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At the end of the year 2016, the population of villages Kovad, Varud, Chikhali is 5x2 – 3y2, 7y2 + 2xy and 9x2 + 4xy respectively. At the beginning of the year 2017, x2 + xy – y2, 5xy and 3x2 + 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?

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Total population of villages at the end of 2016 

= (5x2 – 3y2) + (7y2 + 2xy) + (9x2 + 4xy) 

= 5x2 + 9x2 – 3y2 + 7y2 + 2xy + 4xy 

= 14x2 + 4y2 + 6xy …….(i) 

Total number of persons who went to other village at the beginning of 2017 

= (x2 + xy – y2) + (5xy) + (3x2 + xy) 

= x2 + 3x2 – y2 + xy + 5xy + xy 

= 4x2 – y2 + 7xy … (ii)

Remaining total population of villages = Total population at the end of 2016 – total number of persons who went to other village at the beginning of 2017 

= 14x2 + 4y2 + 6xy – (4x2 – y2 + 7xy) … [From (i) and (ii)] 

= 14x2 + 4y2 + 6xy – 4x2 + y2 – 7xy 

= 14x2 – 4x2 + 4y2 + y2 + 6xy – 7xy = 1 

= 10x2 + 5y2 – xy 

∴ The remaining total population of the three villages is 10x2 + 5y2 – xy.

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