P varies directly as Q and inversely as R. When Q = 36 and R = 16, P = 27. Find the relation between P, Q and R.

1. Analyze the question

We are given a variation problem involving three variables: $P$, $Q$, and $R$. We need to find the mathematical relationship (the formula) connecting them based on the provided values.

2. Explain the concept

• Direct Variation: If $P$ varies directly as $Q$, it means $P \propto Q$ (or $P = kQ$).
• Inverse Variation: If $P$ varies inversely as $R$, it means $P \propto \frac{1}{R}$ (or $P = \frac{k}{R}$).
• Combined Variation: When combined, $P$ varies directly as $Q$ and inversely as $R$, expressed as: $$P = \frac{kQ}{R}$$where$k$ is the constant of proportionality.

3. Show the solution

Step 1: Set up the equation. Using the combined variation formula:

Step 2: Substitute the given values to find $k$. We are given $P = 27$, $Q = 36$, and $R = 16$.

Step 3: Solve for $k$. Multiply both sides by 16:

Divide both sides by 36:

Step 4: Write the final relation. Substitute $k = 12$ back into the original equation:

4. Identify the answer

Comparing our result to the given options, the correct relation is $P = \frac{12Q}{R}$. This matches option B.

5. Exam Tip

When dealing with variation questions, always remember: "Direct" means the variable goes on the numerator (top), and "Inverse" means the variable goes on the denominator (bottom). This simple rule will help you set up your equation in seconds!