If W₁ : W₂ = 2 : 3 and W₁ : W₃ = 1 : 2, then W₂ : W₃ is -

$$\eqalign{ & = \frac{{{{\text{W}}_{\text{2}}}}}{{{{\text{W}}_{\text{1}}}}} = \frac{3}{2}{\text{and}}\frac{{{{\text{W}}_{\text{1}}}}}{{{{\text{W}}_{\text{3}}}}} = \frac{1}{2} \cr & \Rightarrow \frac{{{W_2}}}{{{W_3}}} = \left( {\frac{{{{\text{W}}_{\text{2}}}}}{{{{\text{W}}_{\text{1}}}}} \times \frac{{{{\text{W}}_{\text{1}}}}}{{{{\text{W}}_{\text{3}}}}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{3}{2} \times \frac{1}{2} = \frac{3}{4} \cr & \Rightarrow {W_2}:{W_3} = 3:4 \cr} $$