Consider a string P of length l that is laid out as a straight line segment. Another string K is laid out as a semi-circular arc with string P as diameter, (figure 1). When both the strings are shortened by a length of x they can be re-arranged such that the shortened string K forms a full circle with the shortened string P as diameter (figure 2). The value of x/l is _____.

The correct option is π/2(π-1).

Step 1: Understand the initial setup from Figure 1
From Figure 1, we observe two strings:
1. String P is a straight line segment of length $l$ that acts as the diameter of the semi-circle.
2. String K is laid out as a semi-circular arc with String P as its diameter.
The length of a semi-circular arc with diameter $l$ is given by half the circumference of a full circle of diameter $l$. Therefore, the initial length of String K is:

$L_K=\frac{\pi l}{2}$

Step 2: Express the lengths after shortening
Both strings are shortened by a length of $x$.
- The new length of String P becomes:

$L_{\text{P, new}}=l-x$

- The new length of String K becomes:

$L_{\text{K, new}}=\frac{\pi l}{2}-x$

Step 3: Relate the shortened strings using Figure 2
From Figure 2, after shortening, the shortened String K forms a full circle, and the shortened String P forms the diameter of this new circle.
The circumference of a circle with diameter $D$ is given by $\pi D$. Since the new diameter is $L_{\text{P, new}}$ and the circumference is $L_{\text{K, new}}$, we have:

$L_{\text{K, new}}=\pi ·L_{\text{P, new}}$

Substituting the expressions for the shortened lengths into this equation:

$\frac{\pi l}{2}-x=\pi (l-x)$

Step 4: Solve for the ratio x/l
First, expand the right-hand side of the equation:

$\frac{\pi l}{2}-x=\pi l-\pi x$

Rearrange the terms to group all terms containing $x$ on the left-hand side and all terms containing $l$ on the right-hand side:

$\pi x-x=\pi l-\frac{\pi l}{2}$

Factor out $x$ on the left side and simplify the right side:

$x(\pi -1)=\frac{\pi l}{2}$

To find the value of $\frac{x}{l}$, divide both sides by $l(\pi -1)$:

$\frac{x}{l}=\frac{\pi }{2(\pi -1)}$

This matches the provided correct answer.