Answer: $\sqrt 3 : 2$ (or $\sqrt{12}:4$)

Using one unknown

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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

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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:

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Area scale factor between the two TVs: 

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Enlarge the widescreen TV so that the area scale factor is 1:

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Ratio traditional width : widescreen width is $1:\frac2{\sqrt3} = \sqrt3:2$