When a quantity varies directly as another quantity, it means that the first quantity is proportional to the second quantity. When a quantity varies inversely as another quantity, it means that the product of the two quantities is a constant.

Given that \(P\) varies directly as \(Q\) and inversely as \(R\), we can write the relationship as:

\(P \propto \frac{Q}{R}\)

Where \(k\) is the constant of proportionality. Rearranging the equation gives:

\(P = k \cdot \frac{Q}{R}\)

Now, we are given specific values for \(Q\), \(R\), and \(P\):

When \(Q = 36\), \(R = 16\), and \(P = 27\), we can substitute these values into the equation to find the value of \(k\):

\(27 = k \cdot \frac{36}{16}\)

Solving for \(k\):

\(k = \frac{27 \cdot 16}{36} = 12\)

Now that we have the value of \(k\), we can rewrite the equation:

\(P = 12 \cdot \frac{Q}{R}\)

Simplifying:

\(P = \frac{12Q}{R}\)