If \(C\) is a quantity which is partly constant and partly varies with \(n\), then 

\[C = A + Bn\]

Where \(A\) and \(B\) are constant.

Example: Expenses of a hostel are partly constant and partly vary with number of students. When there are 20 students, total expenses are Rs. 2000. When there are 50 students, total expenses are  Rs. 3500. Find the total expenses when there are 70 students in the hostel.

Solution: Suppose Expenses = \(E\)

\(E = A + Bn\)

Where \(A, B\) are constant and \(n\) is the number of students. Now, 

2000 = \(A + 20B\)

3500 = \(A + 50B\)

Solving these equations, \(30B = 1500  \Rightarrow B = 50\)

Putting value of \(B, A = 1000\).

Now total expenses, when there are 70 students.

\(E = {\rm{ }}1000{\rm{ }} + {\rm{ }}70 \times 50{\rm{ }} = {\rm{ }}4500\)

Example: If \(z\) is directly proportional to \(x\), when \(y\) is constant and directly proportional to \({y^2},\) when \(x\) is constant. When \(x = 3\) and \(y = 4\), then \(z\) is 240. Now find \(z\) if \(x = 4\) and \(y = 3\)

Solution: From given information,

\(z \propto x\)   when \(y\) is constant and 

\(z \propto {y^2}\), when \(x\) is  constant

Combining both the relations, we get,

\(z\; \propto x{y^2}\) or \(z/x{y^2} = k\) or \({z_1}/{x_1}{y_1}^2 = {z_2}/{x_2}{y_2}^2\)

Hence \({z_2} = {\rm{ }}180\)