Data Sufficiency

We'll start by setting up an equation for the average weight of the watermelons:
$$\frac{10x\ +\ Ry}{x+y}=avg.\ weight$$
So we should be able to solve for R if we know the value of x and y.
We should also note that if the average weight of the watermelons is 12, R must be greater than 10.

Statement 1
This means that y = x+5. Plugging this into our equation gives $$\frac{10x\ +\ R\left(x+5\right)}{x+x+5}=avg.\ weight$$ $$\frac{10x\ +\ Rx+5}{2x+5}=avg.\ weight$$ $$\frac{\left(10+R\right)x+5}{2x+5}=avg.\ weight$$ However, since we still don't know the value of x, we can't solve. Insufficient.

Statement 2
This means that R = y. Plugging this into our equation gives: $$\frac{10x\ +\ R^2}{x+R}=avg.\ weight$$ However, like in Statement 1, we still need x to solve. Insufficient.

Both
y=x+5, and R=y, so R=x+5. Rearranging this gives x=R-5. If we plug this into either the equation from Statement 1 or Statement 2, we should be left with an equation containing only R. We should then be able to solve for R. Sufficient.