Equation of inverse proportion

If \(y\) is inversely proportional to \(x\), we write \(y\propto\dfrac{1}{x}\). The product \(xy\) is a constant \(k\), so \(xy=k\) or \(y=\dfrac{k}{x}\), where \(k\ne0\).

Why: dividing both sides of \(xy=k\) by \(x\) gives \(y=\dfrac{k}{x}\). This also shows that \(y\) is directly proportional to \(\dfrac{1}{x}\), because \(y=k\left(\dfrac{1}{x}\right)\).

Method

1. Write \(y=\dfrac{k}{x}\) using the letters in the question.
2. Substitute the given pair and find \(k\): \(k=xy\).
3. Rewrite the equation with the value of \(k\).
4. Substitute the new value and solve; multiply both sides by the denominator when the unknown is on the bottom.

Example

The time, \(t\) minutes, to download a file is inversely proportional to the download speed, \(s\) Mbps. At \(40\) Mbps the download takes \(6\) minutes. Find an equation connecting \(t\) and \(s\), the time at \(60\) Mbps and the speed needed for a \(3\)-minute download.

\(\begin{aligned} t&=\dfrac{k}{s} \\ 6&=\dfrac{k}{40} \\ k&=240 \\ \therefore t&=\dfrac{240}{s} \end{aligned}\)

When \(s=60\): \(t=\dfrac{240}{60}=4\) minutes.

When \(t=3\): \(3=\dfrac{240}{s}\), so \(3s=240\) and \(s=80\) Mbps.

Check: multiply each pair: \(40\times6=60\times4=80\times3=240\). A common error is to find \(k=\dfrac{t}{s}\), which is the direct proportion constant.