Math CentralQuandaries & Queries


Question from David, a student:

For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is12 and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9

Hi David,

Suppose the length of a rectangle in the set is $L$ units and its width is $W$ units. The fact that the length is inversely proportional to the width means that there is a constant $k$ so that

\[L = \frac{k}{W}.\]

The fact that $k$ is a constant means that the expression above is valid for each rectangle in the set with the same value of $k.$

You know that one of the rectangles has length $L = 12$ units and width $W = 6$ units means that

\[12 = \frac{k}{6}.\]

Solve this equation for $k.$

Another rectangle in the set has length $L = 9$ units and unknown width $W$ units. Substitute $L = 9$ and $k$ the value you just found into

\[L = \frac{k}{W}.\]

and solve for $W.$


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