Televisual Technology
If a new and an old television have screens of the same area, what is the ratio of their widths?
Age
14 to 16
Challenge level
Traditional television screens have a width : height ratio of $4 : 3$.
Widescreen televisions have a width : height ratio of $16 : 9$.
If a traditional screen and a widescreen have the same area, what is the ratio of their widths?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
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 $1:\frac2{\sqrt3} = \sqrt3:2$