\:f = x(x + 1) + y(x + x + 1)$

$\rm :\longmapsto\:f = x(x + 1) + (23 - 2x)(2x+ 1)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf{ \because \: y \: = \: 23 - 2x}}$

$\rm :\longmapsto\:f = {x}^{2} + x + 46x + 23 - {4x}^{2} - 2x$

$\rm :\longmapsto\:f = - 3{x}^{2} + 45x+ 23$

$\rm :\longmapsto\: 3{x}^{2} - 45x + f - 23 = 0$

Since, we have to find the value of f and its a quadratic equation.

So for solution to be exist, Discriminant should be greater than or equals to 0.

$\rm :\longmapsto\: {b}^{2} - 4ac \geqslant 0$

$\rm :\longmapsto\: {( - 45)}^{2} - 4(3)(f - 23) \geqslant 0$

$\rm :\longmapsto\: 2025 - 12(f - 23) \geqslant 0$

$\rm :\longmapsto\: 2025 - 12f + 276 \geqslant 0$

$\rm :\longmapsto\: 2301- 12f \geqslant 0$

$\rm :\longmapsto\: - 12f \geqslant - 2301$

$\bf\implies \:f \leqslant \dfrac{2301}{12} \approx \: 191.75$

$\bf\implies \:f \leqslant 191.75$

$\bf\implies \:Maximum \: number \: of \: fights \: = 191$