Step 1: Understand the displacement method for a convex lens.
When an object and screen are fixed at distance $D$ apart, a convex lens can be placed at two positions between them to form a clear image on the screen. The distance between these two positions is called $d$. This is the lens displacement (or conjugate positions) method.
Step 2: Identify the given quantities.
Distance between object and screen: $D = 100\,\text{cm}$. Distance between the two lens positions: $d = 20\,\text{cm}$.
Step 3: Recall the displacement method formula.
The focal length of the lens is given by: \[ f = \frac{D^2 - d^2}{4D} \] This formula can be derived from the lens equation and the constraint that both positions give real images on the screen.
Step 4: Substitute the values.
\[ f = \frac{(100)^2 - (20)^2}{4 \times 100} = \frac{10000 - 400}{400} \]
Step 5: Compute the result.
\[ f = \frac{9600}{400} = 24\,\text{cm} \]
Step 6: State the final answer.
The focal length of the convex lens is: \[ \boxed{24\,\text{cm}} \]