Question

A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When student A takes food for $20$ days, she has to pay $Rs.1000$ as hostel charges whereas student B, who takes food for $26$ days, pays $Rs.1180$ as hostel charges. Find the fixed charges and the cost of food per day.

Answer 1

Step 1:Obtain the system of equations from the given details:

Let the fixed charge be Rs. $x$ and the variable food charge be Rs.$y$.

It is given that when student A takes food for $20$ days, she has to pay $Rs.1000$ as hostel charges.

$\Rightarrow x+20y=1000...\left(1\right)$

Also, student B, who takes food for $26$ days, pays $Rs.1180$ as hostel charges.

$\Rightarrow x+26y=1180...\left(2\right)$

Step 2: Solve the system of linear equations using the elimination method to obtain the fixed charges and the cost of food per day:

Now, subtract equation (1) from equation (2) to get:

$\begin{array}{rcl}\left(x+26y\right)-\left(x+20y\right)& =& 1180-1000\\ & \Rightarrow & 6y=180\\ & \Rightarrow & y=30\end{array}$

Now, substitute the value of $y$ in equation (1) to get:

$\begin{array}{rcl}x+20y& =& 1000\\ & \Rightarrow & x+20\left(30\right)=1000\\ & \Rightarrow & x=1000-600\\ & \Rightarrow & x=400\end{array}$

Hence, the fixed charges are $\text{Rs.}400$ and the cost of food per day is $\text{Rs}.30$.