The product of two numbers is 6250 and their quotient is 10. Find their sum.

Let the two numbers be $x$ and $y$. Assume $x$ is the larger number and $y$ is the smaller number.
From the problem statement, we have two conditions:

1. The product of the two numbers is 6250: $$x \cdot y = 6250 \quad (1)$$
2. The quotient when the larger number is divided by the smaller number is 10: $$\frac{x}{y} = 10 \quad (2)$$

From equation (2), we can express $x$ in terms of $y$:

$$x = 10y$$

Substitute $x = 10y$ into $x \cdot y = 6250$:

$$(10y) \cdot y = 6250$$ $$10y^2 = 6250$$

Divide both sides by 10:

$$y^2 = \frac{6250}{10}$$ $$y^2 = 625$$

Take the square root of both sides:

$$y = \sqrt{625}$$ $$y = 25$$

Substitute the value of $y$ back into $x = 10y$:

$$x = 10 \cdot 25$$ $$x = 250$$

The sum of the numbers is $x + y$:

$$\text{Sum} = 250 + 25$$ $$\text{Sum} = 275$$

The correct option is (c).

The final answer is $\boxed{\text{275}}$.