Math CentralQuandaries & Queries


Question from colum:

Can you work out the length and width of a rectangle where you know only the area and aspect ratio?

Hi Colum,

Yes you can.

Let me do an example first. Suppose you know that the area is 80 square centimeters and the ratio of the length to the width is $5:4.$ Let the length be $l$ centimeters and the width $w$ centimeters. Since $l:w = 5:4$ you know that $4 \times l = 5 \times w$ or $l = \large \frac54 \normalsize \times w.$ Since $l \times w = 80$ square centimeters you have that

\[\frac54 w \times w = 80\]


\[w^2 = 80 \times \frac45 = 64\]

and hence $w = 8$ cm and $l = 10$ cm.

Thus in general if the rectangle dimensions are $l$ and $w$ units, the area is $A$ square units and $l:w = p:q$ then you know that $q \times l = p \times w.$ Solve this equation for $l,$ substitute into the area expression $A = l \times w$ and solve for $w.$

I hope this helps,

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