In a city, 35% of the population is composed of migrants, 20% of whom are from rural areas. Of the local population, 48% is female while this figure for rural and urban migrants is 30% and 40% respectively. What percentage of the total population comprises of females ?

Solution(By Examveda Team)

Let the total population be x
Then, migrant population :
$$\eqalign{ & = 35\% {\text{ of }}x \cr & = \left( {\frac{{35}}{{100}} \times x} \right) \cr & = \frac{{7x}}{{20}} \cr} $$
Local population :
$$\eqalign{ & = \left( {x - \frac{{7x}}{{20}}} \right) \cr & = \frac{{13x}}{{20}} \cr} $$
Number of rural migrants :
$$\eqalign{ & = 20\% {\text{ of }}\frac{{7x}}{{20}} \cr & = \left( {\frac{{20}}{{100}} \times \frac{{7x}}{{20}}} \right) \cr & = \frac{{7x}}{{100}} \cr} $$
Number of urban migrants :
$$\eqalign{ & = \left( {\frac{{7x}}{{20}} - \frac{{7x}}{{100}}} \right) \cr & = \frac{{28x}}{{100}} \cr & = \frac{{7x}}{{25}} \cr} $$
Female population :
$$48\% {\text{ of }}\frac{{13x}}{{20}} + 30\% {\text{ of }}\frac{{7x}}{{100}}$$     $$ +\, 40\% {\text{ of }}\frac{{7x}}{{25}}$$
$$ = \left( {\frac{{48}}{{100}} \times \frac{{13x}}{{20}}} \right) + \left( {\frac{{30}}{{100}} \times \frac{{7x}}{{100}}} \right)$$      $$ + \left( {\frac{{40}}{{100}} \times \frac{{7x}}{{25}}} \right)$$
$$\eqalign{ & = \frac{{39x}}{{125}} + \frac{{21x}}{{1000}} + \frac{{14x}}{{125}} \cr & = \frac{{445x}}{{1000}} \cr} $$
∴ Required percentage :
$$\eqalign{ & = \left( {\frac{{445x}}{{1000}} \times \frac{1}{x} \times 100} \right)\% \cr & = 44.50\% \cr} $$