Total expenses of a hostel are partly constant and partly vary directly as the number of boarders. When the number of boarders are $120$ 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 component be $C$, the variable rate per boarder be $k$, and the total number of boarders be $n$. The equation representing total expense ($E$) is:
$$E=C+k\cdot n$$

From the given data, we can set up two linear equations:

1. For $n=120$, $E=2000$:

$$C+120k=2000\text{— (1)}$$

2. For $n=100$, $E=1700$:

$$C+100k=1700\text{— (2)}$$

Subtracting equation (2) from equation (1):

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

$$20k=300⟹k=15$$

Substitute $k=15$ into equation (2) to solve for $C$:

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

$$C+1500=1700⟹C=200$$

Thus, the complete linear model for the hostel expense is:

$$E=200+15n$$

Now, we calculate the number of boarders ($n$) when $E=1880$:

$$1880=200+15n$$

$$15n=1680$$

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