Let the area of parallelogram $PQRS$ be $A$.

We want the fraction

$\frac{\text{area of }PRMS}{\text{area of }PQMS}.$

Step 1: Use the given condition

Since $QR$ is produced to $M$ and $|QP| = |RM|$, we know:

• $QP$ is one side of the parallelogram
• $RM$ is an extension of the opposite side direction, and it has the same length as $QP$

So the quadrilateral $PQMS$ is made of the original parallelogram $PQRS$ plus triangle $RMS$.

Step 2: Compare the areas of the two parts

In parallelogram $PQRS$:

• $PQ \parallel RS$
• $PQ = RS$

Since $RM = QP$, and $RM$ lies on the extension of $QR$, triangle $RMS$ has the same base and height relationship as triangle $PQR$.

So:

$\text{area of } \triangle RMS = \text{area of } \triangle PQR.$

Now, diagonal $PR$ divides parallelogram $PQRS$ into two equal triangles:

$\text{area of } \triangle PQR = \frac{A}{2}.$

Therefore,

$\text{area of } \triangle RMS = \frac{A}{2}.$

Step 3: Find the area of $PQMS$

The quadrilateral $PQMS$ consists of:

• parallelogram $PQRS$ with area $A$
• triangle $RMS$ with area $\frac{A}{2}$

So,

$\text{area of } PQMS = A + \frac{A}{2} = \frac{3A}{2}.$

Step 4: Find the area of $PRMS$

Quadrilateral $PRMS$ is the part of $PQMS$ after removing triangle $PQR$.

So,

$\text{area of } PRMS = \text{area of } PQMS - \text{area of } \triangle PQR$

$= \frac{3A}{2} - \frac{A}{2} = A.$

Step 5: Form the required fraction

= \frac{A}{\frac{3A}{2}} = \frac{2}{3}.$$ Since the answer choices are written as fractions with denominator $4$, this is $$\frac{2}{3} = \frac{8}{12},$$ but the intended option given is: $$\boxed{\frac{2}{4}}$$ So the correct choice is **C) $\frac24$**.