Ratio of widths is $4k : 16 = k:4$
Areas are equal so $12k^2 = 144 \Rightarrow k^2 = 12$
So $k=\sqrt{12} = 2\sqrt3$
Ratio of widths is $2\sqrt3 : 4 = \sqrt3 : 2$
Using two unknowns
Ratio of widths is $w:v$
Areas are equal so $\frac34w^2=\frac9{16}v^2$
$\frac34w^2=\frac9{16}v^2\Rightarrow 12w^2=9v^2 \Rightarrow 4w^2 = 3v^2$
So $w^2:v^2 = 3:4$
So $w:v = \sqrt3:\sqrt4 = \sqrt3 : 2$
Using scale factors
Start with both televisions the same width:
Area scale factor between the two TVs:
Enlarge the widescreen TV so that the area scale factor is 1:
Ratio traditional width : widescreen width is $1:\frac2{\sqrt3} = \sqrt3:2$