There are 45 personnel in an army camp. If the number of personnel increases by 9, the expenses of mess increase by Rs. 54 per day, while the average expenditure per person diminishes by 1. Find the original expenditure of the mess.

Let’s denote the original expenditure of the mess as \(E\) rupees per day and the original average expenditure per person as \(A\) rupees per person per day.

According to the given information:

– Original number of personnel = 45
– New number of personnel = 45 + 9 = 54
– Increase in expenses of mess = Rs. 54 per day
– Decrease in average expenditure per person = Rs. 1 per person per day

Using the definition of average expenditure per person, we can write the following equations for the original and new scenarios:

1. Original scenario:

\[ A = \frac{E}{45} \]

2. New scenario:

\[ A – 1 = \frac{E + 54}{54} \]

From equation (1), we can express \(E\) in terms of \(A\):

\[ E = 45A \]

Substituting \(E\) in equation (2):

\[ A – 1 = \frac{45A + 54}{54} \]

Multiplying through by 54:

\[ 54A – 54 = 45A + 54 \]

Rearranging:

\[ 9A = 108 \]

\[ A = 12 \]

Now, using the value of \(A\) to find \(E\):

\[ E = 45A = 45 \times 12 = 540 \]

Therefore, the original expenditure of the mess is Rs. 540 per day.