The total expenses of a boarding house are partly fixed and partly variable…

Concept: When a total cost has a fixed part and a part that varies with the number of units, Total Cost = a + kn, where a is the fixed cost, k is the variable cost per unit, and n is the number of units. The cost per unit (charge per head) is then a/n + k, which falls as n grows -- because the fixed cost gets spread over more units -- and approaches k as n becomes very large.

Application: Let the fixed cost be a and the variable cost per boarder be k, so Total Cost = a + kn.

1. With 75 boarders the charge is Rs 48 per head, so the total cost is 75 x 48 = 3600. This gives the equation a + 75k = 3600.

2. With 50 boarders the charge is Rs 52 per head, so the total cost is 50 x 52 = 2600. This gives the equation a + 50k = 2600.

3. Subtracting the second equation from the first eliminates a: (a + 75k) - (a + 50k) = 3600 - 2600, so 25k = 1000, giving k = 40.

4. Substituting k = 40 into a + 50k = 2600 gives a + 2000 = 2600, so a = 600.

5. For 100 boarders, the total cost is a + 100k = 600 + 100 x 40 = 600 + 4000 = 4600, so the charge per head is 4600 / 100 = Rs 46.

Cross-check: Using the per-head formula a/n + k = 600/n + 40 directly: at n = 75 it gives 8 + 40 = 48 (matches); at n = 50 it gives 12 + 40 = 52 (matches); at n = 100 it gives 6 + 40 = 46, confirming the answer independently of the substitution steps above. This also matches the expected trend -- as the number of boarders rises (50, then 75, then 100), the charge per head keeps falling (52, then 48, then 46) since the fixed cost is shared over more people.