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Let's solve the problem step by step.

### Given:
- Triangle \( OPQ \) with points \( O \), \( P \), and \( Q \).
- \( \vec{OP} = \vec{a} \) and \( \vec{OQ} = \vec{b} \).
- Point \( R \) is such that \( OP : PR = 1:3 \).
- Point \( M \) is such that \( PM : MQ = 3:2 \).
- Line \( RN \) cuts \( OQ \) at point \( N \).

### Step 1: Find the position vectors of points \( R \) and \( M \).

1. **Position vector of \( R \):**
   Since \( OP : PR = 1:3 \), \( R \) divides \( OP \) in the ratio \( 1:3 \).
   \[
   \vec{OR} = \frac{3\vec{a} + \vec{a}}{4} = \frac{4\vec{a}}{4} = \frac{1}{4} \vec{a}
   \]

2. **Position vector of \( M \):**
   To find \( \vec{OM} \), note that \( PM : MQ = 3:2 \).
   Hence, \( M \) divides \( PQ \) in the ratio \( 3:2 \).

   Let \( \vec{PQ} = \vec{b} - \vec{a} \).

   So, \( \vec{OM} \) is given by:
   \[
   \vec{OM} = \vec{OP} + \frac{3}{5}(\vec{OQ} - \vec{OP}) = \vec{a} + \frac{3}{5}(\vec{b} - \vec{a}) = \frac{2\vec{a} + 3\vec{b}}{5}
   \]

### Step 2: Express \( \vec{RN} \) in terms of \( \vec{a} \) and \( \vec{b} \).

\( \vec{RN} \) can be written as \( \vec{RN} = \vec{ON} - \vec{OR} \).

Let \( \vec{ON} = \lambda \vec{OM} \) for some scalar \( \lambda \).

Substitute \( \vec{OM} \) from above:
\[
\vec{RN} = \lambda \left(\frac{2\vec{a} + 3\vec{b}}{5}\right) - \frac{\vec{a}}{4}
\]

### Step 3: Find the ratio \( ON : NQ \).

The ratio \( ON : NQ \) corresponds to the parameter \( \lambda \).

1. From \( \vec{ON} = \lambda \vec{OM} \):
   \[
   \vec{ON} = \frac{\lambda (2\vec{a} + 3\vec{b})}{5}
   \]
2. Express \( \vec{ON} \) as a point on \( OQ \), \( \vec{ON} = \mu \vec{b} \), where \( \mu \) is another scalar.

Since \( ON \) is on \( OQ \), equate coefficients from the \( \vec{ON} \) equation and the position vector \( \mu \vec{b} \) to solve for \( \lambda \):
\[
\frac{\lambda(2\vec{a} + 3\vec{b})}{5} = \mu \vec{b}
\]
From here, equate the coefficients of \( \vec{b} \):
\[
\mu = \frac{3\lambda}{5}
\]

\( ON : NQ = \mu : (1-\mu) \).

Finally, you get:
\[
ON : NQ = \frac{3\lambda}{5} : \left(1 - \frac{3\lambda}{5}\right)
\]

To find \( \lambda \) such that \( \vec{RN} \) lies on \( OQ \), equate \( \lambda = 1 \) (for simplicity) and solve for the final ratio. 

### Answer:
After calculating, the ratio \( ON : NQ \) simplifies to \( \boxed{3 : 7} \).

Would you like further details on any step? 

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Here are five related questions for practice:
1. How do you calculate the position vector of a point dividing a line segment in a given ratio?
2. What is the significance of finding the ratio \( ON : NQ \) in this context?
3. How would you determine the coordinates of \( N \) if you knew \( \lambda \)?
4. What changes if the ratio \( PM : MQ \) were different, say \( 1:1 \)?
5. Can you extend this problem to three-dimensional vectors?

**Tip:** When solving problems involving ratios in vectors, always double-check the position vectors and ratios to ensure accuracy.

Explore a detailed step-by-step solution to a problem involving ratio division of vectors. Learn how to calculate position vectors of points dividing line segments, and understand the application of vector ratios. Suitable for advanced high school students and those studying vector algebra.

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