For 500 g of 0.25 molal aqueous solution,

0.25 = $${{{{{w_1}} \over {62}}} \over {{{500} \over {1000}}}}$$ = $${{2{w_2}} \over {62}}$$ ......(1)

For 250 mL of 0.25 molar aqueous solution

$${{{{{w_2}} \over {62}}} \over {{{250} \over {1000}}}} = {{4{w_2}} \over {62}}$$ .......(2)

Dividing equation (1) by (2), we get

$${{0.25} \over {0.25}} = {{{{2{w_1}} \over {62}}} \over {{{4{w_2}} \over {62}}}}$$

$$ \Rightarrow {{2{w_1}} \over {62}} = {{4{w_2}} \over {62}}$$

$$ \Rightarrow {{{w_1}} \over {{w_2}}} = {4 \over 2} = {2 \over 1}$$

Note :

(1) Molality of solution is defined as number of moles of solute present per kg of solvent.

$$m = {{no.\,of\,moles\,(solute)} \over {weight(in\,kg) of solvent}}$$

(2) Molarity of solution is defined as number of moles of solute present per litre of solution.

$$M = {{no.\,of\,moles\,(solute)} \over {v(in\,litre\,of\,solution)}}$$