X varies directly as the product of U and V, and inversely as their sum. If X = 3 when U = 3 and V = 1, find X when U = 3 and V = 3.

The problem states that $X$ varies directly as the product of $U$ and $V$, and inversely as their sum. This can be written as:

$$X = k \frac{UV}{U+V}$$

where $k$ is the constant of proportionality.

We are given that $X = 3$ when $U = 3$ and $V = 1$. Substitute these values into the equation:

$$3 = k \frac{3 \times 1}{3 + 1}$$ $$3 = k \frac{3}{4}$$

To solve for $k$, multiply both sides by $\frac{4}{3}$:

$$k = 3 \times \frac{4}{3}$$ $$k = 4$$

Now that we have the value of $k$, the equation becomes:

$$X = 4 \frac{UV}{U+V}$$

We need to find $X$ when $U = 3$ and $V = 3$. Substitute these values into the complete equation:

$$X = 4 \frac{3 \times 3}{3 + 3}$$ $$X = 4 \frac{9}{6}$$

$$X = 4 \times \frac{3}{2}$$ $$X = 2 \times 3$$ $$X = \boxed{\text{6}}$$