Question

At the end of the year 2016, the population of villages Kovad, Varud, Chikhali is 5x² $-$3 y² , 7 y² + 2 xy and 9 x² + 4 xy respectively. At the beginning of the year 2017, x² + xy $-$ y² , 5 xy and 3 x² + xy persons from each of the three villages respectively went to another village for education then what is the remaining total population of these three villages ?

Answer 1

Total population of the three villages
$$\begin{gathered} =\left(5x^2-3y^2\right)+\left(7y^2+2xy\right)+\left(9x^2+4xy\right) \\ =5x^2+9x^2-3y^2+7y^2+2xy+4xy \\ =14x^2+4y^2+6xy \end{gathered}$$
Total number of persons who went to another village for education
$$\begin{gathered} =\left(x^2+xy-y^2\right)+5xy+\left(3x^2+xy\right) \\ =x^2+3x^2-y^2+xy+5xy+xy \\ =4x^2-y^2+7xy \end{gathered}$$
∴ Remaining total population of the three villages = Total population of the three villages − Total number of persons who went to another village for education
$$\begin{gathered} =\left(14x^2+4y^2+6xy\right)-\left(4x^2-y^2+7xy\right) \\ =14x^2+4y^2+6xy-4x^2+y^2-7xy \\ =14x^2-4x^2+4y^2+y^2+6xy-7xy \\ =10x^2+5y^2-xy \end{gathered}$$
Thus, the remaining total population of these three villages is 10x² + 5y² − xy.