Question

Kamal has \[x\] children by his first wife. Ritu has \[(x+1)\] children by her first husband. They marry and have children of their own. The whole family has \[24\] children. Assuming the two children of the same parents do not fight, then the maximum possible number of fights that can take place is :
1) \[190\]
2) \[191\]
3) \[200\]
4) \[52\]

Answer 1

Hint: In this type of question your approach should be greedy, that is whatever the question is saying you just need to do as it mathematically, and you must know some basics of solving linear equation and quadratic equation, finding discriminant of quadratic equation then you will reach your answer by this.

Complete step by step answer:
First we will know something about mathematical equations and its importance in real life:
In algebra, an equation can be defined as a mathematical statement consisting of an equal symbol between two algebraic expressions that have the same value.
The most basic and common algebraic equations in math consist of one or more variables.
For instance, \[2x+5=7\] is an equation, in which \[2x+5\] and \[7\] are two expressions separated by an ‘equal’ sign.

So, coming to our question it is given that:
Kamal has \[x\] children from his first wife.
Ritu has \[(x+1)\] children from her first husband.
Total number of children after they get married and have their own children \[=24\]
Let \[y\] be the children of Kamal and Ritu .
Now if we have to distinguish between all children according to the children having same parent, it will be pretty much clear that \[x\] children of Kamal, \[(x+1)\] children of Ritu and \[y\] children of both will having different parents as each group of children don’t have common mother and father both i.e. some having same mother but different father and same having same father but different mother.
Above was the full concept of the question as in the question it is asked to find the maximum fights and it is given that fights must only occur between the children having different parents and here we just define such groups of children.
So now according to question we have:
\[x+x+1+y=24\]
\[\Rightarrow 2x+y=23\]
\[\Rightarrow y=(23-2x)\] \[......(1)\]
Let \[z\] denote the number of fights.
Then \[z\] must be:
\[z=x(x+1)+(xy)+(x+1)y\]
Now by using equation \[(1)\]
\[\Rightarrow z={{x}^{2}}+x+\left[ x(23-2x) \right]+(x+1)(23-2x)\]
Now on simplifying further we get,
\[\Rightarrow z={{x}^{2}}+x+23x-2{{x}^{2}}+23x-2{{x}^{2}}+23-2x\]
Now after adding and subtracting the terms having same coefficients we get,
\[\Rightarrow z=-3{{x}^{2}}+45x+23\]
Now on taking every terms on same side we get a quadratic equation as:
\[\Rightarrow z+3{{x}^{2}}-45x-23=0\]
\[\Rightarrow 3{{x}^{2}}-45x+(z-23)=0\]
Since \[x\] is the real number.
Discriminant of above quadratic equation cannot be negative,
Therefore,
\[{{b}^{2}}-4ac\ge 0\]
\[\Rightarrow {{\left( 45 \right)}^{2}}-43\times 3\times \left( z-23 \right)\ge 0\]
\[\Rightarrow 12z\le 2301\]
\[\Rightarrow z\le \left[ 2301\text{ }/\text{ }12 \right]\]
\[\Rightarrow z\le 191\]
Since \[z\] is always an integer as it is the number of fights.
Therefore, maximum number of fights \[=191\]

So, the correct answer is “Option 2”.

Note: Almost any situation where there is an unknown quantity can be represented by a linear equation, like figuring out income over time, calculating mileage rates, or predicting profit. Many people use linear equations every day, even if they do the calculations in their head without drawing a line graph.