Total expenses of a hostel are partly constant and partly vary directly as the number of boarders. When the number of boarders is $120$ and $100$, the total expenses are $\text{Rs. }2000$ and $\text{Rs. }1700$ respectively. Find the total number of boarders if the total expense is $\text{Rs. }1880$.

Let the constant expense be $C$ and the variable expense per boarder be $v$. If $B$ represents the number of boarders, the total expense $E$ can be written as:
$$E=C+B\times v$$

From the given conditions:

1. For $120$ boarders:

$$C+120v=2000\text{— (i)}$$

2. For $100$ boarders:

$$C+100v=1700\text{— (ii)}$$

Subtracting equation (ii) from equation (i):

$$(C+120v)-(C+100v)=2000-1700$$

$$20v=300⟹v=15$$

Substituting $v=15$ into equation (ii):

$$C+100(15)=1700$$

$$C+1500=1700⟹C=200$$

Now, we need to find the number of boarders $B$ when the total expense $E=1880$:

$$200+B\times 15=1880$$

$$15B=1880-200$$

$$15B=1680$$

$$B=\frac{1680}{15}=112$$